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).