166
5 Fast Nonadiabatic Dynamics
so the branching space actually has dimension 5 in the present case. This has an
impact on the geometric phase as well, which shows in this case a more complicated
behavior with respect to Sect. 5.4.3.
Note however that, in the absence of spin–orbit coupling (as in the H 3 molecule
considered in Sect. 5.4.3), H 12 is real and H 1T 2 = 0: in fact, according to (5.69),
the spin of ˆ
T η 2 is reversed with respect to η 2 (so, either H 12 = 0 or H 1T 2 = 0). The
4 × 4 matrix (5.76) reduces to two identical and uncoupled 2 × 2 matrices. We are
therefore back to the case of branching space of dimension 2.
In general, it can be shown that H 1T 2 = 0 if the molecule has C s symmetry, or
higher. In that case, with complex H 12 , the branching space has dimension 3 (three
conditions to impose for U 1 = U 2 ).
Case C. System not symmetric with respect to time reversal ( ˆ
H el ˆ
T = ˆ
T ˆ
H el ). This
is the case, for example, of a molecule in an external magnetic field. The Kramers
degeneracy is removed, but H 12 is, in general, complex. We have therefore a threedimensional branching space.
We want now to evaluate the phase Ω
(C)
k (see Eq. (5.62)) in the present context of a
complex-valued Hamiltonian [12, 13]. We consider for simplicity a three-dimensional
nuclear configurational space so that, due to Stokes’ theorem, the circulation of ˜
g kk
on C corresponds to the flux of a vector V k = ∇ × ˜
g kk across a surface S C bounded
by C
Ω
(C)
k
= −i
C
˜
g kk · dQ = −i
S C
V k · dS .
(5.79)
Clearly ∇ · V k = 0, so the flux of V k across a closed surface vanishes (unless the
enclosed volume contains a degeneracy point, where ˜
g kk has a singularity). Therefore,
the last integral in Eq. (5.79) only depends on the closed path C, and not on the
peculiar choice of the surface S C . We have
V k = ∇ × ˜
g kk
(5.80)
= ∇ ˜
ϕ k |×| ∇ ˜
ϕ k
(5.81)
=
m =k
∇ ˜
ϕ k | ˜
ϕ m × ˜
ϕ m |∇ ˜
ϕ k
(5.82)
=
m =k
ϕ k
∇ ˆ
H el
ϕ m
×
ϕ m
∇ ˆ
H el
ϕ k
(U k − U m ) 2
(5.83)
where we used the relation ∇ × ( f ∇g) = ∇ f × ∇g and Eq. (2.70) for the nonadiabatic couplings. The sum is extended to a complete basis set of adiabatic wavefunctions; the term with m = k is excluded as ∇ ˜
ϕ k | ˜
ϕ k is purely imaginary and the
vector product with its complex conjugate vanishes. It is particularly evident from
Eq. (5.82) that V k is independent on the phases of the electronic functions. It is then
not necessary to use the single-valued ˜
ϕ m . Moreover, V k is imaginary: considering a
real Hamiltonian, we can choose ϕ m as real so that V k = 0 and Ω
(C)
k
vanishes. This
5 Fast Nonadiabatic Dynamics
so the branching space actually has dimension 5 in the present case. This has an
impact on the geometric phase as well, which shows in this case a more complicated
behavior with respect to Sect. 5.4.3.
Note however that, in the absence of spin–orbit coupling (as in the H 3 molecule
considered in Sect. 5.4.3), H 12 is real and H 1T 2 = 0: in fact, according to (5.69),
the spin of ˆ
T η 2 is reversed with respect to η 2 (so, either H 12 = 0 or H 1T 2 = 0). The
4 × 4 matrix (5.76) reduces to two identical and uncoupled 2 × 2 matrices. We are
therefore back to the case of branching space of dimension 2.
In general, it can be shown that H 1T 2 = 0 if the molecule has C s symmetry, or
higher. In that case, with complex H 12 , the branching space has dimension 3 (three
conditions to impose for U 1 = U 2 ).
Case C. System not symmetric with respect to time reversal ( ˆ
H el ˆ
T = ˆ
T ˆ
H el ). This
is the case, for example, of a molecule in an external magnetic field. The Kramers
degeneracy is removed, but H 12 is, in general, complex. We have therefore a threedimensional branching space.
We want now to evaluate the phase Ω
(C)
k (see Eq. (5.62)) in the present context of a
complex-valued Hamiltonian [12, 13]. We consider for simplicity a three-dimensional
nuclear configurational space so that, due to Stokes’ theorem, the circulation of ˜
g kk
on C corresponds to the flux of a vector V k = ∇ × ˜
g kk across a surface S C bounded
by C
Ω
(C)
k
= −i
C
˜
g kk · dQ = −i
S C
V k · dS .
(5.79)
Clearly ∇ · V k = 0, so the flux of V k across a closed surface vanishes (unless the
enclosed volume contains a degeneracy point, where ˜
g kk has a singularity). Therefore,
the last integral in Eq. (5.79) only depends on the closed path C, and not on the
peculiar choice of the surface S C . We have
V k = ∇ × ˜
g kk
(5.80)
= ∇ ˜
ϕ k |×| ∇ ˜
ϕ k
(5.81)
=
m =k
∇ ˜
ϕ k | ˜
ϕ m × ˜
ϕ m |∇ ˜
ϕ k
(5.82)
=
m =k
ϕ k
∇ ˆ
H el
ϕ m
×
ϕ m
∇ ˆ
H el
ϕ k
(U k − U m ) 2
(5.83)
where we used the relation ∇ × ( f ∇g) = ∇ f × ∇g and Eq. (2.70) for the nonadiabatic couplings. The sum is extended to a complete basis set of adiabatic wavefunctions; the term with m = k is excluded as ∇ ˜
ϕ k | ˜
ϕ k is purely imaginary and the
vector product with its complex conjugate vanishes. It is particularly evident from
Eq. (5.82) that V k is independent on the phases of the electronic functions. It is then
not necessary to use the single-valued ˜
ϕ m . Moreover, V k is imaginary: considering a
real Hamiltonian, we can choose ϕ m as real so that V k = 0 and Ω
(C)
k
vanishes. This