3.1 Constant and Time-Dependent Perturbations
81
ψ
(0)
j state is populated (i.e., c j = 0) and V i j = 0. This is the basis of all “selection
rules,” in spectroscopy as in other aspects of molecular dynamics. When V i j = 0 we
say that the perturbation ˆ
V “couples” states ψ
(0)
i
and ψ
(0)
j . The V i j matrix element
is called the “coupling” or “interaction” between the two states.
Notice that, as far as ˆ
H
(0)
= ˆ
H , the quantity
j
c j
2 ε j is not the total energy
of the system; at best, it is an approximation of it, valid for small couplings V i j .
However, if ˆ
V depends on time, there may be times at which ˆ
H
(0) coincides with the
total Hamiltonian. For instance, if the system is perturbed by a light pulse, before the
pulse is switched on and after it dies off we have ˆ
V = 0: so, the
c j
2 probabilities
correctly describe the molecular energy distributions before and after the interaction
with the light pulse.
3.2 Light–Molecule Interaction
Since we are particularly interested in the light absorption process, we shall consider
the case where ˆ
V is the radiation–molecule interaction. Molecules interact with light
mainly through the electric field. Consider, for instance, electric and magnetic dipoles
with the usual orders of magnitude of 1 a.u., i.e., respectively, two proton/electron
charges separated by 1 bohr, and the orbital or spin angular momentum of an electron.
Since the field magnitudes in Eqs. (1.1) and (1.2) are related by E 0 = cB 0 , the fielddipole interaction is about 100 times larger for the electric field than for the magnetic
one. As we shall see, the effect of such perturbations in many circumstances is
proportional to the square of the interaction, so the electric field of light affects a
molecule about 10
4 times more than the magnetic field.
The interaction energy of a molecule with a time- and space-dependent electric
field is, in atomic units
ˆ
V = −
α
Z α r α · E(r α , t) +
i
r i · E(r i , t)
(3.9)
where α numbers the nuclei and i the electrons. For UV, visible or NIR light of
interest in photochemistry, the wavelengths we consider are larger than 100 nm, i.e.,
much larger than many molecules or at least of the part of a molecule where the
excitation is localized. As a consequence, in Eq. (3.9) we can replace E(r i , t) with
its value at an arbitrary location r 0 within the molecule, for instance, its center of
mass. Then, we shall drop the dependence on position and write
ˆ
V = −µ · E(t)
(3.10)
where
µ =
α
Z α r α −
i
r i
(3.11)
81
ψ
(0)
j state is populated (i.e., c j = 0) and V i j = 0. This is the basis of all “selection
rules,” in spectroscopy as in other aspects of molecular dynamics. When V i j = 0 we
say that the perturbation ˆ
V “couples” states ψ
(0)
i
and ψ
(0)
j . The V i j matrix element
is called the “coupling” or “interaction” between the two states.
Notice that, as far as ˆ
H
(0)
= ˆ
H , the quantity
j
c j
2 ε j is not the total energy
of the system; at best, it is an approximation of it, valid for small couplings V i j .
However, if ˆ
V depends on time, there may be times at which ˆ
H
(0) coincides with the
total Hamiltonian. For instance, if the system is perturbed by a light pulse, before the
pulse is switched on and after it dies off we have ˆ
V = 0: so, the
c j
2 probabilities
correctly describe the molecular energy distributions before and after the interaction
with the light pulse.
3.2 Light–Molecule Interaction
Since we are particularly interested in the light absorption process, we shall consider
the case where ˆ
V is the radiation–molecule interaction. Molecules interact with light
mainly through the electric field. Consider, for instance, electric and magnetic dipoles
with the usual orders of magnitude of 1 a.u., i.e., respectively, two proton/electron
charges separated by 1 bohr, and the orbital or spin angular momentum of an electron.
Since the field magnitudes in Eqs. (1.1) and (1.2) are related by E 0 = cB 0 , the fielddipole interaction is about 100 times larger for the electric field than for the magnetic
one. As we shall see, the effect of such perturbations in many circumstances is
proportional to the square of the interaction, so the electric field of light affects a
molecule about 10
4 times more than the magnetic field.
The interaction energy of a molecule with a time- and space-dependent electric
field is, in atomic units
ˆ
V = −
α
Z α r α · E(r α , t) +
i
r i · E(r i , t)
(3.9)
where α numbers the nuclei and i the electrons. For UV, visible or NIR light of
interest in photochemistry, the wavelengths we consider are larger than 100 nm, i.e.,
much larger than many molecules or at least of the part of a molecule where the
excitation is localized. As a consequence, in Eq. (3.9) we can replace E(r i , t) with
its value at an arbitrary location r 0 within the molecule, for instance, its center of
mass. Then, we shall drop the dependence on position and write
ˆ
V = −µ · E(t)
(3.10)
where
µ =
α
Z α r α −
i
r i
(3.11)