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Quantum Mechanics and Force Fields: A Comprehensive Overview of Theoretical Methods and Applications

Explore the intricate world of Quantum Mechanics and its application in Force Fields, including Hartree-Fock methods, basis sets, and post-Hartree-Fock techniques. Dive into semi-empirical methods, atomic charges, and multipole moments, with detailed insights into solving the Schrödinger Equation and achieving accurate QM calculations on solids. Unravel the complexities of Fock operators, molecular orbitals, basis set optimizations, and correlation energies within the Born-Oppenheimer approximation. Learn about important post-HF methods like Configuration Interaction, Møller-Plesset perturbation series, Density Functional Theory, and the Molecular Electrostatic Potential. Understand the concept of electrostatic potential charges, point charges derivation, and various QM codes used for solid-state calculations.

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Quantum Mechanics and Force Fields: A Comprehensive Overview of Theoretical Methods and Applications

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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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