1. Quantum Mechanics and Force Fields Hartree-Fock revisited Semi-Empirical Methods Basis sets Post Hartree-Fock Methods Atomic Charges and Multipoles QM calculations on Solids

2. Schrodinger Equation Within Born-Oppenheimer Approximation

3. Without the electron repulsion term

4. MO = Linear Combination of Atomic Orbitals Fock Operator (example for He)

5. Hartree-Fock Roothaan equations Overlap integral Density Matrix

6. Self Consistent Field Procedure 1.Choose start coefficients for MO’s 2.Construct Fock Matrix with coefficients 3.Solve Hartree-Fock Roothaan equations 4.Repeat 2 and 3 until ingoing and outgoing coefficients are the same

7. SEMI-EMPIRICAL METHODS Number 2-el integrals (  ) is n 4 /8 n = number of basis functions Treat only valence electrons explicit Neglect large number of 2-el integrals Replace others by empirical parameters

8. Approximations Complete Neglect of Differential Overlap (CNDO) Intermediate Neglect of Differential Overlap (INDO/MINDO) Neglect of Diatomic Differential Overlap (NDDO/MNDO,AM1,PM3)

9. Neglected 2-el Integrals 2-el integral CNDOINDONDDO +++ +++ -++ --+ --- -++ --+ --- ---

10. Approximations of 1-el integrals U  from atomic spectra V  value per atom pair   on the same atom One  parameter per element

12. BASIS-SETS Slaters (STO) Gaussians (GTO) Angular part * Better basis than Gaussians 2-el integrals hard : zz 2-el integrals simple Wrong behaviour at nucleus Decrease to fast with r

13. STOnG Split Valence: 3-21G,4-31G, 6-31G Each atom optimized STO is fit with n GTO’s Minimum number of AO’s needed Contracted GTO’s optimized per atom Doubling of the number of valence AO’s

14. STOnG

15. Contracted GTO’s c i contraction coefficients

17. Example 6-31G for Li-F AO’s 1s6 GTO’s 2s,2p x,2p y,2p z 3 GTO per AO 2s`,2p x `,2p y `,2p z `1 GTO per AO

18. Polarization Functions Add AO with higher angular momentum (L) Basis-sets: 3-21G*, 6-31G*, 6-31G**, etc. ElementConfigurationPolarisation Function H1s (L=0)p (L=1) Li-F1s,2s,2p x,2p y,2p z (L=1)d (L=2)

19. Correlation Energy HF does not treat correlations of motions of electrons properly E exact – E HF = E correlation Post HF Methods: –Configuration Interaction (CI,SDCI) –Møller-Plesset Perturbation series (MP2-MP4) Density Functional Theory (DFT)

20. When AB INITIO interaction energy is not accessible Neglecting: Polarization Charge Transfer E int = E vdw + E elec Calculate it with a model potential Approximations to E elec : Interacting partial charges Interacting multipole expansions

21. The Molecular Electrostatic Potential

22. Properties of the MEP: Positive part of one molecule will dock with negative part of another. Directional effect on complexation. Most important aspect of structure activity correlation of proteins. Predicts preferred site of electrophilic /nucleophilic attack. Minima correlate to strengths of hydrogen-bonds, Pka etc.

23. Electrostatic Potential Color Coded on an Isodensity Surface

24. Electrostatic Potential

25. Charges Derived

26. Multipole Derived

28. Methods for obtaining Point Charges Based on Electronegativity Rules (Qeq) From QM calculation: –Schemes that partition electron density over atoms (Mulliken, Hirshfeld, Bader) –Charges are optimized to reproduce QM electrostatic potential (ESP charges)

29. Atoms in Molecules (Bader)

30. Mulliken Populations Electron Density  Integrated Density equals Number of electrons:

31. q x is the contribution due to electron density on atom X N is a sum of atomic and overlap contributions:

32. STO3G 3-21G 6-31G* -0.016+0.016+0.219-0.219+0.318-0.318 -0.260 +0.065 -0.788 +0.197 -0.660 +0.165 +0.157 -0.470-0.838 +0.279+0.331 -0.992 +0.183+0.364+0.433 -0.367 -0.728-0.866

33. Electrostatic Potential derived charges (ESP charges) QM electrostatic potential is sampled at van der Waals surfaces Least squares fitting of q1q1 q2q2 q3q3 ri3ri3 ri2ri2 ri1ri1 i

35. QM Calculations on Solids K-space sampling

37. a Translational Symmetry Adapted Wavefunction: H H H H H H H

39. H 2 H 2

40. Overview of Popular QM codes Gaussian (Ab Initio) Gamess-US/UK,, MOPAC(Semi-Empirical)

41. QM codes for Solids DMol 3 (Atom-centered BF, DFT) SIESTA,, VASP(PlaneWaves, DFT) MOPAC2000(Semi-Empirical) CRYSTAL95 CPMD WIEN