Computational Chemistry

“even less desirability” was perhaps that the computed results would be too com-
plex to interpret; one factor which has obviated this problem is the visual display of
information (Sections 5.5.6 and 6.3.6). The development of improved algorithms
and far faster computers has altered the situation almost out of recognition; for
example, an energy calculation on a moderate-size molecule (1,3,5-triamino-2,4,6-
trinitrobenzene) is faster now (mid-2009) by these factors: compared to 17 years
ago, 1,700; compared to 25 years ago, 90,000, compared to 42 years ago, 10^8 [ 6 ].
Why, then, are semiempirical calculations still used? Because they are still about
100–1,000 times faster than ab initio (Chapter 5) or density functional (Chapter 7)
methods. The increase in computer speed means that we can now routinely examine
by ab initio methods moderately large molecules – up to, say, steroids, with about
30 heavy atoms (non-hydrogen atoms), and by semiempirical methods huge mole-
cules, even proteins and nucleic acids.
In the following presentation of semiempirical methods, the general approach
and the distinction between the various methods is best appreciated by under-
standing the concepts in words, rather than attempting to memorize admittedly
somewhat formidable-looking equations (unless you plan to develop a new
semiempirical method).

6.2 The Basic Principles of SCF Semiempirical Methods..................

6.2.1 Preliminaries......................................................

The semiempirical methods we saw inChapter 4simply construct a Fock matrix
and diagonalize it once to get MO energy levels and MOs (i.e. the coefficients of the
basis functions that make up the MOs). The simple H€uckel method Fock matrix
elements were simply relative energies 0 and"1 (0 and"1|b| units, relative to the
nonbonding levela), and the extended H€uckel method Fock matrix elements were
calculated from ionization energies. In both the simple and extended H€uckel
methods a single matrix diagonalization gave the energy levels and MO coeffi-
cients. This chapter is concerned with semiempirical methods that are closer to the
ab initio method in that the SCF procedure (Section5.2.3.6, particularly Sections
5.2.3.6.4and5.2.3.6.5) is used to refine the energy levels and MO coefficients: basis
set coefficients from a “guess” are improved by repeated matrix diagonalization. As
in ab initio calculations each Fock matrix element is calculated from a core integral
Hcorers , density matrix elementsPtu, and electron repulsion integrals (rs|tu), (ru|ts):

Frs¼Hcorers ð 1 Þþ

Xm

t¼ 1

Xm

u¼ 1

Ptu½ðrsjtuÞ"

1

2

ðrujtsފ (6.1 = 5.82)

As stated above, the following discussion applies to semiempirical methods that,
Iike ab initio, use the SCF procedure and so pay some service to Eq.6.1¼5.82.

6.2 The Basic Principles of SCF Semiempirical Methods 393