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

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

Hartree-Fock Roothaan equations Overlap integral Density Matrix

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

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

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

Neglected 2-el Integrals 2-el integral CNDOINDONDDO

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

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

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

STOnG

Contracted GTO’s c i contraction coefficients

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

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)

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)

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

The Molecular Electrostatic Potential

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.

Electrostatic Potential Color Coded on an Isodensity Surface

Electrostatic Potential

Charges Derived

Multipole Derived

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)

Atoms in Molecules (Bader)

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

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

STO3G 3-21G 6-31G*

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

QM Calculations on Solids K-space sampling

a Translational Symmetry Adapted Wavefunction: H H H H H H H

H 2 H 2

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

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