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Quantum Mechanics and Force Fields

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. Schrodinger Equation. Within Born-Oppenheimer Approximation. Without the electron repulsion term.

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Quantum Mechanics and Force Fields

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

  5. Hartree-Fock Roothaan equations Overlap integral Density Matrix

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

  7. SEMI-EMPIRICAL METHODS • Number 2-el integrals (mu|ls) is n4/8 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

  10. Umm from atomic spectra VAB value per atom pair m,uon the same atom One b parameter per element Approximations of 1-el integrals

  11. 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 BASIS-SETS

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

  13. STOnG

  14. Contracted GTO’s ci contraction coefficients

  15. Example 6-31G for Li-F

  16. Polarization Functions Add AO with higher angular momentum (L) Basis-sets: 3-21G*, 6-31G*, 6-31G**, etc.

  17. Correlation Energy • HF does not treat correlations of motions of electrons properly • Eexact – EHF = Ecorrelation • Post HF Methods: • Configuration Interaction (CI,SDCI) • Møller-Plesset Perturbation series (MP2-MP4) • Density Functional Theory (DFT)

  18. When AB INITIO interaction energy is not accessible Eint = Evdw + Eelec Calculate it with a model potential Neglecting: Polarization Charge Transfer Approximations to Eelec: Interacting partial charges Interacting multipole expansions

  19. The Molecular Electrostatic Potential

  20. 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.

  21. Electrostatic Potential Color Coded on an Isodensity Surface

  22. Electrostatic Potential

  23. Charges Derived

  24. Multipole Derived

  25. 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)

  26. Atoms in Molecules (Bader)

  27. Mulliken Populations Electron Density r: Integrated Density equals Number of electrons:

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

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

  30. q2 ri2 q1 q3 ri1 ri3 i Electrostatic Potential derived charges(ESP charges) • QM electrostatic potential is sampled at van der Waals surfaces • Least squares fitting of

  31. QM Calculations on Solids • K-space sampling

  32. Translational Symmetry Adapted Wavefunction: a H H HH H H H

  33. H2 H2 H2 H2

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

  35. QM codes for Solids • DMol3 (Atom-centered BF, DFT) • SIESTA ,, • VASP (PlaneWaves, DFT) • MOPAC2000 (Semi-Empirical) • CRYSTAL95 CPMD WIEN

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