physical chemistry 2 nd edition

Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 26 Chapter 26 Computational Chemistry

© 2010 Pearson Education South Asia Pte LtdPhysical Chemistry 2nd EditionChapter 26: Computational Chemistry

ObjectivesObjectives• Discover the usage of numerical methods.• Discussion is the Hartree-Fock molecular

orbital model.


OutlineOutline1.The Promise of Computational Chemistry2.Potential Energy Surfaces3.Hartree-Fock Molecular Orbital Theory: A

Direct Descendant of the Schrödinger Equation

4.Properties of Limiting Hartree-Fock Models5.Theoretical Models and Theoretical Model

Chemistry


OutlineOutline6.Moving Beyond Hartree- Fock Theory7.Gaussian Basis Sets8.Selection of a Theoretical Model9.Graphical Models10. Conclusion


26.1 The Promise of Computational Chemistry26.1 The Promise of Computational Chemistry

• Sufficient accuracy can be obtained from computational chemistry.

• Approximations need to be made to realize equations that can be solved.

• No one method of calculation is likely to be ideal for all application.

• Hartree-Fock theory leads to ways to improve on it and to a range of practical quantum chemical models.


26.2.1 26.2.1 Potential Energy Surfaces and Potential Energy Surfaces and Geometry Geometry

• Energy minima give the equilibrium structures of the reactants and products.

• Energy maximum defines the transition state.

• Reactants, products, and transition states are all stationary points on the potential energy diagram.



• In the one-dimensional case, 1st derivative of the potential energy with respect to the reaction coordinate is zero:

• For many-dimensional case, each independent coordinate, Ri, gives rise to 3N-6 second derivatives:

0dRdV

63

2

NiRRV



• Stationary points where all second derivatives are positive are energy minima:

where ζi = normal coordinates• Stationary points where all but one are

positive are saddle points:

where ζi = reaction coordinate

63,...,2,1 02

2

NiV

i

02

2

p

V


26.2.2 Potential Energy Surfaces and Vibrational 26.2.2 Potential Energy Surfaces and Vibrational Spectra Spectra

• The vibrational frequency for a diatomic molecule A-B is

• k is the force constant which is defined as

• And μ is the reduced mass.

kv

21

2

2

dRRVdk

BA

BA

mmmm


26.2.3 26.2.3 Potential Energy Surfaces and Potential Energy Surfaces and Thermodynamics Thermodynamics

• The energy difference between the reactants and products determines the thermodynamics of a reaction.

• The ratio is as follow,

kTEE

nn tsreacproducts

tsreac

products tan

tan

exp


26.2.3 26.2.3 Potential Energy Surfaces and Potential Energy Surfaces and Thermodynamics Thermodynamics

• The energy difference between the reactants and transition state determines the rate of a reaction.

• The rate constant is given by the Arrhenius equation and depends on the temperature:

kT

EEAk tsreacstatetransition tan exp


26.3 Hartree-Fock Molecular Orbital Theory: A 26.3 Hartree-Fock Molecular Orbital Theory: A Direct Direct Descendant of the Schrödinger Equation Descendant of the Schrödinger Equation

• 3 approximations need to realize a practical quantum mechanical theory for multielectron Schrödinger equation:

a) Born-Oppenheimer approximationb) Hartree-Fock approximationc) Linear combination of atomic orbitals

(LCAO) approximation

EH


MATHEMATICAL FORMULATION OF THE MATHEMATICAL FORMULATION OF THE HARTREE-FOCK METHODHARTREE-FOCK METHOD

The Hartree-Fock and LCAO approximations, taken together and applied to the electronic Schrödinger equation, lead to a set of matrix equations now known as the Roothaan-Hall equations:

where c = unknown molecular orbital coefficients ε = orbital energies S = overlap matrix F = Fock matrix

ScFc



For Fock matrix,

where Hcore = core Hamiltonian

Coulomb and exchange elements are given by

vvcorevv KJHF

drrr

ZemhrH v

nuclei

A

A

e

corev

0

22

2

42

vPK

vPJ

functionbasis

v

functionbasis

v

21



P is called the density matrix

The cost of a calculation rises rapidly with the size of the basis set:

ii

occupiedorbitalsmolecular

ccP

i

2

212212

111 drdrrrr

rrv v


26.4 Properties of Limiting Hartree-Fock Models26.4 Properties of Limiting Hartree-Fock Models

• For computation, it is expected to have errors in:

1. Relative energies2. Geometries3. Vibrational frequencies4. Properties such as dipole moments


26.4.1 Reaction Energies26.4.1 Reaction Energies

• Hartree-Fock models is compare with homolytic bond dissociation energies.

• For example in methanol,



• The poor results seen for homolytic bond dissociation reactions do not necessarily carry over into other types of reactions as long as the total number of electron pairs is maintained.


26.4.2 Equilibrium Geometries26.4.2 Equilibrium Geometries

• Systematic discrepancies are also noted in comparisons involving limiting Hartree-Fock and experimental.

• They are geometries and bond distances.• The reason is that limiting Hartree-Fock

bond distances is shorter than experimental values.


26.4.3 Vibrational Frequencies26.4.3 Vibrational Frequencies

• The error in bond distances for limiting Hartree-Fock models calculated frequencies are larger than experimental frequencies.

• The reason is that the Hartree-Fock model does not dissociate to the proper limit of two radicals as a bond is stretched.


26.4.4 Dipole Moments26.4.4 Dipole Moments

• Electric dipole moments are compared, the calculated values are larger than experimental values.


26.5 Theoretical Models and Theoretical Model 26.5 Theoretical Models and Theoretical Model Chemistry Chemistry

• Limiting Hartree-Fock models do not provide results that are identical to experimental results.

• Theoretical model chemistry is a detailed theory starting from the electronic Schrödinger equation and ending with a useful scheme.


26.6 Moving Beyond Hartree-Fock Theory26.6 Moving Beyond Hartree-Fock Theory

• Improvements will increase the cost of a calculation.

• 2 approaches to improve Hartree-Fock theory:

1. Increases the flexibility by combining it with wave functions corresponding to various excited states.

2. Introduces an explicit term in the Hamiltonian to account for the interdependence of electron motions.


26.6.1 Configuration Interaction Models26.6.1 Configuration Interaction Models

• Improvements will increase the cost of a calculation.

• 2 approaches to improve Hartree-Fock theory:

1. Increases the flexibility by combining it with wave functions corresponding to various excited states.

2. Introduces an explicit term in the Hamiltonian to account for the interdependence of electron motions.


26.6.2 Møller-Plesset Models26.6.2 Møller-Plesset Models

• Møller-Plesset models are based on Hartree-Fock wave function and ground-state energy E0 as exact solutions.

where = small perturbation λ = dimensionless parameter

VHH ˆˆˆ0

V


MATHEMATICAL FORMULATION OF MØLLER-MATHEMATICAL FORMULATION OF MØLLER-PLESSET MODELSPLESSET MODELS

Substituting the expansions into the Schrödingerequation and gathering terms in λn yields

...

ˆˆ

ˆˆ

ˆ

02112012

0

0110

01

0

00

0

EEEVH

EEVH

EH



Multiplying each by ψ0 and integrating over all space yields the following expression for the nth-order (MPn) energy:

...

...ˆ...

...ˆ...

...ˆ...

211

02

21001

21000

n

n

n

dddHE

dddVE

dddHE



In this framework, the Hartree-Fock energy is the sum of the zero- and firstorder Møller-Plesset energies:

The first correction, E(2) can be written as follows

ndddVHEE ...ˆˆ... 21

10

10



The integrals (ij || ab) over filled (i and j) and empty (a and b) molecular orbitals account for changes in electron–electron interactions as a result of electron promotion,

in which the integrals (ij | ab) and (ib | ja) involve molecular orbitals rather than basis functions.

The two integrals are related by a simple transformation,

jaijabijabij


26.6.3 Density Functional Models26.6.3 Density Functional Models

• Density functional theory is based on the availability of an exact solution for an idealized many-electron problem.

• The Hartree-Fock energy may be written as

where ET = kinetic energy EV = the electron–nuclear potential energy EJ = Coulomb EK = interaction energy

KJVTHF EEEEE


26.6.3 Density Functional Models26.6.3 Density Functional Models

• For idealized electron gas problem:

where EXC = exchange/correlation energy

• Except for ET, all components depend on the total electron density, p(r):

XCJVTDFT EEEEE

orbitals

ii rr 22


MATHEMATICAL FORMULATION OF DENSITY MATHEMATICAL FORMULATION OF DENSITY FUNCTIONAL THEORYFUNCTIONAL THEORY

Within a finite basis set (analogous to the LCAO approximation for Hartree Fock models), the components of the density functional energy, EDFT, can be written as follows:

drrrfE

vE

drrRr

eZrE

drrmehrE

XC

v

functionsbasis

vJ

vA

Anuclei

Av

functionsbasis

vV

ve

functionsbasis

vT

...,

21

4

2

2

0

2

222



Better models result from also fitting the gradient of the density. Minimizing EDFT with respect to the unknown orbital coefficients yields a set of matrix equations, the Kohn-Sham equations, analogous to the Roothaan-Hall equations

Here the elements of the Fock matrix are given by

ScFc

XCvv

corevv FJHF



FXC is the exchange/correlation part, the form of which depends on the particular exchange/correlation functional used. Note that substitution of the Hartree-Fock exchange, K, for FXC yields the Roothaan-Hall equations.


26.6.4 Overview of Quantum Chemical Models26.6.4 Overview of Quantum Chemical Models

• An overview of quantum chemical models.


26.7 Gaussian Basis Sets26.7 Gaussian Basis Sets

• LCAO approximation requires the use of a finite number of well-defined functions centered on each atom.

• Early numerical calculations use nodeless Slater-type orbitals (STOs),

• If the AOs are expanded in terms of Gaussian functions,

2rkjiijk ezyNxrg


26.7.1 Minimal Basis Sets26.7.1 Minimal Basis Sets

• The minimum number is the number of functions required to hold all the electrons of the atom while still maintaining its overall spherical nature.

• This simplest representation or minimal basis set involves a single (1s) function for hydrogen and helium.

• In STO-3G basis set, basis functions is expanded in terms of three Gaussian functions.


26.7.2 Split-Valence Basis Sets26.7.2 Split-Valence Basis Sets

• Minimal basis set is bias toward atoms with spherical environments.

• A split-valence basis set represents core atomic orbitals by one set of functions and valence atomic orbitals by two sets of functions:

for lithium to neon

for sodium to argon


26.7.3 Polarization Basis Sets26.7.3 Polarization Basis Sets

• Minimal (or split-valence) basis set functions are centered on atoms rather than between atoms.

• The inclusion of polarization functions can be thought about either in terms of hybrid orbitals.


26.7.4 Basis Sets Incorporating Diffuse Functions26.7.4 Basis Sets Incorporating Diffuse Functions

• Calculations involving anions can pose problems as highest energy electrons may only be loosely associated with specific atoms (or pairs of atoms).

• In these situations, basis sets may need to be supplemented by diffuse functions.


26.8 Selection of a Theoretical Model26.8 Selection of a Theoretical Model

• Hartree-Fock models have proven to be successful in large number of situations and remain a mainstay of computational chemistry.

• Correlated models can be divided into 2 categories:

1. Density functional models2. Møller-Plesset models• Transitionstate geometry optimizations are

more time-consuming than equilibrium geometry optimizations, due primarily to guess of geometry.


26.8.1 Equilibrium Bond Distances26.8.1 Equilibrium Bond Distances

• Hartree-Fock double bond lengths are shorter than experimental distances.

• Treatment of electron correlation involves the promotion of electrons from occupied molecular orbitals to unoccupied molecular orbitals.


26.8.2 Finding Equilibrium Geometries26.8.2 Finding Equilibrium Geometries

• An equilibrium structure corresponds to the bottom of a well on the overall potential energy surface.

• Equilibrium structures that cannot be detected are referred to as reactive intermediates.

• Geometry optimization does not guarantee that the final geometry will have a lower energy than any other geometry of the same molecular formula.



• Reaction energy comparisons are divided into three parts:

1. Bond dissociation energies2. Energies of reactions relating structural

isomers3. Relative proton affinities.


26.8.4 Energies, Enthalpies, and Gibbs Energies26.8.4 Energies, Enthalpies, and Gibbs Energies

• Quantum chemical calculations account for thermochemistry by combining the energies of reactant and product molecules at 0 K.

• Residual energy of vibration is ignored.• We would need 3 corrections:1. Correction of the internal energy for finite

temperature.2. Correction for zero point vibrational

energy.3. Corrections of entropy.


26.8.5 Conformational Energy Differences26.8.5 Conformational Energy Differences

• Hartree-Fock models overestimate differences by large amounts.

• Correlated models also typically overestimate energy differences but magnitudes of the errors are much smaller than those seen for Hartree-Fock models.


26.8.6 Determining Molecular Shape26.8.6 Determining Molecular Shape

• The problem of identifying the lowest energy conformer in simple molecules is when the number of conformational degrees of freedom increases.

• Sampling techniques will need to replace systematic procedures for complex molecules, thus Monte Carlo methods is used.


26.8.7 Alternatives to Bond Rotation26.8.7 Alternatives to Bond Rotation

• Single-bond rotation is the most common mechanism for conformer interconversion.

• 2 other processes are known:1. Inversion is associated with pyramidal

nitrogen or phosphorus and involves a planar transition state.

2. Pseudorotation is associated with trigonal bipyramidal phosphorus and involves a square-based-pyramidal transition state.


26.8.8 Dipole Moments26.8.8 Dipole Moments

• Dipole moments from the two Hartree-Fock models are larger than experimental values due to behavior of the limiting Hartree-Fock model.

• Recognize that electron promotion from occupied to unoccupied molecular orbitals takes electrons from “where they are” to “where they are not”.


26.8.9 Atomic Charges: Real or Make Believe?26.8.9 Atomic Charges: Real or Make Believe?

• Charge distributions assess overall molecular structure and stability.

• Mulliken population analysis can be used to formulate atomic charges.


MATHEMATICAL DESCRIPTION OF THE MULLIKEN MATHEMATICAL DESCRIPTION OF THE MULLIKEN POPULATION ANALYSISPOPULATION ANALYSIS

The Mulliken population analysis starts from the definition of the electron density, ρ(r), in the framework of the Hartree-Fock model:

Summing over basis functions and integrating over all space leads to an expression for thetotal number of electrons, n:

rrPr vv

functionsbasis

v

nSPrrPdrr vv

functionsbasis

vvv

functionsbasis

v



where Sμv are elements of the overlap matrix:

It is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements as follows:

drrrS vv

nSPPSP vv

functionsbasis

v

functionsbasis

vv

functionsbasis

v

2



According to Mulliken’s scheme, the gross electron population for basis function is given by

Atomic electron populations, qA, and atomic charges, QA, follow, where ZA is the atomic number of atom A:

qZQ

qq

AA

Aatomonfunctionbasis

A

vv

functionsbasis

v

SPPq


26.8.10 Transition-State Geometries and 26.8.10 Transition-State Geometries and Activation Activation Energies Energies

• Transition-state theory states that all reactants have the same energy, or that none has energy in excess of that needed to reach the transition state.

• Hartree-Fock models overestimate the activation energies by large amounts.


26.8.11 Finding a Transition State26.8.11 Finding a Transition State

• There is less effort (energy) by passing through a “valley” between two “mountains” (pathway B).

• Saddle point referred to a maximum and minimum in the transition state.


26.9 Graphical Models26.9 Graphical Models

• Molecular orbitals, electron density and electrostatic potential can be defined a isovalue surface or isosurface:

• Most common graphical models are on electron density surfaces and electrostatic potential.

constant,, zyxf


26.9.1 Molecular Orbitals26.9.1 Molecular Orbitals

• Molecular orbitals, ψ, are written as

• Highest energy occupied molecular orbital (HOMO) holds the highest energy electrons and is attack by electrophiles, while lowest energy unoccupied molecular orbital (the LUMO) provides the lowest energy space for additional electrons and attack by nucleophiles.

i

functionsbasis

i c


26.9.2 Orbital Symmetry Control of Chemical 26.9.2 Orbital Symmetry Control of Chemical ReactionsReactions

• HOMO and LUMO (frontier molecular orbitals) could be used to rationalize why some chemical reactions proceed easily whereas others do not.


26.9.3 Electron Density26.9.3 Electron Density

• Electron density ρ(r) is written in terms of

• Depending on the value, isodensity surfaces can either serve to locate atoms to delineate chemical bonds or to indicate overall molecular size and shap.


26.9.4 Where Are the Bonds in a Molecule?26.9.4 Where Are the Bonds in a Molecule?

• An electron density surface can be used to know the location of bonds in a molecule.

• Electron density surfaces is also use as the description of the bonding in transition states


26.9.5 How Big Is a Molecule?26.9.5 How Big Is a Molecule?

• The size of a molecule can be defined according to the amount of space that it takes up in a liquid or solid.

• The electron density provides an alternate measure of how much space molecules actually take up.


26.9.6 Electrostatic Potential26.9.6 Electrostatic Potential

• The electrostatic potential,εp, is defined as

• Note that electrostatic potential represents a balance between repulsion of the point charge by the nuclei and attraction of the point charge by the electrons.

dr

rrr

PR

Ze

p

vv

functionbasis

v

nuclei

A AP

Ap

*

0

2

4


26.9.7 Visualizing Lone Pairs26.9.7 Visualizing Lone Pairs

• The octet rule dictates that each main-group atom in a molecule will be surrounded by eight valence electrons.

• A comparison between electrostatic potential surfaces for ammonia in both the observed pyramidal and unstable trigonal planar geometries.


26.9.8 Electrostatic Potential Maps26.9.8 Electrostatic Potential Maps

• Most commonly used property map is the electrostatic potential map.

• It gives the value of the electrostatic potential at locations on a particular surface.


26.9.8 Electrostatic Potential Maps26.9.8 Electrostatic Potential Maps

• Electrostatic potential maps are used to distinguish between molecules in which charge is localized from those where it is delocalized.


26.9.10 Conclusions26.9.10 Conclusions

• Inability of the calculations to deal highly reactive molecules that

1. difficult to synthesize2. with reaction transition states

• Limitations of quantum chemical calculations are:

1. Practical and numerical results not match2. Important quantities cannot be yield3. Calculations apply strictly to isolated

molecules (gas phase)


Example 27.1Example 27.1

a. Are the three mirror planes for the NF3 molecule in the same or in different classes?

b. Are the two mirror planes for H2O in the same or in different classes?


SolutionSolutiona. NF3 belongs to the C3v group, which contains the rotation operators and the vertical mirror planes . These operations and elements are illustrated by this figure:

ECCCC 3

3

1

3233

ˆ and ,ˆˆ,ˆ

3 and ,2,1 vvv

physical chemistry 2 nd edition

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