Computational Chemistry

The SHM and EHM basis sets are shown in Fig.4.26.
2.The EHM Fock matrix interactions i,jdo not have just two values (aorb) as
in the SHM, but are functions of the orbitals (the basis functions)fiandfjand
of the separation of these orbitals, as explained in (3) below.
3.The EHM matrix elements fiH^fi
and fiH^fj

DE

are calculated (rather than set

equal to 0 or"1), although the calculation is a simple one using overlap integrals

and experimental ionization energies; in ab initio calculations (Chapter 5) and

more advanced semiempirical calculations (Chapter 6), the mathematical form

ofHˆtaken into account. Thei,i-type interactions are taken as being proportional

to the negative of the ionization energy [ 58 ] of the orbitalfiand thei,j-type

interactions as being proportional to the overlap integral betweenfiandfjand

the negative of the average of the ionization energiesIiandIjoffiandfj(the

negative of the orbital ionization energy is the energy of an electron in the

orbital, compared to the zero of energy of the electron and the ionized species

infinitely separated and at rest):

fiH^fi

¼"Ii (4.91)

fiH^fj

DE

¼"

1

2

KSijðIiþIjÞ (4.92)

A proportionality constantKof about 2 is commonly used.

For H(1s), C(2s) and C(2p), experiment shows

IðÞ¼H(1s) 13 :6 eV, IðÞ¼C(2s) 20 :8 eV, IðÞ¼C(2p) 11 :3 eV (4.93)

The overlap integrals are calculated using Slater-type (Section 5.3.2) functions

for the basis functions, e.g.

C C

H

H H

H

C C

The simple Hückel method basis set for ethene.

Each carbon has one 2p basis function.

C 2 H 4 has two basis functions

The extended Hückel method basis set for ethene.

Each carbon has one 2 s and three 2p basis functions.

Each H has one 1s basis function.

C 2 H 4 has 12 basis functions.

H

H H

H

Fig. 4.26 The simple H€uckel method normally uses only one basis function per “heavy atom”:
only one 2porbital on each carbon, oxygen, nitrogen, etc., ignoring the hydrogens. The extended
H€uckel method uses for each carbon, oxygen, nitrogen, etc., a 2sand three 2porbitals, and for each
hydrogen a 1sorbital. This is called a minimal valence basis set

4.4 The Extended H€uckel Method 155