Essentials of Computational Chemistry

540 15 ADIABATIC REACTION DYNAMICS
E
U 1
Figure 15.7 Near approach (or avoided crossing) of two electronic states as a function of nuclear
coordinate Q. The inset expands the region of the avoided crossing to facilitate the definition of
quantities appearing in the Landau–Zener surface-hopping-probability model
by the adiabatic state corresponding to that c and integrate. After taking advantage of the
orthonormality of the adiabatic states, and noting that each adiabatic electronic wave function
is an eigenfunction of the electronic Hamiltonian with an associated energy eigenvalue, we
derive
nuclei
k
− 1
⎡
2mk
⎣∇ 2 k +
Q
V 12
2
(2〈ψi|∇k|ψj〉·∇k +〈ψi|∇ 2 k |ψj〉)
⎤
⎦ ci = (Efull − Ei)ci
j=1
U 2
ψ 2
ψ 1
(15.45)
where Ei is the energy eigenvalue for the ith electronic state ψi and the vector operator ∇
is defined as
∂
∇k = ,
∂xk
∂
,
∂yk
∂
∂zk
(15.46)
Note that Eq. (15.45) is itself a Schrödinger equation for nuclear eigenfunction ci. The
Born–Oppenheimer approximation, previously discussed in Section 4.2.3, involves assuming
a value of zero for all of the integrals in Eq. (15.45) involving the nuclear ∇ or ∇ 2 operator
acting on electronic wave functions (cf. Eq. (9.37)). Under that assumption, the nuclear
and electronic wave functions are separable, but spontaneous changes in electronic states,
i.e., surface-to-surface crossings, are not permitted. Any model addressing such state–state
interconversions must instead start from Eq. (15.45) (possibly generalized to a larger number
of electronic states).
Unfortunately, Eq. (15.45) does not admit to simple analytic solutions under realistic sets
of chemical conditions. Moreover, if we now try to extend Eq. (15.43) to its time-dependent