Quantum Mechanics and Force Fields

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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

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Without the electron repulsion term
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Self Consistent Field Procedure

1. Choose start coefficients for MOs
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

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SEMI-EMPIRICAL METHODS

• Number 2-el integrals (muls) 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

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Approximations

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

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Neglected 2-el Integrals
2-el integral CNDO INDO NDDO


-
- -
- - -
-
- -
- - -
- - -
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Approximations of 1-el integrals
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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

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• STOnG
• Split Valence 3-21G,4-31G, 6-31G

• Each atom optimized STO is fit with n GTOs
• Minimum number of AOs needed

• Contracted GTOs optimized per atom
• Doubling of the number of valence AOs

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STOnG
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Contracted GTOs
ci contraction coefficients
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Example 6-31G for Li-F
AOs
1s 6 GTOs
2s,2px,2py,2pz 3 GTO per AO
2s,2px,2py,2pz 1 GTO per AO
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Polarization Functions
Add AO with higher angular momentum (L)
Basis-sets 3-21G, 6-31G, 6-31G, etc.
Element Configuration Polarisation Function
H 1s (L0) p (L1)
Li-F 1s,2s,2px,2py,2pz (L1) d (L2)
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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)

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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

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The Molecular Electrostatic Potential
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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.

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Electrostatic Potential Color Coded on an
Isodensity Surface
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Electrostatic Potential
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Charges Derived
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Multipole Derived
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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)

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Atoms in Molecules (Bader)
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Mulliken Populations
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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
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Electrostatic Potential derived charges(ESP
charges)

• QM electrostatic potential is sampled at van der
  Waals surfaces
• Least squares fitting of

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QM Calculations on Solids

• K-space sampling

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H H H H H H H
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H2 H2 H2 H2
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Overview of Popular QM codes

• Gaussian (Ab Initio)
• Gamess-US/UK ,,
• MOPAC (Semi-Empirical)

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QM codes for Solids

• DMol3 (Atom-centered BF, DFT)
• SIESTA ,,
• VASP (PlaneWaves, DFT)
• MOPAC2000 (Semi-Empirical)
• CRYSTAL95 CPMD WIEN

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