5.3 Landau–Zener Rule
151
which is the celebrated Landau–Zener formula for the adiabatic transition probability.
Examples of application of the Landau–Zener formula are given as problems at the
end of this chapter.
5.4 Conical Intersections
We get back to the topic of Sect. 5.1, considering now the general case of a polyatomic
molecule (s > 1). The nonintersection rule is no longer valid with more than one
internal coordinate. For example, with two coordinates Q 1 and Q 2 the two-equations
system (5.4) in principle can be solved, so we may have one (or several) points where
U 1 = U 2 . With three coordinates we may have U 2 = U 1 on a curve. In general, with
s internal coordinates the set of points where U 1 = U 2 , if any, has dimension s − 2.
Let us consider the Taylor expansion of the diabatic quantities ΔH and H 12 at a
degeneracy point Q x (for later convenience it is useful to consider 2H 12 )
ΔH (Q) = q · (Q − Q x ) + · · ·
2H 12 (Q) = h · (Q − Q x ) + · · ·
(5.35)
where
q = ∇ΔH (Q x )
and
h = 2∇ H 12 (Q x ).
(5.36)
As in the previous sections we assume here H 12 ∈ R. We define the two versors
ˆ
x = q/q and ˆ
y = h/ h, where q and h are the norms of q and h, respectively. Then
Eq. (5.35) becomes, at first order
ΔH (Q) = qx
and
2H 12 (Q) = hy
(5.37)
where x = ˆ
x · (Q − Q x ) and y = ˆ
y · (Q − Q x ) are the displacements from the
degeneracy point along the two versors. We can always assume, without losing
generality, that the two vectors q and h are orthogonal. In fact, the diabatic basis
is not uniquely defined: in particular, it is determined up to a constant orthogonal
transformation, which can be chosen in such a way that q · h = 0.
The Hamiltonian matrix at the first order in the displacement from Q x is
H el =
s
2
· (Q − Q x )
1 0
0 1
+
1
2
−qx hy
hy qx
(5.38)
where s = ∇ H 11 (Q x ) + ∇ H 22 (Q x ) and we have set the energy scale so that H 11
(Q x ) = H 22 (Q x ) = 0. The corresponding adiabatic energies are then (see Eq. (5.3))
U 2,1 =
1
2
s · (Q − Q x ) ±
(qx) 2 + (hy) 2
.
(5.39)
151
which is the celebrated Landau–Zener formula for the adiabatic transition probability.
Examples of application of the Landau–Zener formula are given as problems at the
end of this chapter.
5.4 Conical Intersections
We get back to the topic of Sect. 5.1, considering now the general case of a polyatomic
molecule (s > 1). The nonintersection rule is no longer valid with more than one
internal coordinate. For example, with two coordinates Q 1 and Q 2 the two-equations
system (5.4) in principle can be solved, so we may have one (or several) points where
U 1 = U 2 . With three coordinates we may have U 2 = U 1 on a curve. In general, with
s internal coordinates the set of points where U 1 = U 2 , if any, has dimension s − 2.
Let us consider the Taylor expansion of the diabatic quantities ΔH and H 12 at a
degeneracy point Q x (for later convenience it is useful to consider 2H 12 )
ΔH (Q) = q · (Q − Q x ) + · · ·
2H 12 (Q) = h · (Q − Q x ) + · · ·
(5.35)
where
q = ∇ΔH (Q x )
and
h = 2∇ H 12 (Q x ).
(5.36)
As in the previous sections we assume here H 12 ∈ R. We define the two versors
ˆ
x = q/q and ˆ
y = h/ h, where q and h are the norms of q and h, respectively. Then
Eq. (5.35) becomes, at first order
ΔH (Q) = qx
and
2H 12 (Q) = hy
(5.37)
where x = ˆ
x · (Q − Q x ) and y = ˆ
y · (Q − Q x ) are the displacements from the
degeneracy point along the two versors. We can always assume, without losing
generality, that the two vectors q and h are orthogonal. In fact, the diabatic basis
is not uniquely defined: in particular, it is determined up to a constant orthogonal
transformation, which can be chosen in such a way that q · h = 0.
The Hamiltonian matrix at the first order in the displacement from Q x is
H el =
s
2
· (Q − Q x )
1 0
0 1
+
1
2
−qx hy
hy qx
(5.38)
where s = ∇ H 11 (Q x ) + ∇ H 22 (Q x ) and we have set the energy scale so that H 11
(Q x ) = H 22 (Q x ) = 0. The corresponding adiabatic energies are then (see Eq. (5.3))
U 2,1 =
1
2
s · (Q − Q x ) ±
(qx) 2 + (hy) 2
.
(5.39)