5.2 Diabatic States
147
the other hand, the necessary condition (5.12) is always trivially satisfied in one
coordinate Q α (i.e., for β = α), and in principle Eq. (5.10) restricted to S can be
integrated for a single Q α . Therefore, the strict diabatization can be performed within
S for a diatomic molecule, in principle.
To summarize, it is possible to find a unitary matrix T transforming a truncated
adiabatic basis into a strictly diabatic one only for one internal coordinate at a time
(or along a given path).
In view of these difficulties, the diabatic states are usually defined using less strict
conditions than the exact cancelation of the dynamical couplings. In practice, the
ν kl are requested to be minimal, or however negligibly small. The corresponding η k
are then called quasi-diabatic states. Obviously the quasi-diabatic basis cannot be
uniquely defined, and many different methods have been devised to their evaluation
(for a short review see [5]). To cite only a simple example, one could rely on molecular
properties: for instance, the transition dipole moment can be used to define dark and
bright diabatic states; space localization to define excitonic states, and so on.
The nonadiabatic couplings g
(α)
kl are expressed in terms of the quasi-diabatic basis
in this way
g
(α)
kl = (U l − U k )
−1 C
+
k
∂H
∂ Q α
C l + C
+
k ν
(α) C l
(5.18)
where C k is the kth column of the diabatic-to-adiabatic matrix C (the inverse of
matrix T). The residual derivative coupling between quasi-diabatic functions is usually neglected: anyway, the g
(α)
kl are large in near-degeneracy regions, where the
first term dominates. Equation (5.18) represents a very convenient way to evaluate
the nonadiabatic couplings, as the elements H kl =
η k
ˆ
H el
η l
of matrix H are, by
construction, smooth functions of Q.
Also time evolution is particularly simple in the quasi-diabatic basis. The Born–
Huang expansion of the total wavefunction Ψ (Q, r, t) becomes
Ψ (Q, r, t) =
k
Θ k (Q, t)η k (r; Q)
(5.19)
where here Θ k (Q, t) are the nuclear wavepackets on the quasi-diabatic states. Their
time evolution is given by (compare Eq. (2.78))
i
dΘ k
dt
= ˆ
T n Θ k +
l
H kl Θ l
(5.20)
where we have neglected the residual derivative couplings between quasi-diabatic
functions, which ought to be small. As noted above, the H kl are smooth functions of
Q, while in the adiabatic expression (2.78) both U
k and ˆ
V
B O
kl
contain terms which
diverge at degeneracy points (see Sect. 5.4).
147
the other hand, the necessary condition (5.12) is always trivially satisfied in one
coordinate Q α (i.e., for β = α), and in principle Eq. (5.10) restricted to S can be
integrated for a single Q α . Therefore, the strict diabatization can be performed within
S for a diatomic molecule, in principle.
To summarize, it is possible to find a unitary matrix T transforming a truncated
adiabatic basis into a strictly diabatic one only for one internal coordinate at a time
(or along a given path).
In view of these difficulties, the diabatic states are usually defined using less strict
conditions than the exact cancelation of the dynamical couplings. In practice, the
ν kl are requested to be minimal, or however negligibly small. The corresponding η k
are then called quasi-diabatic states. Obviously the quasi-diabatic basis cannot be
uniquely defined, and many different methods have been devised to their evaluation
(for a short review see [5]). To cite only a simple example, one could rely on molecular
properties: for instance, the transition dipole moment can be used to define dark and
bright diabatic states; space localization to define excitonic states, and so on.
The nonadiabatic couplings g
(α)
kl are expressed in terms of the quasi-diabatic basis
in this way
g
(α)
kl = (U l − U k )
−1 C
+
k
∂H
∂ Q α
C l + C
+
k ν
(α) C l
(5.18)
where C k is the kth column of the diabatic-to-adiabatic matrix C (the inverse of
matrix T). The residual derivative coupling between quasi-diabatic functions is usually neglected: anyway, the g
(α)
kl are large in near-degeneracy regions, where the
first term dominates. Equation (5.18) represents a very convenient way to evaluate
the nonadiabatic couplings, as the elements H kl =
η k
ˆ
H el
η l
of matrix H are, by
construction, smooth functions of Q.
Also time evolution is particularly simple in the quasi-diabatic basis. The Born–
Huang expansion of the total wavefunction Ψ (Q, r, t) becomes
Ψ (Q, r, t) =
k
Θ k (Q, t)η k (r; Q)
(5.19)
where here Θ k (Q, t) are the nuclear wavepackets on the quasi-diabatic states. Their
time evolution is given by (compare Eq. (2.78))
i
dΘ k
dt
= ˆ
T n Θ k +
l
H kl Θ l
(5.20)
where we have neglected the residual derivative couplings between quasi-diabatic
functions, which ought to be small. As noted above, the H kl are smooth functions of
Q, while in the adiabatic expression (2.78) both U
k and ˆ
V
B O
kl
contain terms which
diverge at degeneracy points (see Sect. 5.4).