quantum mechanics and force fields hartree-fock revisited semi-empirical methods basis sets post...
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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
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)
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
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
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.
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)
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 codes for Solids
DMol3 (Atom-centered BF, DFT)
SIESTA ,,
VASP (PlaneWaves, DFT)
MOPAC2000 (Semi-Empirical)
CRYSTAL95 CPMD WIEN