80
3 Electronic Excitation and Decay
A formally similar situation can also be found in the absence of external perturbations,
when the full molecular Hamiltonian is split in two terms, ˆ
H
(0) and ˆ
V , and one can
assume the initial state to be an eigenstate
ψ
(0)
i
of ˆ
H
(0) :
ˆ
H
(0)
ψ
(0)
i
= ε i
ψ
(0)
i
.
(3.2)
Let us write the time-dependent wavefunction as a linear combination of the ψ
(0)
j
eigenstates:
|ψ(t) =
j
c j (t) e
−iε j t/
ψ
(0)
j
.
(3.3)
As we know from Sect. 2.1.2, without the perturbation ˆ
V this expression would
represent the time evolution of the system with constant coefficients: c j (t) = c j (0).
By using this expansion, the time-dependent Schrödinger equation
i
d
dt
|ψ(t) = ˆ
H |ψ(t)
(3.4)
becomes
j
˙
c j −
iε j
c j
e
−iε j t/
ψ
(0)
j
= −
i
j
c j e
−iε j t/
ε j + ˆ
V
ψ
(0)
j
. (3.5)
Thanks to the introduction of the exponential factors exp(−iε j t/) in Eq. (3.3) the
terms −iε j c j / cancel out. We can single out the time derivative of coefficient c i by
multiplying on the left both members of this equation by
ψ
(0)
i
:
˙
c i = −
i
j
c j e
iω i j t V i j ∀ i
(3.6)
where
V i j =
ψ
(0)
i
ˆ
V
ψ
(0)
j
(3.7)
and
ω i j =
ε i − ε j
.
(3.8)
The (3.6) are a set of coupled equations, one for each c i coefficient, the solution of
which yields the full information about the time evolution of the system. The squared
module of each coefficient in the development (3.3), P i = |c i (t)|
2 , is the probability
of finding the system in state ψ
(0)
i
at time t, also called the state population. When
the probability P i increases in time, we say that transitions occur from other states
ψ
(0)
j to state ψ
(0)
i . According to Eq. (3.6) this is possible only when at least one other
3 Electronic Excitation and Decay
A formally similar situation can also be found in the absence of external perturbations,
when the full molecular Hamiltonian is split in two terms, ˆ
H
(0) and ˆ
V , and one can
assume the initial state to be an eigenstate
ψ
(0)
i
of ˆ
H
(0) :
ˆ
H
(0)
ψ
(0)
i
= ε i
ψ
(0)
i
.
(3.2)
Let us write the time-dependent wavefunction as a linear combination of the ψ
(0)
j
eigenstates:
|ψ(t) =
j
c j (t) e
−iε j t/
ψ
(0)
j
.
(3.3)
As we know from Sect. 2.1.2, without the perturbation ˆ
V this expression would
represent the time evolution of the system with constant coefficients: c j (t) = c j (0).
By using this expansion, the time-dependent Schrödinger equation
i
d
dt
|ψ(t) = ˆ
H |ψ(t)
(3.4)
becomes
j
˙
c j −
iε j
c j
e
−iε j t/
ψ
(0)
j
= −
i
j
c j e
−iε j t/
ε j + ˆ
V
ψ
(0)
j
. (3.5)
Thanks to the introduction of the exponential factors exp(−iε j t/) in Eq. (3.3) the
terms −iε j c j / cancel out. We can single out the time derivative of coefficient c i by
multiplying on the left both members of this equation by
ψ
(0)
i
:
˙
c i = −
i
j
c j e
iω i j t V i j ∀ i
(3.6)
where
V i j =
ψ
(0)
i
ˆ
V
ψ
(0)
j
(3.7)
and
ω i j =
ε i − ε j
.
(3.8)
The (3.6) are a set of coupled equations, one for each c i coefficient, the solution of
which yields the full information about the time evolution of the system. The squared
module of each coefficient in the development (3.3), P i = |c i (t)|
2 , is the probability
of finding the system in state ψ
(0)
i
at time t, also called the state population. When
the probability P i increases in time, we say that transitions occur from other states
ψ
(0)
j to state ψ
(0)
i . According to Eq. (3.6) this is possible only when at least one other