Concise Physical Chemistry

c20 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

324 QUANTUM MOLECULAR MODELING

(NDO) andneglect of diatomic differential overlap(NDDO) approximations led to

a series of semiempirical programs denoted CNDO (complete neglect of differential

overlap), INDO, MINDO, and so on. Initially, NDDO programs were parameterized

to reproduceab initiovalues for simple molecules but later semiempirical programs

such as the AM1 programs of Dewar et al. and the PM3 method of Stewart are

parameterized against experimental thermochemical results so as to calculate energies

and enthalpies. Presently, they are both in wide use for this purpose. Along with

thermochemical data, dipole moments, geometries, and isomerization potentials are

also calculated by modern semiempirical programs.

20.5 AB INITIO METHODS

An exact solution of the Schrodinger equation would employ no empirical parameters ̈

beyond mass and charge of the subatomic particles, and it would be an absolute

solution to the problem that could never be changed or revised. In practice, the

absolute properties of a molecule are never found, though they may be approached

by ever-improving approximate methods. Today, what we call “ab initioprocedures”

contain small empirical “corrections,” but a very serious effort has been put forth

to minimize these aspects of the procedure. One condition we wish for the set of

functions{χμ}we choose as a basis set is that they be as nearly complete as possible,

that is, we hope that the set willspanthe vector space.

20.6 THE GAUSSIAN BASIS SET

We saw in Chapter 17 that the single Gaussian functionφ(r)=Ce−αr

2

, whereC=

1 .0 andα= 0 .2829, does not give a very good approximation to the energy of the

hydrogen atom. The result, HF=0.4244Eh, is 84.9% of the true (defined) energy

of the hydrogen atom. The Gaussian, withr^2 in the exponent, drops off faster than

the 1sorbital, which hasrin the exponent (Fig. 20.4). The Gaussian is too “thin” at

larger distances from the nucleus.

031 2

0.5

1

1

0

(r)

(r)

0 r 3

FIGURE 20.4 The 1sSTO (solid line) and a Gaussian approximation (dotted line).