Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009

1. Statistical Mechanics and Multi-Scale Simulation MethodsChBE 591-009 Prof. C. Heath Turner Lecture 04 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm

2. Introduction to Quantum Chemistry • Basis Sets: • Split-valence: 3-21G, 6-31G, 6-311G, cc-pVDZ, cc-pVTZ (contraction coefficients optimized for HF and for e- correlation methods). Allow orbitals to change size but not shape. • Polarization Functions: • Add 1 quantum number of higher angular momentum (than valence orbitals). • In practice: adds next level of unoccupied orbitals: • adds p GTOs to H • adds d GTOs to first and second row elements • adds f GTOs to transition metals • Why? Increases flexibility of the wave function. More important for molecules than for individual atoms – neighbors and bonds can polarize the e- in molecules. • Examples: water, ammonia. Without polarization, HF predicts planar geometry for NH3.

3. Introduction to Quantum Chemistry Polarization of a p-orbital:1 Application to the water molecule:2 • A. R. Leach, Molecular Modelling: Principles and Applications, 2nd ed. Prentice Hall (2001). • C. J. Cramer, Essentials of Computational Chemistry, Wiley (2004).

4. Introduction to Quantum Chemistry Experimental Bond Angle of H2O (gas phase) = 104.5º Lets try calculations with and without polarization functions (using Gaussian03)…

5. Introduction to Quantum Chemistry Input file #1: %chk=che651_water_1.chk %mem=200MB %nproc=1 # opt hf/6-31g geom=connectivity Calculation w/o polarization 0 1 O H 1 B1 H 1 B2 2 A1 B1 1.0 B2 1.0 A1 100.0 1 2 1.0 3 1.0 2 3

6. Introduction to Quantum Chemistry Input file #2: %chk=che651_water_2.chk %mem=200MB %nproc=1 # opt hf/6-31g(d,p) geom=connectivity Calculation w/o polarization 0 1 O H 1 B1 H 1 B2 2 A1 B1 1.0 B2 1.0 A1 100.0 1 2 1.0 3 1.0 2 3

7. Introduction to Quantum Chemistry Experimental Bond Angle of H2O (gas phase) = 104.5º • Lets try calculations with and without polarization functions (using Gaussian03)… • HF/6-31G: bond angle = 111.5º • HF/6-31G(d,p): bond angle = 105.9 º

8. Introduction to Quantum Chemistry Alternative: “FLOGOs” floating Gaussian orbitals (shown on left):1 1. C. J. Cramer, Essentials of Computational Chemistry, Wiley (2004).

9. Introduction to Quantum Chemistry • Polarization (nomenclature): • 6-31G(d) OR 6-31G* = adds d functions to polarize the p functions on the “heavy” atoms. • 6-31G(d,p) OR 6-31G** = adds d functions to polarize the p functions on the “heavy” atoms AND adds p functions to polarize H (and He) atoms. • 6-31G(3d2fg,2pd) = heavy atoms polarized by 3 sets of d functions, 2 sets of f functions, and a set of g functions, and H polarized by 2 sets of p functions, and one set of d functions. (This basis set would be ‘unbalanced’)

10. Gaussian03 Calculation %chk=CH3OH.chk %mem=6MW %nproc=1 # opt rhf/6-31g geom=connectivity Title Card Required 0 1 C H 1 B1 H 2 B2 1 A1 H 1 B3 2 A2 3 D1 H 1 B4 4 A3 3 D2 O 1 B5 4 A4 3 D3 B1 1.07000000 B2 1.51320851 B3 1.07000000 B4 1.97020303 B5 1.43000000 A1 44.99999991 A2 144.73561047 A3 117.34791957 A4 90.00000001 D1 0.00000000 D2 180.00000000 D3 180.00000000 1 2 1.0 3 1.0 4 1.0 6 1.0 2 3 4 5 6 1.0 6

11. Gaussian03 Results

12. Gaussian03 Results

13. Gaussian03 Results

14. Alabama Supercomputer Center There are PC and Macintosh versions of the SSH Secure Shell Client. If secure shell is not installed on your machine, contact your system administrator, or look into the PuTTY software, available at http://www.chiark.greenend.org.uk/~sgtatham/putty/ The PuTTY package allows ssh connections, and PSCP or PSFTP can be used to transfer files to or from the supercomputers. • Once you are logged-in, here are some useful Unix commands: • cd ~ (change to home directory) • pwd (check to see the current directory) • ls (list the files in the current directory) • ls –l (list the files in the current directory and show the details) • cp filename1 filename2 (copy ‘file1’ to ‘file2’) • cp filename1 dir\filename2 (copy ‘file1’ to ‘file2’ in directory ‘dir’) • qstat –u UserID (show the jobs running for UserID) • qdel job# (kill the simulation labeled ‘job#’) • exit (log out of the machine) • vi filename (view the file named ‘filename’), ‘:q!’ will quit • man command (show how to use the command ‘command’), ‘q’ will quit

15. Gaussian03 Keywords For a complete list of the G03 keywords, go to the website http://www.gaussian.com and look under ‘Tech Support’ • For single-point energy calculations, here are some useful commands: • Stable (test for stability of the wavefunction and SCF calculation, tests open shell and relaxes symmetry constraints) • Stable=Opt (test for stability of SCF and reoptimize the wavefunction, if needed. This is a not a geometry optimization.) • Guess=Mix (mix HOMO and LUMO within the wavefunction. Often useful for producing a UHF wavefunction for singlet system.) • UHF (unrestricted Hartree-Fock calculation, open shell)

16. Introduction to Quantum Chemistry • Diffuse Functions: • Account for ‘weakly’ bound electrons (excited states, anions, lone pairs). Normally, e- density is localized near the nucleus in previous basis sets. • 6-31+G = one additional set of s and p Gaussian functions on ‘heavy’ atoms. • 6-31++G = adds one additional diffuse s function on H • Have an exponent about ¼ smaller than the smallest valence exponent. • ‘aug’ prefix in aug-cc-pVnZ indicates addition of diffuse functions (for each angular momentum already present) • Necessary for calculating acidities and electron affinities. • In absence of these specific instances, no strong reason to include diffuse functions.

17. Introduction to Quantum Chemistry • The HF Limit: • Can be extrapolated using the cc-pVnZ and cc-pCVnZ basis sets (Dunning's correlation consistent basis sets): • Other approximations – additivity assumption: • E[HF/6-311++G(d,p)] ≈ E[HF/6-31G] + • {E[HF/6-31G(d,p)] – E[HF/6-31G]} + • {E[HF/6-311G] – E[HF/6-31G]} + • {E[HF/6-31++G] – E[HF/6-31G]} • Time Savings = (2044 + 3784 + 2944)/5424 = 0.43

18. Introduction to Quantum Chemistry • Effective Core Potentials: • Heavy elements are challenging for MO theory (many e-, many basis functions). • Atom nuclear-electronic ‘core’ is approximated. • Relativistic effects can be incorporated (important for core electrons in heavy elements). • How many e- should be included in ‘core’? • large-core ECP = include all but valence e- in the core • small-core ECP = scale back approximation to next lower shell. Usually worth the added computational cost. • Most widely used pseudopotentials – Los Alamos National Labs (LANL) ECPs, Stuttgard/Dresden (SDD)

19. Introduction to Quantum Chemistry • Property Calculations (HF): • Energetics: • Lack of electron correlation – any process that involves the change in the total # of paired e- is not accurately calculated (heats of formation), isomerizations, and most other changes in bonding. • Conformational changes predicted fairly well, such as torsional barriers. • Cancellation of errors – basis set incompleteness and neglecting e- correlation introduce errors of opposite sign, sometimes appearing to give accurate results (in particular, polarized double-zeta basis sets at HF level). • Protonation/deprotonation energies fairly well estimated. • Basis Set Superposition Errors (BSSE) – borrowing basis functions within a complex, resulting in an artificial lowering of the energy.

20. Introduction to Quantum Chemistry • Property Calculations (HF): • Geometry: • With appropriate basis set HF does a good job for minimum-energy structures. • Bond lengths, bond angles, and dihedral angles are reproduced within a few percent. Bond length err by underpredicting. • Polarization functions required in systems with hypervalent bonding, heteroatoms with single lone pairs. • Crowding of non-bonded e- can cause e- to become important • Dative bonds poorly described (both e- in the bonding pair come from only one of the atoms). • TS structures not well reproduced. • Non-bonded complexes not well described – distances unrealistically large. Hydrogen bonding is usually fairly accurate, due to cancellation of errors.

21. Introduction to Quantum Chemistry • Property Calculations (HF): • Charge Distributions: • Dipole moment (e- + nuclear contribution): • Quadrupole moment, octopole moment, etc. • Scaling factors can be used to correct for small basis sets and e- correlation. Overestimated by 10-25%, molecules predicted to be too polar. • Sensitive to optimized geometry. • Fairly insensitive to basis set size beyond valence-double-zeta. • Population analysis – assign partial charges to individual atoms (somewhat arbitrary) – Mulliken population analysis, Löwdin population analysis, natural population analysis.