Chem 126

Computational Chemistry Fall 2011

Lecturer:

Dr. Kalju KahnOffice: PSB-N 2623, Phone: 893-6157E-mail: kalju@chem.ucsb.edu, Website: http://www.chem.ucsb.edu/~kalju

Mission statement

The course focuses on learning the principles of computational chemistry and computer-based molecular design. Both molecular mechanical and quantum mechanical models are covered. Students will learn a variety of commonly used techniques, such as geometry optimization, location of transition states, conformational analysis, and prediction of molecular and spectroscopic properties. Students will learn basics of implementing key algorithms, such as Newton-Rhapson minimization, and normal mode analysis of vibrational motions. Students also will become familiar with different software packages, including MOLDEN for general model building, Gaussian, Firefly, and NWChem for quantum chemical calculations, and BOSS for liquid simulations. Students who complete the course are expected to be able to ask questions that can be solved with modern computational approaches and choose right computational tools to assist in their current or future research.

Course Materials

Syllabus General information about the course PDF

Last Years Computational Chemistry 126 by Dr. Kahn: 2008 Link

Last Years Computational Chemistry 126 by Dr. Kirtman: 2007 Link

Textbook Suggested: Intoduction to Computational Chemistry : Theories and Models Amazon

Textbook Alternative: Essentials of Computational Chemistry : Theories and Models Amazon

Upload Submit your assignments as MS Word or PDF files Link

Exam Midterm Preparation Guide PDF

Exam Final Preparation Guide PDF

Literature

Literature Required or optional reading in PDF Acrobat

Required "Mathematical Methods" by Frank Jensen PDF

Required "Optimization Techniques" by Frank Jensen PDF

Required "Force Field Methods" by Frank Jensen PDF

Required "Biomolecular simulation and modelling: Status ..." by van der Kamp et al PDF

Required "Potential energy functions for atomic-level simulations of water and organic and biomolecular systems" by Jorgensen and Tirado-Rives

PDF

Required Intro Quantum Mechanics: MO Theory PDF

Optional Basis Sets for Ab Initio MO Calculations ... PDF

Required DFT: Performance and Problems PDF

Optional "Performance of B3LYP Density Functional Methods ..." by Tirado-Rives and Jorgensen PDF

Optional Explanation of Thermochemistry at NIST Link

Assignments

The assignments are posted one week before the due date. Answers shall be submitted electronically no later than the midnight of the due date.

Assignments Submit Your Work Link

1 Minimization: Tutorial and Assignments Link

2 Molecules: Building, Minimization, and Conformational Analysis Link

3.a Monte Carlo Simulations Link

3.b Introduction to QM and Semiempirical Methods Link

4 Introduction to An Initio Methods Link

5 Molecular Vibrations Link

5 Reaction Thermodynamics and Kinetics: Transition States Link

6 Not Required: Electronic Spectra and Solvent Effects Link

6 Not Required: QM/MM Modeling of Reactions: Free Energy Perturbation Link

Answer Keys

The answer keys are posted one week after the due date.

Computational Chemistry Links

Computational Chemistry List

Course materials by Dr. Kalju Kahn, Department of Chemistry and Biochemistry , UC Santa Barbara. 2011

Tutorial Example for MethylamineRunning an NBO CalculationThis page provides an introductory quick-start tutorial on running an NBO calculation and interpreting the output. The example chosen is that of methylamine (CH3NH2) in Pople-Gordon idealized geometry, treated at the ab initio RHF/3-21G level. This simple split-valence basis set consists of 28 AOs (nine each on C and N, two on each H), extended by 13 AOs beyond the minimal basis level.

Input files to perform this calculation are given here for Gaussian and GAMESS. The pop=nbo option of the Gaussian program requests default NBO analysis. NBO analysis is requested in the GAMESS calculation by simply including the line

$NBO $END

in the GAMESS input file. This "empty" NBO keylist specifies that NBO analysis should be carried out at the default level.

The default NBO output produced by this example is shown below, just as it appears in your output file. The start of the NBO section is marked by a standard header, citation, job title, and storage info:

*********************************** NBO 5.0 *********************************** N A T U R A L A T O M I C O R B I T A L A N D N A T U R A L B O N D O R B I T A L A N A L Y S I S******************************************************************************* (c) Copyright 1996-2001 Board of Regents of the University of Wisconsin System on behalf of the Theoretical Chemistry Institute. All Rights Reserved.

Cite this program as:

NBO 5.0. E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, J. A. Bohmann, C. M. Morales, and F. Weinhold (Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2001)

Job title: Methylamine...RHF/3-21G//Pople-Gordon standard geometry

Storage needed: 2562 in NPA, 3278 in NBO ( 2000000 available)

Note that all NBO output is formatted to a maximum 80-character width for convenient display. The NBO heading echoes any requested keywords (none for the present default case).

Natural Population AnalysisThe next four NBO output segments summarize the results of natural population analysis (NPA). The first segment is the main NAO table, as shown below:

NATURAL POPULATIONS: Natural atomic orbital occupancies NAO Atom # lang Type(AO) Occupancy Energy --------------------------------------------------------- 1 C 1 s Cor( 1s) 1.99900 -11.04184 2 C 1 s Val( 2s) 1.09038 -0.28186 3 C 1 s Ryd( 3s) 0.00068 1.95506 4 C 1 px Val( 2p) 0.89085 -0.01645 5 C 1 px Ryd( 3p) 0.00137 0.93125 6 C 1 py Val( 2p) 1.21211 -0.07191 7 C 1 py Ryd( 3p) 0.00068 1.03027 8 C 1 pz Val( 2p) 1.24514 -0.08862 9 C 1 pz Ryd( 3p) 0.00057 1.01801

10 N 2 s Cor( 1s) 1.99953 -15.25950 11 N 2 s Val( 2s) 1.42608 -0.71700 12 N 2 s Ryd( 3s) 0.00016 2.75771 13 N 2 px Val( 2p) 1.28262 -0.18042 14 N 2 px Ryd( 3p) 0.00109 1.57018 15 N 2 py Val( 2p) 1.83295 -0.33858 16 N 2 py Ryd( 3p) 0.00190 1.48447 17 N 2 pz Val( 2p) 1.35214 -0.19175 18 N 2 pz Ryd( 3p) 0.00069 1.59492

19 H 3 s Val( 1s) 0.81453 0.13283 20 H 3 s Ryd( 2s) 0.00177 0.95067

21 H 4 s Val( 1s) 0.78192 0.15354 22 H 4 s Ryd( 2s) 0.00096 0.94521

23 H 5 s Val( 1s) 0.78192 0.15354 24 H 5 s Ryd( 2s) 0.00096 0.94521

25 H 6 s Val( 1s) 0.63879 0.20572 26 H 6 s Ryd( 2s) 0.00122 0.99883

27 H 7 s Val( 1s) 0.63879 0.20572 28 H 7 s Ryd( 2s) 0.00122 0.99883

For each of the 28 NAO functions, this table lists the atom to which the NAO is attached, the angular momentum type (s, px, etc.), the orbital type (whether core, valence, or Rydberg, with a conventional hydrogenic-type label), the orbital occupancy (number of electrons, or "natural population" of the orbital), and the orbital energy (in atomic units: 1 a.u. = 627.5 kcal/mol). Note that the occupancies of the Rydberg (Ryd) NAOs are typically much lower than those of the core (Cor) and valence (Val) NAOs of the natural minimum basis (NMB) set, reflecting the dominant role of the NMB orbitals in describing molecular properties.

The principal quantum numbers for the NAO labels (1s, 2s, 3s, etc.) are assigned on the basis of the energy if a Fock or Kohn-Sham matrix is available, or on the basis of occupancy otherwise. A message warning of a "population inversion" is printed if the occupancy and energy ordering do not coincide (of interest, but rarely of concern).

The next segment is an atomic summary showing the natural atomic charges (nuclear charge minus summed natural populations of NAOs on the atom) and total core, valence, and Rydberg populations on each atom:

Summary of Natural Population Analysis: Natural Population Natural ----------------------------------------------- Atom # Charge Core Valence Rydberg Total----------------------------------------------------------------------- C 1 -0.44079 1.99900 4.43848 0.00331 6.44079 N 2 -0.89715 1.99953 5.89378 0.00384 7.89715 H 3 0.18370 0.00000 0.81453 0.00177 0.81630 H 4 0.21713 0.00000 0.78192 0.00096 0.78287 H 5 0.21713 0.00000 0.78192 0.00096 0.78287 H 6 0.35999 0.00000 0.63879 0.00122 0.64001 H 7 0.35999 0.00000 0.63879 0.00122 0.64001======================================================================= * Total * 0.00000 3.99853 13.98820 0.01328 18.00000

This table succinctly describes the molecular charge distribution in terms of NPA charges.

Next follows a summary of the NMB and NRB populations for the composite system, summed over atoms:

Natural Population -------------------------------------------------------- Core 3.99853 ( 99.9632% of 4) Valence 13.98820 ( 99.9157% of 14) Natural Minimal Basis 17.98672 ( 99.9262% of 18) Natural Rydberg Basis 0.01328 ( 0.0738% of 18)--------------------------------------------------------

This reveals the high percentage contribution (typically, > 99%) of the NMB set to the molecular charge distribution.

Finally, the natural populations are summarized as an effective valence electron configuration ("natural electron configuration") for each atom:

Atom # Natural Electron Configuration---------------------------------------------------------------------------- C 1 [core]2s( 1.09)2p( 3.35) N 2 [core]2s( 1.43)2p( 4.47) H 3 1s( 0.81) H 4 1s( 0.78) H 5 1s( 0.78) H 6 1s( 0.64) H 7 1s( 0.64)

Although the occupancies of the atomic orbitals are non-integer in the molecular environment, the effective atomic configurations can be related to idealized atomic states in "promoted" configurations.

Natural Bond Orbital AnalysisThe next segments of the output summarize the results of NBO analysis. The first segment reports on details of the search for an NBO natural Lewis structure:

NATURAL BOND ORBITAL ANALYSIS:

Occupancies Lewis Structure Low High Occ. ------------------- ----------------- occ occ Cycle Thresh. Lewis Non-Lewis CR BD 3C LP (L) (NL) Dev============================================================================= 1(1) 1.90 17.95048 0.04952 2 6 0 1 0 0 0.02-----------------------------------------------------------------------------

Structure accepted: No low occupancy Lewis orbitals

Normally, there is but one cycle of the NBO search. The table summarizes a variety of information for each cycle: the occupancy threshold for a "good" pair in the NBO search; the total populations of Lewis and non-Lewis NBOs; the number of core (CR), 2-center bond (BD), 3-center bond (3C), and lone pair (LP) NBOs in the natural Lewis structure; the number of low-occupancy Lewis (L) and high-occupancy (> 0.1e) non-Lewis (NL) orbitals; and the maximum deviation (Dev) of any formal bond order for the structure from a nominal estimate (NAO Wiberg bond index). The Lewis structure is accepted if all orbitals of the formal Lewis structure exceed the occupancy threshold (default = 1.90 electrons).

Next follows a more detailed breakdown of the Lewis and non-Lewis occupancies into core, valence, and Rydberg shell contributions:

WARNING: 1 low occupancy (<1.9990e) core orbital found on C 1

-------------------------------------------------------- Core 3.99853 ( 99.963% of 4) Valence Lewis 13.95195 ( 99.657% of 14)

================== ============================ Total Lewis 17.95048 ( 99.725% of 18) ----------------------------------------------------- Valence non-Lewis 0.03977 ( 0.221% of 18) Rydberg non-Lewis 0.00975 ( 0.054% of 18) ================== ============================ Total non-Lewis 0.04952 ( 0.275% of 18)--------------------------------------------------------

This shows the general quality of the natural Lewis structure description in terms of the percentage of the total electron density (e.g., in the above case, about 99.7%). The table also exhibits the relatively important role of the valence non-Lewis orbitals (i.e., the six valence antibonds, NBOs 23-28, listed below) relative to the extra-valence orbitals (the 13 Rydberg NBOs 10-22) in the slight departures from a localized Lewis structure model. The table also includes a warning about a carbon core orbital with slightly less than double occupancy.

Next follows the main listing of NBOs, displaying the form and occupancy of the complete set of orbitals that span the input AO space:

(Occupancy) Bond orbital/ Coefficients/ Hybrids------------------------------------------------------------------------------- 1. (1.99858) BD ( 1) C 1- N 2 ( 40.07%) 0.6330* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 59.93%) 0.7742* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 2. (1.99860) BD ( 1) C 1- H 3 ( 59.71%) 0.7727* C 1 s( 25.78%)p 2.88( 74.22%) -0.0002 -0.5077 0.0069 0.1928 0.0098 0.8396 -0.0046 0.0000 0.0000 ( 40.29%) 0.6347* H 3 s(100.00%) -1.0000 -0.0030 3. (1.99399) BD ( 1) C 1- H 4 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 0.7070 -0.0103 ( 38.98%) 0.6243* H 4 s(100.00%) 1.0000 0.0008 4. (1.99399) BD ( 1) C 1- H 5 ( 61.02%) 0.7812* C 1 s( 26.28%)p 2.80( 73.72%) 0.0001 0.5127 -0.0038 -0.3046 -0.0015 0.3800 -0.0017 -0.7070 0.0103 ( 38.98%) 0.6243* H 5 s(100.00%) 1.0000 0.0008 5. (1.99442) BD ( 1) N 2- H 6 ( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 -0.7070 -0.0093 ( 31.88%) 0.5646* H 6 s(100.00%) 1.0000 0.0020 6. (1.99442) BD ( 1) N 2- H 7

( 68.12%) 0.8253* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 0.5062 0.0005 0.3571 0.0171 -0.3405 0.0069 0.7070 0.0093 ( 31.88%) 0.5646* H 7 s(100.00%) 1.0000 0.0020 7. (1.99900) CR ( 1) C 1 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0003 0.0000 -0.0002 0.0000 0.0001 0.0000 0.0000 0.0000 8. (1.99953) CR ( 1) N 2 s(100.00%)p 0.00( 0.00%) 1.0000 -0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 9. (1.97795) LP ( 1) N 2 s( 17.85%)p 4.60( 82.15%) 0.0000 0.4225 0.0002 0.2360 -0.0027 0.8749 -0.0162 0.0000 0.0000 10. (0.00105) RY*( 1) C 1 s( 1.57%)p62.84( 98.43%) 0.0000 -0.0095 0.1248 -0.0305 0.7302 -0.0046 0.6710 0.0000 0.0000 11. (0.00034) RY*( 2) C 1 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0146 0.9999 12. (0.00022) RY*( 3) C 1 s( 56.51%)p 0.77( 43.49%) 0.0000 -0.0023 0.7517 -0.0237 0.3710 -0.0094 -0.5447 0.0000 0.0000 13. (0.00002) RY*( 4) C 1 s( 41.87%)p 1.39( 58.13%) 14. (0.00116) RY*( 1) N 2 s( 1.50%)p65.53( 98.50%) 0.0000 -0.0062 0.1224 0.0063 0.0371 0.0197 0.9915 0.0000 0.0000 15. (0.00044) RY*( 2) N 2 s( 0.00%)p 1.00(100.00%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0132 0.9999 16. (0.00038) RY*( 3) N 2 s( 33.38%)p 2.00( 66.62%) 0.0000 0.0133 0.5776 0.0087 -0.8150 -0.0121 -0.0405 0.0000 0.0000 17. (0.00002) RY*( 4) N 2 s( 65.14%)p 0.54( 34.86%) 18. (0.00178) RY*( 1) H 3 s(100.00%) -0.0030 1.0000 19. (0.00096) RY*( 1) H 4 s(100.00%) -0.0008 1.0000 20. (0.00096) RY*( 1) H 5 s(100.00%) -0.0008 1.0000 21. (0.00122) RY*( 1) H 6 s(100.00%) -0.0020 1.0000 22. (0.00122) RY*( 1) H 7 s(100.00%) -0.0020 1.0000 23. (0.00016) BD*( 1) C 1- N 2 ( 59.93%) 0.7742* C 1 s( 21.71%)p 3.61( 78.29%) -0.0003 -0.4653 -0.0238 -0.8808 -0.0291 -0.0786 -0.0110 0.0000 0.0000 ( 40.07%) -0.6330* N 2 s( 30.88%)p 2.24( 69.12%) -0.0001 -0.5557 0.0011 0.8302 0.0004 0.0443 -0.0098 0.0000 0.0000 24. (0.01569) BD*( 1) C 1- H 3 ( 40.29%) 0.6347* C 1 s( 25.78%)p 2.88( 74.22%) 0.0002 0.5077 -0.0069 -0.1928 -0.0098 -0.8396 0.0046 0.0000 0.0000 ( 59.71%) -0.7727* H 3 s(100.00%) 1.0000 0.0030

25. (0.00769) BD*( 1) C 1- H 4 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 -0.7070 0.0103 ( 61.02%) -0.7812* H 4 s(100.00%) -1.0000 -0.0008 26. (0.00769) BD*( 1) C 1- H 5 ( 38.98%) 0.6243* C 1 s( 26.28%)p 2.80( 73.72%) -0.0001 -0.5127 0.0038 0.3046 0.0015 -0.3800 0.0017 0.7070 -0.0103 ( 61.02%) -0.7812* H 5 s(100.00%) -1.0000 -0.0008 27. (0.00426) BD*( 1) N 2- H 6 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 0.7070 0.0093 ( 68.12%) -0.8253* H 6 s(100.00%) -1.0000 -0.0020 28. (0.00426) BD*( 1) N 2- H 7 ( 31.88%) 0.5646* N 2 s( 25.62%)p 2.90( 74.38%) 0.0000 -0.5062 -0.0005 -0.3571 -0.0171 0.3405 -0.0069 -0.7070 -0.0093 ( 68.12%) -0.8253* H 7 s(100.00%) -1.0000 -0.0020

For each NBO (1-28), the first line of printout shows the occupancy (between 0 and 2 electrons) and unique label of the NBO. This label gives the type (BD for 2-center bond, CR for 1-center core pair, LP for 1-center valence lone pair, RY* for 1-center Rydberg, and BD* for 2-center antibond, the unstarred and starred labels corresponding to Lewis and non-Lewis NBOs, respectively), a serial number (1, 2,... if there is a single, double,... bond between the pair of atoms), and the atom(s) to which the NBO is affixed. The next lines summarize the natural atomic hybrids hA of which the NBO is composed, giving the percentage (cA-squared) of the NBO on each hybrid (in parentheses), the polarization coefficient cA, the atom label, and a hybrid label showing the sp-hybridization (percentage s-character, p-character, etc.) of each hA. Below each NHO label is the set of coefficients that specify how the NHO is written explicitly as a linear combination of NAOs on the atom. The order of NAO coefficients follows the numbering of the NAO tables.

In the CH3NH2 example, the NBO search finds the C-N bond (NBO 1), three C-H bonds (NBOs 2, 3, 4), two N-H bonds (NBOs 5, 6), N lone pair (NBO 9), and C and N core pairs (NBOs 7, 8) of the expected Lewis structure. NBOs 10-28 represent the residual non-Lewis NBOs of low occupancy. In this example, it is also interesting to note the slight asymmetry of the three CH NBOs, and the slightly higher occupancy (0.016 vs. 0.008 electrons) in the CH3 antibond (NBO 24) lying trans to the nitrogen lone pair.

NHO Directional AnalysisThe next segment of output summarizes the angular properties of the natural hybrid orbitals (NHOs):

NHO Directionality and "Bond Bending" (deviations from line of nuclear centers)

[Thresholds for printing: angular deviation > 1.0 degree] hybrid p-character > 25.0% orbital occupancy > 0.10e

Line of Centers Hybrid 1 Hybrid 2 --------------- ------------------- ------------------ NBO Theta Phi Theta Phi Dev Theta Phi Dev=============================================================================== 1. BD ( 1) C 1- N 2 90.0 5.4 -- -- -- 90.0 182.4 3.0 3. BD ( 1) C 1- H 4 35.3 130.7 34.9 129.0 1.0 -- -- -- 4. BD ( 1) C 1- H 5 144.7 130.7 145.1 129.0 1.0 -- -- -- 5. BD ( 1) N 2- H 6 144.7 310.7 145.0 318.3 4.4 -- -- -- 6. BD ( 1) N 2- H 7 35.3 310.7 35.0 318.3 4.4 -- -- -- 9. LP ( 1) N 2 -- -- 90.0 74.8 -- -- -- --

The "direction" of a hybrid is specified in terms of the polar () and azimuthal () angles (in the coordinate system of the calling program) of the vector describing its p-component. For more general spd hybrids the hybrid direction is determined numerically to correspond to the maximum angular amplitude. The hybrid direction is then compared with the direction of the line of centers between the two nuclei to determine the bending of the bond, expressed as the deviation angle (Dev, in degrees) between these two directions. For example, in the CH3NH2 case shown above, the nitrogen NHO of the CN bond (NBO 1) is bent away from the line of C-N centers by 3.0°, whereas the carbon NHO is approximately aligned with the C-N axis (within the 1.0° threshold for printing). The N-H bonds (NBOs 5, 6) are bent even further (by 4.4°). The information in this table is often useful in anticipating the direction of geometry changes resulting from geometry optimization (viz., likely reduced pyramidalization of the -NH2 group to relieve the ~4° nitrogen bond bending found in the tetrahedral Pople-Gordon geometry).

Perturbation Theory Energy AnalysisThe next segment summarizes the second-order perturbative estimates of donor-acceptor (bond-antibond) interactions in the NBO basis:

Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis

Threshold for printing: 0.50 kcal/mol E(2) E(j)-E(i) F(i,j) Donor NBO (i) Acceptor NBO (j) kcal/mol a.u. a.u. ===============================================================================

within unit 1 2. BD ( 1) C 1- H 3 / 14. RY*( 1) N 2 0.84 2.18 0.038 3. BD ( 1) C 1- H 4 / 26. BD*( 1) C 1- H 5 0.52 1.39 0.024

3. BD ( 1) C 1- H 4 / 27. BD*( 1) N 2- H 6 3.03 1.37 0.057 4. BD ( 1) C 1- H 5 / 25. BD*( 1) C 1- H 4 0.52 1.39 0.024 4. BD ( 1) C 1- H 5 / 28. BD*( 1) N 2- H 7 3.03 1.37 0.057 5. BD ( 1) N 2- H 6 / 10. RY*( 1) C 1 0.56 1.78 0.028 5. BD ( 1) N 2- H 6 / 25. BD*( 1) C 1- H 4 2.85 1.51 0.059 6. BD ( 1) N 2- H 7 / 10. RY*( 1) C 1 0.56 1.78 0.028 6. BD ( 1) N 2- H 7 / 26. BD*( 1) C 1- H 5 2.85 1.51 0.059 7. CR ( 1) C 1 / 16. RY*( 3) N 2 0.61 13.11 0.080 7. CR ( 1) C 1 / 18. RY*( 1) H 3 1.40 11.99 0.116 7. CR ( 1) C 1 / 19. RY*( 1) H 4 1.55 11.99 0.122 7. CR ( 1) C 1 / 20. RY*( 1) H 5 1.55 11.99 0.122 8. CR ( 1) N 2 / 10. RY*( 1) C 1 1.51 16.23 0.140 8. CR ( 1) N 2 / 12. RY*( 3) C 1 0.84 16.77 0.106 8. CR ( 1) N 2 / 21. RY*( 1) H 6 0.61 16.26 0.089 8. CR ( 1) N 2 / 22. RY*( 1) H 7 0.61 16.26 0.089 9. LP ( 1) N 2 / 24. BD*( 1) C 1- H 3 8.13 1.13 0.086 9. LP ( 1) N 2 / 25. BD*( 1) C 1- H 4 1.46 1.14 0.037 9. LP ( 1) N 2 / 26. BD*( 1) C 1- H 5 1.46 1.14 0.037

This analysis is carried out by examining all possible interactions between "filled" (donor) Lewis-type NBOs and "empty" (acceptor) non-Lewis NBOs, and estimating their energetic importance by 2nd-order perturbation theory. Since these interactions lead to donation of occupancy from the localized NBOs of the idealized Lewis structure into the empty non-Lewis orbitals (and thus, to departures from the idealized Lewis structure description), they are referred to as "delocalization" corrections to the zeroth-order natural Lewis structure. For each donor NBO (i) and acceptor NBO (j), the stabilization energy E(2) associated with delocalization ("2e-stabilization") i j is estimated as

where qi is the donor orbital occupancy, i, j are diagonal elements (orbital energies) and F(i,j) is the off-diagonal NBO Fock matrix element. [In the example above, the nN CH* interaction between the nitrogen lone pair (NBO 8) and the antiperiplanar C1-H3 antibond (NBO 24) is seen to give the strongest stabilization, 8.13 kcal/mol.] As the heading indicates, entries are included in this table only when the interaction energy exceeds a default threshold of 0.5 kcal/mol.

NBO SummaryNext appears a condensed summary of the principal NBOs, showing the occupancy, orbital energy, and the qualitative pattern of delocalization interactions associated with each:

Natural Bond Orbitals (Summary):

Principal Delocalizations NBO Occupancy Energy (geminal,vicinal,remote)===============================================================================

Molecular unit 1 (CH5N) 1. BD ( 1) C 1- N 2 1.99858 -0.89908 2. BD ( 1) C 1- H 3 1.99860 -0.69181 14(v) 3. BD ( 1) C 1- H 4 1.99399 -0.68892 27(v),26(g) 4. BD ( 1) C 1- H 5 1.99399 -0.68892 28(v),25(g) 5. BD ( 1) N 2- H 6 1.99442 -0.80951 25(v),10(v) 6. BD ( 1) N 2- H 7 1.99442 -0.80951 26(v),10(v) 7. CR ( 1) C 1 1.99900 -11.04131 19(v),20(v),18(v),16(v) 8. CR ( 1) N 2 1.99953 -15.25927 10(v),12(v),21(v),22(v) 9. LP ( 1) N 2 1.97795 -0.44592 24(v),25(v),26(v) 10. RY*( 1) C 1 0.00105 0.97105 11. RY*( 2) C 1 0.00034 1.02120 12. RY*( 3) C 1 0.00022 1.51414 13. RY*( 4) C 1 0.00002 1.42223 14. RY*( 1) N 2 0.00116 1.48790 15. RY*( 2) N 2 0.00044 1.59323 16. RY*( 3) N 2 0.00038 2.06475 17. RY*( 4) N 2 0.00002 2.25932 18. RY*( 1) H 3 0.00178 0.94860 19. RY*( 1) H 4 0.00096 0.94464 20. RY*( 1) H 5 0.00096 0.94464 21. RY*( 1) H 6 0.00122 0.99735 22. RY*( 1) H 7 0.00122 0.99735 23. BD*( 1) C 1- N 2 0.00016 0.57000 24. BD*( 1) C 1- H 3 0.01569 0.68735 25. BD*( 1) C 1- H 4 0.00769 0.69640 26. BD*( 1) C 1- H 5 0.00769 0.69640 27. BD*( 1) N 2- H 6 0.00426 0.68086 28. BD*( 1) N 2- H 7 0.00426 0.68086 ------------------------------- Total Lewis 17.95048 ( 99.7249%) Valence non-Lewis 0.03977 ( 0.2209%) Rydberg non-Lewis 0.00975 ( 0.0542%) ------------------------------- Total unit 1 18.00000 (100.0000%) Charge unit 1 0.00000

This table allows one to quickly identify the principal delocalizing acceptor orbitals associated with each donor NBO, and their topological relationship to this NBO, i.e., whether attached to the same atom (geminal, "g"), to an adjacent bonded atom (vicinal, "v"), or to a more remote ("r") site. These acceptor NBOs will generally correspond to the principal "delocalization tails" of the NLMO associated with the parent donor NBO. [For example, in the table above, the nitrogen lone pair (NBO 9) is seen to be the lowest-occupancy (1.978 electrons) and highest-energy (-0.446 a.u.) Lewis NBO, and to be primarily delocalized into antibonds 24, 25, 26 (the vicinal CH NBOs). The summary at the bottom of the table shows that the Lewis NBOs 1-9 describe about 99.7% of the total electron density, with the remaining non-Lewis density found primarily in the valence-shell antibonds (particularly, NBO 24).]

NBO Tutorials NBO Home

Calculation of NMR SpectraBrief Background on Computational Spectroscopy

Approximate solutions to molecular Schrodinger equation allow useful prediction of various spectral properties. The allowed electronic energy levels of a molecule are related to absorption peak positions in its UV-Vis spectrum (for details, you may consult "Description of Electronically Excited States"); from the knowledge of the wavefunction in ground and excited state we can estiate the peak intensities. Evaluation of second derivatives of molecular energy with respect to nuclear displacements allows to calculate infrared frequencies. Evaluating derivatives with respect to magnetic field, nuclear magnetic moments, and electric field gradients allows to predict the position and splitting pattern of NMR peaks. Such calculations are possible with a variety of quantum chemistry programs including DALTON, Gaussian, and NWChem. The accurate prediction of spin-spin coupling constants is very challenging and requires the use of large uncontracted basis sets (i.e. large expansions of the trial wave function in the gaussian basis) to describe properly the electron distribution near nuclei. Most programs support calculation of NMR coupling constants at the Hartree-Fock (HF) and Density Functional Theory (DFT) level. Only a few programs, such as CFOUR allows modeling NMR spectra using correlated coupled cluster methods.

Prediction of Spin-Spin Coupling Constants with Gaussian

Many computational chemistry programs offer calculation of spin-spin coupling constants in molecules. In practice, the user has to create an input file that specifies molecular geometry either in the XYZ coordinates, or in the so-called Z-matrix (internal connectivity matrix) coordinates. User also has to specify the basis set. Most quantum chemistry programs use basis sets built from many gaussian functions to approximate the molecular wave function. The gaussian basis sets go by names such as STO-3G, 3-21G, 6-31G, 6-31G(d,p), 6-311G(d,p), cc-pVDZ, aug-cc-pVTZ, aug-cc-pCVQZ and so on. Small basis sets, such as STO-3G use fewer gaussians, and produce results quickly, but the results are quantitatively wrong and qualitatively unreliable. Medium-size basis sets, such as 6-311+G(d,p) sometimes produce qualitatively reliable results. Calculations with large basis sets, such as aug-cc-pVQZ, promise to yield reliable results but the calculations on polyatomic molecules will take very long time or exhaust the available memory and hard disk space. Gaussian09 implements a "Mixed" method where a better basis set is used to calculate the most critical contributions while a faster basis set is used for more time-consuming parts. Finally, user has to specify what type of calculation is sought. An example Gaussian input file for a Hartree-Fock spin-spin coupling calculation for an idealized model of allantoin with the arm HN-CN dihedral at 80 degrees is shown below:

%Mem=420MW

# HF/aug-cc-pVTZ NMR=Mixed MaxDisk=2GW

Allantoin w/ H-C5-N6-H at 80 degrees: NMR coupling w/ Mixed method (uTZ-w)

0 1 C 0.000000 0.000000 0.000000 N 0.000000 0.000000 1.445819 N 1.277036 0.000000 -0.629183 C -0.585043 -1.329297 -0.492038 H -0.664818 0.773863 -0.367979 C 0.942023 0.558554 2.231878 H -0.767825 -0.455267 1.880499 N 0.663355 0.393323 3.567138 O 1.912375 1.133905 1.854290 H -0.168148 -0.099700 3.852784 H 1.304261 0.773337 4.246029 C 1.505912 -1.098091 -1.386514 H 1.946597 0.723281 -0.534511 N 0.361060 -1.884143 -1.283853 O 2.466297 -1.369890 -2.019229 H 0.269280 -2.755398 -1.750049 O -1.653831 -1.753376 -0.214391 Analysis

You can examine the result of such a calculation here. The coupling results are listed as a matrix of J-values in Hz units; the axes of the matrix are the atom numbers in the same order as in the input file. You first need to determine the order numbers for atoms that you expect to see a three-bond coupling for. You can convert the Gaussian output file into a mol2 file for visualization. This can be done with the program OpenBabel. To convert with OpenBabel, type babel -ig03 Alla_dih80_NMR.log -omol2 Alla_dih80_NMR.mol2 into Unix shell. Visualize the mol2 file with gOpenMol and write down the number corresponding to each of the three hydrogen atoms we are concerned about.

Tutorial by Dr. Kalju Kahn, Department of Chemistry and Biochemistry, UC Santa Barbara. ©2010.

Description of Electronically Excited StatesExcited States of Molecules

Most molecules have bound higher energy excited electronic states in addition to the ground electronic state E0. These states may be thought of as arising from the promotion of one of the electrons from the occupied orbital in the ground state to a vacant higher energy orbital. The excitation of an electron from the occupied orbital to a higher-energy orbital occurs when a photon with the energy that matches the difference between the two states interacts with the molecule. The classical Franck-Condon principle states that because the rearrangement of electrons is much faster than the motion of nuclei, the nuclear configuration does not change significantly during the energy absorption process. Thus, the absorption spectrum of molecules is characterized by the vertical excitation energies.

Vibrational Fine Structure of Absorption Lines

The ground state (E0) supports a large number of vibrational energy levels. At room temperature, only the lowest vibrational level is populated, and electronic transitions originate from the n=0 vibrational level. The bound excited states (E1) also support several vibrational levels. However, because the excited state potential energy curve is typically shifted, the vertical excitations from the lowest vibrational level of the ground electronic state take the system into one of several vibrational levels of the excited electronic state. Thus, valence transitions, such as n → π* in carbonyl compounds show vibrational fine structure which may be useful for the characterization of the excited state. The quantum mechanical Franck-Condon principle states that the probability of each transition is determined by the extent of overlap between the ground state and excited state vibrational wave functions. These individual vibronic bands

can be sometimes observed in the gas phase absorption spectra of molecules but the assignment of vibronic bands to specific states can be challenging. For example, if the two potential energy curves are significantly shifted, the 0 ← 0 transition may have so low intensity that it can escape detection. The spectra of molecules in solvents that interact strongly with the chromophore are broad and often featureless.

Classification of Absorption Bands

The absorption bands are typically classified as valence bands (for example, the local π → π* transition in many unsaturated organic molecules), Rydberg bands (transitions to very diffuse orbitals around the molecule), and charge transfer bands (involving electron transfer from one part of the molecule to another part). The lower energy bands typically arise from transitions from occupied valence orbitals to unoccupied valence orbitals. Many far-ultraviolet bands (below 200 nm) arise from Rydberg transitions. These bands can be recognized by the lack of vibrational fine structure and by the convergence of their energies toward the ionization potential of the molecule. The lowest energy transition in molecules that contain both lone pairs and π bonds is typically the n → π* transition. However, in some symmetric molecules, the intensity of the n → π* transition is very low because the transition is symmetry forbidden.

Fates of Excited States

Excited states have limited life times (typically in the order of a nanosecond) and they can decay via several modes. Sometimes the excited state is so weakly bound that it will dissociate. For example, ozone undergoes a photodissociation to O2 and atomic oxygen after absorbing a photon of ultraviolet light. A second possibility is that the excited state returns to the ground state without emitting a photon. Such radiationless decay occurs readily when the excited state and the ground state potential energy curves meet via a conical intersection. For example, the excited states of DNA bases in Watson-Crick base paired geometry are very short-lived thanks to efficient radiationless decay; this protects DNA from photochemical damage. The third possibility of return to the ground state is via the emission of photon. Such radiative decay is commonly called fluorescence.

It is usually observed that the fluorescent light from molecules or nanoparticles has a longer wavelength than the exciting light. This immediately suggests that the emission of a photon from the electronically excited state is not a perfect mirror process of the absorption. Instead, the excited vibronic state rapidly relaxes on the excited state potential energy surface: the nuclei adopt a new optimum geometry that is at equilibrium with the excited state electronic wavefunction. In simple words, fluorescence usually takes place from the ground vibrational level of the electronically excited state E1. Again, because the rearrangement of electrons is very fast, the emission is vertical and reaches one of the vibrational levels of the ground electronic state E0. The energy difference between the relaxed excited state energy and the ground state energy is called the adiabatic excitation energy. Adiabatic excitation energies can be measured from the emission spectra of molecules.

Fluorescence occurs very quickly after excitation. There is another radative decay mode, called phosphoresence, which takes the system back to ground state over much longer timescales. In the

case of phosphoresence, the excited state undergoes a change in the spin state. In a typical scenario, the potential energy surface of a singlet excited state crosses with the potential energy surface of a lower-laying triplet energy state, and the system becomes trapped in the triplet excited state. Spontaneous emission from triplet excited state to singlet ground state is spin forbidden, and thus occurs with a low probability. The phenomena of fluorescence and phosphoresence are two manifestations of luminescent emission.

Computational Methods for Excited States

Computational methods could, in principle give accurate information about the excited electronic states. For example, the vertical excitation energy can be obtained in the first approximation as the energy difference between the excited state potential energy curve and the ground state potential energy curve at the ground state minimum energy geometry. Optimization of geometry of the excited electronic state allows to calculate adiabatic excitation energies. Thus, computations could be used to predict absorption and fluorescence emission spectra of molecules. With a little more effort, the absorption of chiral light can be characterized, allowing one to predict the circular dichroism spectra. Such calculations are often valuable for chemists interested in the identification of molecules solely on the basis of their spectra.

A large number of computational methods have been developed for the description of excited states. On one hand, we have easy-to-apply methods, such as Configuration Interaction Singles (CIS) or Time Dependent Density Functional Theory (TDDFT) but these have a generally limited accuracy, and can fail spectacularly for certain situations. On the other hand, multireference configuration interaction methods typically offer an accuracy of about 0.1 eV (2.3 kcal/mol) but require expertise in setting up the calculation. In general, meaningful excited state calculations can be difficult to carry out. There are special cases where a simple method like (CIS) will give useful answers. In general, however, one must be aware of pitfalls such as:

The basis functions that are used to construct molecular orbitals are typically optimized to describe ground states. Calculation with such basis functions are biased to stabilize the ground state over the excited state. Excited states are often more diffuse and basis sets lacking adequate diffuse functions will not describe such states correctly. Typically, small basis sets lead to an overestimation of the transition energy because of poor description of the excited state. Oftentimes, special basis sets are used at the expense of higher computational cost.

Accounting for electron correlation in excited states is not as straightforward as in the ground state. Finding a method that offers a well-balanced treatment of both states is often problematic. The methods needed are computationally demanding and require judgment in their application. For example, one may have to specify beforehand a limited set of occupied and virtual orbitals that are "active" in the transition. Making a good selection is not trivial in many systems.

Material by Dr. Kalju Kahn and Bernie Kirtman, Department of Chemistry and Biochemistry, UC Santa Barbara. ©2007-2012.

Configuration Interaction Singles (CIS)Background about CIS

The simple CIS approach is accurate in certain special cases, in particular for so-called charge transfer transitions. In the CIS approach we use orbitals of the Hartree-Fock solution to generate all singly excited determinants of the configuration interaction expansion. This treatment can be thought of as the Hartree-Fock method for excited states. It allows one to simultaneously solve for a large number of excited states and to optimize the geometry of any (desired) selected state. Both spin singlet and spin triplet states can be generated. The CIS method has some appealing features:

It is relatively fast compared to other methods; CIS can be applied to systems as large as DNA duplex oligomers.

It is easy to set up; typically, the number of desired excited states and the basis set are the only quantities to be specified.

It is possible to discuss the electronic transitions in terms of familiar concepts, such as π → π*. CIS allows geometry optimization of the excited state using analytical gradients. Thus,

calculation of adiabatic excitation energies as well as vertical excitation energies is practical. Properties such as dipole moments, charge densities, and vibrational frequencies of the excited states can be calculated. From the such properties one can compare bonding differences between ground and excited states.

CIS is variational; the lowest energy solution corresponds to the ground electronic state and the next solution corresponds to the first excited state.

CIS is size-consistent; one will get the same excitation energy when considering a single helium atom or two helium atoms separated by a distance so large that the weak interaction between atoms can be neglected.

There are two main problem with the CIS method. First, it is appropriate only for transitions for which the ground state and the excited state are well-described by a single-configuration (such as Hartree-Fock) reference wave function. The ground state of most molecules near their equilibrium geometry is well described by a single reference (ozone is a notable exception). However, some exited states in many important chromophores (such as benzene) have a significant multi-reference character. Also, a close spacing on d-shell electron energy levels in transition metal complexes means that a single-reference description is inappropriate. In such cases, typical errors are 1 eV, which makes it difficult to assign observed spectral lines in the absence of symmetry.

Second, the CIS method, being an analog of the Hartree-Fock method for the excited states, does not include any electron correlation. This would not be a problem if the ground and excited state

were stabilized by electron correlation by the same degree. This is rarely the case, and CIS typically overestimates excitation energies. The obvious solution is to describe electron correlation both in the ground and excited state but a care must be taken to provide a balanced description for all states. Some approaches, such as CISD are completely inappropriate because a closed-shell ground state would enjoy a good electron correlation while the singly excited state would not benefit as much from from virtual double excitations. Thus, CISD energies grossly overestimate the true transition energies. A partially satisfying solution is provided in the CIS-MP2 method that includes a perturbation correction for double excitations. A more balanced description is provided by the CIS(D) method which can be thought of as an analogue of MP2 for the excited states. Recent studies show that CIS(D) is quite reliable (mean error about 0.2 eV) for the description of n → π* valence shell transitions in systems where a single-reference dominates ground and excited states.

If the single reference is not adequate, multi-reference methods offer a way to calculate excitation energies. The simplest multi-reference method is CASSCF, which ignores dynamic electron correlation. More advanced approaches, such as CASPT2 and XMCQDPT add a perturbative electron correlation based on multi-reference wave function (CASSCF); such methods provide a good accuracy in many cases. The main drawback of such multi-reference methods is that their application requires a significant user input by deciding which electrons and orbitals should be included in the multi-reference treatment.

Performance of CIS in the Prediction of UV-Vis Spectra

The table below compares the performance of a CIS calculation with different methods in predicting the UV spectrum of formaldehyde:

EXP CIS CIS-MP2 CIS(D) TDHF TDDFT CASSCF CASPT2 EOM-CCSD CC3Singlet Valence Excited States1 1A2 (n → π*) 3.79* 4.48 4.58 3.98 4.35 3.92 4.62 3.91 4.04 3.881 1B1 (σ → π*) 8.68 9.66 8.47 8.12 9.52 9.03 6.88 9.09 9.26 9.042 1A1 (π → π*) NObs 9.36 7.66 7.26 9.55 8.43 10.24 9.77

10.0 9.18Singlet Rydberg Excited States1 1B2 (n → 3s) 7.11 8.63 6.85 6.44 8.59 6.87 6.88 7.30 7.04 N/A2 1A2 (n → 3p) 8.37 9.78 7.83 7.50 9.74 7.89 8.17 8.32 8.21 8.212 1B2 (n → 4s) 9.26 10.8 8.94 N/A N/A N/A N/A N/A 9.35 N/A

Computational data sources: Gwaltney et al, Chem. Phys. Lett., 248, 189 (1996) using 6-311(2+,2+)G(d,p) basis Wiberg et al J. Phys. Chem. 106, 4192, 2002 (2008) using 6-311(2+,2+)G(d,p) basis Schreiber et al, J. Chem. Phys., 128, 134110 (2008) using bases up to d-aug-cc-pVTZ Head-Gordon et al, Chem. Phys. Lett. 219, 21 (1994) for CIS(D)

Sunanda et al, Spectrosc. Lett. 45, 65 (2011) for some CIS data

* The estimates for the position of 0-0 vertical transition range from 3.5 eV to 4.2 eV. This forbidden transition is seen in absorption spectra as a broad system of peaks due to many vibrational transitions. Analogous vibrational structure is also seen in the energy loss spectra from electron impact.

Note that while the CIS method generally overestimates vertical ionization energies, the n → π* transition in formaldehyde is given fairly well by the CIS, and the MP2 correction improves some results considerably. Another method similar to CIS is TDHF (time-dependent Hartree-Fock), which is a modification of CIS to include some ground state electron correlation. The last of the affordable and easy-to-use methods is the time-dependent density functional theory (TD-DFT). This methods performs well for low-lying electronic states that are not charge transfer transitions but TD-DFT fails to model excited states of linear polyenes. The last four columns refer to higher level methods that are generally more difficult to set up, or very time-consuming to run. Currently, higher level wavefunction-based methods (e.g. CASPT2 or EOM-CCSD) offer the best choice for small molecules.

Running and Analyzing CIS Calculations

The CIS method is implemented in many computer programs including Gaussian and Firefly (PC GAMESS). To run the calculation with Gaussian, specify the keyword CIS. The number of desired excited states can be specified as an option to the CIS keyword: CIS(NStates=8) requests the 8 lowest excited states. The CIS calculation is more resource-consuming than the Hartree-Fock calculation, and calculations with large basis sets, such as aug-cc-pVTZ, may not be possible for larger molecules. Smaller basis such as 6-31+G(d) may be feasible for larger molecules, but one shall worry about loss of accuracy when too small of a basis set is used. If one is only interested in valence-shell transitions, small basis sets can be used. However, for the description of diffuse Rydberg states, basis sets with multiple diffuse functions (e.g. d-aug-cc-pVTZ) are critical.

Perform a CIS calculation of formaldehyde. A sample input file for running this calculation with Gaussian is shown below. Analyze the output and try to identify some excitations that have been observed experimentally based on the excitation energies. The energy-loss from electron impact spectrum for formaldehyde in the gas phase is shown on the right (measured by Walzl, Koerting, and Kuppermann; J. Chem. Phys. 87, 3796 (1987)). Notice that this is easy for the lowest energy state but energy values alone do not allow one to

assign the peaks. In symmetric molecules, the symmetry of excited states is very helpful for identification of transitions. For example, some transitions, such as n → π* are symmetry-forbidden in planar formaldehyde and can be recognized by near-zero oscillator strengths.

%NProc=1%Mem=200MW%Chk=Formald_CIS.chk# CIS(NStates=6)/aug-cc-pVTZ MaxDisk=4GW

Formaldehyde MP2/aug-cc-pVTZ minimum for UV spectra

0 1 C O,1,oc2 H,1,hc3,2,hco3 H,1,hc4,2,hco4,3,dih4,0 Variables: oc2=1.21289414 hc3=1.10017879 hco3=121.7009888 hc4=1.10017879 hco4=121.7009888 dih4=180.

CIS calculations can be readily performed with larger molecules. Below is a sample output from the UV spectrum of a nucleobase uracil in Cs geometry with 6-31+G(2d,p) basis. The experimental spectrum of uracil in water shows two intense bands, centered around 257 nm and 220 nm. Based on the calculated intensities (f values are the oscillator strengths, which are the measures of intensity), these can be identified as transitions to the excited state 2 and to the excited state 8. Notice that excited state 1 has a very small intensity: this transition is nearly forbidden by orbital symmetry considerations. In this case the CIS transition energies are significantly in error. The HOMO is orbital 43; the LUMO is orbital 44. Thus, the main contribution to state 2 (as determined by squaring the given coefficient) arises from the excitation of an electron from the HOMO to LUMO + 5. You can look at the shape of orbitals involved using MOLDEN.

Excited State 1: Singlet-A" 5.7786 eV 214.56 nm f=0.0008 43 -> 44 0.57421 43 -> 45 -0.29806 43 -> 46 0.13312 43 -> 50 0.12401 This state for optimization and/or second-order correction. Copying the CI singles density for this state as the 1-particle RhoCI density.

Excited State 2: Singlet-A' 5.8742 eV 211.06 nm f=0.3761 43 -> 47 -0.10128 43 -> 49 0.66751

Excited State 3: Singlet-A" 6.5816 eV 188.38 nm f=0.0023 43 -> 44 0.35589 43 -> 45 0.48737 43 -> 46 -0.24746

43 -> 48 -0.11210

Excited State 4: Singlet-A" 6.6584 eV 186.21 nm f=0.0002 37 -> 49 0.23353 41 -> 49 0.54352 41 -> 57 -0.20644 41 -> 72 0.10528 41 -> 78 -0.13593

Excited State 5: Singlet-A' 7.0599 eV 175.62 nm f=0.0028 43 -> 47 0.66177 43 -> 52 -0.15451

Excited State 6: Singlet-A" 7.0916 eV 174.83 nm f=0.0037 39 -> 44 0.10592 43 -> 45 0.19574 43 -> 46 0.53360 43 -> 48 -0.12071 43 -> 50 -0.23089 43 -> 51 0.16163 43 -> 55 0.13265

Excited State 7: Singlet-A" 7.5672 eV 163.84 nm f=0.0108 39 -> 44 0.10410 43 -> 45 0.15085 43 -> 46 0.15656 43 -> 48 -0.23182 43 -> 50 0.54108 43 -> 56 0.16348 43 -> 62 -0.11541

Excited State 8: Singlet-A' 7.6361 eV 162.36 nm f=0.5017 39 -> 57 -0.10651 43 -> 52 0.10904 43 -> 54 -0.28671 43 -> 57 0.50535 43 -> 59 -0.19194 43 -> 61 -0.19798Practice Task

Analyze the result of CIS calculation of purine and identify molecular orbitals involved in the lowest energy transition. Then analyze the result of molecular orbital calculation of purine; visually examine the orbitals that give the largest contribution to the lowest energy excitation. Is this a π → π* or n → π* type transition?

Material by Dr. Kalju Kahn and Bernie Kirtman, Department of Chemistry and Biochemistry, UC Santa Barbara. ©2007-2012.

Description of SolvationIntroduction

Many interesting reactions take place in the liquid phase. For example, biochemical reactions occur in the aqueous environment or sometimes at the water-lipid interface. Organic synthesis is typically carried out in solvents. The observation that the environment has a significant effect on reaction outcomes was probably known to medieval alchemists. Studies by Berhelot and Pean de Saint Gilles in 1860s showed that the rates of homogeneous chemical reactions considerably depend on the environment. They noticed, for example, that both the formation and the hydrolysis of ethyl acetate was greatly retarded in the vapor phase as compared to water. Later, in 1890 Menshutkin studied the reaction between trialkylamines and haloalkanes in twenty three different solvents and reported that "solvents are by no means inert in chemical reactions". It is now widely appreciated that molecular properties, such as acid dissociation constants, are strongly solvent dependent. Rates of chemical reactions, especially when polar transition states are involved, can vary many orders of magnitude depending on the solvent. It is clear that computational description of solvent effects is crucial if quantitative agreement with experiments is being sought. In many cases, even qualitative insight requires that solvent effects are accounted for.

Solvatochromism

UV-Vis spectral properties of many molecules depend on the solvent. One of the best understood examples is provided by the n → π* transition in acetone. In the gas phase, this symmetry-forbidden transition is observed as a weak band of peaks near 276 nm (4.49 ev). When dissolved in water, the n → π* transition is observed at 265 nm (4.68 eV). Such changes in the spectrum with solvent are called

solvatochromic shifts. When the spectral maximum shifts to longer wavelengths, we say that a red-shift occurred. In case of acetone, a spectral blue-shift occurred when acetone was moved from the gas phase to water. The solvatochromism in acetone can be understood in terms of dipolar interactions between the polar solute and the solvent. In the gas phase and non-polar solvents, the dipolar interactions are negligible both in the ground and the excited state. In water, the strength of dipolar interactions is significant. More importantly, the ground state and the excited state enjoy different stabilisation by water because the dipole moment of acetone is different in the ground and excited states.

The figure on the right shows the electrostatic potential of acetone, mapped on the electron density surface. This figure illustrates a well-known fact that acetone is a polar molecule with excess negative charge localized around the oxygen atom. The simplest quantitative description of polarity of molecules is provided by its electric dipole moment. The dipole moment can be measured experimentally by studying how the microwave spectral lines of a molecule change in the presence of electric field (the Stark effect). For example, the most recent results suggest that acetone's electric dipole moment is 2.93 Debye (Dorosh & Kisiel, Acta Phys. Polonica A, 112, S-95, 2007). The dipole moment can be also obtained by analyzing the molecular charge distribution from quantum mechanical calculations. The results of such calculations depend somewhat on the level of theory and basis set used. In general, the HF method tends to overestimate dipole moments (HF/aug-cc-pVTZ prediction for acetone is 3.48 Debye) and consideration of electron correlation via MP2 improves the results notably (MP2/aug-cc-pVTZ result for acetone is 2.98). The coupled cluster methods are reliable for the estimation of dipole moments. For example, CCSD-T/aug-cc-pVTZ predicts (via numeric differentiation of energies w.r.t. field) that the dipole moment of acetone in the ground state is 2.94 Debye. The excited state has different electron distribution, and thus different dipole moment. To understand how the dipole moments of the ground and excited state differ, it is instructive to analyze the spatial localization of the orbitals involved in the n → π* transition. The images below show the highest occupied molecular orbital (HOMO), which is the non-bonding orbital, and the lowest unoccupied molecular orbital (LUMO), which is the π* orbital.

The orbital pictures reveal that the n → π* electronic transition involves transfer of one electron from a region near the oxygen atom to a region near the carbonyl carbon. This process is expected to reduce the dipole moment: the excited state of acetone is not as polar as the ground state. Our qualitative reasoning is confirmed by accurate quantum mechanical calculations: the most recent computational results suggest that the dipole moment of the n → π* excited state of acetone is 1.78 Debye (Pašteka, Melichercík, Neogrády & Urban, Mol. Phys. 2012).

As mentioned above, the solvatochromic shifts can be rationalized by considering interactions between the solute and the solvent. A more polar ground state of acetone enjoys greater stabilization by polar water than a less polar excited state. As a result, the energy gap between the two states is greater in the presence of polar solvent in comparison with the energy gap in the gas phase or when dissolved in non-polar solvent. Larger energy gap means that light of higer frequency, or of shorter wavelength, is needed to promote the electron from the n orbital to the π* orbital. In summary, one can rationally predict the direction of the solvatochromic shift when the nature of the electronic transition and the electronic structure of the molecule are well understood. Alternatively, experimental directions and magnitudes of solvatochromic shifts are sometimes used to determine the nature of the electronic transition, and to estimate the dipole moment of the excited state.

The solvatochromic shift in the visible region leads to the change of the color of the solution. For example, the 2-(4'-hydroxystyryl)-N-methyl-quinolinium-betaine absorbs near 585 nm in the nonpolar solvent chloroform but absorbs high-energy light at 410 nm in water. As a result, the

molecule appears blue in chloroform and red in water. In the case of the betaine, chemical intuition suggests that the highly polar ground state (zwitterion!) must be well stabilized by water while the non-polar excited state enjoys less stabilization in water. The result of this differential stabilization is that the energy gap associated with the n → π transition increases as the solvent polarity increases.

The π → π* transitions in nonpolar molecules, such as benzene or toluene, often show small spectral red-shifts when the environment is changed from the gas phase to liquid. Such non-polar molecules have zero (e.g. benzene) or very small (e.g. toluene) dipole moments, and the above-described dipolar interactions are not significant. Non-polar aromatic molecules interact with their surrounding primarily via their permanent quadrupole moment and temporarily induced dipole moments (London dispersion). In symmetric non-polar molecules such as ethylene, butadiene, benzene, toluene, or naphtalene, the excitation of electron from π to π* orbital does not change the quadrupole moment appreciably. However, the polarizability, which determines the strength of the London dispersion interaction, increases noticeably upon π → π* excitation because the electron in the π* orbital is farther away from the attractive nuclear framework. For example, the perpendicular αzz component of benzene increases from 44 atomic units to 52 atomic units upon the lowest-energy π → π* excitation (Christiansen, Hättig & Jørgensen, Spectrochimica Acta, Part A 55, 509 (1999)). Thus, the π* excited state enjoys stronger London dispersion interaction with the surrounding solvent than the less polarizable ground state, and the energy gap in the solvent is smaller than in the gas phase. In the case of benzene, the magnitude of the solvatochromic red shift is rather small.

Analysis of solvatochromic shifts in molecules where both π → π* and n; → π* transitions can take place is more challenging. For example, in conjugated heteroaromatic systems, an excited state can arise via a mixture of π → π* and n; → π* transitions. Furthermore, a π → π* transition in heteroaromatic systems can alter both the dipole moment and the quadrupole moment of the system, so prediction of solvatochromic shifts becomes much more challenging. For example, the methoxy analog of the chromophore found in the green fluorescent protein shows virtually no solvatochromisms in the neutral state (A) but displays complex solvatochromism in the cationic state (B). One possible explanation for the lack of solvatochromism in the neutral form is that long-wavelength

maximum is dominated here by the π → π* transition, which does not alter the dipole moment significantly. The same transition in the cationic species is a mixture of the π → π* and n → π* transitions, the latter being stabilized because transfer of the electron from carbonyl oxygen to π* orbital on carbonyl carbon is favored by the adjacent positive charge. In many cases, computational analysis of ground and exited states can reveal the direction and magnitude of solvatochromic shifts.

Explicit Solvent Models

One approach to model solvent effects is to surround the solute with a small number of explicit solvent molecules during the calculation. For example, one could optimize the structure of the betaine dye in the electronic ground state by placing couple of solvent molecules near the negatively and positively charged centers; such calculation would correctly predict the direction of the solvatochromic shift. However, the magnitude of the solvent effect depends strongly on details: factors such as the number of water molecules and their placement affects results greatly. Addition of a second layer of solvent might mitigate this problem but then the number of solvent molecules is so large that typical QM calculations become unfeasible.

Implicit Solvent Models

Alternative approach is to describe solvation by its average effect that mainly arises from dipolar interactions. In this model, the polar solute polarizes the surrounding dielectric medium. The polarized medium acts as a reaction field that interacts with the solute. The advantage of such approach is simplicity inn setting up the computations but specific interactions, such as strong hydrogen bonding with the solvent is difficult to model. Furthermore, geometry optimization in the presence of polarizable continuum is not as efficient as geometry optimization in the gas phase, and it is not uncommon that optimizations of certain geometries fail when a specific implementation of reaction field is chosen. In Gaussian, the implicit solvent model calculation is invoked via the SCRF keyword.

Practice Problem

The input file for CIS calculation with PC GAMESS for formaldehyde in the gas phase is shown below:

$contrl scftyp=rhf cityp=cis runtyp=energy units=bohr d5=.t. icut=10 inttyp=hondo nosym=1 $end $system mwords=170 timlim=6600 $end $guess guess=huckel $end $basis extfil=.t. gbasis=acc-pvtz $end $cis nstate=6 $end $dataFormaldehyde H-CHO ground stateCNv 2

C 6 0.00000000E+00 0.00000000E+00 -1.00945571E+00 O 8 0.00000000E+00 0.00000000E+00 1.28258204E+00 H 1 0.00000000E+00 1.76884859E+00 -2.10196104E+00 $end

Run this by issuing pcgamess -f -i form_cis.inp -o form_cis.out -b /usr/local/pcgamess/acc-pvtz.lib. Examine the output file with a text editor and with MOLDEN to identify orbitals involved in the lowest energy forbidden transition. Then build a formaldehyde water complex; keep the structure of formaldehyde and water fixed and calculate the CIS spectrum of the complex. Interpret the results.

Tutorial by Dr. Kalju Kahn, Department of Chemistry and Biochemistry, UC Santa Barbara. �2008-2012

AssignmentsLevel 1

1) Read the paper Accuracy of spectroscopic constants of diatomic molecules from ab initio calculations . Carry out the frequency analysis using the Hartree Fock method with cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z basis sets. Answer the following questions.

1. Summarize what you learned from Figure 1 in this paper. 2. Why didn't the authors show SCF results on Figure 2 even though they have made plots for this

data? 3. Create a series of graphs showing the convergence of harmonic frequency with increasing basis

set for each molecule and visually estimate the frequency at the HF limit. 4. Discuss fundamental reasons why HF harmonic frequencies differ from experimental harmonic

frequencies. Is there a quick fix if one wishes to use HF frequencies for the purpose of identification of large organic molecules based on their IR spectra?

2. a) Calculate the syn (Cl-C–C-Cl = 0°) rotational barrier in 1,2-dichloroethane in the gas phase based on MP3 single point energies at HF/6-31+G(d,p) geometries to treat electron correlation. Make sure to use an appropriate basis set in the MP3 calculation. Make a statement about the effect of electron correlation in this case.

2. b) Calculate the syn rotational barrier in 1,2-dichroroethane in water. You may use HF/6-31+G(d,p) calculations with an appropriate implicit solvent model. Rationalize the observed direction of the solvent effect.

Level 2

1) Read the paper "Accuracy of spectroscopic constants of diatomic molecules from ab initio calculations" and a paper "Coupled-cluster connected quadruples and quintuples corrections to the harmonic vibrational frequencies and equilibrium bond distances of HF, N2, F2, and CO". Carry out geometry optimization and harmonic frequency analysis of F2 at the MP3 level using aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets with a program of your choice. The last of these is a rather challenging calculation and you should expolore ways to carry it out with a minimal amount of CPU time spent. Some things to consider include:

1. 1) Gaussian 03 offers analytic second derivatives at the MP2 level but not at the MP3 level. However, Hessian calculated at the MP2 level is an excellent starting point for the BFGS algorithm

2. 2) Gaussian offers analytic first derivatives at the MP3 level. Second derivatives at the MP3 level in Gaussian are evaluated by numeric differentiation of analytic first derivatives.

3. 3) PC GAMESS does not allow automated geometry optimization or frequency analysis at the MP3 level but offers extremely fast single point MP3 energies.

4. 4) Numeric differentiation requires rather accurate energy values. With PC GAMESS / Firefly, the following performance and accuracy-related options are recommended: $contrl icut=10 inttyp=hondo $end$system mwords=60 $end $scf nconv=7 $end $mp3 cutoff=1E-14 $end

5. 5) Speed of electron correlation calculations with PC GAMESS depends on the choice between the the conventional ($scf direct=.f. $end) and direct ($scf direct=.t. $end) method for the SCF part. If you have a very fast disk, it is better to write the atomic integrals to the disk before starting SCF, these integrals will be read in also before the MP3 stage. If the disk is slow, it is better to recalculate AO interals at each SCF cycle as well as during the MP3 stage.

6. 6) With PC GAMESS, specify the basis set on the command line, e.g. -b /usr/local/pcgamess/acc-pvqz.lib

Provide the following with your answer:

1. Discussion about the most efficient (in terms of CPU time) strategy to obtain the desired results. 2. Discussion about the geometry of F2 with MP2, MP3, CCSD, CCSD(T), and CCSDTQ methods near

the basis set limit. Specifically, discuss one fundamental reason why the MP2 bond length is far off from the experimental value

3. Discussion about the harmonic frequency of F2 with HF, MP2, MP3, CCSD, CCSD(T), and CCSDTQ methods near the basis set limit. Specifically, why would authors expect that "all high-order connected contributions to the harmonic frequencies are negative".

4. Evaluate the suggestion that "the harmonic frequency at the MP3 basis set limit can be readily obtained by exponential extrapolation of aug-cc-pVDZ, aug-cc-pTZ, and aug-cc-pVQZ frequency values."

2) Consider the stereoselective synthesis of a methyl ester of 2-[(1S)-1,2,2-trimethylpropyl]-4-pentene(dithioic) acid from (S)-3,4,4-trimethyl-1-(methylthio)-1-(2-propenylthio-(Z)-1-pentene. One of the predictions of the semiempirical PM3 method was that this reaction is thermodynamically unfavorable. Reinvestigate this reaction using correlated ab initio or density functional theory. Each student should individually decide on the appropriate way to generate one conformer for the reactant, and one conformer for the product, and optimize these structures with their method of choice. Calculate the reaction energy and reaction free energy in the gas phase as accurately as you possibly could, given the requirement that none of your calculations should take more than 16 hrs on our local workstations. Then calculate the reaction energy, and reaction free energy in the water using an appropriate implicit solvent model. Note that the free energy calculation in water does not require a frequency calculation. The statistical thermodynamics formulas that allow the calculation of the enthalpy and entropy from vibrational frequencies are strictly valid for isolated molecules.

Level 3

1) Discuss what is the main difference between the MP4(SDQ) and MP4(SDTQ) methods. Determine the minimum energy structure of F2 at MP4(SDQ)/aug-cc-pVTZ and

MP4(SDTQ)/aug-cc-pVTZ levels. Calculate energies on 5-point stencil centered at these structures. Write a computer program that will perform the following tasks:

1. Numerically calculates the first, second, third, and fourth derivative of the potential energy with respect to displacement based on user-supplied energy values via centered finite difference formulas. Notice that if you work with units of Å and Hartrees, the derivatives have units of Hartree*Å-1, Hartree*Å-2, Hartree*Å-3, Hartree*Å-4, respectively.

2. Calculates the harmonic frequency (waven), the equilibrium rotational constant Be (via the moment of inertia), the quartic centrifugal distortion constant, and the vibration-rotation coupling constant; express these in cm-1 units. The quartic centrifugal distortion coefficient for

diatomics is given as and the vibration-rotation coupling constant for diatomics is given as

3. Creates a plot that shows the first five vibrational energy levels for F2 at the MP4(SDTQ)/aug-cc-pVTZ level. The y-axis of this plot should be in cm-1 units, the x-axis could be either in meter or angstroms, you are allowed to consider the minimum energy distance as the origin in this graph.

4. Creates a plot that shows the first five vibratioal wave functions for F2 at the MP4(SDTQ)/aug-cc-pVTZ level. You may scale these wave functions and show them on the same plot together with the vibrational energy levels.

Compare your MP4(SDQ) and MP4(SDTQ) minimum energy structures, harmonic frequencies, centrifugal distortion constants, and vibration-rotation coupling constants with experimental data. Discuss relative merits of MP4(SDTQ) over MP4(SDQ) for spectroscopic description of fluorine.

2) Do one of the following projects:

1. Calculate the visible spectrum of 2-(4'-hydroxystyryl)-N-methyl-quinolinium-betaine in the gas phase with CIS and TDDFT methods using an appropriate basis set. Repeat the calculations in water with a suitable implicit solvent model. Then construct an explicitly solvated structure for 2-(4'-hydroxystyryl)-N-methyl-quinolinium-betaine that contains at least three appropriately placed water molecules. Calculate CIS and TDDFT spectra of this model using the same basis set as earlier. Discuss the performance of CIS vs. TDDFT. Discuss the ability of explicit and implicit solvent models to predict solvatochromic shifts in this case

2. Study the effect of solvation on the Menschutkin reaction between methyl chloride and N,N-dimethylamine or quinuclidine using an appropriate implicit solvent model. Reoptimize the reactants, products, and the transition state with a suitable polarizable continuum model at the HF level using a basis set that you think is appropriate for the description of his reaction. Perform a frequency calculation to verify that the optimized transition state is indeed the first order saddle point. Provide a rationale for the observed solvent effect.

Materials by Dr. Kalju Kahn, Department of Chemistry and Biochemistry, UC Santa Barbara. ©2008.