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

EH

Schrodinger Equation

Within Born-OppenheimerApproximation

ij

n

1i ij

n

1i

N

1 i

2i

2

r

1

r

Z-

2m

h- H

n

ij

n

1i ij

n

1i r

1ih H

n

1i

ih H

Without the electron repulsion term

iiih jjj

n321tot

N

iλc1

i

MO = Linear Combination of Atomic Orbitals

212

2

1

21 d

r

12

r

2-

2

1-1F

Fock Operator (example for He)

SF iii cc

Hartree-Fock Roothaan equations

|S Overlap integral

|

2

1|PHF

ii

occ

i

cc2PDensity 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 n4/8

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 CNDO INDO NDDO

+ + +

+ + +

- + +

- - +

- - -

- + +

- - +

- - -

- - -

AAAA || BBAA ||

AAAA ||

BBAA ||

AABA ||

AAAA ||

BABA ||

B BAA ||

B AAA ||

Approximations of 1-el integrals

AB

ABVUH Ufrom atomic spectra

Vvalue per atom pair

0H on the same atom

SH AB BAAB 21

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

r)exp(

2nml rexp*zyx

• 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

i

L

ii GTOcAO

ci contraction coefficients

Example 6-31G for Li-F

AO’s

1s 6 GTO’s

2s,2px,2py,2pz 3 GTO per AO

2s`,2px`,2py`,2pz` 1 GTO per AO

i

L

ii GTOcAO

Polarization Functions

Add AO with higher angular momentum (L)

Basis-sets: 3-21G*, 6-31G*, 6-31G**, etc.

Element Configuration Polarisation Function

H 1s (L=0) p (L=1)

Li-F 1s,2s,2px,2py,2pz (L=1) d (L=2)

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)

When AB INITIO interaction energy is not accessible

Neglecting:

•Polarization•Charge Transfer

Eint = Evdw + EelecCalculate it with a model potential

Approximations to Eelec:

•Interacting partial charges•Interacting multipole expansions

The Molecular Electrostatic Potential

r

r-r

r

r-r

ZrV

a

a

d

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)

ii

occ

i

cc2P

P

Mulliken Populations

ii2 cc22

iii

Electron Density

SPPN d

Integrated Density equals Number of electrons:

qx is the contribution due to electron density on atom X

SP2PN

N is a sum of atomic and overlap contributions:

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

Electrostatic Potential derived charges(ESP charges)

• QM electrostatic potential is sampled at van der Waals surfaces

• Least squares fitting of 2Modeli

QMi VV

n

j ij

jModeli r

qV

q1

q2

q3

ri3

ri2

ri1

i

QM Calculations on Solids

• K-space sampling

a

nikna

k e Translational Symmetry Adapted Wavefunction:

H H H H H H H

H2 H2 H2 H2

Overview of Popular QM codes

Gaussian (Ab Initio)

Gamess-US/UK ,,

MOPAC (Semi-Empirical)

QM codes for Solids

DMol3 (Atom-centered BF, DFT)

SIESTA ,,

VASP (PlaneWaves, DFT)

MOPAC2000 (Semi-Empirical)

CRYSTAL95 CPMD WIEN