Computational Chemistry

atomsiandj, which has the important consequence that theSvalues depend on

the geometry of the molecule. SinceSis not taken as a unit matrix, we cannot go

directly fromHC¼SC«toHC¼C«and thus we cannot simply diagonalize

the EHM Fock to get thec’s ande’s.

These four points are elaborated on below.

1. Use of a minimal valence basis set in the EHM is more realistic than treating just
   the 2pzorbitals, since all the valence electrons in a molecule are likely to be
   involved in determining its properties. Further, the SHM is largely limited top
   systems, i.e. to alkenes and aromatics and derivatives of these with attachedp
   electron groups, but the EHM, in contrast, can in principle be applied to any
   molecule. The use of a minimal valence basis set makes the Fock matrix much
   larger than in the “corresponding” SHM calculation. For example in an
   SHM calculation on ethene, only two orbitals are used, the 2pzon C 1 and the
   2 pzon C 2 , and the SHM Fock matrix is (using the compact Dirac notation
   fiH^

(^) fj

DE

¼

R

fiH^fjdv



HðSHMÞ¼

C 1 ð 2 pzÞjH^jC 1 ð 2 pzÞ

C 1 ð 2 pzÞjH^jC 2 ð 2 pzÞ

C 2 ð 2 pzÞjH^jC 1 ð 2 pzÞ

C 2 ð 2 pzÞjH^jC 2 ð 2 pzÞ

!

¼

0 " 1

" 10



2 &2 matrix

(4.89)

To write down the EHM Fock matrix, let us label the valence orbitals like this:

H 1 ð 1 sÞf 1 C 1 ð 2 sÞf 5 C 1 ð 2 pxÞf 7 C 1 ð 2 pyÞf 9 C 1 ð 2 pxÞf 11

H 2 ð 1 sÞf 2 C 2 ð 2 sÞf 6 C 2 ð 2 pxÞf 8 C 2 ð 2 pyÞf 10 C 2 ð 2 pzÞf 12

H 3 ð 1 sÞf 3

H 4 ð 1 sÞf 4

Then

HðEHMÞ¼

f 1 jH^jf 1

f 1 jH^jf 2

... f 1 jH^jf 12

f 2 jH^jf 1

f 2 jH^jf 2

... f 2 jH^jf 12

... ... ... ...

f 12 jH^jf 1

f 12 jH^jf 2

... f 12 jH^jf 12

0

B

B

B

BB

B

B

@

1

C

C

C

CC

C

C

A

12 & 12 matrix

(4.90)

154 4 Introduction to Quantum Mechanics in Computational Chemistry