160
5 Fast Nonadiabatic Dynamics
Because of the geometric phase, in the presence of a conical intersection an adiabatic electronic state ϕ k is a multivalued function. Of course, the total wavefunction
has to be single-valued, and as a consequence the nuclear part must be multivalued,
too. Hence, the presence of the conical intersection induces a coupling between the
electronic and the nuclear motion, even when the nonadiabatic effects are negligible.
Actually, we could exploit the fact that the adiabatic wavefunctions are defined up
to Q-dependent phase factors, by setting
˜
ϕ k (r; Q) = e
iΩ k (Q)
ϕ k (r; Q)
(5.59)
in such a way that ˜
ϕ k is single-valued. The nonadiabatic couplings are not invariant
in this “gauge” transformation
˜
g kl (Q) = e
i(Ω l −Ω k ) g kl (Q) + iδ kl ∇Ω l (Q) .
(5.60)
Note in particular that, while with a real Hamiltonian ϕ k can always be chosen realvalued so that g kk = 0, ˜
ϕ k is complex and we have ˜
g kk = i∇Ω k . We know from (2.64)
that ˜
g kk is imaginary; therefore Ω k is real.
The geometric phase was first discovered by Longuet–Higgins [14] for a two-state
system (as the one considered above) and then generalized by Berry [12]. According
to Berry, we take Q as a set of time-dependent external parameters (similarly to Sect.
5.3). Let us consider a closed path C in the Q space, which is covered in the time
interval [0, T ], so that Q(0) = Q(T ). Using the relation ˜
g kk = i∇Ω k and the chain
derivative rule we obtain
˙
Ω k (t) = ∇Ω k · ˙
Q = −i˜ g kk · ˙
Q
(5.61)
Then, the accumulated phase Ω
(C)
k
in the closed path C is
Ω
(C)
k
= Ω k (T ) − Ω k (0) = −i
T
0
˜
g kk · ˙
Qdt = −i
C
˜
g kk · dQ.
(5.62)
Given that ˜
ϕ k is single-valued, it follows from from Eq. (5.59) that the phase factor
acquired by ϕ k in the closed path C is exp(−iΩ
(C)
k ). Note that Ω
(C)
k has two important
properties: it depends on the shape of the path C and not on the velocity at which C
is covered, and it is invariant in the gauge transform referred above (see Eq. (5.60)).
In fact, for any continuous and derivable f , both ˜
g kk and ˜
g kk + i∇ f have the same
circulation. It is for these reasons that the phase Ω k is called geometric.
In the two-state case considered above we can choose Ω 1 = Ω 2 = γ /2. Moreover, assuming for simplicity q = h, we have g 12 = −∇γ /2 = −∇Ω 1 = i˜ g 11 . Then,
in a small circle around a conical intersection we get, according to Eq. (5.44):
exp(−iΩ
(C)
k ) = e
±iπ
= −1.
In Fig. 5.6 and in the Animation 5.1 we show the time evolution of a wavepacket
going through a conical intersection. The two-dimensional model system considered
5 Fast Nonadiabatic Dynamics
Because of the geometric phase, in the presence of a conical intersection an adiabatic electronic state ϕ k is a multivalued function. Of course, the total wavefunction
has to be single-valued, and as a consequence the nuclear part must be multivalued,
too. Hence, the presence of the conical intersection induces a coupling between the
electronic and the nuclear motion, even when the nonadiabatic effects are negligible.
Actually, we could exploit the fact that the adiabatic wavefunctions are defined up
to Q-dependent phase factors, by setting
˜
ϕ k (r; Q) = e
iΩ k (Q)
ϕ k (r; Q)
(5.59)
in such a way that ˜
ϕ k is single-valued. The nonadiabatic couplings are not invariant
in this “gauge” transformation
˜
g kl (Q) = e
i(Ω l −Ω k ) g kl (Q) + iδ kl ∇Ω l (Q) .
(5.60)
Note in particular that, while with a real Hamiltonian ϕ k can always be chosen realvalued so that g kk = 0, ˜
ϕ k is complex and we have ˜
g kk = i∇Ω k . We know from (2.64)
that ˜
g kk is imaginary; therefore Ω k is real.
The geometric phase was first discovered by Longuet–Higgins [14] for a two-state
system (as the one considered above) and then generalized by Berry [12]. According
to Berry, we take Q as a set of time-dependent external parameters (similarly to Sect.
5.3). Let us consider a closed path C in the Q space, which is covered in the time
interval [0, T ], so that Q(0) = Q(T ). Using the relation ˜
g kk = i∇Ω k and the chain
derivative rule we obtain
˙
Ω k (t) = ∇Ω k · ˙
Q = −i˜ g kk · ˙
Q
(5.61)
Then, the accumulated phase Ω
(C)
k
in the closed path C is
Ω
(C)
k
= Ω k (T ) − Ω k (0) = −i
T
0
˜
g kk · ˙
Qdt = −i
C
˜
g kk · dQ.
(5.62)
Given that ˜
ϕ k is single-valued, it follows from from Eq. (5.59) that the phase factor
acquired by ϕ k in the closed path C is exp(−iΩ
(C)
k ). Note that Ω
(C)
k has two important
properties: it depends on the shape of the path C and not on the velocity at which C
is covered, and it is invariant in the gauge transform referred above (see Eq. (5.60)).
In fact, for any continuous and derivable f , both ˜
g kk and ˜
g kk + i∇ f have the same
circulation. It is for these reasons that the phase Ω k is called geometric.
In the two-state case considered above we can choose Ω 1 = Ω 2 = γ /2. Moreover, assuming for simplicity q = h, we have g 12 = −∇γ /2 = −∇Ω 1 = i˜ g 11 . Then,
in a small circle around a conical intersection we get, according to Eq. (5.44):
exp(−iΩ
(C)
k ) = e
±iπ
= −1.
In Fig. 5.6 and in the Animation 5.1 we show the time evolution of a wavepacket
going through a conical intersection. The two-dimensional model system considered