Physical Chemistry Third Edition

952 23 Optical Spectroscopy and Photochemistry

operator of the molecule in the presence of radiation is written in a way similar to

Eq. (19.3-1):^2

̂ĤH(0)+Ĥ′ (23.1-3)

The zero-order HamiltonianĤ(0)is not the same as the zero-order Hamilton in

Section 19.3. It is the complete time-independent Hamiltonian operator of the molecule

in the absence of radiation. The perturbation term̂H′describes the interaction between

the molecule and the electric field of the radiation, and is time-dependent because

of the oscillation of the radiation.

We assume that the zero-order time-independent Schrödinger equation has been

solved to a usable approximation:

Ĥ(0)ψ(0)j E(0)j ψ(0)j (23.1-4)

The wave functionψ(0)j is one of the energy eigenfunction of the molecule in the

absence of radiation. Inclusion of the perturbation produces a time-dependent wave

function, which is written as a linear combination of the zero-order wave functions:

Ψ(q,t)

∑

j

aj(t)ψ

(0)

j (q) (23.1-5)

whereqstands for all of the coordinates of the particles in the molecule. Since the

zero-order wave functions are time-independent, theajcoefficients must depend on

time. They contain all of the time dependence of the wave function.

In order to observe transitions we specify that at timet0 the wave function is

equal to one of the zero-order functions,ψ(0)n ,

Ψ(q,0)ψ(0)n(q) (23.1-6)

so that att0 onlyanis nonzero:

aj(0)δjn

{

1ifjn

0ifjn

(23.1-7)

whereδjnis theKronecker delta, introduced in Eq. (16.4-43). If at later times another

coefficient,aj, becomes nonzero, this corresponds to a nonzero probability of a transi-

tion from the stateψ(0)n to the stateψ(0)j.

Time-dependent perturbation theory provides an approximate formula that gives the

coefficients as functions of time. If the radiation is polarized with its electric field in the

zdirection,|aj(t)|^2 is proportional to the intensity of the radiation of the wavelength that

satisfies the Bohr frequency rule and is also proportional to the square of the following

integral:^3

(μz)jn

∫

ψ

(0)∗

j ̂μzψ

(0)

n dq

∫

̂μzψ

(0)∗

j ψ

(0)

n dq (23.1-8)

wherêμzis operator for thezcomponent of the operator for the electric dipole of

the atom or molecule, as in Eq. (20.4-9). The operator̂μzis a multiplication operator,

(^2) J. C. Davis,Advanced Physical Chemistry, The Ronald Press, New York, 1965, p. 243ff.
(^3) J. C. Davis,loc. cit.(note 2).