Spartan Physical Chemistry EditionTutorial and Activities

WAVEFUNCTION

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Copyright © 2005 by Wavefunction, Inc.

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ISBN 1-890661-31-7

Printed in the United States of America

Preface I

PrefaceOver the last decade, molecular modeling has evolved from a specialized research tool of limited availability to an important means with which to explore chemistry. The obvious catalyst has been the explosion in computer technology. Today’s personal computers are more powerful than yesterday’s supercomputers. Computer-based models are now routinely able to supply quantitative information about the structures, stabilities and reactivities of molecules. Computer graphics has made modeling easy to learn and easy to do.

Molecular modeling has an increasingly significant role in the teaching of chemistry. It offers a natural companion to both traditional lecture/textbook and laboratory approaches. Modeling, like lectures and textbooks, not only facilitates communication of both concepts and content, but also allows for the discovery of “new chemistry”, very much in the same way as a laboratory. Molecular models offer a rich source of visual and quantitative information. They can be used by instructors to enhance and liven traditional lectures and classroom discussions, and by students on both personal and school computers to learn and explore chemistry.

These opportunities prompted us to develop a version of the Spartan molecular modeling program for use by students in their study of chemistry, in particular, organic chemistry. We now follow this with a second “Student” version of Spartan that is appropriate for the physical chemistry curriculum. It extends the capabilities of the earlier Student version with density functional and Møller-Plesset models as well as larger basis sets to more accurately explore the limits of these models. While size limits have been placed on the methods and some of the more advanced features of the full “research” version of Spartan have been omitted, this version offers students access to state-of-the-art modeling tools.

This guide is intended to help students and faculty get started. Following a brief introduction, a series of “tutorials” provide fundamental protocols for building molecules, specifying calculations and analyzing results. A series of “essays” follows the tutorials. These describe the

II

origins of the molecular mechanics and quantum chemical methods available in Spartan and assess their ability to calculate important chemical quantities. They also provide guidelines for determining equilibrium and transition-state geometries and for using Spartan’s graphical analysis tools. Next is a series of “hands-on” problems that focus primarily on “chemistry” and not on the molecular models or modeling techniques. These span a variety of topics covered in the physical chemistry curriculum, and complement the modeling problems found in the recent textbook Physical Chemistry by Thomas Engle and Philip Reid of the University of Washington (Benjamin Cummings, San Francisco, CA, 2006). Some of these problems require only a few minutes to complete while others may demand several hours. The guide concludes with a glossary of terms and acronyms common to molecular modeling and designation of units for the quantities arising from the calculations.

This guide does not provide detailed information about the workings of the functions under Spartan’s menus. This information is included as part of the on-line User’s Guide available under the Help menu.

The development of Spartan for use in teaching chemistry has been influenced by many individuals, including Wim Buijs (of DSM Corporation and of the Technical University of Delft), Richard Johnson and Barbara Hopkins (of the University of New Hampshire), Tom Gardner (of Boston Scientific), Alan Shusterman (of Reed College), Philip Bays (of St. Mary’s College), and Thomas Engle (of the University of Washington).

TOC III

Table of ContentsA. Introduction .............................................................................. 1

B. Tutorials..................................................................................... 5

1. Basic Operations................................................................. 62. Acrylonitrile: The Organic Model Kit................................ 173. Sulfur Tetrafl uoride: The Inorganic Model Kit .................. 234. Infrared Spectrum of Acetone ............................................ 275. Benzene and Pyridine: Electrostatic Potential Maps.......... 296. Weak vs. Strong Acids........................................................ 317. Internal Rotation in n-Butane ............................................. 358. Ene Reaction....................................................................... 39

C. Essays......................................................................................... 43

1. Potential Energy Surfaces................................................... 45 One Dimensional Energy Surfaces................................ 45 Many Dimensional Energy Surfaces ............................. 47

2. Theoretical Models............................................................. 50 The Schrödinger Equation............................................. 50 Hartree-Fock Molecular Orbital Models....................... 51 Correlated Models ......................................................... 52 Semi-Empirical Molecular Orbital Models................... 53 Molecular Mechanics Models ....................................... 53

3. Choosing a Theoretical Model ........................................... 54 In Terms of Task ............................................................ 56 In Terms of Model ......................................................... 57

4. Total Energies and Thermodynamic and Kinetic Data....... 58

5. Finding and Verifying Equilibrium and Transition- State Geometries................................................................. 63 Equilibrium Geometries ................................................ 64 Transition-State Geometries .......................................... 65 Reactions Without Transition States ............................. 67 Calculations Using Approximate Geometries ............... 68

IV TOC

6. Atomic and Molecular Orbitals .......................................... 69 Orbital Surfaces............................................................. 69 Atomic Orbitals ............................................................. 70 Orbitals and Chemical Bonds........................................ 71 Molecular Orbitals of Singlet Methylene...................... 73

7. Electron Densities: Sizes and Shapes of Molecules ........... 76 Space-Filling Models .................................................... 76 Electron Density Surfaces ............................................. 77 Bond Density Surfaces .................................................. 79

8. Electrostatic Potential Maps: Charge Distributions ........... 81

9. Local Ionization Potential Maps and LUMO Maps: Electrophilic and Nucleophilic Reactivity .................... 86 Local Ionization Potential Maps and Electrophilic Reactivity................................................. 86 LUMO Maps and Nucleophilic Reactivity ................... 88

D. “Hands-On” Problems ............................................................. 89

Structure and Bonding................................................................ 91Thermochemistry...................................................................... 100Hydrogen Bonding and Intermolecular Interactions ................ 105Conformation............................................................................ 108Dipole Moments and Charges .................................................. 111Reactivity and Selectivity......................................................... 116Infrared and UV/Visible Spectra .............................................. 121

E. Common Terms and Acronyms............................................. 131

F. Units ......................................................................................... 139

Index................................................................................................ 141

Section A 1

Section AIntroduction

Why does computer-based molecular modeling play an important role in chemical research? Is it that, at the same time the cost of experimental laboratory science has skyrocketed, the cost of modeling has sharply decreased? Is it that computational methods, and the software and hardware needed to implement these methods, have matured to the point where useful results can be obtained for real systems in a practical time period? Could it be that the famous quote,

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the diffi culty is only that the exact application of these laws leads to equations much too complicated to be soluble.

P.A.M. Dirac 1902-1984

made by one of the founders of quantum mechanics at a time when that science was still in its infancy, is now only half true? Could it be that “the whole of chemistry” is now open to computation?

All of these factors contribute. Effective and accurate theoretical models, combined with powerful and “user friendly” software, and inexpensive, powerful computer hardware, have made molecular modeling an affordable and widely accessible tool for solving real chemical problems. The time is at hand for modeling tools to be used on a par with experiment. Both provide legitimate and practical means for exploring chemistry. What is needed now is for mainstream chemists to be trained in the practical use of modeling techniques. Only then will they adopt molecular modeling in the same way that they have adopted other research tools, NMR, lasers, X-ray crystallography, once deemed the exclusive province of highly trained “experts.”

Molecular modeling offers two major benefits over experiment as a tool for exploration of chemistry. First, modeling tools can be used to investigate a much wider variety of chemical species than are normally accessible to the experimental chemist. Different conformers of a

2 Section A

flexible molecule, weakly-bound complexes, highly unstable or reactive molecules, and even transition states for chemical reactions, can all be easily investigated using quantum chemical models. Moreover, the computational effort required to identify and characterize each of these species is essentially the same. This contrasts with experiment, where procedures for isolating and characterizing molecules become increasingly more difficult as its lifetime and/or concentration decreases. And, because transition states “do not exist” in the sense that sizeable populations may be established, they are not subject to direct experimental characterization.

Equally important, molecular modeling allows chemists to think clearly about issues that are central to chemistry: structure, stability and reactivity. Computer-generated models “look” and “behave” like “real molecules”. Good models can be produced even when a student is unable to make an accurate drawing. Starting from this model, quantum chemical methods provide quantitative information about actual three-dimensional structure and conformation.

Models based on quantum mechanics such as those produced by Spartan are not limited to molecular structure. They can be easily and routinely used to calculate and display a myriad of chemical observables, among them, stability (“thermodynamics”), reactivity (“kinetics”), spectra and charge distribution. In short, a chemistry student or a research chemist, working with a computer, can explore “new areas of chemistry”.

Section A 3

This edition of Spartan is intended to provide a realistic impression of the capabilities and limitations of molecular modeling as a tool for exploring chemistry. While the program restricts the choice of theoretical models and limits the size of the systems that can be treated (relative to the full version of Spartan), it is well suited to supplement both the lecture and laboratory components of a physical chemistry course, or to form the basis of an “advanced” course in computational chemistry.

Features and capabilities are provided below.

Available tasks: energies and wavefunctionsequilibrium and transition-state geometriesenergy profi les

Not available: conformational searchingintrinsic reaction coordinates

Available theoretical models: MMFF molecular mechanics PM3 semi-empirical 3-21G, 6-31G* and 6-311+G** Hartree-Fock B3LYP/6-31G* and B3LYP/6-311+G** MP2/6-31G* and MP2/6-311+G**

Not available: SYBYL molecular mechanicsMNDO and AM1 semi-empiricaldensity functional models other than B3LYPMøller-Plesset models other than MP2CCSD and higher-order correlated modelsCIS, CISD and density functional models for excited statesbasis sets other than 3-21G, 6-31G* and 6-311+G**pseudopotentials

Available properties and spectra: dipole moments electrostatic charges infrared spectra

Not available: Mulliken and NBO chargesNMR spectraUV/visible spectra

Size limitations: PM3 50 atoms3-21G, 6-31G* and 6-311+G** Hartree-Fock 30 atomsB3LYP/6-31G* and B3LYP/6-311+G* 20 atomsMP2/6-31G* and MP2/6-311+G** 20 atoms

4 Section A

This guide takes a pragmatic approach, in that its focus is almost entirely on the use of quantum chemical models to solve problems in chemistry. An analogy might be made between molecular modeling and spectroscopy. Most chemists think of the latter in terms of the identification and characterization of compounds using spectroscopic techniques, and not of the quantum mechanical description of the interaction of “light” with matter or of the detailed workings of a spectrometer. Readers who are interested in treatment of the theories underlying molecular models, or the algorithms used to construct these models, should consult one of the many excellent texts that have been written on these topics.

Section B 5

Section BTutorials

This section comprises a series of “tutorials” for Spartan Student Physical Chemistry Edition, covering both Windows and Mac OS X versions. These are intended to provide a first exposure to the workings of the program, as well as to illustrate use of its various calculation and analysis tools including graphical modeling tools. Focus is not on the chemistry but rather on the basic workings of Spartan’s interface and on the use of diverse modeling tools. For this reason, and in order to save computer time, relatively simple theoretical models have been employed in these tutorials. The user may wish to repeat some of the tutorials with “better” models in order to get an idea of differences both in the results and in the computer time required to achieve these results.

The fi rst tutorial does not involve either molecule building or calculation, but rather employs “prebuilt” and “precalculated” models. It is primarily intended to acquaint the fi rst-time user with the “basics”: molecule display and manipulation, molecular structure, energy and property query and display, manipulation and query of graphical models. This tutorial should be completed fi rst. The remaining tutorials involve both molecule building and calculation. Each brings focus to a particular aspect of molecular modeling, for example, construction and interpretation of electrostatic potential maps.

The menus, dialogs and other images shown in the tutorials are from the Windows version. The corresponding Macintosh menus, dialogs and images are very similar. The exception is that both sets of icons are provided (Windows icon/Macintosh icon). Note, that the tutorials have been written assuming a two-button mouse for Macintosh, although a one-button mouse can also be used.

Files associated with the tutorials are grouped in the “tutorials” directory on the CD-ROM. File names are those specifi ed in the individual tutorials. Only for the fi rst tutorial are these fi les needed. The remainder have been provided only to show the “proper outcome”.

6 Tutorial I

1Basic Operations

This tutorial introduces a number of basic operations in the Spartan Student Physical Chemistry Edition that are required for molecule manipulation and property query. Specifi cally it shows how to: i) open mol e cules, ii) view different models and manipulate molecules on screen, iii) measure bond distances, angles and dihedral angles, iv) display en er gies, di pole moments, atomic charges and infrared spectra, and v) dis play graph i cal surfaces and property maps. Molecule building is not il lus trat ed and no calculations are performed.

1. Start Spartan. For Windows: Click on the Start button, then clickAll Programs and fi nally click on click on click Spartan. For Macintosh: Double click on the icon on the desktop. Spartan’s window will appear with click on the icon on the desktop. Spartan’s window will appear with clicka menu bar at the top of the screen.

File Allows you to create a new molecule or read in a mol e cule which you have previously created.

Model Allows you to control the style of your model.

Geometry Allows you to measure bond lengths and angles.

Build Allows you to build and edit molecules.

Setup Allows you to specify the task to be performed and the theoretical model to be employed, to spec i fy graph i cal surfaces and property maps and to sub mit jobs for calculation.

Display Allows you to display text output, molecular and atomic properties, surfaces and prop er ty maps and infrared spectra. Also allows you to pre sen t data in a spreadsheet and make plots from these data.

Search Allows you to “guess” a transition-state geometry based on a library of reactions. This guess may then be used as the basis for a quantum chemical calculation of the actual reaction transition state.

Tutorial I 7

2. Click with the left mouse button on File from the menu bar.

Click on Open.... Alternatively, click on the / icon.

Functions in Spartan’s menus may also be accessed from icons.

/ New / Add Fragment / Measure Distance/ Open / Delete / Measure Angle/ Close / Make Bond / Measure Dihedral/ Save As / Break Bond / Transition States/ View / Minimize

Locate the “tutorials” directory in the dialog that appears, click on “tutorial 1” and click on Open (or double click on “double click on “double click tutorial 1”). Ball-and-spoke models for ethane, acetic acid dimer, propene, ammonia, hydrogen peroxide, acetic acid, water, cyclohexanone, ethylene, benzene and aniline appear on screen. You can select a molecule by clicking on it with the left mouse button. Once selected, a molecule may be manipulated (rotated, translated and scaled).

Windowsselect molecule click (left mouse button)rotate molecule press the left button and move the mousetranslate molecule press the right button and move the mousescale molecule press both the Shift key and the right button and

move the mouse “up and down”

Macintosh (One-Button Mouse)select molecule clickrotate molecule press the button and move the mousetranslate molecule press the button and the option key and move the

mousescale molecule press both the and option keys in addition to the

button and move the mouse “up and down”

8 Tutorial I

Macintosh (Two-Button Mouse)select molecule click (left mouse button)rotate molecule press the left button and move the mousetranslate molecule press the right button and move the mousescale molecule press both the key and the right button and move

the mouse “up and down”

3. Identify ethane on the screen, and click on it (left button) to make it the selected molecule. Practice rotating (move the mouse while pressing the left button) and translating (pressing the left button) and translating (pressing move the mouse while pressing the right button) pressing the right button) pressing ethane. Click on a different molecule, and Click on a different molecule, and Clickthen rotate and translate it.

4. Return to ethane. Click on Model from the menu bar.

Wire Ball-and-Wire Tube Ball-and-Spoke

One after the other, select Wire, Ball and Wire, Tube and fi nally Ball and Spoke from the Model menu. All four models for ethane show roughly the same information. The wire model looks the most like a conventional line formula. It uses color to distinguish different atoms, and one, two and three lines between atoms to indicate single, double and triple bonds, respectively.

Tutorial I 9

Atoms are colored according to type:Hydrogen white Sodium yellowLithium tan Magnesium dark blueBeryllium green Aluminum violetBoron tan Silicon greyCarbon black Phosphorous tanNitrogen blue-gray Sulfur sky blueOxygen red Chlorine tanFluorine green

Atom colors (as well as bond colors, background color, etc.) may be changed from their defaults using Colors under the Options menu. An atom may be labelled with a variety of different quantities including chirality designation and charge using Confi gure... under the Model menu. Labels are then automatically turned “on” and may be turned “off” by selecting Labels under the Model menu.

The ball-and-wire model is identical to the wire model, except that atom positions are represented by small spheres, making it easy to identify atom locations. The tube model is identical to the wire model, except that bonds are represented by solid cylinders. The tube model is better than the wire model in conveying three-dimensional shape. The ball-and-spoke model is a variation on the tube model; atom positions are represented by colored spheres, making it easy to see atom locations.

Select Space Filling from the Model menu.

Space-Filling

This model is different from the others in that bonds are not shown. Rather, each atom is displayed as a colored sphere that represents its approximate “size”. Thus, the space-fi lling model for a molecule provides a measure of its size. While lines between atoms are not drawn, the existence (or absence) of bonds can be inferred from the extent to which spheres on neighboring atoms overlap. If two spheres substantially overlap, then the atoms are almost certainly bonded, and conversely, if two spheres hardly overlap, then the atoms are

10 Tutorial I

not bonded. Intermediate overlaps suggest “weak bonding”, for example, hydrogen bonding.

Select acetic acid dimer. Switch to a space-fi lling model and look for overlap between the (OH) hydrogen on one acetic acid molecule and the (carbonyl) oxygen on the other. Return to a ball-and-spoke model and select Hydrogen Bonds from the Model menu.

Ball-and-Spoke model for acetic acid dimer with hydrogen bonds displayed

The two hydrogen bonds, that are responsible for holding the acetic acid molecules together, will be drawn.

Use the 3 key to toggle between stereo 3D and regular display. To view in 3D you will need to wear the red/blue glasses provided with Spartan.

5. Distances, angles, and dihedral angles can easily be measured with Spartan using Measure Distance, Measure Angle, and Measure , and Measure , andDihedral, respectively, from the Geometry menu.

Alternatively, measurement functions may be accessed from the / , / and / icons.

a) Measure Distance: This measures the distance between two atoms. First select propene from the molecules on screen, and then select Measure Distance from the Geometry menu (or click on the click on the click / icon). Click on a bond or on two atoms (the Click on a bond or on two atoms (the Clickatoms do not need to be bonded). The distance (in Ångstroms) will be displayed at the bottom of the screen. Repeat the process

Tutorial I 11

for several atoms. When you are fi nished, select View from the Build menu.

Alternatively, click on the / icon.

b) Measure Angle: This measures the angle around a central atom. Select ammonia from the molecules on screen, and then select Measure Angle from the Geometry menu (or click on click on clickthe / icon). Click fi rst on H, then on N, then on another H. Alternatively, click on two NH bonds. The HNH angle (in degrees) will be displayed at the bottom of the screen. Click on Click on Click

/ when you are fi nished.

c) Measure Dihedral: This measures the angle formed by two intersecting planes, one containing the fi rst three atoms selected and the other containing the last three atoms selected. Select hydrogen peroxide from the molecules on screen, then select Measure Dihedral from the Geometry menu (or click on the click on the click/ icon) and then click in turn on the four atoms (HOOH) that click in turn on the four atoms (HOOH) that clickmake up hydrogen peroxide. The HOOH dihedral angle will be displayed at the bottom of the screen. Click on Click on Click / when you are fi nished.

6. Energies, dipole moments and atomic charges among other calculated properties, are available from Properties under the Display menu.

a) Energy: Select acetic acid from the molecules on screen and then select Properties from the Display menu. The Molecule Properties dialog appears.

12 Tutorial I

This provides the energy for acetic acid in atomic units (au). See the essay “Total Energies and Thermodynamic and Kinetic Data” for the defi nition of energy and for a discussion of energy units.

b) Dipole Moment: The magnitude of the dipole moment (in debyes) is also provided in the Molecule Properties dialog. A large dipole moment indicates large separation of charge. You can attach the dipole moment vector, “ ” where the lefthand side “+” refers to the positive end of the dipole, to the model on the screen, by checking Dipole near the bottom of the dialog. The vector will not be displayed if the magnitude of the dipole moment is zero, or if the molecule is charged.

c) Atomic Charges: To display the charge on an atom, click on it with the Molecule Properties dialog on the screen. The Atom Properties dialog replaces the Molecule Properties dialog.

Atomic charges are given in units of electrons. A positive charge indicates a defi ciency of electrons on an atom and a negative charge, an excess of electrons. Repeat the process as necessary by clicking on other atoms. Confi rm that the positively-charged atom(s) lie at the positive end of the dipole moment vector. When you are fi nished, remove the dialog from the screen by clicking on the / in the top right-hand corner.

Tutorial I 13

d) Infrared Spectra: Molecules vibrate (stretch, bend, twist) even if they are cooled to absolute zero. This is the basis of infrared spectroscopy, where absorption of energy occurs when the frequency of a particular molecular motion matches the frequency of the “light”. Infrared spectroscopy is important for identifying molecules as different functional groups vibrate at noticeably different and characteristic frequencies.

Select water from the molecules on screen. To animate a vibration, select Spectra from the Display menu. This leads to the Spectra dialog.

This displays the three vibrational frequencies for the water molecule, corresponding to bending and symmetric and antisymmetric stretching motions. One after the other, click on each frequency and examine the motion. Turn “off” the animation when you are fi nished.

No doubt you have seen someone “act out” the three vibrations of water using his/her arms to depict the motion of hydrogens.

Not only does this provide a poor account of the actual motions, it is clearly not applicable to larger molecules.

14 Tutorial I

Select cyclohexanone. The Spectra dialog now lists its 45 vibrational frequencies. Examine each in turn (click on the entry in the dialog) until you locate the frequency corresponding to the CO (carbonyl) stretch. Next, click on Draw IR Spectrum at the bottom of the dialog. The infrared spectrum of cyclohexanone will appear.

You can move the spectrum around the screen by fi rst clickingon it to select it and then moving the mouse while pressing the pressing the pressingright button. You can size it by moving the mouse “up and down” moving the mouse “up and down” movingwhile pressing both the pressing both the pressing Shift key ( key on Macintosh) and the right button. On Macintosh you can also use the scroll button to size the spectrum.

Identify the line in the spectrum associated with the CO stretch (a small gold ball moves from line to line as you step through the frequencies in the Spectra dialog). Note that this line is separated from the other lines in the spectrum and that it is intense. This makes it easy to fi nd and is the primary reason why infrared spectroscopy is an important diagnostic for carbonyl functionality. When you are fi nished, click on click on click / at the top of the Spectra dialog to remove it from the screen.

7. Spartan permits display, manipulation and query of a number of important “graphical” quantities resulting from quantum chemical calculations. Most important are the electron density (that reveals “how much space” a molecule actually takes up; see the essay “E“E“ lectron Densities: Sizes and Shapes of Moleculeslectron Densities: Sizes and Shapes of Moleculesl ”), the bond density (that reveals chemical “bonds”; see the essay on electron densities), and key molecular orbitals (that provide insight into both bonding and chemical reactivity; see the essay “Atomic and both bonding and chemical reactivity; see the essay “Atomic and both bonding and chemical reactivity; see the essay “Molecular Orbitals”). In addition, the electrostatic potential map, an overlaying of a quantity called the electrostatic potential (the attraction or repulsion of a positive charge for a molecule) onto the

Tutorial I 15

electron density, is valuable for describing overall molecular charge distribution as well as anticipating sites of electrophilic addition. Further discussion is provided in the essay “Electrostatic Potential Further discussion is provided in the essay “Electrostatic Potential Further discussion is provided in the essay “Maps: Charge Distributions”. Another indicator of electrophilic addition is provided by the local ionization potential map, an overlaying of the energy of electron removal (“ionization”) onto the electron density. Finally, an indicator of nucleophilic addition is provided by the LUMO map, an overlaying of the lowest-unoccupied molecular orbital (the LUMO) onto the electron density. Both of these latter graphical models are described in the essay “Local these latter graphical models are described in the essay “Local these latter graphical models are described in the essay “Ionization Potential Maps and LUMO Maps”.

Select ethylene from among the molecules on screen, and then select Surfaces from the Display menu. The Surfaces dialog appears.

Click inside the box to the left of the line “homo...” inside the dialog. This will result in the display of ethylene’s highest-occupied molecular orbital as an opaque solid. This is a π orbital, equally concentrated above and below the plane of the molecule. The colors (“red” and “blue”) give the sign of the orbital. Changes in sign correlate with bonding or antibonding character (see the essay “Atomic and Molecular Orbitals“Atomic and Molecular Orbitals“ ”. You can if you wish, turn “off” the graphic by again clicking inside the box to the left of the line clicking inside the box to the left of the line clicking“homo . . .”.

Next, select benzene from among the molecules on screen and click on “density potential...” inside the Surfaces dialog. An electrostatic potential map for benzene will appear. For Windows: Click on the map. The Style menu will appear at the bottom right of the screen. Select Transparent from this menu. For Macintosh: Position the cursor over the map, hold down either the left or right button and select Transparent from the menu that appears. Making the map

16 Tutorial I

transparent allows you to see the molecular skeleton underneath. The surface is colored “red” in the π system (indicating negative potential and the fact that this region is attracted to a positive charge), and “blue” in the σ system (indicating positive potential and the fact that this region is repelled by a positive charge).

Select aniline from the molecules on screen, and click on “density click on “density clickionization...” inside the Surfaces dialog. The graphic that appears, a so-called local ionization potential map, red regions on the density surface indicate areas from which electron removal (ionization) is relatively easy, meaning that they are subject to electrophilic attack. These are easily distinguished from regions where ionization is relatively diffi cult (colored in blue). Note that the ortho and pararing carbons are more red than the meta carbons, consistent with the known directing ability of the amino substituent.

Finally, select cyclohexanone from the molecules on screen , and click on “lumo...” in the Surfaces dialog. The resulting graphic portrays the lowest-energy empty molecular orbital (the LUMO) of cyclohexanone. This is a so-called π* orbital that is antibonding between carbon and oxygen. Note that the LUMO is primarily localized on carbon, meaning that this is where a pair of electrons (a nucleophile) will “attack” cyclohexanone.

A better portrayal is provided by a LUMO map, that displays the (absolute) value of the LUMO on the electron density surface. Here, the color blue is used to represent maximum value of the LUMO and the color red, minimum value. First, remove the LUMO from your structure (click on “lumo...” in the click on “lumo...” in the click Surfaces dialog) and then turn on the LUMO map (click on “density lumo...” in the dialog). Note that click on “density lumo...” in the dialog). Note that clickthe blue region is concentrated directly over the carbonyl carbon. Also, note that the so-called axial face shows a greater concentration axial face shows a greater concentration axialof the LUMO than the equatorial face. This is consistent with the equatorial face. This is consistent with the equatorialknown stereochemistry of nucleophilic addition.

8. When you are fi nished, close all the molecules on screen by selecting Close from the File menu or alternatively by clicking on / .

Tutorial II 17

2Acrylonitrile:

The Organic Model KitThis tutorial illustrates use of the organic model kit, as well as the steps involved in examining and querying different molecular models and in carrying out a quantum chemical calculation.

The simplest building blocks incorporated into Spartan’s organic model kit are “atomic fragments”. These constitute specifi cation of atom type, e.g., carbon, and hybridization, e.g., sp3. The organic model kit also contains libraries of common functional groups and hydrocarbon rings, the members of which can easily be extended or modifi ed. For example, the carboxylic acid group in the library may be modifi ed to build a carboxylate anion (by deleting a free valence from oxygen), or an ester (by adding tetrahedral carbon to the free valence at oxygen).

RC

OH

C

carboxylic acidR

CO–

O

carboxylate anionR

CO

CH3

O

ester

Acrylonitrile provides a good opportunity to illustrate the basics of molecule building, as well as the steps involved in carrying out and analyzing a simple quantum chemical calculation.

C CC CC CC CC CC CC CC CC CC CC CC CC CC CC CC CH

H H

CCN

acrylonitrile

1. Click on File from the menu bar and then click on New from the menu which appears (or click on the click on the click / icon in the File toolbar). The “organic” model kit appears.

18 Tutorial II

Click on trigonal planar spClick on trigonal planar spClick 2 hybridized carbon from the library of atomic fragments. The fragment icon is highlighted, and a model of the fragment appears at the top of the model kit. Bring the cursor anywhere on screen and click. Rotate the carbon fragment (move the mouse while holding down the left button) so that you can clearly see both the double free valence (“=”) and the two single free valences (“-”).

Spartan’s model kits connect atomic fragments (as well as groups, rings and ligands) through free valences. Unless you “use” them or delete them, free valences will automatically be converted to hydrogen atoms.

2. sp2 carbon is still selected. Click on the double free valence. The Click on the double free valence. The Clicktwo fragments are connected by a double bond, leaving you with ethylene.

Spartan’s model kits allow only the same type of free valences to be connected, e.g., single to single, double to double, etc.

3. Click on Click on Click Groups in the model kit, and then select Cyano from among the functional groups available from the menu.

Tutorial II 19

Click on one of the free valences on ethylene, to make acrylonitrile.Click on one of the free valences on ethylene, to make acrylonitrile.Click *

If you make a mistake, you can select Undo from the Edit menu to “undo” the last operation or Clear (Edit menu) to start over.

4. Select Minimize from the Build menu (or click on the click on the click / icon). The “strain energy” and symmetry point group (CS) for acrylonitrile are provided at the bottom right of the screen.

5. Select View from the Build menu (or click on the click on the click / icon). The model kit disappears, leaving only a ball-and-spoke model of acrylonitrile on screen.

6. Select Calculations... from the Setup menu.

The Calculations dialog appears. This will allow you to specify what task is to be done with your molecule and what theoretical model Spartan will use to accomplish this task.

* You could also have built acrylonitrile without using the Groups menu. First, clear the screen by selecting Clear from the Edit menu. Then build ethylene from two sp2 carbons (as above), select sp hybridized carbon from the model kit and then click on the tip of one of the free click on the tip of one of the free clickvalences on ethylene. Next, select sp hybridized nitrogen from the model kit and click on the click on the clicktriple free valence on the sp carbon. Alternatively, you could have built the molecule entirely from groups. First, clear the screen. Then click on click on click Groups, select Alkene from the menu and click anywhere on screen. Then select Cyano from the same menu and click on one of the free valences on ethylene. In general, molecules can be constructed in many ways.

20 Tutorial II

Select Equilibrium Geometry from the menu to the right of “Calculate”. This specifi es optimization of equilibrium geometry. Next, select Hartree-Fock/3-21G from the menu to the right of “with”. This specifi es a Hartree-Fock calculation using the 3-21G basis set (referred to as an HF/3-21G calculation). This method generally provides a reliable account of geometries (see the essay “Choosing a Theoretical Model”).Choosing a Theoretical Model”).Choosing a Theoretical Model

7. For Windows: Click on Submit at the bottom of the dialog. A fi le browser appears.

Type “acrylonitrile” in the box to the right of “File name”, and click on Save*. You will be notifi ed that the calculation has been submitted. Click on OK to remove the message.

After a molecule has been submitted, and until the calculation has completed, you are not permitted to modify information associated with it. You can monitor your calculation as well as abort it if necessary using Monitor under the Options menu (Spartan Monitor under the Spartan ST menu on Macintosh).

* On Windows, default names spartan1, spartan2, will be supplied. On Macintosh; a name will be supplied if the molecule is present in Spartan’s database. You may change this if desired.

Tutorial II 21

8. You will be notifi ed when the calculation has completed. Click OKto remove the message. Select Output from the Display menu. A window containing “text output” for the job appears.

You can scan the output from the calculation by using the scroll bar at the right of the window. Information provided includes the task, basis set, number of electrons, charge and multiplicity, as well as the point group of the molecule. A series of lines, each beginning with “Cycle no:”, tell the history of the optimization process. Each line provides results for a particular geometry; “Energy” gives the energy in atomic units (1 atomic unit = 2625 kJ/mol) for this geometry, “Max Grad.” gives the maximum gradient (“slope”), and “Max Dist.” gives the maximum displacement of atoms between cycles. The energy will monotonically approach a minimum value for an optimized geometry, and Max Grad. and Max Dist. will each approach zero. Near the end of the output is the fi nal total energy (-168.82040 atomic units). Click on Click on Click / in the top right-hand corner of the dialog to remove the dialog from the screen.

9. You can obtain the fi nal total energy and the dipole moment from the Molecule Properties dialog, without having to go through the text output. Select Properties from the Display menu. You can “see” the dipole moment vector (indicating the sign and overall direction of the dipole moment), by checking Dipole near the bottom of this dialog. (A tube model provides the clearest picture.)

22 Tutorial II

When you are fi nished, turn “off” display of the dipole moment vector by unchecking the box.

10.Click on an atom. The (Click on an atom. The (Click Molecule Properties) dialog will be replaced by the Atom Properties dialog. This gives the charge on the selected atom. To obtain the charge on another atom, simply click on it. Clickon / at the top right of the Atom Properties dialog to remove it from the screen.

11.Atomic charges can also be attached as “labels” to your model. Select Confi gure... from the Model menu, and check Charge under “Atom” in the Confi gure dialog which appears.

Click OK to remove the dialog.

12.Click on / to remove “acrylonitrile” from the screen. Also, close any dialogs that may still be open.

Tutorial III 23

3Sulfur Tetrafl uoride:

The Inorganic Model KitThis tutorial illustrates the use of the inorganic model kit for molecule building. It also shows how molecular models may be used to quantify concepts from more qualitative treatments.

Organic molecules are made up of relatively few elements and generally obey conventional valence rules. They may be easily built using the organic model kit. However, many molecules incorporate other elements, do not conform to normal valence rules, or involve ligands. They cannot be constructed using the organic model kit. Sulfur tetrafl uoride is a good example.

SF

FF

Fsulfur tetrafluoride

The unusual “see-saw” geometry observed for the molecule is a consequence of the fact that the “best” (least crowded) way to position fi ve electron pairs around sulfur is in a trigonal bipyramidal arrangement. The lone pair assumes an equatorial position so as to least equatorial position so as to least equatorialinteract with the remaining electron pairs. The rationale behind this is that a lone pair is “bigger” than a bonding electron pair.

Sulfur tetrafl uoride provides the opportunity to look at the bonding and charges in a molecule which “appears” to have an excess of electrons around its central atom (ten instead of eight), as well as to look for evidence of a lone pair.

1. Bring up the inorganic model kit by clicking on / and then clicking on the Inorganic tab at the top of the organic model kit.

24 Tutorial III

The inorganic model kit comprises a Periodic Table followed by a selection of “atomic hybrids” and then bond types. Further down the model kit are the Rings, Groups and Ligands menus, the fi rst two of which are the same as found in the organic model kit.

2. For Windows: Select (click on) S in the Periodic Table. For Macintosh: Position the cursor inside the box to the right of “Element” and hold down on the left button to bring up a Periodic Table. Slide the cursor over S in the Periodic Table and release the button. Select the fi ve-coordinate trigonal bipyramid structure from the list of atomic hybrids. Click anywhere in the main window. Click anywhere in the main window. ClickA trigonal bipyramid sulfur will appear.

3. Select F in the Periodic Table and the one-coordinate entry from the list of atomic hybrids. One after the other, click on both axial free valences of sulfur, and two of the three equatorial free valences.*

4. It is necessary to delete the remaining free valence (on an equatorial position); otherwise it will become a hydrogen upon leaving the builder. Select Delete from the Build menu (or click on the click on the click /icon) and then click on the remaining click on the remaining click equatorial free valence.

5. Click on Click on Click / . Molecular mechanics minimization will result in a structure with C2v symmetry. Click on / .

6. Select Calculations... from the Setup menu. Specify calculation of equilibrium geometry using the HF/3-21G model. Click on Click on Click OK.

* This step and the following two steps could also be accomplished using the organic model kit. To bring it up, click on the click on the click Organic tab at the top of the inorganic model kit.

Tutorial III 25

7. Select Surfaces from the Setup menu. Click on Click on Click Add... at the bottom of the Surfaces dialog and select HOMO from the Surface menu in the (Add Surface) dialog which appears.

Click on Click on Click OK. Leave the Surfaces dialog on screen.

8. Select Submit from the Setup menu, and supply the name “sulfur tetrafl uoride see-saw”.

9. After the calculations have completed, select Properties from the Display menu to bring up the Molecule Properties dialog. Next, click on sulfur to bring up the click on sulfur to bring up the click Atom Properties dialog. Is sulfur neutral or negatively charged, indicating that more than the normal complement of (eight) valence electron surrounds this atom, or is it positively charged, indicating “ionic bonding”?

SF–

F

F

F

+

10.Click on “homo...” inside the Click on “homo...” inside the Click Surfaces dialog to examine the highest-occupied molecular orbital. Does it “point” in the expected direction? It is largely localized on sulfur or is there signifi cant concentration on the fl uorines? If the latter, is the orbital “bonding” or “antibonding”? (For a discussion of non-bonding, bonding and antibonding molecular orbitals, see the essay “Atomic and “Atomic and “Molecular Orbitals”.)

11.Build square planar SF4 as an alternative to the “see-saw” structure. Bring up the inorganic model kit ( / ), select S from the Periodic Table and the four-coordinate square-planar structure from the list of atomic hybrids. Click anywhere on screen. Select Click anywhere on screen. Select Click F in the Periodic Table and the one-coordinate entry from the list of

26 Tutorial III

atomic hybrids. Click on all four free valences on sulfur. Click on all four free valences on sulfur. Click Click on Click on Click/ and then on / .

12.Enter the Calculations dialog (Calculations from the Setup menu) and specify calculation of equilibrium geometry using the HF/3-21G model (the same level of calculation that you used for the “see-saw” structure*). Click on Click on Click Submit at the bottom of the dialog.

13.After the calculation has completed, bring up the Molecule Properties dialog (Properties from the Display menu) and note the energy. Is it actually higher (more positive) than that for the “see-saw” structure?

14.Close both molecules as well as any remaining dialogs.

* You need to use exactly the same theoretical model in order to compare energies or other properties for different molecules.

Tutorial IV 27

4Infrared Spectrum of Acetone

This tutorial illustrates the steps required to calculate and display the infrared spectrum of a molecule.

Molecules vibrate in response to their absorbing infrared light. Absorption occurs only at specifi c wavelengths, which gives rise to the use of infrared spectroscopy as a tool for identifying chemical structures. The vibrational frequency is proportional to the square root of a quantity called a “force constant” divided by a quantity called the “reduced mass”.

frequency α force constantreduced mass

The force constant refl ects the “fl atness” or “steepness” of the energy surface in the vicinity of the energy minimum. The steeper the energy surface, the larger the force constant and the larger the frequency. The reduced mass refl ects the masses of the atoms involved in the vibration. The smaller the reduced mass, the larger the frequency.

This tutorial shows you how to calculate and display the infrared spectrum of acetone, and explore relationships between frequency and both force constant and reduced mass. It also demonstrates that infrared spectroscopy is of particular value in identifying carbonyl functionality in a molecule.

1. Click on / to bring up the organic model kit. Select sp2 carbon ( ) and click anywhere on screen. Select spclick anywhere on screen. Select spclick 2 oxygen ( ) and click on the double free valence on carbon to make the carbonyl group. Select sp3 carbon ( ) and, one after the other, click on the two click on the two clicksingle free valences on carbon. Click on Click on Click / and then on / .

2. Enter the Calculations dialog (Calculations under the Setup menu) and request calculation of an equilibrium geometry using the HF/3-21G model. Check IR to the right of “Compute” to specify calculation of vibrational frequencies. Click on the Click on the Click Submit button at the bottom of the dialog. Provide the name “acetone”.

28 Tutorial IV

3. After the calculation has completed, bring up the Spectra dialog (Spectra under the Display menu). This contains a list of vibrational frequencies for acetone. First click on the top entry (the smallest click on the top entry (the smallest clickfrequency) and, when you are done examining the vibrational motion, click on the bottom entry (the largest frequency).click on the bottom entry (the largest frequency).click

The smallest frequency is associated with torsional motion of the methyl rotors. The largest frequency is associated with stretching motion of CH bonds. Methyl torsion is characterized by a fl at potential energy surface (small force constant), while CH stretching is characterized by a steep potential energy surface (large force constant).

Display the IR spectrum (click on click on click Draw IR Spectrum at the bottom of the Spectra dialog). Locate the frequency corresponding to the CO stretch. The experimental frequency is around 1740 cm-1, but the calculations will yield a higher value (around 1940 cm-1).

The CO stretching frequency is a good “chemical identifi er” because it “stands alone” in the infrared spectrum and because it is “intense”.

4. Change all the hydrogens in acetone to deuteriums to see the effect of increasing mass on vibrational frequencies. First make a copy of “acetone” (Save As... from the File menu or click on the click on the click / icon). Name the copy “acetone d6” Select Properties from the Displaymenu and click on one of the hydrogens. Select click on one of the hydrogens. Select click 2 deuterium from the Mass Number menu. Repeat for the remaining fi ve hydrogens.

5. Submit for calculation. When completed, examine the vibrational frequencies. Note that the frequencies of those motions that involve the hydrogens (in particular, the six vibrational motions corresponding to “CH stretching”) are signifi cantly reduced over those in the non-deuterated system.

6. Close all molecules on screen in addition to any remaining dialogs.

Tutorial V 29

5Benzene and Pyridine:

Electrostatic Potential MapsThis tutorial illustrates the calculation, display and interpretation of electrostatic potential maps. It also illustrates the use of “documents” comprising two or more molecules.

While benzene and pyridine have similar geometries and while both are aromatic, their “chemistry” is different. Benzene’s chemistry is dictated by the molecule’s π system, while the chemistry of pyridine is a consequence of the lone pair on nitrogen. This tutorial shows how to use electrostatic potential maps to highlight these differences.

1. Build benzene. Click on / . Select Benzene from the Ringsmenu and click on screen. Click on / .

2. Build pyridine. In order to put both benzene and pyridine into the same “document”, select New Molecule (not New) from the Filemenu. Benzene (Rings menu) is still selected. Click anywhere on Click anywhere on Clickscreen. Select aromatic nitrogen from the model kit and double click on one of the carbon atoms (not a free valence). click on one of the carbon atoms (not a free valence). click Click on /and then click on / . To go between the two molecules in your document, use the and keys at the bottom left of the screen (or use the slider bar).

3. Select Calculations... (Setup menu) and specify calculation of equilibrium geometry using the HF/3-21G model. Click on Click on Click OK to OK to OKdismiss the dialog. Select Surfaces (Setup menu). Click on Click on Click Add...at the bottom of the Surfaces dialog to bring up the Add Surfacedialog. Select density from the Surface menu and potential from the Property menu. Click on Click on Click OK. Leave the Surfaces dialog on screen. Select Submit (Setup menu) and supply the name “benzene and pyridine”.

4. When completed, bring up the spreadsheet (Spreadsheet under the Display menu). For Macintosh, click on click on click at the bottom left of the

30 Tutorial V

screen (it will become ). Check the box to the right of the molecule Check the box to the right of the molecule Check“Label” (leftmost column) for both entries. This allows benzene and pyridine to be displayed simultaneously. However, the motions of the two molecules will be “coupled” (they will move together). Select (uncheck) uncheck) uncheck Coupled from the Model to allow the two molecules to be manipulated independently. In turn, select (click on) each and click on) each and clickorient such that the two are side-by-side.

5. Click on “density potential...” inside the Click on “density potential...” inside the Click Surfaces dialog. Electrostatic potential maps for both benzene and pyridine will be displayed. Change the scale so that the “neutral” color is “green”. Select Properties (Display menu) and click on one of the maps to bring up the Surface Properties dialog.

Type “-35” and “35” inside the boxes underneath “Property Range” (Press (Press ( the Enter key following each data entry, return key for Macintosh). The “red” regions in benzene, which are most attractive to an electrophile, correspond to the molecule’s π system, while in pyridine they correspond to the σ system in the vicinity of the nitrogen. Note that the π system in benzene is “more red” than the π system in pyridine (indicating that it is more susceptible to electrophilic attack here), but that the nitrogen in pyridine is “more red” than the π system in benzene (indicating that pyridine is overall more susceptible to attack by an electrophile).

Further discussion of the use of such maps is provided in the essay “Electrostatic Potential Maps: Charge Distributionsessay “Electrostatic Potential Maps: Charge Distributionsessay “ ”.

6. Remove “benzene and pyridine” and any dialogs from the screen.

Tutorial VI 31

6Weak vs. Strong Acids

This shows how electrostatic potential maps may be used to distinguish between weak and strong acids, and quantify subtle differences in the strengths of closely-related acids. It also shows how information can be retrieved from Spartan’s database.

Nitric and sulfuric acids are strong acids, acetic acid is a weak acid, and that ethanol is a very weak acid. What these compounds have in common is their ability to undergo heterolytic bond fracture, leading to a stable anion and a “proton”. What distinguishes a strong acid from a weak acid is the stability of the anion. NO3

– and HOSO– and HOSO–3– are very stable anions, – are very stable anions, –

CH3CO2– is somewhat less stable and CH– is somewhat less stable and CH–

3CH2O– is even less so.– is even less so.–

One way to reveal differences in acidity is to calculate the energy of deprotonation for different acids, e.g., for nitric acid.

HONO2 H+ + NO3–

This involves calculations on both the neutral acid and on the resulting anion (the energy of a proton is zero). An alternative approach, illustrated in this tutorial, involves comparison of electrostatic potential maps for different acids, with particular focus on the potential in the vicinity of the “acidic hydrogen”. The more positive the potential, the more likely dissociation will occur, and the stronger the acid.

1. Build nitric acid. Click on / to bring up the organic model kit. Select Nitro from the Groups menu and click anywhere on screen. Add sp3 oxygen to the free valence on nitrogen. Click on Click on Click /

. Build sulfuric acid. Select New Molecule (not New) from the File menu. Select Sulfone from the Groups menu and clickanywhere on screen. Add sp3 oxygen to both free valences on sulfur. Click on Click on Click / . Build acetic acid. Again select New Molecule. Select Carboxylic Acid from the Groups menu and click anywhere click anywhere clickon screen. Add sp3 carbon to the free valence at carbon. Click on Click on Click

/ . Finally, build ethanol. Select New Molecule and construct from two sp3 carbons and an sp3 oxygen. Click on Click on Click / , and then on / .

32 Tutorial VI

2. Bring up the Calculations dialog and specify calculation of equilibrium geometry using the HF/6-31G* model. Click on Click on Click OK. Bring up the Surfaces dialog and click on click on click Add... (at the bottom of the dialog). Select density from the Surface menu and potentialfrom the Property menu in the Add Surface dialog which appears. Click on Click on Click OK. Leave the Surfaces dialog on screen. Submit for calculation with the name “acids”.

3. When completed, bring up the spreadsheet and check the box check the box checkimmediately to the right of the molecule label for all four entries. (For Macintosh, you fi rst need to click on click on click at the bottom left of the screen.) The four molecules will now be displayed simultaneously on screen. Select (uncheck) uncheck) uncheck Coupled from the Model menu so that they may be independently manipulated, and arrange on screen such that the “acidic” hydrogens are visible.

Mouse operations normally refer only to the selected molecule. To rotate/translate molecules together, hold down the Ctrl (Control) key in addition to the left/right buttons, while moving the mouse.

4. Click on “density potential...” inside the Click on “density potential...” inside the Click Surfaces dialog. Electrostatic potential maps for all four acids will be displayed. Examine the potential in the vicinity of the acidic hydrogen (one of the two equivalent acidic hydrogens for sulfuric acid). Change the scale (color) to highlight differences in this region. Select Properties (Display menu) and clickon one of the maps. Type “0” and “90” inside the boxes underneath “Property Range” in the Surface Properties dialog. Press the Enter key (return key for Macintosh) following each data entry. “Blue” regions identify acidic sites, the more blue the greater the acidity. On this basis, rank the acid strength of the four compounds.

5. Remove “acids” and any open dialogs from the screen.

6. One after the other, build trichloroacetic, dichloroacetic, chloroacetic, formic, benzoic, acetic and pivalic acids. Put all into the same document (New Molecule instead of New following the fi rst molecule). Click on Click on Click / when you are fi nished.

Tutorial VI 33

acid pKa acid pKa

trichloroacetic (Cl3CCO2H) 0.7 benzoic (C6H5CO2H) 4.19dichloroacetic (Cl2CHCO2H) 1.48 acetic (CH3CO2H) 4.75chloroacetic (ClCH2CO2H) 2.85 pivalic ((CH3)3CCO2H) 5.03formic (HCO2H) 3.75

7. Note that the name of the presently selected molecule in the document appears at the bottom of the screen. This indicates that a calculated structure is available in Spartan’s database. Click on Click on Click to the left of the name, select “3-21G” from the entries and then click on click on click Replace All in the dialog which results. (For Macintosh, you need to select each molecule in turn and click on click on click Replace.) Structures obtained from HF/3-21G calculations will replace those you have built.

8. Enter the Calculations dialog (Calculations under the Setup menu) and specify a single-point-energy HF/3-21G calculation. Click on Click on ClickOK. Enter the Surfaces dialog (Surfaces under the Setup menu). Click on Click on Click Add..., select density from the Surface menu and potential from the Property menu in the Add Surface dialog which appears and then click on click on click OK. Leave the Surfaces dialog on screen. Submit for calculation. Name it “carboxylic acids”.

9. Bring up the spreadsheet. Expand it so that you can see all seven molecules, and that three data columns are available. Click inside Click inside Clickthe header cell for a blank column. Click on Click on Click Add... at the bottom of the spreadsheet, select Name from the list of entries and click on click on clickOK. The name of each molecule will appear. Next, click inside the click inside the clickheader cell of an available data column, type “pKa” and press the Enter key (return key for Macintosh). Enter the experimental pKa’s (see above) into the appropriate cells under this column. Press the Enter key (return key for Macintosh) following each entry. Finally, click inside the header cell of the next available data column and click inside the header cell of the next available data column and clicktype “potential”. Press the Enter key (return key for Macintosh).

10.After all calculations have completed, arrange the molecules such that the “acidic hydrogen” is visible. You need to check the box to check the box to checkthe right of the “Label” column in the spreadsheet for each entry, and select (uncheck) uncheck) uncheck Coupled from the Model menu. For Macintosh, you fi rst need to click on at the bottom left of the screen.

34 Tutorial VI

11.Click on “density potential...” inside the Click on “density potential...” inside the Click Surfaces dialog to turn on the electrostatic potential map for each molecule. Bring up the Properties dialog (Properties under the Display menu), remove the checkmark from Global Surfaces (Apply Globally for Macintosh), and click on the click on the click Reset button at the top of the dialog. The property range will now apply to the individual molecules. Enter the maximum value (most positive electrostatic potential) into the appropriate cell of the spreadsheet (under “potential”), and pressthe Enter key (return key for Macintosh).

12.Plot experimental pKa vs. potential. Bring up the Plots dialog (Plots under the Display menu), select pKa under the X Axis menu and potential from the Y Axes list, and click onclick onclick OK. The data points are connected by a cubic spline. For a least squares fi t, select Propertiesfrom the Display menu, click on the curve, and select click on the curve, and select click Linear LSQfrom the Fit menu in the Curve Properties dialog.

13.Close “carboxylic acids” and remove any remaining dialogs from the screen.

Tutorial VII 35

7Internal Rotation in n-Butane

This tutorial illustrates the steps required to calculate the energy of a molecule as a function of the torsion angle about one of its bonds, and to produce a conformational energy diagram.

Rotation by 1800 about the central carbon-carbon bond in n-butane gives rise to two distinct “staggered” structures, anti and gauche.

HHCH3

H HH HH HH HH HH HH HH HH HH HH HH HH HH HH HCH3

CH3CH3

CH3

H H

anti gauche

H

Both of these should be energy minima (conformers). The correct description of the properties of n-butane is in terms of a Boltzmann average of the properties of both conformers (for discussion see the essay “Total Energies and Thermodynamic and Kinetic Data”).

This tutorial shows you how to calculate the change in energy as a function of the torsion angle in n-butane, place your data in a spreadsheet and make a conformational energy diagram.

1. Click on / to bring up the organic model kit. Make n-butane from four sp3 carbons. Click on / to dismiss the model kit.

2. Set the CCCC dihedral angle to 00 (syn conformation). Click on Click on Click/ then, one after the other, click on the four carbon atoms in click on the four carbon atoms in click

sequence. Type “0” (00) into the text box to the right of “dihedral...” at the bottom right of the screen and press the Enter key (returnkey for Macintosh).

3. For Windows: Select Constrain Dihedral from the Geometrymenu. Click again on the same four carbons you used to defi ne the Click again on the same four carbons you used to defi ne the Clickdihedral angle, and then click on click on click at the bottom right of the screen. The icon will change to indicating that a dihedral constraint is to be imposed. For Macintosh: Click on Click on Click at the bottom right of the screen. The icon will change to indicating that a dihedral

36 Tutorial VII

constraint is to be imposed. Select Properties (Display menu) and click on the constraint marker on the model on screen. This leads to click on the constraint marker on the model on screen. This leads to clickthe Constraint Properties dialog.

4. Check DynamicCheck DynamicCheck inside the dialog. This extends the Constraint Properties dialog to allow the single (dihedral angle) constraint value to be replaced by a “range” of constraint values.

Leave the value of “0” (0°) in the box to the right of Value as it is, but change the contents of the box to the right of to to “180” (1800). Be sure to press the Enter key (return key for Macintosh) after you type in the value. The box to the right of Steps should contain the value “10”. If it does not, type “10” in this box and press the Enterkey (return key for Macintosh). What you have specifi ed is that the dihedral angle will be constrained fi rst to 0°, then to 20°*, etc. and fi nally to 180°. Click on Click on Click / to dismiss the dialog.

5. Bring up the Calculations dialog (Calculations... under the Setupmenu) and select Energy Profi le from the menu to the right of “Calculate”, and Semi-Empirical from the menu to the right of “with”. Click on Click on Click Submit at the bottom of the dialog. Provide the name “n-butane”.

* The difference between constraint values is given by: (fi nal-initial)/(steps-1).

Tutorial VII 37

6. When the calculations on all conformers have completed, they will go into a document named “n-butane.Profi le1” (“n-butane_prof” for -butane_prof” for -butane_profMacintosh). For Windows: Open this document ( / ). (You might wish to fi rst close “n-butane” to avoid confusion.) For Macintosh: The list of conformers is already open on screen and the “generator” has already been closed. Align the conformers to get a clearer view of the rotation. Select Align Molecules from the Geometry menu and, one after the other, click on either the fi rst three carbons or the click on either the fi rst three carbons or the clicklast three carbons. Then click on the click on the click Align button at the bottom right of the screen, and fi nally click on click on click / . Bring up the spreadsheet (Display menu), and enter both the energies relative to the 180° or anti conformer, and the CCCC dihedral angles. First, click on the click on the clicklabel (“M010”) for the bottom entry in the spreadsheet (this should be the anti conformer), then click on the header cell for the left click on the header cell for the left clickmost blank column, and fi nally, click on click on click Add... at the bottom of the spreadsheet. Select rel. E from among the selections in the dialog that results, kJ/mol from the Energy menu and click on click on click OK. Enter the dihedral angle constraints. For Windows: Click on Click on Click , click on the click on the clickconstraint marker and click on click on click at the bottom of the screen (to the right of the value of the dihedral angle constraint). For Macintosh: Click on Click on Click , click on the constraint marker and click on the constraint marker and click drag the name of the drag the name of the dragconstraint (Dihedral Constraint 1) at the bottom of the screen onto the spreadsheet. Finally, click on / .

7. Select Plots... (Display menu). Select Dihedral (Con1) from the items in the X Axis menu and rel. E(kJ/mol) from the Y Axes list. Click on Click on Click OK to dismiss the dialog and display a plot.OK to dismiss the dialog and display a plot.OK

38 Tutorial VII

You can see that the curve contains two minima, one at 180° (the anti form) and one around 60° (the gauche form). The former is lower in energy.

Further discussion of the potential energy surface for n-butane among other systems is provided in the essay “Potential Energy among other systems is provided in the essay “Potential Energy among other systems is provided in the essay “Surfaces”.

A common practice is to follow a calculation obtained from a simple theoretical model by a calculation using a “better” model. Furthermore, in order to save computer time, it is often assumed that the equilibrium geometry obtained from the simple model will be acceptable.

8. Make a copy of n-butane.Profi le1 ( / ). Name it “n-butane B3LYP”. Using this copy, specify calculation of B3LYP”. Using this copy, specify calculation of B3LYP Energy using the B3LYP/6-31G* model. Submit the job. Ten separate calculations are involved, the calculations will require a few minutes.

9. When completed, remake the plot of relative energy vs. dihedral angle. Note any changes to the shape of the curve and the positions and relative energies of the minima and maxima.

10.Remove any molecules and any remaining dialogs from the screen.

Tutorial VIII 39

8Ene Reaction

This tutorial illustrates the steps involved in fi rst guessing and then obtaining a transition state for a simple chemical reaction. Following this, it shows how to produce a “reaction energy diagram”.

The ene reaction involves addition of an electrophilic double bond to an alkene with an allylic hydrogen. The (allylic) hydrogen is transferred and a new carbon-carbon bond is formed, e.g.

HH1

23

5

4

The ene reaction belongs to the class of pericyclic reactions that includes such important processes as the Diels-Alder reaction and the Cope and Claisen rearrangements.

Spartan may be used to locate the transition-state for the ene reaction of ethylene and propene and then show the detailed motions that the atoms undergo during the course of reaction. It is easier to start from 1-pentene, the product of the ene reaction, rather than from the reactants.

1. Build 1-pentene. Click on / to bring up the organic model kit. Click on the Click on the Click Groups button, select Alkene from the menu and click anywhere on screen. Select spclick anywhere on screen. Select spclick 3 carbon ( ) and build a three-carbon chain onto one of the free valences on ethylene. Adjust the conformation of the molecule such that all fi ve carbons and one of the “hydrogens” (free valences) on the terminal methyl group form a “6-membered ring.” You can rotate about a bond by fi rst clickingon it (a red torsion marker appears on the bond) and then moving the mouse “up and down” while holding down on both the left button and the Alt key (spacebar for Macintosh). Do not minimize. Click on / .

2. Select Transition States from the Search menu (or click on the click on the click/ icon). Orient the molecule such that both the CH bond involving

40 Tutorial VIII

the hydrogen which will “migrate” and the C4-C5 bond are visible (see fi gure on previous page for numbering). Click on the CH bond Click on the CH bond Clickand then on the C4-C5 bond. An “arrow” will be drawn.* Orient the molecule such that both the C3-C4 and the C2-C3 bonds are visible. Click on the CClick on the CClick 3-C4 bond and then on the C2-C3 bond. A second arrow will be drawn. Orient the molecule such that the C1=C2 bond, C1 and the hydrogen (on C5) which will migrate are all visible. Click on the Click on the ClickC1=C2 bond and, while holding down on the Shift key, click on Cclick on Cclick 1

and then on the hydrogen. A third arrow will be drawn.

If you make a mistake, you can delete an arrow. Select Delete from the Build menu (or click on click on click / ), then click on the arrow and click on the arrow and clickfi nally click on click on click / . (For Windows, you can also delete an arrow by clicking on it while holding down on the clicking on it while holding down on the clicking Delete key.) When all is in order, click on click on click / at the bottom right of the screen. Your structure will be replaced by a guess at the transition state.

3. Select Calculations... (Setup menu). Select Transition State Geometry from the menu to the right of “Calculate” and Semi-Empirical from the menu to the right of “with”. Check IR to the right of “Compute”. For Windows: Click on Submit. For Macintosh: Click on Click on Click OK and then select OK and then select OK Submit from the Setup menu. Supply the name “ene transition state”.

4. When the calculation has completed animate the motion of atoms along the reaction coordinate. Select Spectra under the Displaymenu. Click on the top entry of the list in the Click on the top entry of the list in the Click Spectra dialog. It corresponds to an imaginary frequency, and will be designated with an “i” in front of the number**. Make certain that the associated

* Formally, this corresponds to migration of a pair of electrons from the tail of the arrow to the head. However, Spartan treats arrows simply as a convenient and familiar nomenclature with which to search its database of transition states.

** Recall from the tutorial “Infrared Spectrum of Acetone”, that frequency is proportional to the square root of the force constant divided by the reduced mass. The force constant associated with the reaction coordinate is negative because the energy “curves downward” at the transition state (see the essay “Potential Energy Surfaces”). Since the reduced mass is positive, the ratio of force constant to reduced mass is a negative number, meaning that its square root is an imaginary number.

Tutorial VIII 41

vibrational motion is consistent with the reaction of interest and not with some other process.

5. Controls at the bottom of the Spectra dialog allow for changing both the amplitude of Vibration (Amp) and the number of steps which make up the motion (Steps). The latter serves as a “speed control”. Change the amplitude to “0.3”. Type “0.3” in the box to the right of Amp and press the Enter key (return key for Macintosh). Next, click on click on click Make List at the bottom of the dialog. This will give rise to a document containing a series of structures which follow the reaction coordinate down from the transition state both toward reactant and product. To avoid confusion, it might be better to remove original molecule from the screen. Click on “Click on “Click ene transition state” (the vibrating molecule) and close it. Also remove the Spectra dialog by clicking on clicking on clicking / .

6. Enter the Calculations dialog (Calculations... under the Setupmenu) and specify a single-point energy calculation using the semi-empirical model (the same theoretical model used to obtain the transition state and calculate the frequencies). Make certain that Global Calculations (Apply Globally for Macintosh) at the bottom of the dialog is checked before you exit the dialog. Next, checked before you exit the dialog. Next, checkedenter the Surfaces dialog (Surfaces under the Setup menu) and specify evaluation of two surfaces: a bond density surface and a bond density surface onto which the electrostatic potential has been mapped. Click on Add . . ., select density (bond) for Surfaceand none for Property and click on click on click OK. Click on Click on Click Add . . . again, select density (bond) for surface and potential for Property and click on click on click OK. Make certain that Global Surfaces (Apply Globallyfor Macintosh) at the bottom of the dialog is checked. Leave the checked. Leave the checkedSurfaces dialog on screen.

7. Submit for calculation (Submit under the Setup menu). Name it “ene reaction”. Once the calculations have completed, enter the Surfacesdialog and, one after the other, select the surfaces which you have calculated. For each, step through the sequence of structures ( and

) keys at the bottom of the screen) or animate the reaction ( ). Note, in particular, the changes in bonding revealed by the bond density surface. Also pay attention to the “charge” on the migrating atom as revealed by the sequence of electrostatic potential maps. Is

42 Tutorial VIII

it best described as a “proton” (blue color), hydrogen atom (green color) or “hydride anion” (red color)?

Further discussion of the use of electrostatic potential maps to investigate charge distributions is provided in the essay “Electrostatic Potential Maps: Charge Distributions”.

9. Close “ene reaction” as well as any remaining dialogs.

Section C 43

Section CEssays

The following section comprises a series of “Essays” on topics related to the underpinnings of quantum chemical models in general and to the specific models available in the Spartan Physical Chemistry Edition. They are deliberately brief and non-mathematical and are intended primarily as a first exposure.

The first essay describes potential energy surfaces, and defines what is meant by an equilibrium and transition-state structure. The next two essays address the origins of the quantum chemical models available in the Spartan Physical Chemistry Edition, and provide broad guidelines for selecting a particular model for the task at hand. The fourth essay discusses the use of energy data that comes out of quantum chemical calculations to provide reaction thermochemistry and kinetics. The fifth essay outlines techniques for locating and verifying equilibrium and transition state structures. The last four essays discuss graphical models available in the Spartan Physical Chemistry Edition: molecular orbitals, electron densities, electrostatic potential maps, and local ionization potential and LUMO maps. These illustrate how each of the different graphical models can be employed to provide insight in molecular structure and bonding and chemical reactivity.

Associated with Essays 1, 6, 7, 8 and 9 are a series of Spartan files grouped in the “essays” directory on the CD-ROM. File names are specified in the individual essays.

Essay I 45

1Potential Energy Surfaces

One Dimensional Energy Surfaces

Consider a plot depicting the change in energy of ethane as a function of the angle of torsion (dihedral angle) around the carbon-carbon bond.

0° 60° 120° 180° 240° 300° 360°

12 kJ/mol

H

H HH HH HH HH HH HH HH HH HH HH HH H

H HH HH HH HH HH HH HH HH HH H

H

H

H HH HH HH HH HH HH HH HH HH HH HH H

H HH HH HH HH HH HH HH HH HH H

H

H

H HH HH HH HH HH HH HH HH HH HH HH H

H HH HH HH HH HH HH HH HH HH H

H

HH

H HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH H

HH

H HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH H

HH

H HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH H

energy

HCCH dihedral angle

Full 360° rotation leads to three identical “staggered” structures that are energy minima, and three identical “eclipsed” structures that are energy maxima. The difference in energy between eclipsed and staggered structures of ethane, termed the barrier to rotation, is known experimentally to be 12 kJ/mol. Note, that any physical measurements on ethane pertain only to its staggered structure, or more precisely the set of three identical staggered structures. Eclipsed ethane “does not exist” in the sense that it cannot be isolated and characterized.

Open “ethane rotation”. The image that appears is one frame in a sequence depicting rotation about the carbon-carbon bond in ethane. Click on Click on Clickand at the bottom left of the screen to look at other frames. Verify that the staggered structures correspond to minima on the energy plot and that the eclipsed structures correspond to maxima. Click on Click on Click to animate the sequence. Close “ethane rotation” when you are fi nished.

46 Essay I

Next, consider a plot of energy vs. the dihedral angle involving the central carbon-carbon bond in n-butane.

energy

CCCC dihedral angle

CH3CH3

H HH H

CH3

H H

H CH3

H

0° 60° 120° 180° 240° 300° 360°

19kJ/mol

3.8 kJ/mol

16kJ/mol

gauche anti gauche

HCH3

H HH CH3

HCH3

H HCH3 H

CH3

H H

H H

CH3

CH3

H H

CH3 H

H

This plot also reveals three energy minima, corresponding to staggered structures, and three energy maxima, corresponding to eclipsed structures. In n-butane, however, the three structures in each set are not all identical. Rather, one of the minima, corresponding to a dihedral angle of 180° (anti structure), is lower in energy and distinct from the other two gauche minima (dihedral angles around 60° and 300°), which are identical. Similarly, one of the energy maxima corresponding to a dihedral angle of 0°, is distinct from the other two maxima (with dihedral angles around 120° and 240°), which are identical. As with ethane, eclipsed forms of n-butane do not exist. Unlike ethane, which is a single compound, any sample of n-butane is made up of two distinct compounds, anti n-butane and gauche n-butane. The relative abundance of the two compounds as a function of temperature is given by the Boltzmann equation (see the essay “Total Energies and Thermodynamic and Kinetic Data”).

Essay I 47

Open “n-butane rotation”. The image that appears is one frame of a sequence depicting rotation about the central carbon-carbon bond in n-butane. Click on Click on Click and at the bottom left of the screen to look at other frames. Verify that the staggered structures correspond to minima on the energy plot and that the eclipsed structures correspond to maxima. Also, verify that the anti structure is lower in energy than the gauche structure. Click on Click on Click to animate the sequence. Close “n-butane rotation” when you are fi nished.

Both of these processes have been depicted in terms of the change in energy with change in a single geometrical coordinate. This is an oversimplifi cation. While the “important” geometrical coordinate may clearly be identifi ed as a torsion involving one particular carbon-carbon bond, bond lengths and angles no doubt also change during rotation.

Many Dimensional Energy Surfaces

It will usually not be possible to identify a single geometrical coordinate to designate a conformational change or a chemical transformation. Consider the potential energy surface for interconversion of chair forms of cyclohexane.

reaction coordinate

transitionstate

transitionstate

twist boatchair chair

energy

In this case, the geometrical coordinate connecting stable forms is not identifi ed, but is referred to simply as the “reaction coordinate”. Also note that the energy maxima have been designated as “transition states”. This refl ects the fact that their structures may not be as simply described as the energy maxima for rotation in ethane and ndescribed as the energy maxima for rotation in ethane and ndescribed as the energy maxima for rotation in ethane and -butane.

The energy surface for chair interconversion in cyclohexane, like that for n-butane, contains three distinct energy minima, two of lower energy

48 Essay I

identifi ed as “chairs”, and one of higher energy identifi ed as a “twist boat”. The energy difference between the chair and twist boat structures is around 23 kJ/mol, large enough that only the former can be observed at normal temperatures. There are two identical transition states identifi ed as “boats”. These lead from the chair to the twist boat structures.

Open “cyclohexane chair interconversion”. The image that appears is one frame in a sequence depicting chair interconversion in cyclohexane. Click on Click on Click

and at the bottom left of the screen to look at other frames. Verify that the three minima on the energy plot correspond to staggered structures and that the two maxima correspond to eclipsed structures. Also, verify that the twist boat structure is higher in energy than the chair structures. Click on Click on Clickto animate the sequence. Note that the overall process appears to occur in two steps, one step leading up to the twist boat and the other step leading away from it. Close “cyclohexane chair interconversion” when you are fi nished.

Ethane, n-butane and cyclohexane exemplify the types of motions that molecules may undergo. Their potential energy surfaces are special cases of a general type of plot in which the variation in energy is given as a function of reaction coordinate. Such a plot is known as a reaction coordinate diagram.

energy

reaction coordinate

transition state

reactant

product

The positions of the energy minima along the reaction coordinate give the equilibrium structures of the reactant and product. Similarly, the position of the energy maximum gives the structure of the transition state. Both energy minima (“stable molecules”) and the energy maximum (transition state) are well defi ned. However, there are many possible paths connecting reactants and products. Liken this to climbing a mountain. The starting and ending points are well defi ned as is the summit, but there can be many possible routes.

Essay I 49

Equilibrium structure (geometry) may be determined from experiment, given that the molecule can be prepared and is suffi ciently long-lived to be subject to measurement. On the other hand, the geometry of a transition state may not be experimentally established. This is simply because a transition state is not a “molecule” that may be “trapped” in an “energy well”. Therefore, it is impossible to establish a population of molecules on which measurements may be performed.

Both equilibrium and transition-state structures may be determined from quantum chemical calculations. The fact that the former is an energy minimum on a potential surface while the latter is not is of no consequence. Equilibrium and transition-state structures can be distinguished from one another not only from a reaction coordinate diagram, but more generally by determining the complete set of vibrational motions and associated vibrational frequencies. (The latter are the same quantities measured by infrared spectroscopy.) Systems for which all frequencies are real numbers are “stable” molecules (energy minima), while transition states are characterized by having one (and only one) vibrational frequency which is an imaginary number. The coordinate (vibrational motion) associated with this imaginary frequency is the reaction coordinate.

50 Essay II

2Theoretical Models

A wide variety of procedures or “models” have been developed to calculate molecular structure, energetics and other properties. These may be broken down into two categories, quantum chemical models, that ultimately derive from the Schrödinger equation, and molecular mechanics models.

The Schrödinger Equation

Practical quantum chemical methods seek an approximate solution to the deceptively simple-looking Schrödinger equation.

HΨ = εΨˆ

In this equation, H is termed the “Hamiltonian operator”; it describes both the kinetic energies of the nuclei and electrons that make up the molecule, as well as the electrostatic interactions felt between the nuclei and electrons. Nuclei are positively charged and repel other nuclei. Electrons are negatively charged and repel other electrons. However, nuclei and electrons attract each other. The quantity ε in the Schrödinger equation is the energy of the system, and Ψ is termed a “wavefunction”. While the wavefunction has no obvious physical meaning, the square of the wavefunction times a small volume element gives the probability of finding the system at a particular set of coordinates.

The Schrödinger equation has been solved exactly for the hydrogen atom (a one-electron system), where its solutions are actually quite familiar to chemists as the s, p, d atomic orbitals.

s orbital p orbital d orbital

They correspond to the ground and various excited states of the hydrogen atom.

ˆ

Essay II 51

Hartree-Fock Molecular Orbital Models

Although the Schrödinger equation may easily be written down for many-electron atoms as well as for molecules, it cannot be solved. Approximations must be made. So-called Hartree-Fock molecular orbital models, or simply, molecular orbital models, start from the Schrödinger equation and then make three approximations:

1. Separation of nuclear and electron motions (“Born-Oppenheimer approximation”). What this says is that “from the point of view of the electrons, the nuclei are stationary”. This removes the mass from what is now referred to as the electronic Schrödinger equation.

2. Separation of electron motions (“Hartree-Fock approximation”). This represents the many-electron wavefunction as a sum of products of one-electron wavefunctions, the spatial parts of which are termed “molecular orbitals”. This replaces the problem of simultaneously accounting for the motions of all the electrons to the much simpler problem of accounting for the motion of a single electron in an environment made up of the nuclei and all the remaining electrons.

3. Representation of the molecular orbitals as linear combinations of atom-centered basis functions or “atomic orbitals” (“LCAO approximation”). This replaces the problem of fi nding the best functional form for the molecular orbitals to the much simpler problem of fi nding the best set of linear coeffi cients. The limit of a “complete” set of basis functions is the so-called “Hartree-Fock limit”.

Hartree-Fock models differ in the number and kind of atomic basis functions, and their computational cost increases as the fourth power of the number of basis functions. The simplest models utilize a “minimal basis set” of atomic orbitals, that includes only those functions required to hold all the electrons on an atom and to maintain spherical symmetry. Minimal basis sets are too restrictive to properly describe molecular properties, and “split-valence basis sets” like the 3-21G basis set, that incorporate two sets of valence atomic orbitals, or “polarization basis sets”, like the 6-31G* basis set, that also include atomic orbitals of higher angular type than are occupied in the atom in its ground state, for example, d-type atomic orbitals in the case of main-group elements, are needed. Hartree-Fock molecular orbital models using split-valence or

52 Essay II

polarization basis sets, have become a mainstay for routine and reliable descriptions of the structures, stabilities and other molecular properties. The 6-311+G** basis set, that also includes diffuse functions, mimics the properties of the Hartree-Fock limit.

Correlated Models

Hartree-Fock models treat the motions of individual electrons independently, in that they replace “instantaneous interactions” between electrons by interactions between one electron and the average field created by all other electrons. As a consequence, electrons “get in each others way” to a greater extent than they should. This leads to overestimation of the electron-electron repulsion energy and to a total energy that is too high. Electron correlation accounts for coupling or “correlation” of electron motions, and leads to reduction of the electron-electron repulsion energy and to a lowering of the total energy. The correlation energy is defined as the difference between the Hartree-Fock energy and the experimental energy.

Two practical “correlated” approaches are provided by density functional models and Møller-Plesset models. Density functional models explicitely introduce an “approximate” correlation term. They are not much more “costly” in terms of computation time than Hartree-Fock models, although it is not apparent how to improve on a particular choice of correlation correction. Møller-Plesset models extend the flexibility of Hartree-Fock models by mixing ground-state and excited-state wavefunctions. They are significantly more costly than both Hartree-Fock and density functional models, but offer the advantage over the latter of a clear path to improvement. In the limit of “complete mixing” Møller-Plesset models lead to the “exact result”, although in practice this limit cannot be reached.

The B3LYP/6-31G* density functional model and the MP2/6-31G* density functional model are among the most popular correlated models. Both are effective in addressing the known deficiencies of Hartree-Fock models but are still simple enough for routine use. B3LYP/6-311+G** and MP2/6-311+G** models are not practical for routine use on all but very small molecules. However, they mimic the properties of limiting B3LYP and MP2 models.

Essay II 53

Semi-Empirical Molecular Orbital Models

The principal disadvantage of Hartree-Fock and correlated models is their computational cost. Additional approximations may be introduced to significantly reduce cost while still retaining the underlying quantum mechanical formalism. “Semi-empirical” molecular orbital models follow in a straightforward way from Hartree-Fock models:

1. Elimination of overlap between functions on different atoms (“NDDO approximation”). In effect, this means that atoms “don’t see each other”. This is drastic but reduces computation by more than an order of magnitude over Hartree-Fock models.

2. Restriction to a “minimal valence basis set” of atomic functions, meaning that inner-shell (core) functions are removed. Because of this, the cost of doing a calculation involving a second-row element, for example, silicon, is no more than that incurred for the corresponding fi rst-row element, for example, carbon.

3. Introduction of parameters to reproduce specifi c experimental data, for example, equilibrium geometries and heats of formation. This distinguishes among semi-empirical models. The PM3 model is perhaps the most widely parameterized semi-empirical model.

Molecular Mechanics Models

The alternative to quantum chemical models are molecular mechanics models. These start from a simple but “chemically reasonable” picture of molecular structure, a so-called “force field”. In this picture, molecules are made up of atoms (as opposed to nuclei and electrons), some of which are connected (“bonded”). Both crowding (“van der Waals”) and charge-charge (“Coulombic”) interactions between atoms are then considered, and atom positions are adjusted to best match known structural data (bond lengths and angles).

Molecular mechanics requires an explicit description of “chemical bonding”, as well as a large amount of information about the structures of molecules. This biases results and seriously limits its predictive value. Nevertheless, molecular mechanics has become an important tool to establish equilibrium geometries of large molecules and equilibrium conformations of highly-flexible molecules.

54 Essay III

3Choosing a Theoretical Model

No single method of calculation is likely to be ideal for all applications. A great deal of effort has been expended to defi ne the limits of different molecular mechanics and quantum chemical models, to judge the degree of success and to point out failures of different models. Success or failure can be assessed by comparison of model results with known (experimental) data. Molecular mechanics models are restricted to determination of geometries and conformations of stable molecules. Experimental structure data are widely available, primarily from X-ray crystallography, whereas detailed information about the preferred conformation of isolated molecules is quite limited. X-ray diffraction provides information about conformations of molecules as crystalline solids and about molecules complexed to proteins (also in the crystalline state). The relevance of solid phase data to the conformations of isolated molecules is, however, questionable, given that crystal packing and binding energies are of the same order of magnitude as conformational energies. Quantum chemical models also provide energy data, that may in turn be directly compared with experimental thermochemical data, as well as infrared spectra and properties such as dipole moments, that may be compared directly with the corresponding experimental quantities. Quantum chemical models may also be applied to transition states. While there are no experimental structures with which to compare, experimental kinetic data may be interpreted to provide information about activation energies.

“Success” is not an absolute. Different properties and different problems may require different accuracy to be of value. Neither is success suffi cient. A model also needs to be “practical” for the task at hand. The nature and size of the system needs to be taken into account, as do the available computational resources and the experience and “patience” of the practitioner. Practical models are not likely to provide the “best possible” treatments that have been formulated. Compromise is an essential component of model selection.

Essay III 55

Specifi cation of a quantum chemical model is in terms of a method, Hartree-Fock, B3LYP (density functional) or MP2, and a “basis set”, that is, a set of atom-centered functions (atomic orbitals) from which delocalized functions (molecular orbitals) are to be built. The numbers “3” and “6” to the left of the “-” in the 3-21G and 6-31G* basis sets indicate that 3 and 6 functions (“primitives”) are used to describe each inner-shell atomic function. The numbers “21” and “31” to the right of the “-” indicate that two groups of 2 and 1 and 3 and 1 primitives are used to describe each valence-shell atomic function. “G” is used to specify that the primitives are Gaussian type functions, and “*” designates that additional (higher-order) valence functions are provided for heavy (non-hydrogen) atoms. The second “*” for the 6-311+G** basis set indicates that additional (higher-order) are also provided for hydrogens, and the “+” indicates that diffuse functions are provided on heavy atoms.

The table below provides an overview of the performance of the molecular mechanics model and fi ve of the eight quantum chemical models available in your edition of Spartan, for the calculation of equilibrium and transition-state geometries, conformation and thermochemistry. Hartree-Fock, B3LYP and MP2 models with the 6-311+G** basis set (also available in this edition) have not been considered. These models have been made available primarily to establish “near limits” for these three types of calculations rather than to function as practical calculation methods. Three different “grades” have been assigned: G is good, C is good with cautious application and P is poor. NA signifi es “not applicable”.

molecular PM3 semi- Hartree-Fock B3LYP MP2task mechanics empirical 3-21G 6-31G* 6-31G* 6-31G*

geometry C G G G G G(organic)

geometry NA G P P G P(metals)

transition-state NA C G G G Ggeometry

conformation G P C G G G

thermochemistry NA P C C G G

computation time low ---------------------------------------------------- > high

56 Essay III

In Terms of Task

i) All models provide a good account of equilibrium geometries for organic and main-group inorganic molecules and, where they are applicable, of transition-state geometries. (Transition-state geometries can only be judged by assuming that “very good” quantum chemical calculations provide a truthful account.) HF/3-21G, HF/6-31G* and MP2/6-31G* models do not provide a reliable account of the geometries of compounds incorporating transition metals, but the PM3 semi-empirical model, that has been especially parameterized for this task, generally provides a reasonable account. The B3LYP/6-31G* model also provides a good account.

ii) Molecular mechanics models generally provides a good account of conformational energy differences in organic compounds. The PM3 semi-empirical model and the HF/3-21G model are suitable for identifying conformational minima, and for determining the geometries of these minima, but they are not suitable for providing accurate relative conformer energies. The HF/6-31G*, B3LYP/6-31G* and MP2/6-31G* models generally provide a good description of conformational energy differences. As previously mentioned, experimental data for isolated molecules are limited to very simple systems.

iii)The HF/6-31G* model generally provides an acceptable account of the energetics of reactions that do not involve bond making or breaking. The HF/3-21G model is less satisfactory. Neither model provides an acceptable account of the energetics of reactions that involve bond breaking. B3LYP/6-31G* and MP2/6-31G* models generally provide good accounts of all types of reactions, including reactions where bonds are made or broken.

Neither the HF/3-21G nor Hartree-Fock/6-31G* model provides an acceptable account of absolute activation energies, but both models generally provide an excellent description of relative activation energies. The B3LYP/6-31G* and MP2/6-31G* models provide good descriptions of both absolute and relative activation energies.

The PM3 semi-empirical model is unsatisfactory in describing the energetics of all types of processes.

Essay III 57

In Terms of Model

Molecular mechanics models are restricted to the description of equilibrium geometry and conformation. They perform reasonably well.

The PM3 semi-empirical model is suitable for calculation of:

i) Equilibrium and transition-state geometries, where the cost of Hartree-Fock and correlated models may be prohibitive.

ii) Equilibrium and transition-state geometries involving transition metals, where Hartree-Fock models are known to produce poor results and the cost of B3LYP models may be prohibitive.

The PM3 semi-empirical model is unsuitable for calculation of:

i) Reaction energies. ii) Conformational energy differences.

The HF/3-21G and HF/6-31G* models are suitable for calculation of:

i) Equilibrium and transition-state geometries of organic and main-group inorganic molecules.

ii) Reaction energies (except for reactions involving bond making or breaking).

The HF/3-21G and HF/6-31G* models are unsuitable for calculation of:

i) Reaction energies that involve bond making or breaking and absolute activation energies.

ii) Equilibrium and transition-state geometries for transition-metal inorganic and organometallic molecules.

The B3LYP/6-31G* and MP2/6-31G* models are suitable for calculation of:

i) Equilibrium and transition-state geometries for all classes of molecules (except MP2/6-31G* for molecules with transition metals).

ii) Reaction energies including those for reactions that involve bond making and breaking.

The MP2/6-31G* model is unsuitable for calculation of:

i) Equilibrium and transition-state geometries for transition-metal inorganic and organometallic molecules.

58 Essay IV

4Total Energies and

Thermodynamic andKinetic Data

In addition to geometry, another important quantity to come out of a quantum chemical calculation is the energy. Energy can be used to reveal which of several isomers is most stable, to determine whether a chemical reaction will have a thermodynamic driving force (an “exothermic” reaction) or be thermodynamically uphill (an “endothermic” reaction), and to ascertain how fast a reaction will proceed.

There is more than one way to express the energy of a molecule. Most familiar to chemists is as a heat of formation, ∆Hf. This is the enthalpy of a hypothetical reaction that creates a molecule from well defined but arbitrary “standard states” of its constituent elements. For example, ∆Hf for methane is the enthalpy required to create it from graphite and f for methane is the enthalpy required to create it from graphite and f

H2, the “standard states” of carbon and hydrogen, respectively. The heat of formation cannot be directly measured, but it must be obtained indirectly. Heats of formation may be either positive or negative and are typically reported in units of kJ/mol.

An alternative, total energy or more simply energy, is the energy of a hypothetical reaction that creates a molecule from a collection of separated nuclei and electrons. Unlike the heat of formation, the “zero” in the energy scale has a physical meaning as that appropriate for fully separated particles. An important special case is the proton that has a total energy of zero. Like the heat of formation, energy cannot be measured directly, and is used solely to provide a standard method for expressing and comparing energies. Energies are always negative numbers and, in comparison with the energies of chemical bonds, are very large. By convention, they are expressed in “so-called” atomic units*:

* The “exact” energy of hydrogen atom is -0.5 atomic units.

Essay IV 59

1 au = 2625 kJ/mol

It makes no difference which “standard” (heat of formation or energy) is used to calculate the thermochemistry of balanced chemical reactions (reactant 1 + reactant 2 + . . . → product 1 + product 2 + . . .):

∆E(reaction) = Eproduct 1 + Eproduct 2 + … – Ereactant 1 - Ereactant 2 – …

We will use energies. A negative ∆E indicates an exothermic (thermodynamically favorable) reaction, whereas a positive ∆E indicates an endothermic (thermodynamically unfavorable) reaction.

A special case involves differences in isomer stability. This is the energy of a chemical reaction in which the “reactant” is one isomer and the “product” is another isomer (isomer 1 → isomer 2):

∆E(isomer) = Eisomer 2 – Eisomer 1

A negative ∆E means that isomer 2 is more stable than isomer 1.

Total energies may also be used to calculate activation energies, ∆E‡:

∆E‡ = Etransition state – Ereactant 1 – Ereactant 2 – …

Here, Etransition state is the energy of the transition state. Activation energies are always positive numbers*, meaning that the transition state is less stable that reactants.

Reaction and activation energies tell us whether a reaction is exothermic or endothermic or whether it proceeds with a small or a large activation barrier. There are, however, other situations where energies need to be replaced by “Gibbs energies” in order to take account of entropy. For example, a proper account of the equilibrium concentrations of reactants and products requires calculation of the equilibrium constant, Keq, which according to the Boltzmann equation, is related to the Gibbs energy of reaction, ∆Grxn:

Keq = exp(–∆Grxn/RT)

* Note, however, that some reactants proceed with zero activation energy, meaning no transition state can be identifi ed. Further discussion is provided in the essay “Finding and Verifying Equilibrium and Transition-State Geometries”.

60 Essay IV

Here R is the gas constant and T is the temperature (in K). At room temperature (298K) and for ∆Grxn in au, this is given by:

Keq = exp(–1060 ∆Grxn)

∆Grxn has two components, the enthalpy of reaction, ∆Hrxn, and the entropy of reaction, ∆Srxn. These are defined as follows:

∆Grxn = ∆Hrxn – T∆Srxn

∆Hrxn ≈ ∆Erxn = Eproduct 1 + Eproduct 2 + … – Ereactant 1 – Ereactant 2 – …

∆Srxn = Sproduct 1 + Sproduct 2 + … – Sreactant 1 – Sreactant 2 – …

Although ∆Grxn depends on both enthalpy and entropy, there are many reactions for which the entropy contribution is small, and can be ignored. Further assuming that ∆Hrxn ≈ ∆Erxn

*, equilibrium constants can then be estimated from the Boltzmann equation:

Keq ≈ exp(–∆Erxn/RT) ≈ exp(-1060 ∆Erxn)

This Boltzmann equation may also be used to establish the equilibrium composition of a mixture of isomers:

Isomer 1 Isomer 2 Isomer 3 . . .

% Isomer i =100 exp (-1060 EIsomer i)

kΣ exp (-1060 EIsomer k)

Isomer energies, Eisomer, are given in atomic units relative to the energy of the lowest-energy isomer. An important special case is that involving an equilibrium between two isomers:

Isomer 1 Isomer 2

= exp [-1060 (Eisomer2 – Eisomer1)][ Isomer 2 ][ Isomer 1 ]

Reaction rate constants, krxnReaction rate constants, krxnReaction rate constants, k , are also related to Gibbs energies. As before, if entropy contributions can be neglected, the rate constant can be obtained directly from the activation energy, ∆E‡, according to the Arrhenius equation:

* The difference, ∆(PV), is likely to be insignifi cant for reactions carried out under “normal” conditions.

Essay IV 61

krxn ≈ (kBT/h)[exp(-∆E‡/RT)]

Here kB and h are the Boltzmann and Planck constants, respectively. At room temperature and for ∆E‡ in au, krxn in au, krxn in au, k is given by:

krxn = 6.2x1012 exp(-1060 ∆E‡)

Another way to describe reaction rates is by half-life, t1/2, the amount of time it takes for the reactant concentration to drop to one half of its original value. When the reaction follows a first-order rate law, rate = -krxn-krxn-k [reactant], t1/2 is given by:

t1/2 = ln2/krxn = 0.69/krxn

It is useful to associate reaction energies and reaction rates with potential energy diagrams (see the essay “Potential Energy Surfacespotential energy diagrams (see the essay “Potential Energy Surfacespotential energy diagrams (see the essay “ ”). The connections are actually quite simple.

The thermodynamics of reaction is given by the relative energies of the reactant and product on the potential surface, while the kinetics of reaction is given by the relative energies of the reactant and transition state.*

energy

reaction coordinate

"thermodynamics"

transition state

reactant

product

"kinetics"

The above diagram depicts an exothermic reaction. According to thermodynamics, equilibrium favors product. Chemical reactions can also be endothermic, meaning that the energy of the products is higher than that of the reactants. Here, equilibrium favors reactant over product.

* While absolute reaction rate also depends on the concentrations of the reactants and on such factors as the “likelihood” that encounters between molecules will actually lead to reaction, generally speaking, the lower the activation energy, the faster the reaction.

62 Essay IV

Where two or more different products may form in a reaction, thermodynamics tells us that if “we wait long enough”, the product formed in greatest abundance will be that with the lowest energy irrespective of pathway.

reaction coordinate

energy

thermodynamic product

In this case, the product is referred to as the “thermodynamic product” and the reaction is said to be “thermodynamically controlled”.

The product formed in greatest amount in a “kinetically controlled” reaction is referred to as the “kinetic product”. It is the product that forms from the lowest energy transition state, irrespective of whether or not this is lowest energy product (the thermodynamic product).

reaction coordinate

energy

kinetic product

Kinetic product ratios depend on activation energy differences just as thermodynamic product ratios depend on the difference in reactant and product energies.

Essay V 63

5Finding and Verifying

Equilibrium andTransition-State Geometries

The energy of a molecule depends on its geometry. Even small changes in geometry can lead to quite large changes in energy. Proper choice of molecular geometry is therefore quite important in carrying out quantum chemical calculations. Experimental geometries would certainly be suitable, and are available for many stable molecules*. However, experimental geometries for reactive or otherwise short-lived molecules are scarce, and experimental data for transition states are completely lacking. Here, there is no alternative to obtaining geometries from calculation.

Determination of geometry (“geometry optimization”) is an iterative process. The energy and energy “gradient” (fi rst derivatives of the energy with respect to all geometrical coordinates) are calculated for some initial geometry, and this information is then used to project a new geometry. This process continues until three criteria are satisfi ed: First, the gradient must closely approach zero. This insures that the optimization is terminating in a “fl at region” of the potential surface (either the “bottom” in the case of equilibrium geometry or the “top” in the case of transition-state geometry). Second, successive iterations must not change any geometrical parameter by more than specifi ed (small) value. Third, successive iterations must not change the energy by more than a specifi ed (small) value.

* The vast majority of experimental structures derive from X-ray diffraction on crystalline solids, and might be expected to differ from gas-phase geometries due to requirements of crystal packing. However, numerous comparisons with experimental gas-phase geometries (primarily from microwave spectroscopy) show that these differences are small.

64 Essay V

Equilibrium Geometries

In order for a geometry to correspond to an energy minimum, the curvature of the energy surface must be positive, that is, the structure must lie at the “bottom” of an energy well. The surface’s curvature is defi ned by the “Hessian” (the matrix of second derivatives of the energy with respect to geometrical coordinates).

In practice, one needs to fi nd a set of geometrical coordinates (“normal coordinates”) in which the Hessian will be diagonal, that is, all off-diagonal elements will be zero. In this representation, all (diagonal) elements must be positive for the geometry to correspond to an energy minimum. “Normal coordinate analysis” as it is termed is required for the calculation of vibrational frequencies that relate directly to the square root of the elements of the (diagonal) Hessian. Positive Hessian elements yield real frequencies; negative Hessian elements yield imaginary frequencies.

Geometry optimization does not guarantee that the fi nal structure has a lower energy than any other structure of the same molecular formula. All that it guarantees is a “local minimum”, that is, a geometry the energy of which is lower than that of any similar geometry, but which may still not be the lowest energy geometry possible for the molecule. Finding the absolute or “global minimum” requires repeated optimization starting with different initial geometries. Only when all local minima have been located is it possible to say with certainty that the lowest energy geometry has been identifi ed.

In principle, geometry optimization carried out in the absence of symmetry, that is, in C1 symmetry, must result in a local minimum. On the other hand, imposition of symmetry may result in a geometry that is not a local minimum. For example, optimization of ammonia constrained to a planar trigonal geometry (D3h symmetry) will result in a structure that corresponds to an energy maximum in the direction of motion toward a puckered trigonal geometry (C3v symmetry). This is the transition state for inversion at nitrogen. The most conservative tactic is always to optimize geometry in the absence of symmetry. If this is not done, it is always possible to verify that the structure located indeed corresponds to a local minimum by performing a normal-coordinate analysis on the fi nal (optimized) structure. This analysis should yield all real frequencies.

Essay V 65

Transition-State Geometries

A transition state is a structure that lies “at the top of” a potential energy surface connecting reactant and product (see the essay “Potential surface connecting reactant and product (see the essay “Potential surface connecting reactant and product (see the essay “Energy Surfaces”).

transition state

reactant

product

Energy

reaction coordinate

More precisely, a transition state is a point on the potential energy surface for which the gradient is zero (just as it is for an equilibrium geometry; see preceding discussion), but for which the diagonal representation of the Hessian has one and only one negative element, corresponding to the “reaction coordinate” (see diagram above). All the other elements are positive. In other words, a transition state is a structure that is an energy minimum in all dimensions except one, for which it is an energy maximum. Such a point is referred to as a fi rst-order saddle point.

The geometries of transition states on the pathway between reactants and products may not be as easily anticipated as the equilibrium geometries of the reactants and products. This is not to say that they do not exhibit systematic properties as do “normal” molecules, but rather that there is not suffi cient experience to identify what systematics do exist, and how to capitalize on structural similarities. It needs to be recognized that transition states cannot even be detected let alone characterized experimentally, at least not directly. While measured activation energies relate to the energies of transition states above reactants, and while activation entropies and activation volumes as well as kinetic isotope effects may be interpreted in terms of transition-state structure, no experiment can actually provide direct information about the detailed geometries and/or physical properties of transition states. Quite simply, transition states do not exist in terms of a stable population of molecules on which experimental measurements may be

66 Essay V

made. Experimental activation parameters may act as a guide, although here too it should be pointed out that their interpretation is in terms of a theory (“transition state theory”). This assumes that all molecules proceed over a single transition state (the high point along the reaction coordinate) on their way to products. Even then, experiments tell little about what actually transpires in going from reactants to products.

Lack of experience about “what transition states look like” is one reason why their detailed geometries are more diffi cult to obtain than equilibrium geometries. Other reasons include:

1. Techniques for locating transition states are less well developed than procedures for fi nding equilibrium structures. Minimization is an important task in diverse fi elds of science and technology, whereas saddle point location has few if any important applications outside of chemistry.

2. It is likely that the energy surface in the vicinity of a transition state is more “fl at” than the surface in the vicinity of a local minimum. After all, transition states represent a delicate balance of bond breaking and bond making, whereas overall bonding is maximized in equilibrium structures. As a consequence, the energy surface in the vicinity of a transition state may be less well described in terms of a quadratic function (assumed in all common optimization procedures) than the surface in the vicinity of a local minimum.

3. Transition states incorporate partially (or completely) broken bonds, and it is to be expected that simple theoretical models will not be able to provide satisfactory descriptions.

The same iterative procedure used for optimization of equilibrium geometry applies to transition states. However, the number of “steps” required for completion is likely to be much larger.

Having found a transition-state geometry, two “tests” need to be performed in order to verify that it actually corresponds to a “proper” transition state, and further that it actually corresponds to the transition state for the process of interest, i.e., it smoothly connects energy minima corresponding to reactant and product:

1. Verify that the Hessian yields one and only one imaginary frequency. This requires that a normal mode analysis be carried out. The

Essay V 67

imaginary frequency will typically be in the range of 400-2000 cm-1, quite similar to real vibrational frequencies. In the case of fl exible rotors, for example, methyl groups, or “fl oppy rings”, the analysis may yield one or more additional imaginary frequencies with very small (<100 cm-1) values. While these can usually be ignored, verify what motions these small imaginary frequencies actually correspond to before doing so. Be wary of structures that yield only very small imaginary frequencies, as none of these may correspond to the particular reaction of interest.

2. Verify that the normal coordinate corresponding to the imaginary frequency smoothly connects reactants and products. A simple way to do this is to “animate” the normal coordinate corresponding to the imaginary frequency, that is, to “walk along” this coordinate without any additional optimization.

Reactions Without Transition States

Not all chemical reactions have transition states, and the rates of some reactions depend only on the speed with which reactants diffuse together (“diffusion controlled” reactions). Radicals will combine without activation, for example, two methyl radicals to form ethane.

H3C• + •CH3 H3C CH3

Radicals will add to double bonds with little or no activation, for example, methyl radical and ethylene, to form 1-propyl radical.

H3C• + H2C CH2 H3C–CH2–CH2•

Exothermic ion-molecule reactions may require activation in solution, but probably not in the gas phase. Any “complex” of an ion and a neutral molecule is likely to be lower in energy than the separated species and the entire reaction coordinate for an ion-molecule reaction might lie below the energy of the separated reactants e.g., nucleophilic attack by OH– on CH– on CH–

3Cl to give CH3OH and Cl–.

Failure to fi nd a transition state does not necessarily mean failure of the theoretical model. It may simply mean that there is no transition state.

68 Essay V

Calculations Using Approximate Geometries

Given that semi-empirical models and even molecular mechanics models often provide geometries that are quite close to those obtained from Hartree-Fock and correlated models, it is legitimate to ask whether or not structures from these simple techniques may be used for energy and property calculations with better models. This is important as geometry optimization is a major “cost” in any modeling investigation. The answer depends on what “property” is being calculated and the accuracy required. Experience suggests that, except for “unusual molecules”, use of either semi-empirical or molecular mechanics geometries in lieu of “exact” Hartree-Fock or correlated geometries has very little effect on relative energetics. Errors in reaction energies of 4 - 10 kJ/mol, that may arise from the use of “approximate geometries”, must be balanced against the large savings in computer time. Other properties such as dipole moments, may show greater sensitivity to choice of structure.

Semi-empirical techniques are not as successful in reproducing the results of Hartree-Fock and correlated models for transition-state geometries as they are for equilibrium geometries. However, the energy surface is typically very “fl at” in the vicinity of the transition state, and any errors (in energy) incurred because of the use of approximate transition-state geometries may be manageable. Molecular mechanics techniques are not applicable to the description of transition states.

“Exact” geometries must be used for frequency (infrared spectra) calculations. The reason for this is that frequencies are related to the fi rst fi nite term in a Taylor series expansion of the energy (as a function of geometry). This is assumed to be the second-derivative term, which will not be true if the gradient (the fi rst derivative term) is not zero.

Essay VI 69

6Atomic and Molecular Orbitals

Among the methods for describing electrons in molecules, Lewis structures are the most familiar. These drawings assign pairs of electrons either to single atoms (lone pairs) or pairs of atoms (bonds). The quantum mechanical “equivalents” of Lewis structures are atomic and molecular orbitals that arise from solutions of (approximate) Schrödinger equations for atoms and molecules, respectively. Because molecular orbitals are spread throughout the entire molecule, that is, they are “delocalized”, they are typically more diffi cult to interpret than Lewis structures.

Orbital Surfaces

Like Lewis structures, molecular orbitals may provide important clues about molecular structure and properties and about chemical reactivity. Before we can use this information we fi rst need to understand how to visualize and interpret their three-dimensional shapes. The following fi gure shows two representations, a “hand” drawing and a Spartan image of an unoccupied molecular orbital of hydrogen molecule, H2.

H HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH HH H

unoccupied molecular orbital in hydrogen

Open “hydrogen empty”. Note that except for the colors (sign of the orbital) the two sides of the graphic are identical. The junction between “red” and “blue” regions is where the value of the orbital is zero. Close “hydrogen empty” when you are fi nished.

The hand drawing shows the orbital as two circles and a dashed line. The circles identify regions where the orbital takes on a signifi cant value, either positive (shaded) or negative (shaded) or negative (shaded unshaded). The dashed line unshaded). The dashed line unshadedidentifi es locations where the orbital’s value is exactly zero (a “node”). The drawing is useful, but it is also limited. We only obtain information

70 Essay VI

about the orbital in two dimensions, and we only learn the location of “signifi cant” regions and not how the orbital builds and decays inside and outside of these regions.

The Spartan image depicts the same orbital as a “surface” of constant value. The surface is “accurate” in that it is derived from an authentic (but approximate) calculated solution to the quantum mechanical equations of electron motion. The image is three-dimensional, and can be manipulated and looked at from different perspectives.

Note that what we are calling an “orbital surface” actually consists of two distinct surfaces represented by different colors. The two surfaces have the same meaning as the two circles in the orbital drawing. They identify regions where the orbital takes on a specifi c positive (blue) or negative (red) value. The node is not shown, but we can deduce that it lies midway between the two surfaces (the orbital’s value can only change from positive to negative by passing through zero).

Atomic Orbitals

Atomic orbitals, “descriptions of atoms”, are the fundamental building blocks from which molecular orbitals “descriptions of molecules” are assembled. The familiar atomic orbitals for the hydrogen atom are in fact exact solutions of the Schrödinger equation for this one-electron system. They form an infi nite collection (a “complete set”), the lowest-energy member representing the “best” location for the electron, and higher-energy members representing alternative locations. Orbitals for many-electron atoms are assumed to be similar to those of hydrogen atom, the only difference being that more than the lowest-energy atomic orbital is utilized. In practical quantum chemical calculations, atomic orbitals for many-electron atoms are made up of sums and differences of a fi nite collection of hydrogen-like orbitals.

It is common practice to divide the full set of atomic orbitals into core and valence orbitals, and further to “ignore” the former. Valence orbitals for an element in the fi rst long row of the Periodic Table are 2s, 2px, 2py and 2pz, and for the second long row are 3s, 3px, 3py and 3pz. In the case of fi rst-row elements, a single orbital, 1s, lies underneath (is a core orbital) while in the case of second-row elements, a set of fi ve orbitals, 1s, 2s, 2px, 2py and 2pz, lie underneath.

Essay VI 71

Open “fl uoride and chlorideOpen “fl uoride and chlorideOpen “ ”. On the top row are the four valence orbitals of fl uoride anion and on the bottom row the four valence orbitals of chloride anion. You can select among them by clicking (left mouse button) on each in turn. First note that the three 2p orbitals in fl uoride are identical except for the direction in which they point. The same is true for the three 3p orbitals in chloride. Next, note that the valence orbitals in chloride are larger than those in fl uoride. Atoms further down in the Periodic Table are generally larger than analogous atoms further up. Close “fl uoride and chloridelarger than analogous atoms further up. Close “fl uoride and chloridelarger than analogous atoms further up. Close “ ” when you are fi nished.

Molecular Orbitals and Chemical Bonds

Molecular orbitals and Lewis structures are both used to describe electron distributions in molecules. Lewis structures are used to count the number of bonding and non-bonding electrons around each atom. Molecular orbitals are not useful as counting tools, but orbital surfaces and orbital energies are useful tools for describing chemical bonding and reactivity.

If the orbital surface is confi ned to a single atom or to atoms that are not “close together”, the orbital is regarded as “non-bonding”. If the orbital contains a surface that extends continuously over two neighboring atoms, the orbital is regarded as “bonding” with respect to these atoms. Adding electrons to such an orbital will strengthen the bond between these atoms and cause them to draw closer together, while removing electrons will have the opposite effect. Two different kinds of bonding orbitals are depicted below. The drawing and surface on the left correspond to a σ bond while the drawing and surface on the right σ bond while the drawing and surface on the right σcorrespond to a π bond.

A B A B

σ bondingσ bondingσ π bonding

72 Essay VI

Open “nitrogen bonding”. The image on the left corresponds to the σbonding orbital of N2, while that on the right corresponds to one of two equivalent π bonding orbitals. Switch to a transparent or mesh model to see the underlying molecular skeleton. For Windows: Click on the graphic and Click on the graphic and Clickselect the appropriate entry from the menu that appears at the bottom right of the screen. For Macintosh: Position the cursor over the graphic, hold down either the left or right button and select from the menu which appears alongside. Note that the σ orbitals is drawn in a single color (insofar as NN σ orbitals is drawn in a single color (insofar as NN σbonding is concerned) while the π orbital is made up of “red” and “blue” parts. This indicates a “node” or a break in the latter, although not involving the NN bond. Close “nitrogen bonding” when you are fi nished.

If an orbital contains a node that divides the region between two neighboring atoms into separate “atomic” regions, it is regarded as “antibonding” with respect to these atoms. Adding electrons to an antibonding orbital weakens the bond and “pushes” the atoms apart, while removing electrons from such an orbital has the opposite effect. The following pictures show drawings and orbital surfaces for two different kinds of antibonding orbitals. As above, the left and right-hand sides correspond to σ and π type arrangements, respectively.

node

A B

node

A B

σ bondingσ bondingσ π bonding bonding

Open “nitrogen antibonding”. The image on the left corresponds to the σ antibonding (σ*) orbital of N2 while that on the right corresponds to one of the two equivalent π antibonding (π*) orbitals. Switch to a mesh or transparent surface to see the underlying molecular skeleton. Note that the σ* orbital has a single node (change in color from “red” to “blue”) in the middle of the NN bond, while the π* orbital has two nodes (one in the middle of the NN bond and to the other along the bond). Close “nitrogen antibonding” when you are fi nished.

Bonds can be strengthened in two different ways, by adding electrons to bonding orbitals, or by removing electrons from antibonding orbitals. The converse also holds. Bonds can be weakened either by removing electrons from bonding orbitals or by adding electrons to antibonding orbitals.

Essay VI 73

Molecular Orbitals of Singlet Methylene

Molecular orbitals are typically spread throughout the molecule (they are “delocalized”). They may have complicated shapes and contain multiple bonding, non-bonding, antibonding interactions, or any mixture of all three. Nevertheless, these shapes can still be broken down into two-atom interactions and analyzed using the principles outlined earlier.

Analysis of molecular orbitals is illustrated for a triatomic molecule, “singlet” methylene, CH2. (“Singlet” refers to the fact that the eight electrons are organized into four pairs, and that each pair of electrons occupies a different molecular orbital. The lowest-energy state of methylene is actually a triplet with three electron pairs and two unpaired electrons.)

The lowest energy molecular orbital of singlet methylene is essentially a 1s atomic orbital on carbon. The electrons occupying this orbital restrict their motion to the immediate region of the carbon nucleus and do not signifi cantly affect bonding. Because of this, the orbital is referred to as a “core” orbital and its electrons are referred to as “core” electrons.

core orbital for methylene

The next orbital is much higher in energy and consists of a single surface that is delocalized over all three atoms. This means that it is simultaneously (σ) bonding with respect to each CH atom pair.

CH bonding orbital for methylene

The next higher energy orbital is described by two surfaces, a positive surface that encloses one CH bonding region and a negative surface

74 Essay VI

that encloses the other CH bonding region*. Since each surface encloses a bonding region, this orbital is also (σ) bonding with respect to each CH atom pair. This reinforces the bonding character of the previous orbital. The node that separates the two surfaces passes through the carbon nucleus, but not through either of the CH bonding regions, so it does not affect bonding.

CH bonding orbital for methylene

Thus, the two CH bonds in the Lewis structure for singlet methylene have been “replaced” by two “bonding” molecular orbitals.

The highest-occupied molecular orbital of methylene (the HOMO) is also described by two orbital surfaces. One surface extends from carbon into the region opposite the two hydrogens. It is non-bonding. The other surface encompasses the two CH bonding regions. Although it is hard to track the exact path of the orbital node in this picture, it happens to pass almost exactly through the carbon. This means that this particular orbital possesses only weak CH bonding character (it is H---H bonding). It turns out that the non-bonding character of the orbital is much more important than the bonding character, in that it leads to the fact that singlet methylene is able to behave as an electron-pair donor (a “nucleophile”).

HOMO of methylene

The above analysis shows that, while the occupied orbitals of singlet methylene are spread over all three atoms, they are interpretable. The orbitals divide into two groups, a single low-energy “core” orbital and three higher-energy “valence” orbitals. The latter consist of two CH

* Note that the absolute signs of a molecular orbital are arbitrary, but that the relative signs indicate bonding and antibonding character.

Essay VI 75

bonding orbitals and a non-bonding orbital on carbon. There is no one-to-one correspondence between these orbitals and the Lewis structure. The bonding orbitals are not associated with particular bonds, and the non-bonding orbital involves bonding interactions as well.

Open “methylene bonding”. Four images appear corresponding to the core and three valence orbitals of singlet methylene. Switch to a mesh or transparent surface to see the underlying molecular skeleton. Close “methylene bonding” when you are fi nished.

Singlet methylene also possesses unoccupied molecular orbitals. These have higher (more positive) energies than the occupied orbitals, and because they are unoccupied, these orbitals do not describe the electron distribution in singlet methylene.* Nevertheless, the shapes of unoccupied orbitals, in particular, the lowest-unoccupied molecular orbital (LUMO), provides valuable insight into the “chemistry” of singlet methylene.

The LUMO in methylene has non-bonding character, and is essentially a 2p atomic orbital on carbon. This suggests that singlet methylene should be able to behave as an electron-pair acceptor (an “electrophile”). Were the molecule to accept electrons, these would go into non-bonding orbital. Carbon would become more electron-rich, but the CH bonds would not be much affected.

LUMO of methylene

Open “methylene LUMO” and switch to a mesh or transparent surface to see the underlying skeleton. Close “methylene LUMO” when you are fi nished.

* There is no direct analogy between unoccupied molecular orbitals and Lewis structures. The latter attempt to describe only electron pair bonds and non-bonding electron pairs.

76 Essay VII

7Electron Densities:

Sizes and Shapes of MoleculesHow “big” is an atom? How big is a molecule? Atoms and molecules require a certain amount of space, but how much? A gas, where the atoms or molecules are widely separated, can be compressed into a smaller volume but only so far. Liquids and solids, where the atoms or molecules are already packed closely, cannot be easily compressed.

Space-Filling Models

Space-fi lling or CPK models offer a simple approach to the question of atomic and molecular size. The space-fi lling model for an atom is simply a sphere, the radius of which has been chosen to reproduce certain experimental observations, such as the compressibility of a gas, or the spacing between atoms in a crystal. Space-fi lling models for molecules consist of interpenetrating spheres centered on the individual atoms. This refl ects the idea that the chemical bonds that hold a molecule together cause the atoms to move close to each other. If two spheres in a space-fi lling model strongly interpenetrate then the atoms must be bonded.

space-fi lling models for ammonia (left), trimethylamine (center) and 1-azaadamantane (right)

Space-fi lling models for ammonia, trimethylamine and 1-azaadamantane show how “big” these molecules are, and also show that the nitrogen in ammonia is more “exposed” than the corresponding nitrogen atoms in trimethylamine and 1-azaadamantane.

Essay VII 77

Open “amines space fi lling”. Space-fi lling models for ammonia, trimethylamine and 1-azaadamantane all appear on screen. Carbon atoms are colored dark grey, hydrogen atoms white and nitrogen blue. Note that the models clearly reveal the extent to which the nitrogen is “exposed”. Close “amines space fi lling” when you are fi nished.

Electron Density Surfaces

An alternative approach for portraying molecular size and shape relies on the molecule’s electron cloud. Molecules are made up of positively-charged nuclei surrounded by a negatively-charged electron cloud. It is the size and shape of the electron cloud and not the nuclei that defi nes the size and shape of a molecule. The electron cloud is described by the “electron density”, or the number of electrons per unit volume. Consider fi rst a graph of electron density in hydrogen atom as a function of distance from the nucleus.

distance from nucleus

electrondensity

The cloud lacks a clear boundary. While electron density decays rapidly with distance from the nucleus, it never falls to zero. Therefore, when atoms approach other atoms, their electron clouds overlap and merge to a small extent.

electron densityfor second molecule

molecular "boundary"

electron densityfor first molecule

In order to defi ne atomic and molecular size, it is necessary to pick a value of the electron density, and then connect together all the points

78 Essay VII

that have this value. The criteria for selecting this value is the same as that for selecting atomic radii in space-fi lling models.* The result is an electron density surface that, just like a space-fi lling model, is intended to depict overall molecular size and shape.

electron density surfaces for ammonia (left), trimethylamine (center)and 1-azaadamantane (right)

Open “amines electron density”. Electron density surfaces for ammonia, trimethylamine and 1-azaadamantane all appear on screen. Switch to a mesh or transparent surface in order to see the underlying skeletal model. For Windows: click on one of the surfaces, and select click on one of the surfaces, and select click Mesh or Transparentfrom the menu that appears at the bottom right of the screen. For Macintosh: Position the cursor over the graphic, hold down either the left or right button and select from the menu that appears alongside. With “mesh” selected, change the model to Space Filling (Model menu). This allows you to see how similar the electron density is to a space-fi lling model. Close “amines electron density” when you are fi nished.

It is not surprising that both space-fi lling and electron density models yield similar molecular volumes, and both show differences in overall size and shape among molecules. Because electron density surfaces provide no discernible boundaries between atoms, they may appear to be less informative than space-fi lling models in helping to decide to what extent a particular atom in a molecule is “exposed”. This “weakness” raises an important point: electrons cannot be associated with the individual atoms in a molecule. The space-fi lling representation of a molecule with its “discrete” atoms does not refl ect reality.

* The difference is that a space-fi lling model requires specifi cation of a radius for each type of atom, whereas an electron density surface requires specifi cation only of the number of electrons per unit volume.

Essay VII 79

Bond Density Surfaces

Another useful electron density surface, termed the “bond density surface”, marks points corresponding to a much higher value of the electron density*. Since points of high electron density are located closer to the atomic nuclei, bond density surfaces enclose smaller volumes, and do not give a correct account of overall molecular size. On the other hand, because bond density surfaces identify regions corresponding to bonding electron density, their size may be roughly correlated with the number of electrons that participate in bonding. Thus, bond density surfaces are analogous to conventional line drawings and skeletal models.

The bond density surface for hex-5-ene-1-yne clearly shows which atoms are connected. The “size” of the surface associated with the CC triple bond is clearly larger than that associated with the corresponding single and double bonds.

C CH C

C

HH

CHH C H

H

H

bond density surface for hex-5-ene-1-yne

Open “hex-5-ene-1-yne bond density”, and switch to a mesh or transparent surface to see the connection between the chemical bonds in a conventional model and the electron density. Close “hex-5-ene-1-yne bond density” when you are fi nished.

Bond density surfaces can “assign bonding” in molecules where appropriate Lewis structure may not be apparent. For example, the bond density surface for diborane hints that there are relatively few electrons between the two borons. Apparently there is no boron-boron bond in this molecule.

* An even higher value of the electron density leads to a surface in which only “spheres” of electrons around the non-hydrogen atoms are portrayed. This serves to locate the positions of these atoms and is the basis of the X-ray diffraction experiment.

80 Essay VII

HB

HB

H

H

H

H

bond density surface for diborane

Open “diborane bond density”, and switch to a mesh or transparent surface to see how few electrons are actually collected in the region between the two borons. Close “diborane bond density” when you are fi nished.

Perhaps the most important application of bond density surfaces is to transition states. Consider, for example, pyrolysis of ethyl formate, leading to formic acid and ethylene.

O OH+

O

HO O

H

O

bond density surfaces for the reactant, ethyl formate (left), pyrolysis transition state

(center) and for the products, formic acid and ethylene (right)

The CO single bond of the reactant is clearly broken in the transition state. Also, the migrating hydrogen seems more tightly bound to oxygen (as in the product) than to carbon (as in the reactant). It can be concluded that the transition state more closely resembles the products than the reactants, and this provides an example of what chemists call a “late” or “product-like” transition state.

To see the smooth change in electron density throughout the course of the ethyl formate pyrolysis reaction, open “pyrolysis bond densityethyl formate pyrolysis reaction, open “pyrolysis bond densityethyl formate pyrolysis reaction, open “ ”. Click on at the bottom left of the screen to animate the graphic (click on to stop the animation). Switch to a mesh or transparent surface to follow the change in bonding. Close “pyrolysis bond densitybonding. Close “pyrolysis bond densitybonding. Close “ ” when you are fi nished.

Essay VIII 81

8Electrostatic Potential Maps:

Charge DistributionsThe charge distribution in a molecule can provide critical insight into its behavior. For example, molecules that are charged, or highly polar, tend to be water-soluble and may “stick together” in specifi c geometries, such as the “double helix” in DNA. Charge distribution also affects chemical reactivity. The most highly-charged molecule, or the most highly-charged site in a molecule, is often the most reactive. Positively-charged sites invite attack by bases and nucleophiles, while negatively-charged sites invite attack by acids and electrophiles.

The electrostatic potential provides one measure of molecular charge distrbution. This is the energy of interaction of a point positive charge with the nuclei and electrons of a molecule. The value of the electrostatic potential depends on the location of the point charge. If it is placed in an electron-poor region, the net point charge-molecule interaction will be repulsive and the electrostatic potential will be positive. Conversely, if the point charge is placed in an electron-rich region, the net interaction will be attractive and the electrostatic potential will be negative. Thus, by moving the point charge around the molecule, the molecular charge distribution can be surveyed.

Electrostatic potentials can be depicted either in terms of a “surface”, that is, by connecting all of the points in space where the electrostatic potential matches some particular value, or as a “map” on top of an electron density surface corresponding to a space-fi lling model. In the latter case, different colors are used to represent the different values of the electrostatic potential. Spartan uses the “rainbow”. The color red depicts regions of most negative electrostatic potential, while the color blue depicts the regions of most positive electrostatic potential. Intermediate colors represent intermediate values: red < orange < yellow < green < blue.

82 Essay VIII

The connection between a molecule’s electron density surface, its electrostatic potential surface, and an electrostatic potential map is illustrated here for benzene. The electron density surface defi nes molecular shape and size, and just as a space-fi lling model, indicates how close two benzene molecules can get.

space-fi lling model for benzene electron density surface for benzene

Open “benzene electron density”. Two different images, a space-fi lling model and an electron density surface, appear side by side. You can switch between the two models (in order to manipulate them) by clicking on each clicking on each clickingin turn. Notice how similar they are. To get an even clearer impression, switch to a mesh surface. For Windows: Click on the graphic and select Click on the graphic and select ClickMesh from the menu which appears at the bottom right of the screen. For Macintosh: Position the cursor over the graphic, hold down either the left or right button and select from the menu which appears alongside. Switch to a space-fi lling model (Space Filling from the Model menu). The two are now superimposed. Close “benzene electron density” when you are fi nished.

An electrostatic potential surface corresponding to points where the potential is negative is divided into two regions, one above the face of the ring and the other below. This is consistent with the notion that electrophilic addition occurs on the π faces. An electrostatic potential surface corresponding to points where the potential is positive has a disk-shape and wrapped fairly tightly around the nuclei. A point positive charge is repelled by this region.

negative (left) and positive (right) electrostatic potential surfaces for benzene

Essay VIII 83

Open “benzene electrostatic potential”. Two different images appear side benzene electrostatic potential”. Two different images appear side benzene electrostatic potentialby side. The one on the left depicts a surface of constant negative potential, while the one on the right depicts a surface of equal positive potential. You can switch between the two models (in order to manipulate them) by clicking on each in turn. Close “clicking on each in turn. Close “clicking benzene electrostatic potential” when you benzene electrostatic potential” when you benzene electrostatic potentialare fi nished.

Next, combine the electron density and electrostatic potential surfaces to produce an electrostatic potential map. The size and shape of the map are identical to that of the electron density surface, and indicate what part of the molecule is easily accessible to other molecules. The colors reveal the overall charge distribution. The faces of the ring, the π system, are “red” (electron rich), while the plane of the molecule and especially the hydrogens are “blue” (electron poor). Thus, the map indicates that, as expected, electrophilic attack will occur on the π faces of benzene.

electrostatic potential map for benzene

Open “benzene electrostatic potential map”. Manipulate the image to convince yourself that the “red” regions are on the π faces and the “blue” regions are around the edges. Close “benzene electrostatic potential map” when you are fi nished.

Electrostatic potential maps go a step beyond “steric models” for interpreting the way molecules fi t together. A good example of this follows from the electrostatic potential map for benzene which, as shown above, is “negative” on the π faces and “positive” around the periphery. The benzene dimer would, therefore, be expected to exhibit a “perpendicular” geometry, to best accomodate Coulombic interactions, instead of a parallel geometry that might be expected to be sterically favored.

84 Essay VIII

electrostatic potential maps for parallel (left) and perpendicular (right) benzene dimers

Open “benzene dimer electrostatic potential map”. Note that the parallel arrangement forces the negative region of one benzene onto the negative region of the other, while the perpendicular structure associates the negative region of one benzene with a positive region of the other. Close “benzene dimer electrostatic potential map” when you are fi nished.

These same arguments carry over to the structure of benzene in solid state. “Intuition” suggests a parallel stack. Afterall, benzene is “fl at” and fl at things (plates, pancakes, hamburgers) make stacks. However, Coulombs law strongly favors a perpendicular arrangement.

perpendicularparallel

The experimental X-ray crystal structure shows a perpendicular arrangement, but in three dimensions. There are two “lessons” here. The fi rst is that intermolecular interactions go beyond steric interactions. The second is that sometimes our simple “one-dimensional” view of the world leads us to the wrong conclusion.

Essay VIII 85

X-ray crystal structure of benzene

Open “benzene crystal”. Manipulate in order to see the packing of benzene benzene crystal”. Manipulate in order to see the packing of benzene benzene crystalmolecules. Close “benzene crystal” when you are fi nished.benzene crystal” when you are fi nished.benzene crystal

Electrostatic potential maps may also be used to describe the “workings” of chemical reactions. For example, a map may be used to show the transfer of charge during Sn2 reaction of cyanide with methyl iodide.

N C N C CH3 + ICH3 I– –

Open “Sn2 cyanide+methyl iodide”. One “frame” of a sequence of electrostatic potential maps for the Sn2 reaction will appear. Animate by clicking on clicking on clicking at the bottom left of the screen (stop the animation by clicking on clicking on clicking ). Note that the negative charge (red color) fl ows smoothly from cyanide to iodide during the reaction. Note also, that cyanide (as the reactant) is “more red” than iodide (as the product). Iodide is better able to carry negative charge, i.e., it is the better leaving group. Switch to mesh or transparent map to “see” the making and breaking of “bonds”. Close “Sn2 cyanide+methyl iodide” when you are fi nished.

86 Essay IX

9Local Ionization Potential

Maps and LUMO Maps: Electrophilic and

Nucleophilic ReactivitiesThe Hammond Postulate states that the transition state in a one-stepreaction will more closely resemble the side of the reaction that is higher in energy. Thus, the transition state of an endothermic reaction will more closely resemble the products while that of an exothermic reaction will more closely resemble the reactants. The Hammond Postulate provides a conceptual basis for the use of graphical models, and more generally the examination of reactants instead of transition states as a way to assess chemical reactivity and selectivity. In this context, two models stand out as being particularly useful: the local ionization potential map for electrophilic reactions and the LUMO map for nucleophilic reactions.

Local Ionization Potential Maps and Electrophilic Reactivity

The local ionization potential provides a measure of the relative ease of electron removal (“ionization”) at any location around a molecule. For example, a surface of “low” local ionization potential for sulfur tetrafl uoride demarks the areas that are most easily ionized, is clearly recognizable as a lone pair on sulfur.

local ionization potential for sulfur tetrafl uoride

Essay IX 87

A map of the local ionization potential onto an electron density surface reveals those regions from which electrons are most easily ionized.

"electron density"

Local ionization potential maps reveal sites that are susceptible to electrophilic attack. For example, they show both the positional selectivity in electrophilic aromatic substitution (NH2 directs ortho/para, and NO2 directs meta), and the fact that π-donor groups (NH2) activate benzene while electron-withdrawing groups (NO2) deactivate benzene.

NH2 NO2

local ionization potential maps for benzene (left), aniline (center) and nitrobenzene (right)

Open “benzene, aniline, nitrobenzene local ionization potential maps”. Here, the color red corresponds to regions of lowest ionization potential (most accessible to electrophiles). Note, that the π system in aniline is more accessible than the π system in benzene (the standard) and that the orthoand para positions are more accessible than meta positions. Note also, that the π system in nitrobenzene is less accesible than the π system in benzene and that here the meta positions are more accessible than the ortho and parapositions, in accord with observation.

88 Essay IX

LUMO Maps and Nucleophilic Reactivity

The LUMO map encodes the (absolute value) of the lowest-unoccupied molecular orbital (the LUMO) onto an electron density surface. This reveals where an electron pair (a nucleophile) is likely to attack.

"electron density"

LUMOelectron pair

A good example is provided by the LUMO map for cyclohexenone, that reveals two regions susceptible to nucleophilic attack.

O

α

β

LUMO map for cyclohexenone showing positions for nucleophilic attack

One region is over the carbonyl carbon, consistent with the observation that carbonyl compounds undergo nucleophilic addition at the carbonyl carbon, while the other region is over the β carbon, again consistent with the known chemistry of α,β-unsaturated carbonyl compounds, in this case conjugate or Michael addition.

OCH3HO

CH3Li (CH3)2CuLi

CH3

carbonyl addition Michael addition

O

Open “cyclohexenone LUMO map” and click on the resulting surface. click on the resulting surface. clickSwitch to a mesh or transparent surface in order to see the underlying skeletal model. For Windows: Click on the graphic and select Click on the graphic and select Click Mesh or Transparent from the menu that appears at the bottom right of the screen. For Macintosh: Position the cursor over the graphic, hold down on either the left or right button and select from the menu that appears alongside. Orient the molecule such that the two “blue regions” are positioned over the appropriate carbons and then switch back to a solid surface.

Section D 89

Section D“Hands-On” Problems

This section comprises a selection of “hands-on” problems for individual or group study. These are divided into seven categories: Structure and Bonding (1-14), Thermochemistry (15-25), Hydrogen Bonding and Intermolecular Interactions (26-31), Conformation (32-35), Dipole Moments and Charges (36-46), Reactivity and Selectivity (47-56) and Infrared and UV/Visible Spectra (56-67). Each problem is intended to illustrate how molecular modeling can contribute to a student’s understanding of a particular topic. Wherever possible, the problems have been kept “open ended” allowing different choices of molecules and/or theoretical models.

The problems are brief and “focused”, and most can be completed in twenty minutes to one hour and require only a few minutes of computer time. A working knowledge of the basic operation of Spartan is assumed. Students should complete the tutorials provided in Section B before attempting this material.

Structure and Bonding 91

Structure and Bonding

1 While the Hartree-Fock model is remarkably successful in accounting for the structures of main-group compounds, Hartree-

Fock geometries exhibit a number of systematic errors. The most conspicuous is that bond lengths are almost always shorter than experimental values. The magnitude of the error generally increases as the elements involved in the bond move from left to right in the Periodic Table.

a. Optimize the geometries of ethane, methyl fl uoride and fl uorine molecule using the HF/6-311+G** model. While your results will not fully represent the so-called Hartree-Fock limit, the 6-311+G** basis set is fl exible enough to approach this limit. Compare calculated CC, CF and FF bond distances with experimental values (1.531Å, 1.383Å and 1.412Å, respectively). Which if any of the calculated bond lengths fall within 0.02Å of the experimental distance (a typical error for a bond distance obtained by X-ray crystallography)?

b. Repeat your calculations using the HF/6-31G* model. This model is simple enough to allow its widespread application to sizeable molecules. Are the differences in bond lengths (from HF/6-311+G** results) relatively constant, or do they change from one molecule to another? Are HF/6-31G* bond lengths inside the 0.02Å error limits?

2 Errors in (limiting) Hartree-Fock bond distances increase from single to double and triple bonds. For example, while Hartree-Fock carbon-

carbon single bond lengths are quite close to experimental distances, the corresponding double and triple bond lengths are typically too short. This can be rationalized by recognizing that approaches beyond the Hartree-Fock model, in one way or another, “mix” ground and excited-state descriptions. Bond distances in excited states will tend to be longer than those in the ground state, meaning that any “mixing” of ground and excited states will lead to bond lengthening. Furthermore, as excited states will generally be more accessible (lower in energy) for unsaturated systems compared to saturated systems, it is reasonable to expect that changes from Hartree-Fock results will be greater.

92 Structure and Bonding

a. Optimize the geometries of ethane, ethylene and acetylene using the HF/6-311+G** model. Compare calculated C–C, C=C and C≡C bond distances with experimental values (1.531Å, 1.339Å and 1.203Å, respectively). Is the error in the single bond distance smaller than the errors in the double and triple bond lengths?

b. Display the HOMO and LUMO for ethylene. The fi rst excited-state of ethylene might be viewed as resulting from excitation of an electron from the HOMO to the LUMO. Is the HOMO carbon-carbon bonding, non-bonding or antibonding? (See the essay “Atomic and bonding, non-bonding or antibonding? (See the essay “Atomic and bonding, non-bonding or antibonding? (See the essay “Molecular Orbitals” for a description of terminology.) Is the LUMO carbon-carbon bonding, non-bonding or antibonding? Would you expect the carbon-carbon bond in the fi rst excited state of ethylene to be longer, shorter or unchanged from that in the ground state? Elaborate. What effect, if any, on the carbon-carbon bond length in ethylene would be expected from mixing of excited states?

3 Diazomethane is usually described as a composite of two Lewis structures, both of which involve separated charges.

N N N N– ++ –

Optimize the geometry of diazomethane using the HF/6-31G* model. Also optimize the geometries of methylamine, CH3NH2, and methyleneimine, H2C=NH, as examples of molecules incorporating “normal” CN single and double bonds, respectively, and of transdiimide, HN=NH, and nitrogen, N≡N, as examples of molecules incorporating “normal” NN double and triple bonds, respectively. Which Lewis structure provides the better description for diazomethane or are both required for adequate representation?

4 Draw a Lewis structure for cyanide anion, CN–, and assign formal charges. (See page 133 for the “recipe”.) Does it incorporate a double

bond like in methyleneimine, H2C=NH, or a triple bond like in hydrogen cyanide, HC≡N? On which atom does the negative charge reside?

To see if your Lewis structure presents a “realistic” picture, obtain equilibrium geometries for cyanide anion, methyleneimine and

Structure and Bonding 93

hydrogen cyanide using the HF/6-31G* model. According to your calculations, is the CN bond in cyanide anion closer to double or triple (compare it to bond lengths in methyleneimine and hydrogen cyanide)? Which atom bears the negative charge, or is it distributed over both carbon and nitrogen?

5 Draw two different “reasonable” geometries for ozone, O3. For each, provide Lewis structures and assign formal charges to the

oxygen atoms. Obtain the equilibrium geometries for both using the HF/6-31G* model. Which structure is lower in energy? Is it in accord with the experimentally known equilibrium geometry? Is the higher-energy structure actually an energy minimum? Elaborate. If the preferred structure has more than one distinct oxygen atom, which is most positively charged? Most negatively charged? Is your result consistent with formal charges?

6 Pyridine and pyridazine are each represented by two Lewis structures.

pyridine

NN

pyridazine

NN

NN

While the two structures are the same for pyridine, they are different for pyridazine. Compare carbon-nitrogen bond distances in pyridazine (using those in pyridine as a “reference”) obtained from optimized structures using the HF/6-31G* model. Should its two Lewis structures be given equal weight? If not, which structure is the more important? Elaborate.

7 Acyl cation, CH3CO+, adds to benzene and other aromatics. According to B3LYP/6-31G* calculations, which atom in acyl

cation is the most positively-charged? Draw the Lewis structure for acyl cation that is most consistent with its charge distribution. Is the calculated geometry of acyl cation consistent with its Lewis structure? Which atom (carbon or oxygen) would you expect to add to benzene? Draw a Lewis structure of the acyl cation - benzene adduct.

94 Structure and Bonding

8 Nitronium cation, NO8 Nitronium cation, NO8 2+, is the active reagent in the nitration

of benzene and other aromatics. According to B3LYP/6-31G* calculations, is it linear or bent? What common neutral molecule has the same number of electrons as NO2

+? Is this molecule linear or bent? Examine the charges on the nitrogen and oxygen atoms in NO2

+. Is nitrogen or oxygen more positive? Draw a Lewis structure for nitronium cation that is most consistent with the calculated geometry and charges. Display an electrostatic potential map for NO2

+. What atom (nitrogen or oxygen) would you expect to add to benzene? Draw a Lewis structure of the nitronium cation - benzene adduct.

9 Molecular geometry depends not only on the constituent atoms, but also on the total number of electrons. Molecules with the 9 but also on the total number of electrons. Molecules with the 9

same stoichiometry but with varying numbers of electrons may prefer different geometries. Optimize geometries of 2-methyl-2-propyl cation, radical and anion using the HF/6-31G* model. What changes, if any, to the local geometry of the central carbon do you observe with increasing number of valence electrons? What is the origin of these changes?

10 What happens to electron pairs that are “left over” after all bonds have been formed? Is each electron pair primarily associated

with a single atom or is it “spread out”?

a. Optimize the geometries of ammonia, water and hydrogen fl uoride using the HF/6-31G* model and examine electrostatic potential surfaces. (A value of -80 kJ/mol for electrostatic potential isosurfaces will demark highly electron-rich regions.) Describe the three surfaces and relate them to the Lewis structures.

N O

HHH HH

F

H

Rationalize the unusual shape of the potential for water, and clarify the difference in the shapes of the ammonia and hydrogen fl uoride potentials (that might fi rst appear to you to be nearly identical).

b. Optimize the geometries of methyl anion, ammonia and hydronium cation using the HF/6-31G* model and examine electrostatic potential surfaces. For which does the potential extend furthest away from the nuclei? For which is the extension the least? What

Structure and Bonding 95

do the relative sizes (extensions) of the potential tell you about the relative abilities of these three molecules to act as electron sources (“nucleophiles”)?

c. Optimize the geometries of anti and gauche conformers of hydrazine using the HF/6-31G* model. Note the energies of the two highest-energy occupied molecular orbitals (the HOMO and the orbital immediately below the HOMO) and examine electrostatic potential surfaces.

N NH

HH N N

HH

anti hydrazine gauche hydrazine

H H

H

Is there a noticeable difference in the extent to which the two electron pairs interact (“delocalize”) between the two conformers? Interaction should result in the “spreading out” of the potential over both nitrogens and in the “splitting” of the energies of the two highest-energy occupied orbitals. If there is a difference, is the “more delocalized” conformer lower or higher in energy than the “less delocalized” conformer?

11 Examine structures obtained from HF/3-21G calculations for cycloalkynes from cyclohexyne, C6H8, to cycloundecyne,

C11H18. What is the minimum ring size needed to allow a nearly linear geometry (within 10°) of the incorporated C–C≡C–C structural unit? Optimize geometries for the corresponding cis-cycloalkenes and calculate energies for addition of hydrogen to the cycloalkynes.

C C C CH H

H2(CH2)n (CH2)nn = 4 – 9

Is there a relationship between hydrogenation energy and C–C≡C bond angle?

96 Structure and Bonding

12 The HNH bond angle in ammonia is 106.7°, somewhat less than the tetrahedral value (109.5°). So too is the HOH bond angle

in water (104.5°). The usual rationale is that the lone pair on nitrogen and the two lone pairs on oxygen “take up more space” than NH and OH bonds, respectively. As seen from the experimental data below, HXH bond angles in second-row and heavier main-group analogues of ammonia and water deviate even more from tetrahedral.

NH3 106.7 PH3 93.3 AsH3 92.1 SbH3 91.6H2O 104.5 H2S 92.1 H2Se 90.6 H2Te 90.3

Is this further reduction in bond angle due to increased size of lone pairs on the heavy elements or are other factors involved? Do electrostatics (Coulomb’s law) or changes in orbital hybridization play a role?

a. Optimize the geometries of the eight hydrides shown above using the HF/3-21G model and also calculate electrostatic potential surfaces. (Set the isovalue for each to -10.) These surfaces demark the “most available” electrons that may loosely be interpreted as the lone pair electrons. What is the ordering of sizes of lone pairs (as indicated by the electrostatic potential surfaces) in the series NH3, PH3, AsH3, SbH3? In the series H2O, H2S, H2Se, H2Te? Is the size ordering consistent with the observed bond angles? Elaborate.

b. Examine hydrogen charges in ammonia and its analogues. Do they increase (hydrogen becoming more positive), decrease or remain about the same in moving to heavier analogues? Rationalize your result in terms of the electronegativities of nitrogen and its heavier analogues (relative to the electronegativity of hydrogen). Use Coulomb’s law to predict the trend in HXH bond angle in the series NH3, PH3, AsH3, SbH3. Repeat your analysis for water and its analogues.

c. The bonds in ammonia, water and their heavier analogues may be described in terms of sp3 hybrids. The “p contribution” to these bonds should increase as the energy of the (atomic) p orbitals move closer to the energy of the s orbital and should result in a decrease in bond angle. In order to get a measure of relative valence s and p orbitals (2s, 2p for fi rst-row elements, 3s, 3p for second-row elements, etc.) perform HF/3-21G calculations on Ne, Ar, Kr and Xe. Do valence s

Structure and Bonding 97

and p orbitals move closer together, move further apart or are their relative positions unchanged in going from Ne to Xe? If they do change their relative positions, how would you expect the HXH bond angles to change in moving from NH3 to SbH3 and from H2O to H2Te? Elaborate.

13 Carbon monoxide is the most common molecule to appear in organometallic compounds. CO bonds “end on” from carbon,

and contributes two electrons to the metal.

O C M:

As the electrons from carbon monoxide are non-bonding, it might be expected that their loss will not have signifi cant consequences. However, it is well known that the infrared stretching frequency of “complexed” CO is smaller than that in free carbon monoxide.

a. Optimize the geometry of carbon monoxide using the B3LYP/6-31G* model and examine both the HOMO and LUMO. Is the HOMO bonding, antibonding or essentially non-bonding between carbon and oxygen? What, if anything, would you expect to happen to the CO bond strength as electrons are donated from the HOMO to the metal? Elaborate. Is this consistent with the changes seen in the infrared stretching frequency of carbon monoxide?

The LUMO is where the next (pair of) electrons will go. Is it bonding, antibonding or essentially non-bonding between carbon and oxygen? What if anything would you expect to happen to the CO bond strength were electrons to be donated (from the metal) into this orbital? Elaborate. Is this consistent with the changes seen in the infrared stretching frequency of carbon monoxide?

To see if the metal center incorporates a high-energy fi lled molecular orbital properly disposed to donate electrons into the LUMO of CO, you need to perform calculations on a molecule from which carbon monoxide has been removed. Iron tetracarbonyl, resulting from loss of CO from iron pentacarbonyl is one such molecule.

98 Structure and Bonding

b. Optimize the geometry of iron pentacarbonyl (a trigonal bipyramid) using the B3LYP/6-31G* model.* Then, delete one of the equatorial CO ligands to make iron tetracarbonyl and perform a single-point energy calculation using the B3LYP/6-31G* model.

Is the HOMO of the iron tetracarbonyl fragment properly disposed to interact with the LUMO in CO? Elaborate. Would you expect electron donation to occur?

14 Two “limiting” structures can be drawn to represent ethylene “bonded” to a transition metal. The fi rst may be thought of as

a “weak complex” in that it maintains the carbon-carbon double bond, while the second destroys the double bond in order to form two new metal-carbon σ bonds, leading to a three-membered ring (a so-called σ bonds, leading to a three-membered ring (a so-called σ“metallacycle”).

M M

The difference between the two representations is only one of degree, and “real” metal-alkene complexes might be expected to span the full range of possible structures.

a. Optimize the geometry of ethylene using the B3LYP/6-31G* model and examine both the HOMO and LUMO. Is the HOMO bonding, antibonding or non-bonding between the two carbons? What if anything should happen to the carbon-carbon bond as electrons are donated from the HOMO to the metal? Do you expect the carbon-carbon bond length to decrease, increase or remain about the same? Elaborate.

The LUMO is where the next (pair of) electrons will go. Is this orbital bonding, antibonding or non-bonding between the two carbons? What, if anything, should happen to the carbon-carbon bond as electrons are donated (from the metal) into the LUMO? Is the expected change in the carbon-carbon bond due to this interaction

* To save computer time, you could use the PM3 model to optimize the geometry of iron pentacarbonyl.

Structure and Bonding 99

in the same direction or in the opposite direction as any change due to interaction of the HOMO with the metal? Elaborate.

To see if the metal center incorporates appropriate unfi lled and fi lled molecular orbitals to interact with the HOMO and LUMO of ethylene, respectively, perform B3LYP/6-31G* calculations on iron tetracarbonyl, arising from loss of ethylene from ethylene iron tetracarbonyl.

b. Optimize the geometry of ethylene iron tetracarbonyl (a trigonal bipyramid with ethylene occupying an equatorial position with the CC bond in the equatorial plane) using the B3LYP/6-31G* model.*

Then, delete the ethylene ligand and perform a single-point B3LYP/6-31G* calculation on the resulting (iron tetracarbonyl) fragment. Examine both the HOMO and LUMO of this fragment.

Is the LUMO of the iron tetracarbonyl fragment properly disposed to interact with the HOMO of ethylene? Elaborate. Would you expect electron donation from ethylene to the metal to occur? Is the HOMO of the fragment properly disposed to interact with the LUMO of ethylene? Elaborate. Would you expect electron donation from the metal to ethylene to occur?

* To save computer time, you could use the PM3 model to optimize the geometry of ethylene iron tetracarbonyl.

100 Thermochemistry

Thermochemistry

15 Which step in the hydrogenation of acetylene to ethane is the more exothermic?

H C C H C CH

H H

HC C

H

H H

H

H

H

H2 H2

Is a triple bond as strong as two double bonds? Use energies based on optimized geometries from B3LYP/6-31G* calculations.

16 Compare the B3LYP/6-31G* energy of the hypothetical molecule, 1,3,5-cyclohexatriene, with alternating single and double bonds,

with that of benzene. 1,3,5-cyclohexatriene is not an energy minimum (it collapses to benzene). To calculate its energy, you need to set the carbon-carbon bonds to alternating single (1.54Å) and double (1.32Å) lengths. (Do not optimize.(Do not optimize.( ) Is the difference in energy roughly the same as the aromatic stabilization afforded benzene (~ 160 kJ/mol), or signifi cantly smaller? Does your result suggest that aromatic stabilization occurs at least in part without structural change? Elaborate.

17 Addition of one equivalent of hydrogen to thiophene (X=S) not only breaks one of the double bonds but also destroys any

aromaticity. On the other hand, addition of a second equivalent only breaks the remaining double bond. Therefore, the difference in energy between the fi rst and second hydrogenation steps provides a measure of the aromatic stabilization afforded thiophene.

X X XH2 H2

X = S thiopheneX = NH pyrroleX = O furan

X = S tetrahydrothiopheneX = NH tetrahydropyrroleX = O tetrahydrofuran

a. Optimize geometries for hydrogen, for thiophene and for its fi rst and second hydrogenation products using the B3LYP/6-31G* model, and calculate the difference in hydrogenation energies between thiophene (leading to dihydrothiophene) and dihydrothiophene (leading to tetrahydrothiophene). Is this difference comparable to that between the fi rst and second hydrogenation energies of benzene

Thermochemistry 101

(~160 kJ/mol from calculations using the B3LYP/6-31G* model) or is it signifi cantly smaller or greater?

b. Repeat for pyrrole (X=NH) and furan (X=O). Order the aromatic stabilities of the three molecules.

18 Small-ring cycloalkanes are less stable than the correspondingn-alkanes. This is usually attributed to CCC bond angles

that deviate from tetrahedral and eclipsing interactions between CH bonds. The destabilization or “ring strain” of cycloalkanes, relative to n-alkanes, is provided by the energy of a hypothetical reaction in which one equivalent of hydrogen is added leading to the analogousn-alkane, relative to the corresponding hydrogenation of cyclohexane (assumed to be an unstrained molecule).

(CH2)4

CH3CH3

(CH2)n

CH3CH3+ +

H2C

(CH2)n

CH2 H2C

(CH2)4

CH2

a. Use this reaction to obtain strain energies for cyclopropane, cyclobutane and cyclopentane from geometries optimized using the HF/6-31G* model. Which is the most strained cycloalkane? Is any cycloalkane less strained than cyclohexane? If so, why?

b. Obtain the strain energy for cycloheptane. Is it signifi cantly less or greater than the strain energy of cyclohexane? If so, suggest why.

19 HF is a much stronger acid than H2O, that in turn is a stronger acid than NH3. This parallels a decrease in the electronegativity

of the atom bonded to hydrogen (F > O > N) and presumably a decrease in bond polarity. Hydrogen in HF is more positive than the hydrogens in H2O, that are in turn more positive than the hydrogens in NH3. Therefore, acid strength would be expected to decrease from HF to HI, paralleling the decrease in electronegativity of the halogen.

F > Cl > Br > I4.0 3.2 3.0 2.7

The opposite is true, HI is the strongest acid and HF is the weakest.

Acid strength relates to the energy of heterolytic bond dissociation into separated positive and negative ions.

102 Thermochemistry

HX H+ + X–

Gas-phase heterolytic bond dissociation energies are much larger than the corresponding energies in a solvent such as water. This is because the solvent acts to stabilize the charged dissociation products much more than it does the uncharged reactants.

a. Optimize geometries for HF, HCl, HBr and HI using the HF/3-21G model. Also, perform single-point energy calculations on F–, Cl–, Br–Br–Br and I– and I– –. Compute heterolytic bond dissociation energies for the four molecules. (The energy of the proton is 0.) Is the ordering of calculated bond dissociation energies the same as the ordering of acidities observed for these compounds?

Heterolytic bond dissociation in these compounds leads to separated ions, one of which, H+, is common to all. Is it reasonable that bond energy will follow the ability of the anion to stabilize the negative charge. One measure is provided by an electrostatic potential map.

b. Calculate electrostatic potential maps for F–, Cl–, Br–, Br–, Br and I– and I– –. Which ion, best accommodates the negative charge? Which most poorly accommodates the charge? Is there a correlation between the “size” of the ion and its ability to accommodate charge? Is the ability to accommodate charge in the anion refl ected in the heterolytic bond dissociation energy of the corresponding hydride?

20 Is the energy required to deprotonate acetylene less or greater than the energy required to deprotonate ethylene? Which

molecule is the stronger acid? Use optimized geometries from B3LYP/6-31G* calculations to decide.

21 What anion results from deprotonation of the following alkynes? Optimize geometries using the B3LYP/6-31G* model.

OHOH

a. b. c. d.

Thermochemistry 103

22 Aldehydes and ketones add water to form hydrates.

H2OC O

R´

RC

R´

R

OHOH

Equilibrium constants vary over an enormous range. Some compounds exist solely as the carbonyl, while others exist entirely as the hydrate.

a. Use the HF/6-31G* model to calculate reaction energies for addition of water to acetaldehyde, acetone and hexafl uoroacetone. Which is the least likely to exist as a hydrate? Which is the most likely? Explain your results in terms of your knowledge of electron donating/accepting effects of methyl and trifl uoromethyl substituents.

b. For trichloroacetaldehyde (chloral) the equilibrium lies almost entirely in favor of the hydrate, chloral hydrate. Does the trichloromethyl group destabilize the carbonyl, stabilize the hydrate or both?

C O + H2OH

Cl3CC

H

Cl3C

OHOH

To tell, use the HF/6-31G* model to calculate the energies of the reactions:

C O + CH4

H

Cl3C

CH

Cl3C

OHOH

C O + Cl3CH

HCH3

+ CH4 CH

H

OHOH

+ Cl3C CH3

These compare the effect of a trichloromethyl group relative to that of hydrogen on both chloral and its hydrate. What do you conclude is the reason for the preference for hydrate formation?

23 Formaldehyde cannot be purchased as a pure substance. Instead it needs to be prepared in situ by “cracking” either 1,3,5-trioxane

or paraformaldehyde.

104 Thermochemistry

H2C OO O

O

1,3,5-trioxane

or

paraformaldehyde

XCH2O CH2O CH2OHn

Is “cracking” 1,3,5-trioxane endothermic or exothermic? Use B3LYP/6-31G* optimized geometries to tell. Is the entropy change for the cracking reaction likely to be positive or negative? Explain.

24 The equilibrium between ketones and their corresponding alcohols (“enols”), for example, between acetone and propen-2-

ol, almost always favors the ketone.

O

CH3C CH3

OH

CH2C CH3

acetone propen-2-ol

a. Use the Boltzmann equation and energies from optimized B3LYP/6-31G* structures to calculate the room temperature distribution of acetone and its enol.

b. Repeat your calculations for 1,1,1-trifl uoroacetone and methyl acetate and their respective enol forms. For which system is the equilibrium abundance of the enol form the highest? For which is it the lowest? Rationalize your result.

O

CH3C CF3

O

CH3C OCH3

1,1,1-trifluoroacetone methyl acetate

25 Use B3LYP/6-31G* optimized geometries to calculate the equilibrium abundance of 2-hydroxypyridine and its “keto

form” (2-pyridone) at room temperature.

2-hydroxypyridine 2-pyridone

NHO NO

H

Hydrogen Bonding and Intermolecular Interactions 105

Hydrogen Bonding and Intermolecular Interactions

26 Water incorporates an equal number of electron pairs (electron-donors) and “acidic hydrogens” (electron-acceptors).

two electron pairs OHH

two acidic hydrogens

Water molecules use these two “complementary resources” fully to form a three-dimensional network of hydrogen bonds.

a. What is the maximum number of hydrogen bonds that each water molecule can make to its neighbors in liquid water? Build a cluster of 30-50 water molecules*, optimize using molecular mechanics and display hydrogen bonds. Are the molecules “in the middle” involved in the “maximum” number of hydrogen bonds? Do hydrogen bond lengths fall in a narrow range (± 0.05Å)? Is this consistent with the fact that hydrogen bonds are much weaker than covalent bonds? Display the cluster as a space-fi lling model. How much “empty space” is there in liquid water?

b. Add a molecule of ammonia into (the center of) your water cluster and reoptimize. How many hydrogen bonds are there to ammonia? On average, how many hydrogen bonds are there to the water molecules immediately surrounding ammonia? Does the situation appear to be similar or different from that for the “pure” water cluster? Has your cluster noticeably expanded or contracted in the vicinity of ammonia? (Examine it as a space-fi lling model.) Would you expect water to dissolve ammonia? Elaborate.

b. Replace ammonia by methane and reoptimize. Has the cluster noticeably expanded or contracted in the vicinity of methane? Rationalize your result in terms of changes in hydrogen bonding (relative to the “pure” water cluster). Would you expect water to dissolve methane? Elaborate.

* With sp3 oxygen selected, hold down the Insert key (option key on Macintosh) and click at click at clickdifferent locations on screen. Turn the cluster every few molecules to obtain a three-dimensional structure.

106 Hydrogen Bonding and Intermolecular Interactions

27 Acetic acids forms a symmetrical hydrogen-bonded dimer.

OCH3C

O H

OC CH3

O

H

Compare its structure with that of “free” acetic acid. Use the HF/6-31G* model. Point out any signifi cant changes in bond lengths and angles. Have the hydrogens involved in the hydrogen bonds moved to positions halfway between the oxygens or have they remained with one oxygen (as in acetic acid)? Do the structural changes (or lack of structural changes) suggest that hydrogen bonds are comparable to normal (covalent) bonds or are they weaker? Elaborate.

28 Explain the difference in boiling points of ethylamine (17° C) and ethanol (79° C) in terms of hydrogen bonding. Examine the

geometries of hydrogen-bonded dimers of both molecules and compare hydrogen-bond energies. Use the HF/6-31G* model.

29 Examine the geometry, atomic charges and electrostatic potential map for lithium aluminum hydride, LiAlH4. Use the B3LYP/

6-31G* model. Is it better described as an ion pair, that is, a “loose complex” between lithium cation and aluminum hydride anion or as a molecule where the two components are covalently bonded? Compare its structure and atomic charges with aluminum hydride anion, AlH4

–.

Repeat your analysis for sodium borohydride, NaBH4.

30 The mixture of antimony pentafl uoride, SbF5, and hydrogen fl uoride, HF, turns out to be a very strong acid (a so-called

“superacid”). Optimize the geometry of the SbF5/HF system using the HF/3-21G model, and calculate vibrational frequencies to insure that the structure you have found is an energy minimum. (See the essay “Finding and Verifying Equilibrium and Transition State Geometries” for a discussion.) Is the system better described as a complex between neutral antimony pentafl uoride and neutral hydrogen fl uoride or between SbF6 anion and a proton? (Compare with geometries of SbF5, SbF6

– and HF.) Compare charges at hydrogen and electrostatic potential – and HF.) Compare charges at hydrogen and electrostatic potential –

Hydrogen Bonding and Intermolecular Interactions 107

maps for the SbF5/HF system and for “free” HF. Is the hydrogen (in HF) “more positive” because of its association with SbF5?

31Optimize the geometry of cyclopentadienyl sodium using the PM3 model. Is the charge on sodium consistent with

representation of the system as a complex between sodium cation and cyclopentadienyl anion?

Na+

–

cyclopentadienyl sodium

Does the geometry of the incorporated cyclopentadienyl fragment also fi t such a description? (Compare with cyclopentadienyl anion.)

Repeat your calculation for ferrocene. (Again use the PM3 model.)

ferrocene

Fe2+

–

–

What is the charge on iron?

108 Conformation

Conformation

32 Ethane adopts a geometry whereby CH bonds stagger each other. The structure in which CH bonds eclipse is an energy

maximum.

C C

H

H

staggered eclipsed

HH

H H

C C

H H

HHH

H

Similarly, single (CH and CC) bonds in n-butane stagger each other leading to two distinct minimum energy forms (anti and gaucheconformers), and two distinct maximum energy forms (syn and skew).

C C

H3C

CH3

HH

H H

C C

H3C

HH

C C

H3C

H H

C C

H3C H

CH3HH

H

CH3

HH

H

CH3H

anti gauche syn skew

The observed conformations in these and in other molecules involving sp3 hybridized centers have been codifi ed “single bonds stagger”. Does the “staggered rule” extend to bonds involving sp2 hybridized elements, most important, sp2 hybridized carbon?

a. Use the HF/3-21G model to calculate the energy of 1-butene with change in the C=CCC dihedral angle from 0° to 180° in 20° steps. Plot the energy of 1-butene as a function of the C=CCC dihedral angle. How many energy minima are there? How many energy minima would there be if you had varied the dihedral angle from 0° to 360° instead of from 0° to 180°? Elaborate. Characterize the structures of the energy minima as “staggered” or “eclipsed” relative to the CC double bond. Characterize the structures of the energy maxima. Formulate a “rule” covering what you observe.

b. Build cis-2-butene. Lock both HCC=C dihedral angles to 0° (eclipsed). You will see that hydrogens on the two methyl groups are in close proximity. Next, defi ne a range of values for only one of these dihedral angles from 0° to 180° in 20° steps. As with 1-butene,

Conformation 109

obtain the energy of cis-2-butene as a function of this dihedral angle using the HF/3-21G model, and construct a plot. Characterize the structure of the energy minima as “staggered” or “eclipsed” relative to the CC double bond. Have other structural parameters signifi cantly altered in response to rotation?

33 The central CC bond in 2,2,3,3-tetramethylbutane is ~75 kJ/mol weaker than the CC bond in ethane. One explanation is that

bond cleavage relieves the crowding of the methyl groups. A measure of this should be provided by the energy of the reaction:

Me3C CMe3 + H3C CH3 Me3C CHMe2 + H3C CH2Me

a. According to HF/6-31G* calculations, is the reaction exothermic as written? If so, is the reaction energy large enough to account for the difference in bond strengths between the 2,2,3,3-tetramethylbutane and ethane? If not, or if the energy change is too small, suggest another reason for the difference in bond strengths.

b. Compare the calculated geometries of ethane and 2,2,3,3-tetra-methylbutane. Do both systems prefer geometries in which all single bonds are staggered? Be careful not to start your optimization of 2,2,3,3-tetramethylbutane with a staggered structure.

34 Use optimized geometries from the B3LYP/6-31G* model to assign the preferred conformation of acetic acid, with the OH

bond syn or anti to the CO bond.

syn anti

C OO

CH3

H

C OO

CH3

H

Rationalize your result. Hint: look at the dipole moments of both conformers. Is the difference small enough that the minor conformer might be observable at room temperature (> 1%)?

110 Conformation

35 To what extent does λmax (the “color”) for a diene depend on its conformation? To what extent does λmax parallel the energy of

the ground-state molecule with change in conformation?

Calculate the energy of 1,3-butadiene, 2-methyl-1,3-butadiene and 2-tert-butyl-1,3-butadiene with change in C=C–C=C dihedral angle starting from 0° going to 180° in 20° steps. Use the HF/3-21G model. For each diene, plot both the energy and the HOMO/LUMO gap as a function of dihedral angle. (To a reasonable approximation, changes in the HOMO/LUMO gap should mirror changes in λmax.)

For each diene: At what dihedral angle is the HOMO/LUMO gap the largest? At what dihedral angle is it the smallest? Is there much difference in the HOMO/LUMO gap between cis and trans-planar diene conformers? Is there much difference among the three dienes? For each diene: Does the variation in total energy closely follow the HOMO/LUMO gap or are the two uncorrelated? What (if anything) does your result say about the importance of conjugation (double bonds coplanar) on diene conformation?

Dipole Moments and Charges 111

Dipole Moments and Charges

36 Atomic charges obtained from Hartree-Fock models are generally larger than those from MP2 models. While it is not

possible to say which charges are more “realistic”, it is possible to say which model provides the better description of overall polarity as characterized by the dipole moment.

a. Optimize the geometry of formaldehyde using the HF/6-311+G** model. Is the calculated dipole moment smaller, larger or about the same as the experimental moment (2.34 debyes)?

b. Repeat your calculations using the MP2/6-311+G** model. Relative to the Hartree-Fock calculations, do you see a reduction, an increase, or no change in the calculated dipole moment? Is the dipole moment calculated from the MP2 model in better or poorer agreement with the experimental value? Compare Hartree-Fock and MP2 charges (on CH2 as a unit vs. O). Is the change consistent with the change in dipole moment?

c. Examine the HOMO and LUMO of formaldehyde (from the Hartree-Fock calculation). Where is the HOMO more concentrated, on the CH2 group or on oxygen? Where is the LUMO more concentrated? Would electron promotion from HOMO to LUMO be expected to lead to an increase or a decrease in charge separation? An increase or decrease in dipole moment? Given that the MP2 model involves a “mixing” of ground and excited states, is your result consistent with the change in dipole moment in going from Hartree-Fock to MP2 models? Elaborate.

37 The dipole moment provides a measure of the extent to which charge is distributed in a molecule. In a molecule like H2, with

both “sides” the same, the charge on the two atoms is equal and the dipole moment is zero. Increasing the difference in charge increases the dipole moment. The magnitude of the dipole moment also depends on the extent to which charge is separated. The larger the separation of charge, the larger the dipole moment. For a diatomic molecule, the dipole moment is proportional to the product of the absolute difference in charge between the two atoms, |qA – qA – qA B|, and the bond length, rAB|, and the bond length, rAB|, and the bond length, r .

112 Dipole Moments and Charges

dipole moment α qA–qB rAB

a. Obtain dipole moments for hydrogen fl uoride, hydrogen chloride, hydrogen bromide and hydrogen iodide using geometries optimized with the HF/3-21G model. Use the electronegativities provided below to correlate calculated dipole moments with the product of bond length and electronegativity difference.

H 2.2 F 4.0 Li 1.0 Cl 3.2 Na 0.9 Br 3.0 I 2.7

Does the correlation line reproduce the fact that the dipole moment of a homonuclear diatomic is zero?

b. Repeat your analyses for the series: lithium hydride, lithium fl uoride, lithium chloride, lithium bromide and lithium iodide and for the series: sodium hydride, sodium fl uoride, sodium chloride, sodium bromide and sodium iodide.

38 Optimize geometries for compounds X–CN, where X is halogen (F, Cl, Br, I) using the HF/3-21G model, and assign the direction

of the dipole moment in each. Given that the more electronegative “group” (CN or X) will be at the negative end of the dipole, what can you say about the electronegativity of the cyano group?

Repeat for X–NO2 compounds (X = F, Cl, Br, I). What can you say about the electronegativity of the nitro group?

39 Optimize geometries for the six heteronuclear diatomics X–Y, where X, Y are halogens (F, Cl, Br, I) using the HF/3-21G

model, and assign the direction of the dipole moment in each. Given that the more electronegative element will be at the negative end of the dipole, order the electronegativities of the halogens. Is the ordering you obtain the same as usually given, that is, F > Cl > Br > I?

Repeat for compounds X–C≡C–Y where X, Y are (different) halogens. Do you reach the same conclusions with regard to the ordering of electronegativities of the halogens?

Dipole Moments and Charges 113

40 Determine the sign of the dipole moment in propene using an optimized structure from the B3LYP/6-31G* model.

Is the electronegativity of the methyl group more or less than the electronegativity of a vinyl group? Repeat for propyne and for toluene.

41 Is hydrogen unique among the elements in that it is able to change its “personality” to refl ect its chemical environment? Are the

changes as large as the nomenclature suggests “protic to hydridic”?

a. Optimize geometries for H2, LiH, BeH2 (linear), BH3 (trigonal planar), CH4, NH3, H2O and HF using the HF/6-31G* model. Is there a correlation between calculated hydrogen charges and the difference in electronegativities between hydrogen and the element to which it is attached? (Electronegativities are provided below.)

H 2.2 Li 1.0 Be 1.6 B 2.0

C 2.6 N 3.0 O 3.4 F 4.0

b. Compare electron density surfaces for the molecules. Is there a trend in the “size” of hydrogen as you move from lithium hydride to hydrogen fl uoride? If so, for which molecule is hydrogen the smallest? For which is it the largest? Are your results consistent with the previous characterization of hydrogen as taking on “protic” or “hydridic” identity depending on the environment? Elaborate.

While you cannot calculate the electron density for proton (there are no electrons), you can for the hydride anion, H–. Calculate an electron density surface for H– using the HF/6-31G* model. – using the HF/6-31G* model. –

How does its size compare with the “largest hydrogen” in your compounds? What does this say about the extent to which the bond in this compound has dissociated into ions?

c. Compare electrostatic potential maps for the molecules, and “measure” the maximum (or minimum) value of the electrostatic potential at hydrogen in each. Is there a correlation between maximum (minimum) potential at hydrogen and difference in electronegativity between the two atoms that make up the bond?

114 Dipole Moments and Charges

42 Optimize geometries for acetylene, propyne and 3,3,3-trifl uoropropyne using the B3LYP/6-31G* model. For which is

the hydrogen (on the alkyne) most positive? For which is it least positive? What does this say about the role of methyl and trifl uoromethyl groups in affecting the acidity of acetylene? Which group has the stronger infl uence?

43 Optimize geometries for methyl fl uoride and methyl lithium using the B3LYP/6-31G* model. What is the charge on the

methyl group in methyl fl uoride? In methyl lithium? Are the charge distributions well represented by electrostatic potential maps for the two molecules? Which molecule more closely approaches its “ionic dissociation limit”? Is the difference in electronegativities between C and F/Li larger in this molecule than in the other?

44 Optimize geometries for ethane, ethylene and acetylene using the B3LYP/6-31G* model and compare their electrostatic

potential maps. For which are the hydrogens most positive? For which are they least positive? Is the ordering what you expected based on the known acidities of hydrocarbons: alkanes < alkenes < alkynes?

Repeat for propane and cyclopropane. Are the hydrogens in cyclopropane more or less positive than the (most positive) hydrogen in propane? What does your result suggest about the acidity of cyclopropane relative to that of propane?

45 Electrostatic potential maps provide a measure of overall charge distribution and may be used to distinguish molecules that

are likely to be soluble in non-polar media from those that will only dissolve in polar media.

a. Calculate electrostatic potential maps for hexane (a non-polar solvent) and water (a polar solvent) based on optimized structures from the HF/3-21G model. (Be careful to put the two maps on the same scale with the zero in potential at the center.) In very rough terms, what percentage of the surface of each is “negative” (red color), what percentage is “positive” (blue color) and what percentage is neutral (green color)? Overall, what percentage of the surface is “polar”?

Dipole Moments and Charges 115

b. Calculate and compare electrostatic potential maps (based on optimized geometries from HF/3-21G calculations) for methanol, ethanol, propanol, 1-pentanol, 1-heptanol and 1-decanol. (Again, you need to make certain that all the maps are on the same scale.) Divide the surface for each into negative, positive and neutral regions. Predict which of the alcohols is likely to be most water soluble and which is likely to be least water soluble.

46 Trimethylamine is much more basic than ammonia in the gas phase, whereas in water, the two are of roughly equal strength.46 phase, whereas in water, the two are of roughly equal strength.46

Me3N + NH4+ Me3NH+ + NH3

∆Ggas = –84 kJmol∆Gaq = –4 kJ/mol

How many “acidic hydrogens” are there in ammonium ion to coordinate with the solvent? How many are there in trimethylammonium ion? Optimize geometries for the two ions using the B3LYP/6-31G* model and compare hydrogen (NH) charges and electrostatic potential maps. For which are the hydrogens more acidic? Based on number and strength of acidic sites, rationalize the change in basicity from the gas phase into water.

116 Reactivity and Selectivity

Reactivity and Selectivity

47 The transition state for Diels-Alder reaction of 1,3-butadiene and ethylene is commonly drawn as a six-membered ring

incorporating six π electrons. Benzene also accomodates six π electrons in a six-membered ring.

+

Obtain and compare structures for the Diels-Alder transition state and for benzene. Use the HF/3-21G model. In what ways are they similar? In what ways are they different?

48 The rates of SN2 reactions involving methyl halides decrease with each successive replacement of hydrogen by methyl, for

example.

C BrH

H

H

C BrH

CH3

H

C BrH

CH3

CH3

fast > > > slowC BrCH3

CH3

CH3

The usual explanation is that these reactions involve transition states in which carbon has fi ve “neighbors” instead of four in the reactants and products, for example, for reaction of bromide and methyl bromide.

Br– + CH3BrH

CBrHH

Br

–

CH3Br + Br–

This implies that steric effects will be “amplifi ed” in the transition state, consistent with the trend in reaction rate with increased substitution.

a. Obtain transition states for addition of bromide to both methyl bromide and to tert-butylbromide using the HF/3-21G model. Examine these as space-fi lling models. Is the transition state for addition to tert-butyl bromide noticeably more crowded than that for addition to methyl bromide? If not, why not? Hint: examine carbon-bromine bond distances in the two transition states.

Reactivity and Selectivity 117

b. Calculate and compare electrostatic potential maps for the two SN2 transition states. Is the extent of charge separation comparable or does it differ between the two transition states? If the latter, what is the relationship between carbon-bromine distance and charge separation? Speculate on the origin of the change in rate of SN2 reactions with change in substitution at carbon.

49 Which hydroboration reaction is faster?

CH3

CH3

+ Me2BH

+ Me2BH

Use the HF/3-21G model to establish equilibrium and transition state geometries and follow these with single-point energy calculations using the B3LYP/6-31G* model to obtain activation energies. Rationalize your result based on the mechanism of hydroboration and “electronic” differences between alkynes and alkenes.

50 Singlet dichlorocarbene adds to ethylene to yield 1,1-dichlorocyclopropane.

CCl2 + H2C CH2

Cl Cl

a. Examine the electrostatic potential map of singlet dichlorocarbene obtained from an optimized structure using the HF/6-31G* model. Where is the “electrophilic site”, in the ClCCl plane or out of the plane? How would you expect dichlorocarbene to approach ethylene?

b. Obtain the transition state for addition of dichlorocarbene to ethylene using the HF/6-31G* model. Is your structure consistent with the conclusions reached regarding the electrophilic character of dichlorocarbene? Elaborate.

51 Which molecule, trans-2-butene or 2-butyne, is more susceptible to electrophilic attack? Compare electrostatic potential maps for

the two molecules using optimized structures from the B3LYP/6-31G*

118 Reactivity and Selectivity

model. Is the result the same if you use local ionization potential maps instead of electrostatic potential maps?

52 Compare local ionization potential maps for ethylene and substituted alkenes, H2C=CHX: propene (X=Me), vinyl chloride

(X=Cl), acrylonitrile (X=CN) and 3,3,3-trifl uoropropene (X=CF3). Use optimized structures from B3LYP/6-31G* calculations. Which would you expect to be more susceptible to electrophilic attack than ethylene? For these molecules, which carbon is more likely to be attacked? Which would you expect to be less susceptible to electrophilic attack than ethylene? For these molecules, which carbon is likely to be attacked?

53 Electron donor groups (EDG) “activate” benzene toward electrophilic substitution and lead predominately to ortho (1, 2)

and para (1,4) products, whereas electron withdrawing groups (EWG) “deactive” benzene and lead predominately to meta (1,3) products.

EDG

E+

EDG EDG

E

E

EWG

E+

EWG

E

metaortho para

+

Optimize geometries for benzene, toluene, aniline, phenol, chlorobenzene and nitrobenzene using the HF/3-21G model. Compare local ionization potential maps. Which substituents appear to be electron donor groups and which appear to be electron withdrawing groups? Elaborate.

54 A simple model for nucleophilic addition to carbonyl compounds considers the interaction of a high-energy non-bonded electron

pair on the nucleophile and the lowest-energy unoccupied molecular orbital (LUMO) on the carbonyl compound.

Reactivity and Selectivity 119

LUMO oncarbonyl compound

non-bondedelectron pairon nucleophile

Interaction increases with decrease in the energy gap between the non-bonded electron pair and the LUMO. Thus, for a given nucleophile, the lower the energy of the LUMO, the more susceptible the carbonyl compound should be toward nucleophilic attack.

Use B3LYP/6-31G* optimized geometries to rank the reactivity toward nucleophilic addition of: acetamide, CH3CONH2, acetic acid, CH3CO2H, methyl acetate, CH3CO2Me, acetyl chloride, CH3COCl, and acetic anhydride, CH3CO2COCH3, based on LUMO energy.

55 With the exception of so-called “phosphorous ylides”, compounds incorporating a double bond between carbon and

a second-row element are quite rare. Most notably, only a few stable compounds incorporating a carbon-silicon double bond have been characterized, among them 1 - 4. Their structures provide a “hint” as to why molecules incorporating carbon-silicon double bonds have proven to be so elusive.

Si CSiMe3

Me SiMe(Cme3)2

MeSi C

OSiMe3

Me3Si

Me3Si

1 2(1-adamantyl)

Si CC

(CMe3)Me2Si C

(CMe3)Me2Si

3

CMe3

CMe3

Si C(CMe3)Me2Si

Me3Si

4

a. Build one or more of the compounds, 1 - 4, and display as a space-fi lling model. Given that the “chemistry” of olefi ns (and presumably of silaolefi ns) is associated with the π bond, do you anticipate any “problems” that the molecule might have reacting? Elaborate.

120 Reactivity and Selectivity

b. Optimize the geometry of tetramethylsilaethylene, Me2Si=CMe2, along with that of its carbon analog 2,3-dimethyl-2-butene, Me2C=CMe2, using the HF/6-31G* model, and display both a local ionization potential map and a LUMO map for both molecules. (Make certain that the scale of the maps is the same for both molecules.) A local ionization potential map indicates the relative ease of electron removal (“ionization”) at the “accessible surface” of a molecule. This would be expected to correlate with the relative ease of addition of an electrophile. A LUMO map indicates the extent to which the lowest-unoccupied molecular orbital (the LUMO) can be “seen” at the “accessible surface” of a molecule. This indicates the “most likely” regions for electrons to be added, and would be expected to correlate with the likelihood of nucleophilic attack.

On the basis of comparison of local ionization potential maps, would you conclude that the silaolefi n is likely to be more or less reactive than the olefi n toward electrophiles? Elaborate. On the basis of comparison of the LUMO maps, would you conclude that the silaolefi n is likely to be more or less reactive than the olefi n toward nucleophiles? Elaborate. How do your conclusions fi t with the known silaolefi ns? Elaborate.

IR UV/vis Spectra 121

Infrared and UV/Visible Spectra

56 The frequency (energy) of vibration of a diatomic molecule is proportional to the square root of the ratio of the “force constant” 56 proportional to the square root of the ratio of the “force constant” 56

and the “reduced mass”.

frequency α force constantreduced mass

The force constant corresponds to the curvature of the energy surface in the vicinity of the minimum (it is the second derivative of the energy with respect to change in distance away from the equilibrium value). In effect, the magnitude of the force constant tells us whether the motion is “easy” (shallow energy surface meaning a low force constant) or “diffi cult” (steep energy surface meaning a high force constant).

Analysis of vibrational motions and energies (frequencies) for polyatomic molecules follows from the same general principles. Note, however, that the motions that polyatomic molecules undergo seldom correspond to changes in individual bond lengths, bond angles, etc. Rather the involve combinations of these motions, referred to as “normal modes”.

a. Calculate the geometry and infrared spectrum of dimethylsulfoxide, (CH3)2S=O, using the HF/3-21G model. Characterize the motion associated with each vibrational frequency as being primarily bond stretching, angle bending or a combination of the two. Is bond stretching or angle bending “easier”? Do the stretching motions each involve a single bond or do they involve combinations of all three bonds?

Most elements occur naturally as a mixture of isotopes. Aside from their difference in mass, molecules incorporating different isotopes have virtually identical physical and chemical properties. However, mass does enter into vibrational frequencies, and into thermodynamic quantities such as entropy that depend on these frequencies.

b. Replace all six hydrogens in dimethylsulfoxide with six deuteriums and repeat the calculations. Compare the resulting vibrational frequencies with those obtained for the non-deuterated molecule.

122 IR UV/vis Spectra

Rationalize any differences based on what you know for diatomic molecules.

Note that the heavy atoms in dimethylsulfoxide do not lie in a plane as they do in a molecule like acetone, (CH3)2C=O, that incorporates a carbon-oxygen rather than a sulfur-oxygen double bond. This is because sulfur assumes a “tetrahedral” geometry in order to accomodate four electron pairs (in addition to the two electrons associated with the sulfur-oxygen π bond) whereas carbon assumes a trigonal planar geometry to accomodate three electron pairs (excluding the π electrons).

c. Calculate the equilibrium geometry and infrared spectrum of dimethylsulfoxide in a geometry with C2v symmetry where all four heavy atoms are in the same plane, and where one CH bond in each methyl group eclipses the sulfur-oxygen double bond. Use the HF/3-21G* model. The spectrum should contain an imaginary frequency (indicate by the letter “i”). Given what you know from diatomic molecules, and given that reduced mass is necessarily a positive quantity, what does this tell you the sign of the force constant for this particular vibrational motion? What does this tell you about the disposition of “planar” dimethylsulfoxide on the potential energy surface? Describe the motion. Identify the analogous motion in “pyramidal” dimethylsulfoxide. Is it a low or high frequency (energy) motion?

57 Vibrational frequencies calculated from Hartree-Fock models are typically 10-12% larger than measured frequencies. About

half of the discrepancy is due to the fact that the calculation assumes a harmonic (quadratic) form to the potential energy in the vicinity of the minimum. The remaining error is consistent with the behavior of Hartree-Fock models in producing bond lengths that are almost always shorter than experimental distances (see problems 1 and 2). This implies bonds that are too strong which in turn leads to frequencies that are too large.

a. Optimize the geometry of acetone using the HF/6-31G* model and calculate the infrared spectrum. Locate the line in the spectrum corresponding to the carbonyl stretch. What is the ratio of the Hartree-Fock sketching frequency to the experimental frequency (1731 cm-1)? Locate the line in the spectrum corresponding to the

IR UV/vis Spectra 123

fully symmetric combination of carbon-hydrogen stretching motions. Is the ratio of this frequency to its experimental value (2937 cm-1) similar to that for the ratio of calculated and experimental carbon-oxygen stretching frequencies?

b. Repeat your calculations using the B3LYP/6-31G* model. Are the ratios of calculated to experimental stretching frequencies considered above closer to unity?

58 The infrared spectrum of trans-1,2-dichloroethylene in the region of 500 to 3500 cm-1 comprises only four lines, while nine

lines can be seen in the spectrum of the corresponding cis isomer.

C C

H

Cl H

Cl

C C

H

Cl Cl

H

trans-dichloroethylene cis-dichloroethylene

Both molecules might be expected to exhibit twelve infrared lines. While a few of the “missing lines” are below the 500 cm-1 measurement range, the primary reason for the discrepancy is that some vibrational motions give rise to absorptions that are too weak to be observed or be “infrared inactive”, meaning that they have “zero intensity”. In order for a particular vibrational motion to be “infrared active”, it needs to lead the change in the dipole moment of the molecule. In fact, the intensity of absorption is proportional to the change in dipole moment.

a. Calculate the geometry and infrared spectrum for both cis and transisomers of 1,2-dichloroethylene using the B3LYP/6-31G* model. Examine the spectrum for the cis isomer, and associate each vibration falling between 500 and 3500 cm-1 and having an intensity 1% or greater of the maximum intensity with the experimental frequency. Use the (experimentally assigned) “description of vibration” as well as the labels describing the “symmetry of vibration” to assist you.

124 IR UV/vis Spectra

description of vibration symmetry of vibration frequency

CCCl deformation b1 571CH bend b2 697CCl stretch a1 711CCl stretch b1 857CH bend a1 1179CH bend b1 1303CC stretch a1 1587CH stretch b1 3072CH stretch a1 3077

Repeat your analysis with trans-1,2-dichloroethylene.

description of vibration symmetry of vibration frequency

CCl deformation b1 571CH bend b2 697CCl stretch a1 711CCl stretch b1 857

Do the calculations properly assign those lines that are intense enough to actually be observed?

b. Identify the most intense line in the calculated infrared spectrum of trans-dichloroethylene. Distort the structure along the “normal coordinate” corresponding to this particular vibration such that the largest distortion of any atom from its equilibrium position is roughly 0.1Å.* What is the dipole moment of trans-dichloroethylene in its equilibrium geometry? What is the change in dipole moment as the molecule is distorted?

c. Repeat your analysis with the vibrational frequency corresponding to the stretching of the carbon-carbon bond. It should have an intensity of zero. Is your result different? How does your result here together with that from the previous step fi t in with the fact that intensity depends on change in dipole moment?

* Use Make List inside the Spectra dialog with Amp set to 0.1 and Steps set to 5. Delete all but the last molecule in the list that results and run a single-point energy calculation.

IR UV/vis Spectra 125

59 Which (if any) of the following molecules has one or more vibrational modes that are infrared inactive: water, carbon

dioxide, acetone, 2-butyne? Describe each infrared inactive mode. Calculations using the HF/3-21G model are suffi cient for this purpose.

60 Optimize the geometry of ethanol using the HF/6-31G* model and calculate its infrared spectrum. How many lines in the

spectrum in the range of 500 - 4000 cm-1 involve stretching of carbon-hydrogen bonds? Does each involve only one carbon-hydrogen bond or several carbon-hydrogen bonds? Is the line in the spectrum that involves oxygen-hydrogen stretching of the oxygen-hydrogen of lower or higher frequency than the lines that involve carbon-hydrogen stretching? Given that the mass of oxygen is greater than that of carbon, what (if anything) does this tell you about the “stiffness” of an oxygen-hydrogen bond relative to a carbon-hydrogen bond? Does this parallel the relative strengths of the bonds?

61 The experimental infrared spectrum of 1-octyne is given below.

Optimize the geometry of 1-octyne using the HF/3-21G* model and calculate its infrared spectrum. Assign each of the major bands in the experimental spectrum that are > 1400 cm-1 to specifi c vibrational motions, for example, carbon-hydrogen and carbon-carbon stretches and HCH and CCC bends. Remember that the vibrational frequencies calculated using Hartree-Fock models will typically be 10-12% larger than experimental values.

62 The infrared spectrum of a dilute solution of 1-butanol in carbon tetrachloride contains two bands attributed to stretching of

oxygen-hydrogen bonds. This has been taken as evidence that 1-butanol forms hydrogen-bonded dimers (or larger hydrogen-bonded clusters).

126 IR UV/vis Spectra

Calculate infrared spectra for both 1-butanol and 1-butanol dimer using optimized geometries from the HF/3-21G model. Are one or both of the lines in the spectra of the dimer associated with oxygen-hydrogen stretching at higher or lower frequency relative to the corresponding line in 1-butanol? Is what you observe consistent with hydrogen bond formation? Elaborate.

63 The infrared spectrum of an “unknown” C4H4 isomer shows strong absorptions at 215, 854, 1608, 2994 and 3080 cm-1. “Reasonable”

C4H4 structures include but-1-yne-3-ene, butatriene, singlet and triplet electronic states of cyclobutadiene, methylenecyclopropene and tetrahedrane.

H

H

H

H

H

H

H

H HH

H H

H H H

HH

H

HH

but-1-yne-3-ene butatriene cyclobutadiene

methylenecyclopropene tetrahedrane

Calculate geometries and infrared spectra for each using the B3LYP/6-31G*. Are all of these structures actually energy minima, or can you eliminate one or more? Elaborate. Which structure best fi ts the experimental infrared spectrum? Is it the lowest-energy C4H4 isomer? Assign the vibrational frequencies in the experimental spectrum using the data from the structure you selected.

64 The earth and all the other planets can be thought of as blackbodies that radiate into the universe. This provides a means to dissipate

the energy that falls on the planets due to the sun. Because of their low ambient temperatures, so-called “blackbody radiation” occurs primarily

IR UV/vis Spectra 127

in the infrared. The earth is actually warmer than such a picture would predict, due to the fact that some of the blackbody radiation is absorbed by its gaseous atmosphere. This is known as the “greenhouse effect”. The magnitude of the effect, “greenhouse warming”, depends on both the extent and chemical makeup of the atmosphere.

To be an effective “greenhouse gas”, a molecule must absorb in the infrared. Neither nitrogen nor oxygen, that together make up 99% of the earth’s atmosphere, satisfy this requirement. However, several “minor” atmospheric components, carbon dioxide, water and ozone most important among them, absorb in the infrared and contribute directly to greenhouse warming. These “subtract” from the earth’s blackbody radiation leaving actual radiation profi le (in the range of 500 to 1500 cm-1) that is given below.

1

2

3

0600 700 900 1100 1300 1500

Thus, to be an effective greenhouse gas, a molecule must absorb in a frequency region where blackbody radiation is intense.

Two further requirements are that the molecule needs to get into the atmosphere and have a reasonable lifetime in the atmosphere. These factors are addressed in a beautiful paper proposing an experimental investigation suitable for physical chemistry laboratory (M.J. Elrod, J. Chem. Ed., 76, 1702, 1999).

a. Calculate the geometry and infrared spectrum of carbon dioxide using the B3LYP/6-31G* model. How do the calculated infrared frequencies compare with the experimental values given below? How do the calculated infrared intensities compare with the (qualitative) experimental designations? Can you see evidence in

128 IR UV/vis Spectra

the “experimental” blackbody profi le for the presence of carbon dioxide in the atmosphere? Elaborate.

description of mode frequency cm-1 intensity

carbon dioxide bend 667 strong symmetric stretch 1333 inactive asymmetric stretch 2349 very strong

Propose a multiplicative factor that, when applied to the calculated frequencies, will bring them into best agreement with the experimental values.

Methane and nitrous oxide (N2O) are both introduced into the atmosphere from agriculture. Methane is also released as a result of oil recovery and fuel production. 1,1,1,2-tetrafl uoroethane is now widely used in automobile air conditioners and is likely also introduced into the atmosphere in signifi cant quantities.

b. Calculate geometries and infrared spectra for methane, nitrous oxide and 1,1,1,2-tetrafl uoroethane using the B3LYP/6-31G* model. Using as a selection criterion the presence of a (modestly) strong infrared absorption in the range of 500 - 1500 cm-1 (use frequencies corrected by the multiplicative factor determined in part a), comment as to whether each might be an effective greenhouse gas.

65 d1-Dichloromethane can react with chlorine atoms in either of two ways: by hydrogen abstration (producing HCl) or by

deuterium abstraction (producing DCl).

CClCl

H

D

Cl•

Cl•

C•ClCl

D + HCl

C•ClCl

H + DCl

hydrogenabstraction

deuteriumabstraction

The two reactions follow the “same” reaction coordinate, pass through identical transition states and lead to the same products except for the location of deuterium. The location of the deuterium is important, however, because it affects the zero-point energies of the transition

IR UV/vis Spectra 129

state and products, and can produce both equilibrium and kinetic isotope effects.

Zero-point energy differences for reaction products and reaction transition states provide an approximate account of equilibrium and kinetic isotope effects, respectively. To calculate the zero-point energy (in kJ/mol) sum the vibrational frequencies and multiply by 0.00598 (this corresponds to a unit conversion of inverse centimeters to kJ/mol). Ignore the imaginary frequency for transition states.

a. Optimize the geometries of the reaction products, dichloromethyl radical and hydrogen chloride, using the HF/3-21G model. Calculate the zero-point energy for each of the four species involved in the equilibrium,

Cl2CH + DCl Cl2CD + HCl••

and use the Boltzmann equation to calculate the equilibrium distribution of products at 25°C.

b. Starting from the following “guess”, optimize the structure of the transition state for hydrogen abstration from dichloromethane by chlorine atom. Use the HF/3-21G model.

CClCl

H

HCl

1.36Å

1.5Å

Calculate zero-point energies for the two possible monodeuterated transition-states. Use the Boltzmann equation (with transition state zero-point energies instead of reaction product zero-point energies) to calculate the kinetic distribution of products at 25°C.

66 Examination of highest-occupied and lowest-unoccupied molecular orbitals (the HOMO and LUMO) for molecules

in their electronic ground state is often suffi cient to characterize the lowest-energy (longest wavelength) electronic transition. Use the HF/6-31G* model to:

a. Examine the HOMO and LUMO of nitrogen, N2. Is the longest-wavelength (lowest-energy) electronic transition n → π* or π → π*?

130 IR UV/vis Spectra

b. Examine the HOMO and LUMO of acetone, (CH3)2CO. Is the longest-wavelength (lowest-energy) electronic transition n → π* or π → π*?

67 A good “guess” at the change in color of a molecule with change in molecular structure may be made by simply looking at the

change in the energy difference between the highest-occupied and lowest-unoccupied molecular orbitals, the so-called HOMO/LUMO gap. The underlying idea is that light absorption causes an electron to jump from the HOMO to the LUMO. Structural changes that lower the energy of the jump shift color toward the red.

Use the HF/3-21G model to optimize geometries for for azobenzene (R=R´=H), 4-hydroxyazobenzene (R=OH, R´=H) and 4-amino-4´-nitroazobenzene (R=NH2, R´=NO2).

N NR R´

Azobenzene is orange. Predict whether the color of each of the other molecules will be shifted toward the red or blue end of the spectrum.

Section B 131

Section ECommon Terms and Acronyms

3-21G. A Split-Valence Basis Set in which each Core Basis Function is written in terms of three Gaussians, and each Valence Basis Function is split into two parts, written in terms of two and one Gaussians, respectively.

6-31G*. A Polarization Basis Set in which each Core Basis Function is written in terms of six Gaussians, and each Valence Basis Function is split into two parts, written in terms of three and one Gaussians, respectively. Non-hydrogen atoms are supplemented by a set of single Gaussian d-type functions.

6-311+G**. A Polarization Basis Set in which each Core Basis Function is written in terms of six Gaussians, and each Valence Basis Function is split into three parts, written in terms of three, one and one Gaussians, respectively. Non-hydrogen atoms are supplemented by d-type Gaussians and hydrogen atoms are supplemented by p-type Gaussians (Polarization Functions). Non-hydrogen atoms are also supplemented by Diffuse Functions.

Ab Initio Models. The general term used to describe non-empirical methods seeking approximate solutions to the Electronic Schrödinger Equation.

Activation Energy. The energy of a Transition State above that of reactants. Activation energy is related to reaction rate by way of the Arrhenius Equation.

Antibonding Molecular Orbital. A Molecular Orbital that has a “break” or Node between particular atomic centers. Adding electrons to such an orbital weakens bonding interactions between these centers while removing electrons strengthens them.

Arrhenius Equation. An equation governing the rate of a chemical reaction as a function of the Activation Energy and the temperature.

Atomic Orbital. A Basis Function centered on an atom. Atomic orbitals typically take on the form of the solutions to the hydrogen atom (s, p, d, f... type orbitals).

Atomic Units. The set of units that removes constants from the Schrödinger Equation. The Bohr is the unit of length and the Hartree is the unit of energy.

B3LYP Model. A Hybrid Density Functional Model that improves on the Local Density Model by accounting explicitly for non-uniformity in electron distributions, and that also incorporates the Exchange Energy from the Hartree-Fock Model. The B3LYP model involves three adjustable parameters.

Basis Functions. Functions centered on atoms that are linearly combined to make up the set of Molecular Orbitals. Except for Semi-Empirical Models where basis

132 Section B

functions are Slater type, basis functions are Gaussian type.

Basis Set. The entire collection of Basis Functions.

Bohr. The Atomic Unit of length. 1 bohr = 0.529167Å.

Boltzmann Equation. The equation governing the distribution of products in Thermodynamically-Controlled Reaction.

Bonding Molecular Orbital. A Molecular Orbital that has positive overlap between two particular atomic centers. Adding electrons to such an orbital strengthens bonding interactions between these centers, while removing electrons weakens them.

Bond Surface. An Isodensity Surface used to elucidate the bonding in molecules. The value of the density is typically taken as 0.1 electrons/bohr.3

Born-Oppenheimer Approximation. An approximation based on the assumption that nuclei are stationary. Applied to the Schrödinger Equation, it leads to the Electronic Schrödinger Equation.

Closed Shell. An atom or molecule in which all electrons are paired.

Conformation. The arrangement about single bonds and of flexible rings.

Core. Electrons that are primarily associated with individual atoms and do not participate significantly in chemical bonding (1s electrons for first-row elements, 1s, 2s, 2px, 2py, 2pz electrons for second-row elements, etc.).

Correlated Models. Models that take account of the Correlation of electron motions. Møller-Plesset Models and Density Functional Models are correlated models.

Correlation. The coupling of electron motions not explicitly taken into account in Hartree-Fock Models.

Correlation Energy. The difference in energy between the Hartree-Fock Energyand the experimental energy.

Coulomb Energy. The electron-electron repulsion energy according to Coulomb’s law.

Coulombic Interactions. Interactions between charges that follow Coulomb’s law.

CPK Model. A molecular model in which atoms are represented by spheres, the radii of which correspond to van der Waals Radii. Intended to portray molecular size and shape. Also known as a Space-Filling Model.

Density Functional Models. Models in which the energy is a function of the Electron Density. Electron Correlation is taken into account explicitly by terms based on exact solutions of idealized many-electron systems.

Diffuse Functions. Functions added to a Basis Set to allow description of electron distributions far away from atomic positions. Important for descriptions of anions.

Section B 133

Diffusion-Controlled Reactions. Reactions without Transition States, the rates of which are determined solely by the speed in which molecules encounter each other and how likely these encounters are to lead to reaction.

Dipole Moment. A measure of the overall polarity of a molecule, taking into account differences in nuclear charges and electron distribution.

Doublet. A molecule with one unpaired electron.

Electron Correlation. See Correlation.

Electron Density. The number of electrons per unit volume at a point in space. This is the quantity that is measured in an X-ray diffraction experiment.

Electronic Schrödinger Equation. The equation that results from incorporation of the Born-Oppenheimer Approximation to the Schrödinger Equation.

Electrostatic Charges. Atomic charges chosen to best match the Electrostatic Potential at points surrounding a molecule, subject to overall charge balance.

Electrostatic Potential. The energy of interaction of a positive point charge with the nuclei and fixed electron distribution of a molecule.

Electrostatic Potential Map. A graph that shows the value of Electrostatic Potentialon an Electron Density Isosurface. Useful for assessing overall charge distribution.

Energy(∆E). The heat given off (negative energy) or taken in (positive energy) in a chemical reaction at constant volume.

Equilibrium Geometry. A Local Minimum on a Potential Energy Surface.

Excited State. An electronic state for an atom or molecule that is not the lowest-energy or Ground State.

Exchange Energy. An “attractive” (negative) component of the electron-electron interaction energy. Arises due to an overestimation of the Coulomb energy.

Force Field. The set of rules underlying Molecular Mechanics Models. Comprises terms that account for distortions from ideal bond distances and angles and for Non-Bonded van der Waals and Coulombic Interactions.

Frontier Molecular Orbitals. The HOMO and LUMO.

Formal Charge. A “recipe” to assign charges to atoms: formal charge = # of valence electrons - # of electrons in lone pairs - # of single bond equivalents.

Gaussian Basis Set. A Basis Set made up of Gaussian Basis Functions.

Gaussian Function. A function of the form xlymzn exp (αr2) where l, m, n are integers (0, 1, 2 . . .) and α is a constant. Used in the construction of Basis Sets.

Global Minimum. The lowest energy Local Minimum on a Potential Energy Surface.

134 Section B

Ground State. The lowest energy electronic state for an atom or molecule.

Hammond Postulate. The hypothesis that the Transition State for an exothermicreaction more closely resembles reactants than products. This provides the basis for “modeling” properties of Transition States in terms of the properties of reactants.

Harmonic Frequency. A Vibrational Frequency that has been corrected to remove all non-quadratic (non-harmonic) components. Calculated Vibrational Frequencies correspond to harmonic frequencies.

Hartree. The Atomic Unit of energy. 1 hartree = 2725 kJ/mol.

Hartree-Fock Approximation. Separation of electron motions in many-electron systems into a product form of the motions of the individual electrons.

Hartree-Fock Energy. The energy resulting from Hartree-Fock Models.

Hartree-Fock Limit. Properties resulting from Hartree-Fock Models with complete (infinite) basis sets.

Hartree-Fock Models. Methods in which the many-electron wavefunction is written terms of a product of one-electron wavefunctions. Electrons are assigned in pairs to functions called Molecular Orbitals.

Hessian. The matrix of second energy derivatives with respect to geometrical coordinates. The Hessian together with the atomic masses lead to the Vibrational Frequencies of molecular systems.

Heterolytic Bond Dissociation. A process in which a bond is broken and a cation and anion result. The number of electron pairs is conserved, but a non-bonding electron pair (on the anion) has been substituted for a bond.

HOMO. Highest Occupied Mccupied Mccupied olecular Oolecular Oolecular rbital. The highest-energy molecular orbital that has electrons in it.

Homolytic Bond Dissociation. A process in which a bond is broken and two Radicals result. The number of electron pairs is not conserved.

Hybrid Density Functional Models. Density Functional Models that incorporate the Exchange Energy from the Hartree-Fock Model. The B3LYP Model is a hybrid density functional model.

Imaginary Frequency. A frequency that results from downward (negative) curvature of the Potential Energy Surface. Transition States are characterized by the presence of one and only one imaginary frequency.

Infrared Spectrum. The set of Energies corresponding to the vibrational motions that molecules undergo upon absorption of infrared light.

Ionic Bond. A chemical bond in which the pair of electrons is not significantly shared by the two atoms.

Section B 135

Isodensity Surface. An Electron Density Isosurface. A value of 0.002 electrons/bohr3 is used to provide a measure of molecular size and shape.

Isopotential Surface. An Electrostatic Potential Isosurface. Used to elucidate regions in a molecule that are electron rich and subject to electrophilic attack and those that are electron poor, subject to nucleophilic attack.

Isosurface. A three-dimensional surface defined by the set of points in space where the value of the function is constant.

Isotope Effect. Dependence of molecular properties and chemical behavior on atomic masses.

Kinetically-Controlled Reaction. A reaction that has not gone all the way to completion, and the product ratio is given by the relative Activation Energies.

Kinetic Product. The product of a Kinetically-Controlled Reaction.

LCAO Approximation. Linear Combination of Atomic Orbitals approximation. Approximates the unknown Hartree-Fock Wavefunction (Molecular Orbitals) by linear combinations of atom-centered functions (Atomic Orbitals).

Local Density Models. Density Functional Models that are based on the assumption that the Electron Density is (nearly) constant throughout all space.

Local Ionization Potential. A measure of the relative ease of electron removal (“ionization”) as a function of location.

Local Ionization Potential Map. A graph of the value of the Local Ionization Potential on an Isodensity Surface corresponding to a van der Waals Surface. Useful for identifying sites for electrophilic reactivity.

Local Minimum. Any Stationary Point on a Potential Energy Surface for which all coordinates are at energy minima.

Lone Pair. A Non-Bonded Molecular Orbital.

LUMO. Lowest Unoccupied Molecular Orbital. The lowest-energy molecular orbital that does not have electrons in it.

LUMO Map. A graph of the absolute value of the LUMO on an Isodensity Surface. Useful for identifying sites for nucleophilic reactivity.

Minimal Basis Set. A Basis Set that contains the fewest functions needed to hold all the electrons on an atom and still maintain spherical symmetry.

Molecular Mechanics Models. Methods for structure and conformation based on bond stretching, angle bending and torsional distortions, together with Non-Bonded Van der Waals and Coulombic Interactions, and parameterized to fit experimental data.

Molecular Orbital. A function made of contributions of Basis Functions on individual atoms (Atomic Orbitals) and delocalized throughout the molecule.

136 Section B

Molecular Orbital Models. Methods based on writing the many-electron solution of the Electronic Schrödinger Equation in terms of a product of one-electron solutions (Molecular Orbitals).

Møller-Plesset Energy. The energy resulting from Møller-Plesset Modelsterminated to a given order, for example, the MP2 energy is the energy of the second-order Møller-Plesset Model (or MP2).

Møller-Plesset Models. Methods that partially account for Electron Correlation by way of the perturbation theory of Møller and Plesset.

MP2 Energy. The energy resulting from MP2 Models.

MP2 Model. A Møller-Plesset Model terminated to second order in the energy.

Multiplicity. The number of unpaired electrons: 1=singlet; 2=doublet; 3=triplet, etc.

NDDO Approximation. Neglect of Diatomic Differential Overlap approximation. Restriction that atomic orbitals on different atoms do not overlap. The basis for Semi-Empirical Models.

Node. A change in the sign of a Molecular Orbital. Nodes between particular atoms indicate that a Molecular Orbital is Antibonding with respect to these atoms.

Non-Bonded Interactions. Interactions between atoms that are not directly bonded. van der Waals Interactions and Coulombic Interactions are non-bonded interactions.

Non-Bonded Molecular Orbital. A molecular orbital that does not show any significant Bonding or Antibonding characteristics.

Open Shell. An atom or molecule in which one or more electrons are unpaired. Doublets, Triplets and Radicals are open-shell molecules.

PM3. Parameterization Method 3. A Semi-Empirical Model.

Point Group. A classification of the Symmetry Elements in a molecule.

Polarization Basis Set. A Basis Set that contains functions of higher angular quantum number (Polarization Functions) than required for description of the Ground State of the atom, in particular, d-type functions for non-hydrogen atoms. 6-31G* and 6-311+G** are polarization basis sets.

Polarization Functions. Functions of higher angular quantum than are required for the Ground State atomic description. Added to a Basis Set to allow displacement of Valence Basis Functions away from atomic positions.

Potential Energy Surface. A multidimensional function of the energy of a molecule in terms of the geometrical coordinates of the atoms.

Property Map. A representation or “map” of a “property” on top of an Isosurface, typically an Isodensity Surface.

Section B 137

Pseudorotation. A mechanism for interconversion of equatorial and equatorial and equatorial axial sites axial sites axialaround trigonal bipyramidal centers, for example, the fluorines in phosphorous pentafluoride.

Quantum Chemical Models. Methods based on approximate solution of the Schrödinger Equation.

Radical. An Open Shell molecule with one or more unpaired electrons.

Reaction Coordinate. The coordinate that connects the Local Minima corresponding to the reactant and product, and that passes through a Transition State.

Reaction Coordinate Diagram. A plot of energy vs. Reaction Coordinate.

Schrödinger Equation. The quantum mechanical equation that accounts for the motions of nuclei and electrons in atomic and molecular systems.

Semi-Empirical Models. Quantum Mechanics methods that seek approximate solutions to the Electronic Schrödinger Equation, but involve empirical parameters. PM3 is a semi-empirical model.

Singlet. A molecule in which all electrons are paired.

Slater Function. A function of the form xlymzn exp (-n exp (-n ζr) where l, m, n are integers (0, 1, 2...) and ζ is a constant. Similar to the exact solutions to the Schrödinger Equationfor the hydrogen atom. Used as Basis Functions in Semi-Empirical Models.

SOMO. Singly Occupied Molecular Orbital. An orbital that has only a single electron in it. The HOMO of a Radical.

Space-Filling Model; See CPK Model.

Spin Density. The difference in the number of electrons per unit volume of “up” spin and “down” spin at a point in space.

Spin Density Map. A graph that shows the value of the Spin Density on an Isodensity Surface corresponding to a van der Waals Surface.

Split-Valence Basis Set. A Basis Set in which the Core is represented by a single set of Basis Functions (a Minimal Basis Set) and the Valence is represented by two or more sets of Basis Functions. This allows for description of aspherical atomic environments in molecules. 3-21G is a split-valence basis set.

Stationary Point. A point on a Potential Energy Surface for which all energy first derivatives with respect to the coordinates are zero. Local Minima and Transition States are stationary points.

Strain Energy. The energy resulting from a calculation using a Molecular Mechanics Model.

Symmetry Elements. Operations that reflect the equivalence of different parts of a molecule. For example, a plane of symmetry reflects the fact that atoms on both sides are equivalent.

138 Section E

Theoretical Model. A “recipe” leading from the Schrödinger Equation to a general computational scheme. A theoretical model needs to be unique and well defined and, to the maximum extent possible, be unbiased by preconceived ideas. It should lead to Potential Energy Surfaces that are continuous.

Theoretical Model Chemistry. The set of results (“chemistry”) following from application of a particular Theoretical Model.

Thermodynamically-Controlled Reaction. A chemical reaction that has gone all the way to completion, and the ratio of different possible products is related to their thermochemical stabilities according to the Boltzmann Equation.

Thermodynamic Product. The product of a reaction that is under Thermodynamic Control.

Transition State. A Stationary Point on a Potential Energy Surface for which all but one of the coordinates is at an energy minimum and one of the coordinates is at an energy maximum. Corresponds to the highest-energy point on the Reaction Coordinate.

Transition-State Geometry. The geometry of a Transition State.

Triplet. An Open Shell molecule with two unpaired electrons.

Transition State Theory. The hypothesis that all molecules necessarily pass through a single well-defined Transition State.

Valence Electrons. Electrons that are delocalized throughout the molecule and participate in chemical bonding (2s, 2px, 2py, 2pz for first-row elements, 3s, 3px, 3py, 3pz for second-row elements, etc.).

van der Waals Interactions. Interactions that account for short-range repulsion of non-bonded atoms as well as for weak long-range attraction.

van der Waals Radius. The radius of an atom (in a molecule), that is intended to reflect its overall size.

van der Waals Surface. A surface formed by a set of interpreting spheres (atoms) with specific van der Waals radii, and that is intended to represent overall molecular size and shape.

Vibrational Frequencies. The energies at which molecules vibrate. Vibrational frequencies correspond to the peaks in an Infrared and Raman Spectrum.

Wavefunction. The solutions of the Electronic Schrödinger Equation. In the case of the hydrogen atom, a function of the coordinates that describes the motion of the electron as fully as possible. In the case of a many-electron system a function that describes the motion of the individual electrons.

Zero-Point Energy. The residual energy of vibration at 0K.

Section D 139

Section FUnits

Geometries

Cartesian coordinates are given in Ångstroms (Å).

Bond distances are given in Å. Bond angles and dihedral angles are given in degrees (°).

Surface areas are given in Å2 and volumes in Å3.

1 Å = 0.1 nm=1.889762 au

Energies, Heats of Formation and Strain Energies

Total energies from Hartree-Fock, B3LYP and MP2 calculations are given in au.

Heats of formation from semi-empirical calculations are given in kJ/mol.

Strain energies from molecular mechanics calculations are given in kJ/mol.

Solvation energies are given in the same units as those for the underlying (gas-phase) calculations.

Orbital Energies

Orbital energies are given in eV.

au kcal/mol kJ/mol eV

1 au – 627.5 2625 27.211 kcal/mol 1.593 (-3) – 4.184 4.337 (-2)1 kJ/mol 3.809 (-4) 2.390 (-1) – 1.036 (-2)1 eV 3.675 (-2) 23.06 96.49 –

a) exponent follows in parenthesis, e.g., 1.593 (-3) = 1.593 x 10-3

140 Section D

Electron Densities, Spin Densities, Dipole Moments, Charges, Electrostatic Potentials and Local Ionization Potentials

Electron densities and spin densities are given in electrons/au3.

Dipole moments are given in debyes.

Atomic charges are given in electrons.

Electrostatic potentials are given in kcal/mol.

Local ionization potentials are given in eV.

Vibrational Frequencies

Vibrational frequencies are given in wavenumbers (cm-1).

Index 141

3 key........................................................ 10

3-21G basis set................................. 51,131

6-31G* basis set............................... 51,131

6-311+G** basis set......................... 52,131

A

acetaldehyde.......................................... 103

acetaldehyde hydrate............................. 103

acetamide .............................................. 119

acetic acid................. 11,31,32,106,109,119

acetic acid dimer .............................. 10,106

acetic anhydride .................................... 119

acetone ...................................... 27,103,104

acetone enol .......................................... 104

acetone hydrate ..................................... 103

acetone, perdeutero ................................. 28

acetyl chloride....................................... 119

acetylene ............................ 92,100,102,114

aciditycalculating............................................ 31of carboxylic acids............................... 32relationship to bond energy ............... 101relationship to electronegativity......... 102relationship to electrostaticpotential .......................................... 32,33

acyl cation ............................................... 93

acrylonitrile ...................................... 17,118

activation energy.............................. 59,131

Add Surface dialog ................................. 25

Align Molecules ..................................... 37

aligning molecules .................................. 37

alt key...................................................... 39

aluminum hydride anion ....................... 106

ammonia..................... 11,78,94,96,113,115

ammonium ion ...................................... 115

aniline.......................................... 16,87,118

antimony hexafl uoride anion................. 106

antimony pentafl uoride ......................... 106

antibonding orbital ........................... 72,131

apple key ................................................ 7,8

apply globally.......................................... 34

argon ....................................................... 97

aromaticitycalculating.......................................... 100of benzene.......................................... 100of furan .............................................. 100of pyrrole ........................................... 100of thiophene ....................................... 100relation to geometry........................... 100

Arrhenius equation........................... 60,131

arsine ....................................................... 96

atom colorschanging................................................. 9default settings....................................... 9

atom labels ................................................ 9

atomic chargesdisplaying on model............................... 9effect of correlation on....................... 111electrostatic ........................................ 133formal................................................... 93for acetic acid....................................... 12for acyl cation ...................................... 94for acetylene ...................................... 114for aluminum hydride anion .............. 106for ammonia......................................... 96for ammonium ion ............................. 115for antimony pentafl uoride/hydrogen fl uoride system................... 106for arsine .............................................. 96for borohydride anion ........................ 106for cyanide anion ................................. 93for cyclopentadienyl sodium ............. 107

IndexText in bold refer to entries under Spartan’s menus

142 Index

for ferrocene ...................................... 107for formaldehyde ............................... 111for hydrogen fl uoride ......................... 106for hydrogen selenide .......................... 97for hydrogen sulfi de............................. 97for hydrogen telluride .......................... 97for lithium aluminum hydride............ 106for methyl fl uoride ............................. 114for methyl lithium.............................. 114for nitronium cation ............................. 94for ozone .............................................. 93for phosphine ....................................... 96for propyne ........................................ 114for sodium borohydride anion ........... 106for stibene ............................................ 96for sulfur tetrafl uoride.......................... 25for 3,3,3-trifl uoropropyne .................. 114for trimethylammonium ion............... 115for water............................................... 97reporting............................................... 12units ................................................... 141

atomic orbitalscore ...................................................... 70for chloride anion................................. 71for fl uoride anion ................................. 71for hydrogen ................................... 50,70for many-electron atoms...................... 70valence ................................................. 70

atomic size, and charge ......................... 102

atomic units...................................... 58,131

Atom Properties dialog ........................... 12

1-azaadamantane..................................... 78

B

B3LYP models ................................. 52,131

B3LYP/6-31G* model ............................ 52

B3LYP/6-311+G** model ...................... 52

barrier to rotation, in ethane.................... 45

Ball and Spoke......................................... 8

Ball and Wire........................................... 8

basicityof alkylamines.................................... 115solvent effects on ............................... 115

basis functions.................................. 51,131

basis sets3-21G ............................................ 51,1316-31G* .......................................... 51,1316-311+G**.................................... 52,131complete............................................... 51diffuse ........................................... 52,132Gaussian functions........................ 55,133minimal ......................................... 51,135nomenclature ....................................... 55minimal valence................................... 53polarization ................................... 51,136split valence .................................. 51,137valence ................................................. 53

benzene ............... 15,29,82,87,100,116,118

benzene crystal structure......................... 85

benzene dimer ......................................... 84

benzoic acid ............................................ 32

beryllium hydride.................................. 113

blackbody radiation............................... 126

Boltzmann equation ......................... 59,132

bond anglescorrelation with electronegativity.. ...... 96correlation with size of lone pairs........ 96correlation with orbital hybridization........................................ 97in main-group hydrides........................ 96

bond density ............................................ 78

bond density surfacefor diborane.......................................... 80for ene reaction .................................... 41for ethyl formate .................................. 80for ethylene .......................................... 80for formic acid ..................................... 80for hex-5-ene-1-yne ............................. 79for transition state for ethyl formate pyrolysis ................................. 80relationship to skeletal model .............. 79

bond dissociation energyheterolytic .......................................... 101of hydrogen bromide ......................... 101of hydrogen chloride.......................... 101of hydrogen fl uoride .......................... 101of hydrogen iodide............................. 101

bonding orbital ................................. 71,132

Index 143

bonds, rotating about............................... 39

borane.................................................... 113

Born-Oppenheimer approximation ..51,132

borohydride anion ................................. 106

bromide anion ....................................... 102

bromide anion + tert-butylbromide SN2 transition state.................. 116

bromide anion + methyl bromideSN2 transition state ................................ 116

bromine chloride ................................... 112

bromine fl uoride.................................... 112

bromine iodide ...................................... 112

bromochloroacetylene........................... 112

bromofl uoroacetylene ........................... 112

bromoiodoacetylene.............................. 112

Build menu........................................... 6,11

1,3-butadiene......................................... 110

1,3-butadiene + ethyleneDiels-Alder transition state ................... 116

butane................................... 35,46,101,108

butatriene............................................... 126

1-butene................................................. 109

cis-2-butene........................................... 109

trans-2-butene ....................................... 117

2-tert-butyl-1,3-butadiene..................... 110

2-butyne ................................................ 117

but-1-yne-3-ene.............................. 102,126

butynol .................................................. 102

C

Calculations ........................................... 19

Calculations dialog.................................. 20

carbon dioxide....................................... 127

carbon monoxide..................................... 97

chair-chair interconversion incyclohexane............................................. 47

chloral ................................................... 103

chloral hydrate ...................................... 103

chloride anion........................................ 102

chorine fl uoride ..................................... 112

chlorine iodide ...................................... 112

chloroacetic acid ..................................... 32

chlorobenzene ....................................... 118

chlorofl uoroacetylene............................ 112

chloroiodoacetylene .............................. 112

Clear ....................................................... 19

Close ....................................................... 16

Colors ....................................................... 9

computation times................................... 55

Confi gure ................................................. 9

Confi gure dialog...................................... 22

conformational energy differences, inacetic acid .......................................... 1091,3-butadiene ..................................... 110n-butane .......................................... 35,461-butene ............................................. 109cis-2-butene ....................................... 1092-tert-butyl-1,3-butadiene.................. 110cyclohexane ......................................... 47ethane................................................... 452-methyl-1,3-butadiene...................... 1102,2,3,3-tetramethylbutane .................. 108

Constrain Dihedral ............................... 35

constraining dihedral angle ..................... 35

constraint marker .................................... 37

Constraint Properties dialog.................... 36

control key; See ctrl key

core................................................... 53,132

Correlated modelsB3LYP........................................... 52,131density functional ......................... 52,132Møller-Plesset ...................................... 52MP2 ..................................................... 52

correlation ........................................ 52,132

correlation energy ............................ 52,132

Coulombic interactions .................... 53,132

Coupled .................................................. 30

CPK model....................................... 76,132

144 Index

ctrl key .................................................... 32

Curve Properties dialog........................... 34

cyanide anion .......................................... 93

cyclobutane ........................................... 101

cyclobutadiene ...................................... 126

cyclodecene............................................. 95

cyclodecyne............................................. 95

cycloheptane ......................................... 101

cycloheptene ........................................... 95

cycloheptyne ........................................... 95

cyclohexaneboat form.............................................. 48chair-chair interconversion .................. 47chair form ..................................... 47,101twist boat form..................................... 47

cyclohexanone.................................... 14,16

1,3,5-cyclohexatriene............................ 100

cyclohexene............................................. 95

cyclohexenone......................................... 14

cyclohexyne ............................................ 95

cyclononene ............................................ 95

cyclononyne ............................................ 95

cyclooctene ............................................. 95

cyclooctyne ............................................. 95

cyclopentadienyl anion ......................... 107

cyclopentadienyl sodium ...................... 107

cyclopentane ......................................... 101

cyclopropane ......................................... 101

cycloundecene......................................... 95

cycloundecyne......................................... 95

D

1-decanol............................................... 115

Delete ...................................................... 24

density; See electron density

density functional models ................ 52,132

diazomethane .......................................... 92

diborane................................................... 79

dichloroacetic acid .................................. 32

dichlorocarbene..................................... 117

dichlorocarbene + ethylenetransition state ....................................... 117

cis-dichloroethylene.............................. 123

trans-dichloroethylene .......................... 123

diffusion-controlled reaction............ 67,133

dihedral anglesetting................................................... 35stepping through .................................. 36

dihydrofuran.......................................... 101

dihydropyrrole....................................... 101

dihydrothiophene .................................. 100

trans-diimide........................................... 92

dimethylborane ..................................... 117

dimethylborane + propenehydroboration transition state ............... 117

dimethylborane + propynehydroboration transition state ............... 117

2,3-dimethyl-2-butene........................... 120

dimethylsulfoxide ................................. 121

dimethylsulfoxide, perdeutero .............. 121

dipole momentdependence on bond distance ............ 111dependence on difference inatomic charges ................................... 111of acetic acid................................. 12,109of acrylonitrile ..................................... 21of cyano bromide ............................... 112of cyano chloride ............................... 112of cyano fl uoride................................ 112of cyano iodide .................................. 112of formaldehyde................................. 111of hydrogen bromide ......................... 112of hydrogen chloride.......................... 112of hydrogen fl uoride .......................... 112of hydrogen iodide............................. 112of lithium bromide ............................. 112of lithium chloride ............................. 112of lithium fl uoride.............................. 112of lithium hydride .............................. 112of lithium iodide ................................ 112of nitro bromide ................................. 112

Index 145

of nitro chloride ................................. 112of nitro fl uoride.................................. 112of nitro iodide .................................... 112of propene .......................................... 113of propyne.......................................... 113of sodium bromide............................. 112of sodium chloride ............................. 112of sodium fl uoride.............................. 112of sodium hydride.............................. 112of sodium iodide ................................ 112of toluene ........................................... 113reporting............................................... 12sign of ................................................ 113units ................................................... 139

dipole moment vector ............................. 12

Display menu ....................................... 6,11

documentsadding new molecules ......................... 29aligning molecules ............................... 37animating molecules ............................ 41coupling/uncoupling motions of molecules......................................... 30going between molecules..................... 29making from energy profi le ................. 36making from vibrational motion.......... 41simultaneously displayingmolecules ............................................. 30

E

Edit menu................................................ 19

electron correlation; See correlation

electron densityidentifi cation of bonds ......................... 79location of atomic positions................. 79on hydrogen ....................................... 113relationship to molecular size and shape ............................................. 78units ................................................... 140

electron density surfacefor ammonia.................................. 78,113for 1-azaadamantane............................ 78for benzene .......................................... 82for beryllium hydride......................... 113for borane........................................... 113for hydrogen fl uoride ......................... 113for lithium hydride............................. 113

for methane ........................................ 113for trimethylamine ............................... 78for water............................................. 113relationship to molecular size. ...... 76,113relationship to space-fi lling models.................................................. 76

electronic Schrödinger equation ...... 51,133

electrophilic reactions, examinationusing local ionization potential maps...... 87

electrostatic potential ....................... 81,133

electrostatic potential mapchanging property range ...................... 30colors ......................................... 15,32,81for acetic acid.................................. 32,33for acetylene ...................................... 114for ammonia....................................... 113for benzene ................................ 15,30,83for benzene dimer ................................ 84for benzoic acid ................................... 33for beryllium hydride......................... 113for borane........................................... 113for bromide anion .............................. 102for trans-2-butene .............................. 117for 2-butyne ....................................... 117for chloride anion............................... 102for chloroacetic acid ............................ 33for cyclopropane ................................ 114for 1-decanol...................................... 115for ene reaction .................................... 41 for ethane ........................................... 114for dichloroacetic acid ......................... 33for dichlorocarbene............................ 117for ene reaction .................................... 41for ethanol..................................... 32,115for ethylene ........................................ 114for fl uoride anion ............................... 102for formic acid ..................................... 33for 1-heptanol .................................... 115for hexane .......................................... 114for hydrogen fl uoride ......................... 113for iodide anion.................................. 102for lithium hydride............................. 113for methane ........................................ 113for methanol....................................... 115for methyl fl uoride ............................. 114for methyl lithium.............................. 114for nitric acid ....................................... 32

146 Index

for nitronium cation ............................. 94for pivalic acid ..................................... 33for propane......................................... 114for propanol ....................................... 115for pyridine .......................................... 30for SN2 reaction of cyanide andmethyl iodide ....................................... 85for sulfuric acid.................................... 32for transition state of SN2 reactionof bromide and tert butyl bromidetert butyl bromidetert ..... 117for transition state of SN2 reactionof bromide and methyl bromide ........ 117for trichloroacetic acid......................... 33for water...................................... 113,114use in anticipating acid strength ..... 32,33use in assessing electrophilicreactivity ......................................... 30,83use in assessing solubility........... 114,115units ................................................... 140

electrostatic potential surface, forammonia .................................... 94,95,96arsine.................................................... 96benzene ................................................ 83hydrazine ............................................. 95hydrogen cyanide................................. 93hydrogen fl uoride.......................... 94,113hydrogen selenide ................................ 96hydrogen sulfi de .................................. 96hydrogen telluride................................ 96hydronium cation................................. 95methyl anion ........................................ 95phosphine............................................. 96SN2 reaction of cyanide and methyl iodide ....................................... 85stibene.................................................. 96water ............................................... 94,96

endothermic reaction............................... 59

energyconversions ........................................ 139reporting............................................... 12units .............................................. 58,139

enthalpy................................................... 58

entropy .................................................... 59

equatorial rule....................................... 109

equilibrium constant................................ 59

equilibrium geometrydependence on number ofvalence electrons.................................. 94fi nding.................................................. 64gas vs. solid.......................................... 63using approximate geometries. ....... 68,45

ethane ........................... 8,45,91,92,100,108

ethanol....................................... 32,106,115

ethanol dimer ........................................ 106

ethylamine............................................. 106

ethylamine dimer .................................. 106

ethylene ............. 15,80,92,100,102,117,118

ethylene iron tetracarbonyl ..................... 99

ethyl formate ........................................... 80

ethyl formate, transition state for pyrolysis ............................................ 80

exothermic reaction................................. 59

F

ferrocene ............................................... 107

fi le browser.............................................. 20

File menu ............................................... 6,7

fl uoride anion ........................................ 102

fl uorine .................................................... 91

fl uorine iodide ....................................... 112

fl uoroiodoacetylene............................... 112

fl uoromethane ......................................... 91

force constant ................................... 27,121

force fi eld ......................................... 53,133

formal chargefor cyanide anion ................................. 93for ozone .............................................. 93recipe ................................................. 133

formaldehyde ................................. 103,111

formic acid ......................................... 32,80

frequency; See vibrationalfrequency; infrared spectrum

furan ...................................................... 101

Index 147

G

Geometry menu.................................... 6,10

Gibbs energy ........................................... 59

global minimum............................... 64,133

global surfaces ........................................ 34

glossary of common terms andacronyms............................................... 131

graphical modelsatomic orbitals ..................................... 69bond densities ...................................... 78electron densities ................................. 76electrostatic potential maps ................. 81electrostatic potential surfaces............. 82local ionization potential maps ............ 87local ionization potential surfaces ..86,87LUMO maps ........................................ 88molecular orbitals ................................ 69

greenhouse gases................................... 126

H

half-life.................................................... 61

Hamiltonian operator in Schrödinger equation .............................. 50

Hammond postulate ......................... 86,134

Hartree; See atomic unit

Hartree-Fock approximation............ 51,134

Hartree-Fock limit............................ 51,134

Hartree-Fock modelsapproximations .................................... 51errors in bond lengths ..................... 91,92errors in dipole moments ................... 111errors in vibrational frequencies. ....... 122

heat of formationexperimental ........................................ 58units ................................................... 141

heptane .................................................. 101

heptanol................................................. 115

Hessian............................................. 64,134

hexafl uoroacetone ................................. 103

hexafl uoroacetone hydrate .................... 103

hexane ................................................... 101

HF/3-21G model ..................................... 51

HF/6-31G* model ................................... 51

HF/6-311+G** model ............................. 52

HOMO .................................................. 134

HOMO, ofacetone ............................................... 130carbon monoxide ................................. 97ethylene...................................... 15,92,98formaldehyde ..................................... 111iron tetracarbonyl............................ 98,99methylene............................................. 74nitrogen.............................................. 129sulfur tetrafl uoride ............................... 25

HOMO energy, of hydrazine................... 95

HOMO-LUMO gapin 4-amino-4´-nitroazobenzene.......... 130in azobenzene .................................... 130in 1,3-butadiene ................................. 110in 2-tert-butyl-1,3-butadiene ............. 110in 4-hydroxyazobenzene.................... 130in 2-methyl-1,3-butadiene ................. 110relationship with λmax ......................... 110

hydrazine................................................. 95

hydride anion ........................................ 113

hydrogen ................................. 100,101,113

hydrogen bondingdisplaying............................................. 10effect on infrared spectrum................ 125in acetic acid dimer............................ 106in ethylamine ..................................... 106in ethanol ........................................... 106in water .............................................. 105

Hydrogen Bonds.................................... 10

hydrogen bromide ................................. 102

hydrogen chloride ................................. 102

hydrogen cyanide .................................... 92

hydrogen fl uoride ...................... 94,102,106

hydrogen iodide .................................... 102

hydrogen peroxide .................................. 11

hydrogen selenide ................................... 96

hydrogen, size in molecules.................. 113

hydrogen sulfi de...................................... 96

148 Index

hydrogen telluride ................................... 96

hydronium cation .................................... 94

2-hydroxypyridine................................. 104

I

icons .......................................................... 7

imaginary frequency ........................ 40,134

infrared active ....................................... 123

infrared inactive ............................. 123,125

infrared spectrumdisplaying............................................. 14intensity of absorption ....................... 123for acetone ............................. 27,122,125for acetone, perdeutero ........................ 28for 1-butanol ...................................... 125for 1-butanol dimer............................ 125for butatriene...................................... 126for 2-butyne ....................................... 125for but-1-yne-3-ene............................ 126for carbon dioxide.............................. 125for cyclobutadiene ............................. 126for cyclohexanone................................ 14for cis-dichloroethylene..................... 123for trans-dichloroethylene ................. 123for dimethylsulfoxide ........................ 121for dimethylsulfoxide, perdeutero ..... 122for ethanol.......................................... 125for methane ........................................ 128for methylenecyclopropene ............... 126for nitrous oxide................................. 128for 1-octyne........................................ 125for 1,1,2,2-tetrafl uoroethane .............. 128for tetrahedrane.................................. 126for transition state for addition of dichlorocarbene to ethylene.. ........ 117for transition state for ene reaction of 1-pentene ........................... 39for water........................................ 13,125vibrational motions .............................. 13

inorganic model kit ................................. 24

insert key............................................... 105

iodide anion........................................... 102

iron pentacarbonyl .................................. 98

iron tetracarbonyl ............................... 98,99

isotope; See mass number

isotope effectfor hydrogen abstraction frommethylene chloride ............................ 128kinetic ................................................ 129thermodynamic .................................. 129

K

kinetically-controlled reaction ......... 62,135

krypton .................................................... 97

L

Labels ....................................................... 9

LCAO approximation ...................... 51,135

leaving group, relationship to electrostatic potential................................................... 85

Lewis structureschoosing appropriate weights. ........ 92,93for acyl cation ...................................... 94for acyl cation-benzene adduct ............ 94for cyanide anion ................................. 93for diazomethane ................................. 92for nitronium cation ............................. 94for nitronium cation-benzeneadduct................................................... 94for ozone .............................................. 93for pyridazine....................................... 93for pyridine .......................................... 93relationship to molecular orbitals ........ 69

lists of molecules; See documents

lithium aluminum hydride..................... 106

lithium bromide..................................... 112

lithium cation ........................................ 106

lithium chloride..................................... 112

lithium fl uoride...................................... 112

lithium hydride...................................... 112

lithium iodide ........................................ 112

local ionization potential....................... 135

local ionization potential mapcolors ................................................... 16for acrylonitrile .................................. 118for aniline................................. 16,87,118for benzene ................................... 87,118

Index 149

for trans-2-butene .............................. 118for 2-butyne ....................................... 118for chlorobenzene .............................. 118for 2,3-dimethyl-2-butene.................. 120for ethylene ........................................ 118for nitrobenzene............................ 87,118for phenol........................................... 118for propene......................................... 118for tetramethylsilaethylene ................ 120for toluene.......................................... 118for 3,3,3-trifl uoropropene .................. 118for vinyl chloride ............................... 118use in assessing electrophilicreactivity .............................................. 87

local ionization potential surface, for sulfur tetrafl uoride ............................. 86

local ionization potential, units ............. 140

local minimum ................................. 64,135

LUMO................................................... 135

LUMO, ofacetone ............................................... 130carbon monoxide ................................. 97cyclohexanone ..................................... 16ethylene........................................... 92,98formaldehyde ..................................... 111hydrogen .............................................. 69iron tetracarbonyl............................ 98,99methylene............................................. 75nitrogen.............................................. 129

LUMO energy, foracetamide ........................................... 119acetic acid .......................................... 119acetic anhydride................................. 119acetyl chloride.................................... 119methyl acetate .................................... 119

LUMO mapcolors ................................................... 16for cyclohexanone................................ 16for cyclohexenone................................ 88for 2,3-dimethyl-2-butene.................. 120for tetramethylsilaethylene ................ 120use in assessing nucleophilic reactivity .............................................. 88

M

map; See electrostatic potential map;

local ionization potential map;LUMO map

mass number, changing.................... 28,121

Measure Angle ....................................... 11

Measure Dihedral.................................. 11

Measure Distance .................................. 10

measuringbond angles .......................................... 11bond distances...................................... 10dihedral angles ..................................... 11

methane .......................................... 113,128

methyl acetate ................................ 104,119

methyl acetate enol ............................... 104

methylamine............................................ 92

methyl anion............................................ 94

2-methyl-1,3-butadiene......................... 110

methylene................................................ 73

methylenecyclopropene ........................ 126

methyleneimine....................................... 92

methyl fl uoride ................................. 91,114

2-methyl-2-propyl anion ......................... 94

2-methyl-2-propyl cation ........................ 94

2-methyl-2-propyl radical ....................... 94

minimal basis set.............................. 51,135

minimal valence basis set........................ 53

Minimize ................................................ 19

modelball-and-spoke ....................................... 8ball-and-wire.......................................... 8space-fi lling ........................................... 8tube ........................................................ 8wire ........................................................ 8

model kitsgoing between...................................... 23groups menu ........................................ 19inorganic .............................................. 24organic ................................................. 18

Model menu ........................................... 6,8

molecular mechanics models ........... 53,135

molecular orbital models.................. 51,135

150 Index

molecular orbitalsantibonding ................................... 72,131bonding ......................................... 71,132for hydrogen ........................................ 69for methylene.................................... 73fffor nitrogen ..................................... 71,72nodes............................................. 69,136non-bonding.................................. 71,136

Molecule Properties dialog ..................... 12

Møller-Plesset models...................... 52,136

Monitor .................................................. 20

mouse operations ................................... 7,8

MP2 models ..................................... 52,136

MP2/6-31G* model ................................ 52

MP2/6-311+G** model .......................... 52

N

names, default molecule.......................... 20

NDDO approximation...................... 53,136

neon......................................................... 97

New ......................................................... 17

New Molecule ........................................ 29

nitric acid ................................................ 31

nitrobenzene..................................... 87,118

nitrogen ................................................... 92

nitronium cation ...................................... 94

nitrous oxide.......................................... 128

nodes ................................................ 69,136

non-bonding orbital.......................... 74,136

normal modes........................................ 121

nucleophilic reactions, examinationusing LUMO maps.................................. 88

O

Open ......................................................... 7

option key.............................................. 105

Options menu............................................ 9

orbital energies, units ............................ 139

organic model kit .................................... 18

Output .................................................... 21

Output dialog .......................................... 21

ozone ....................................................... 93

P

pentanol................................................. 115

1-pentene................................................. 39

pent-1-yne-3-ene ................................... 102

Periodic Table ......................................... 24

phenol.................................................... 118

phosphine ................................................ 96

pivalic acid .............................................. 32

pKa, correlation with electrostatic potential .............................. 33

play button .............................................. 41

Plots ........................................................ 34

plotsmaking ................................................. 34specifying least-squares fi t line............ 34

PM3 model....................................... 53,136

polarization basis set ........................ 51,136

potential energy surfacesfor chair interconversion incyclohexane. ........................................ 47for rotation in ethane............................ 45for rotation in n-butane ........................ 46kinetic vs. thermodynamic products ....62many dimensional................................ 47one dimensional ................................... 45reaction coordinate .............................. 47transition state...................................... 47

propane........................................... 101,108

propanol ................................................ 115

propargyl alcohol .................................. 102

propene............................... 10,113,117,118

propen-2-ol............................................ 104

Properties............................................... 11

propyne ............................ 102,113,114,117

pyridazine................................................ 93

pyridine .............................................. 29,93

2-pyridone ............................................. 104

Index 151

pyrrole ................................................... 101

R

rate constant ............................................ 60

reaction coordinate........................... 47,137

reaction coordinate diagram............. 48,137

reduced mass.................................... 27,121

ring strain .............................................. 101

rotate molecules ..................................... 7,8

S

Save As ................................................... 28

Save As dialog......................................... 20

scale molecule........................................ 7,8

Schrödinger equation ....................... 50,137

Search menu......................................... 6,39

select molecule....................................... 7,8

semi-empirical models ..................... 53,137

setting up calculations............................. 20

Setup menu........................................... 6,19

shift key................................................ 7,40

sodium borohydride .............................. 106

sodium bromide .................................... 112

sodium chloride..................................... 112

sodium fl uoride ..................................... 112

sodium hydride...................................... 112

sodium iodide........................................ 112

spacebar................................................... 39

Space-Filling ............................................ 9

space fi lling model, forammonia .............................................. 761-azaadamantane ................................. 76benzene ................................................ 82trimethylamine..................................... 76

Spartanavailable models .................................... 3limitations .............................................. 3

Spartan’s databasereplace one molecule ........................... 33

replace all molecules ........................... 33

Spartan Monitor.................................... 20

Spartan ST menu..................................... 20

Spectra ................................................... 40

Spectra dialog.......................................... 13

split-valence basis set....................... 51,137

Spreadsheet............................................ 30

spreadsheetentering numerical data ....................... 33making plots from........................... 33,37posting data to...................................... 37

staggered rule ........................................ 109

starting SpartanMac ........................................................ 6Windows ................................................ 6

step buttons ............................................. 29

stereo (3D) display.................................. 10

steric effects on SN2 reactions............... 116

stibene ..................................................... 96

Style menu .............................................. 15

Submit .................................................... 20

sulfur tetrafl uoride.............................. 23,86

sulfuric acid............................................. 31

Surface Properties dialog ........................ 30

Surfaces .................................................. 15

Surfaces dialog........................................ 15

surfaces, display styles............................ 15

T

1,1,2,2-tetrafl uoroethane ....................... 128

tetrahedrane........................................... 126

2,2,3,3-tetramethylbutane ..................... 108

tetramethylsilaethylene ......................... 120

theoretical modelsB3LYP........................................... 52,131computation time ................................. 55correlated ...................................... 52,132density functional ......................... 52,132Hartree-Fock................................. 51,134molecular mechanics .................... 53,135

152 Index

Møller-Plesset ............................... 52,136MP2 .............................................. 52,136nomenclature ....................................... 55performance .................................... 56,57selection criteria................................... 54semi-empirical .............................. 53,137

thermodynamically-controlledreaction............................................. 62,137

thiophene............................................... 100

toluene............................................ 113,118

total energy.............................................. 58

Transition States.................................... 39

transition statefi nding.................................................. 65for addition of dichlorocarbeneto ethylene.......................................... 117for Diels-Alder reaction of1,3-butadiene and ethylene ................ 116for ene reaction of 1-pentene ............... 39for hydroboration reaction of dimethylborane and propene ............. 117for hydroboration reaction of dimethylborane and propyne ............. 117for hydrogen abstraction frommethylene chloride ............................ 128for pyrolysis of ethyl formate .............. 80for SN2 reaction of bromide and tert-butyl bromide.............................. 116for SN2 reaction of bromide and methyl bromide.................................. 116imaginary vibrational frequency.......... 49non-existance ....................................... 49reactions without transition states........ 67generating ............................................ 39using approximate transition-stategeometries............................................ 68verifying............................................... 66

transition state theory....................... 66,138

translate molecule .................................. 7,8

trichloroacetaldehyde............................ 103

trichloroacetaldehyde hydrate............... 103

trichloroacetic acid.................................. 32

trifl uoroacetone ..................................... 104

trifl uoroacetone enol ............................. 104

3,3,3-trifl uoropropene ........................... 118

3,3,3-trifl uoropropyne ........................... 114

trimethylamine ................................. 78,115

trimethylammonium ion........................ 115

1,3,5-trioxane ........................................ 103

Tube .......................................................... 8

U

Undo ....................................................... 19

V

valence basis set...................................... 53

van der Waals interactions ............... 53,138

vibrational frequenciesimaginary ............................................. 40units ................................................... 140verifying equilibrium andtransition state geometries .............. 64,67

vibrational motionsanimating ............................................. 40changing amplitude.............................. 41making list of ....................................... 41

View ........................................................ 11

vinyl alcohol.......................................... 102

vinyl chloride ........................................ 118

W

water............................................ 94,96,113

water-ammonia cluster.......................... 105

water cluster .......................................... 105

water-methane cluster ........................... 105

wavefunction.................................... 50,138

Wire .......................................................... 8

X

Xeon........................................................ 97

Z

zero-point energy ........................... 129,138