# Molecular Modeling: Semi-Empirical Methods

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```					 Molecular Modeling:
Semi-Empirical Methods

C372
Introduction to Cheminformatics II
Kelsey Forsythe
Semi-Empirical Methods
 Faster than ab initio
 Less sensitive to parameterization than MM
methods
   Accuracy depends upon parameterization
Semi-Empirical Methods

 Ignore Core Electrons
 Approximate part of HF integration
Estimating Energy
   Recall Eclassical               <E>quantal
   
  (r )O(r )dr  average/ expectation value of observableO
*

 O                
  * (r ) (r )dr

         
 x 
  * (r ) x (r )dr
*           average / expectation value of x
  (r )(r )dr
Approximate Methods
   SCF (Self Consistent Field) Method (a.ka. Mean
Field or Hartree Fock)
   Pick single electron and average influence of remaining
electrons as a single force field (V0 external)
   Then solve Schrodinger equation for single electron in
presence of field (e.g. H-atom problem with extra force
field)
   Perform for all electrons in system
   Combine to give system wavefunction and energy (E)
   Repeat to error tolerance (Ei+1-Ei)
Estimating Energy
 ˆ  
 E 
  * (r ) H (r )dr
*          average / expectation value of energy
  ( r )  ( r ) dr
F
                  
 ( r )   cm   m ( r )
m

   Want to find c’s so that                dE
 0
dc
 ˆ                              
 m
i
cm (   * (r ) H j (r )dr  E   * (r ) j (r )dr )  0 m
mi
          
H mj                       S mj
                     ˆ                               
M  c  0 M ij    * (r ) H j (r )  E
i                       i
 * ( r ) j ( r )
M ij  H ij  E S ij
Non - trivial solutions for det(M) = 0
Estimating Energy
 ˆ                                
 m
i
cm (   * (r ) H j (r )dr  E    mi 
 * (r ) j (r )dr )  0 m
            
H mj                       S mj

 M 11M 12M 13......... 1N 
M
 F simulataneous                                  
 M 21M 22                  
equations gives a  M 31 M 33



M ij  i th row, j th column

matrix equation       
              

  M N1              M NN 

M c  0
 ˆ                          
                      
M ij   * (r ) H j (r )  E  * (r ) j (r )
i                       i

M ij  H ij  E Sij
Non - trivial solutions for det(M) = 0
Matrix Algebra
•    Finding determinant akin to rotating matrix
until diagonal ( M  0(i  j) )
ij

M 11M 12M 13......... 1N 
M
                         
M 21M 22                 

M 31 M 33                   M  i th row, j th column
ij
                         
                         
                         
M N1               M NN 


Matrix Algebra

 M 11M 12  M 11M 12
 M M   M M = ( M 11M 22) - ( M 12M 21)
det          
 21 22            21  22

1 3 
 2 4   (1* 4)  (2 * 3)  2
det     
     
Huckel Theory
   Assumptions
   Atomic basis set - parallel 2p orbitals
   No overlap between orbitals, ( Sij   ij )
   2p Orbital energy equal to ionization potential of
methyl radical (singly occupied 2p orbital)
   The  stabilization energy is the difference between the
2p-parallel configuration and the 2p perpendicular
configuration
E  stabilization  2 E p  E  2  E
E
       H ij
2
   Non-nearest interactions are zero
Ex. Allyl (C3H5)
   One p-orbital per carbon atom -
 basis size = 3
   Huckel matrix is

 E    0
    -E      0,   0
0        -E
E    2 ,  ,   2
   Resonance stabilization same for
Ex. Allyl (C3H5)
   Huckel matrix (determinant
   Huckel matrix (determinant                      form)-resonance (beta represents
form)-no resonance                              overlap/interaction between
orbitals) In matrix (determinant
E     0       0                               form)
0       -E    0  0,   0                      E    0
0       0      -E
    -E      0,   0
E   , ,
0        -E
   Energy of three isolated                        E    2 ,  ,   2
methylene sp2 orbitals

Overlap between orbital 1
   Energy of resonance system.
                                                       Note the lowest energy is less
and orbital 2
(hence matrix element           than the isolated orbital/AO due
H12)                            (this is resonance stabilization)
Extended Huckel Theory
(aka Tight Binding
Approximation)
 Includes non-nearest neighbor orbital
interactions
 Experimental Valence Shell Ionization
Potentials used to model matrix elements
 Generally applicable to any element
 Useful for calculating band structures in
solid-state physics
Beyond One-Electron
Formalism
   HF method                    hi  hartree hamiltonia n
Ignores electron               1 2 M Zk
 -  i    Vi  j

correlation                    2      k 1 rik
2
   Effective interaction                    j               j
Vi  j            dr                dr
potential                         j i   rij       j i   rij

   Hatree Product-
Fock introduced
H   hi separability
i
exchange –
N                  (relativistic quantum
   i                    mechanics)
i
HF-Exchange
   For a two electron system
   a (1) (1) * b (2) (2)
ˆ
P  Permutivity operator
ˆ
P a (1) (1) * b (2) (2)   a (2) (2) * b (1) (1) NO CHANGE IN SIGN

   Fock modified wavefunction
   a (1) (1) * b (2) (2)  a (2) (2) * b (1) (1)
ˆ
P   (2) (2) * (1) (1)  (1) (1) * (2) (2)
a            b            a            b

 -
Slater Determinants
   Ex. Hydrogen molecule
   a (1) (1) * b (2)  (2)
   a (1) (1) * b (2)  (2)  a (2) (2) * b (1)  (1)
ˆ
P   (2) (2) * (1)  (1)  (1) (1) * (2)  (2)
a             b            a            b

 -

 a (1) (1)  b (1)  (1) 
                            Slater Determinant
 a (2) (2)  b (2)  (2) 
Beyond One-Electron
Formalism
   HF method                    hi  hartree hamiltonia n
Ignores electron               1 2 M Zk
 -  i    Vi  j

correlation                    2      k 1 rik
2
   Effective interaction                    j               j
Vi  j            dr                dr
potential                         j i   rij       j i   rij

   Hatree Product-
Fock introduced
H   hi separability
i
exchange –
N                  (relativistic quantum
   i                    mechanics)
i
Neglect of Differential
Overlap (NDO)
   STO-basis (/S-spectra,/2 d-orbitals)
   CNDO (1965, Pople et al)

   MINDO (1975, Dewar )          /1/2/3, organics

   /d, organics, transition metals
   MNDO (1977, Thiel)

   INDO (1967, Pople et al)      Organics

   Electronic spectra, transition metals
   ZINDO

   1-3 row binding energies,
   SINDO1                         photochemistry and transition
metals
Semi-Empirical Methods
   SAM1
   Closer to # of ab initio basis functions (e.g. d orbitals)
   Increased CPU time
   G1,G2 and G3
   Extrapolated ab initio results for organics
   “slightly empirical theory”(Gilbert-more ab initio than
semi-empirical in nature)
Semi-Empirical Methods
   AM1
 Modified nuclear repulsion terms model to
account for H-bonding (1985, Dewar et al)
 Widely used today (transition metals, inorganics)

   PM3 (1989, Stewart)
 Larger data set for parameterization compared to
AM1
 Widely used today (transition metals, inorganics)
General
Reccommendations
 More accurate than empirical methods
 Less accurate than ab initio methods
 Inorganics and transition metals
 Pretty good geometry OR energies
 Poor results for systems with diffusive
interactions (van der Waals, H-bonded,
Complete Neglect of
Differential Overlap
(CNDO)
2
j               j
Vi  j            dr                dr
j i   rij       j i   rij

   Overlap integrals, S, is
assumed zero
Neglect of Differential
Overlap (NDO)
2
j               j
Vi  j            dr                dr
j i   rij       j i   rij

Gives rise to overlap between
electronic basis functions of
different types and on different
atoms
Complete Neglect of
Differential Overlap
(CNDO)
 One-electron overlap integral for different
electrons is zero (as in Huckel Theory)
 Two-electron integrals are zero if basis
functions not identical
Intermediate Neglect of
Differential Overlap
(CNDO)
   Overlap integrals, S, is assumed zero
Eigenvalue Equation
 Matrix * Vector = Matrix (diagonal) * Vector
 Schrodinger’s equation!
ˆ
H  E
F
(r )   c m   m (r )
m
F                    F
H  c m   m (r )  E  c m  m (r )
ˆ
m                    m

 solutions to this differential equation are equal
The
to the solutions to the matrix eigenvalue equation
M ij  H ij  E j Sij

Solutions for det(M)= 0

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