so that the Hamiltonian matrix H0 is

diagonal on that basis set (see Chapter 14). We write the solution of the

time-dependent equation

‚ iˆ ˆ

Ψ(r, t) = ’ (H 0 + H 1 (t))Ψ (5.22)

‚t

as a linear combination of time-independent basis functions:

Ψ(r, t) = cn (t)φn (r). (5.23)

n

Note that cn (t) contains an oscillating term exp(’iωn t), where ωn = En / .

120 Dynamics of mixed quantum/classical systems

The time-dependent Schr¨dinger equation now implies

o

‚ i ˆ

cm φm = ’

cm φm = ™ cm Hφm , (5.24)

‚t m m

which, after left-multiplying with φn and integrating, results in

i i

ˆ

cn = ’ cm φn |H|φm = ’ (Hc)n ,

™ (5.25)

m

or in matrix notation:

i

c = ’ Hc.

™ (5.26)

Since H is diagonal, there are two terms in the time-dependence of cn :

i

cn = ’iωn cn ’ (H1 c)n .

™ (5.27)

The ¬rst term simply gives the oscillating behavior exp(’iωn t) of the un-

perturbed wave function; the second term describes the mixing-in of other

states due to the time-dependent perturbation.

Often a description in terms of the density matrix ρ is more convenient, as

it allows ensemble-averaging without loss of any information on the quantum

behavior of the system (see Chapter 14). The density matrix is de¬ned by

ρnm = cn c— (5.28)

m

and its equation of motion on a stationary basis set is the Liouville-Von

Neumann equation:

i i

[ρ, H0 + H1 (t)]

ρ=

™ [ρ, H] = (5.29)

The diagonal terms of ρ, which do not oscillate in time, indicate how much

each state contributes to the total wave function probability Ψ— Ψ and can

be interpreted as a population of a certain state; the o¬-diagonal terms

ρnm , which oscillate because they contain a term exp(’i(ωn ’ ωm )t), reveal

a coherence in the phase behavior resulting from a speci¬c history (e.g., a

recent excitation). If averaged over an equilibrium ensemble, the o¬-diagonal

elements cancel because their phases are randomly distributed. See also

Section 14.8.

In Section 5.2.4 of this chapter it will be shown for a two-level system

how a randomly ¬‚uctuating perturbation will cause relaxation of the density

matrix.

5.2 Quantum dynamics in a non-stationary potential 121

5.2.3 Time-dependent basis set

We now consider the case that the basis functions are time-dependent them-

selves through the parametric dependence on nuclear coordinates R: φn =

φn (r; R(t)). They are eigenfunctions of the time-independent Schr¨dinger

o

equation, in which the time dependence of R is neglected. The total wave

function is expanded in this basis set:

Ψ= cn φn . (5.30)

n

Inserting this into the time-dependent Schr¨dinger equation (5.22) we ¬nd

o

(see also Section 14.7 and (14.54))

i

™ ˆ

cm φm + R · cm ∇R φm = ’

™ cm Hφm . (5.31)

m m m

After left-multiplying with φ— and integrating, the equation of motion for

n

the coe¬cient cn is obtained:

i

™

cn + R · cm φn |∇R |φm = ’ (Hc)n ,

™ (5.32)

m

which in matrix notation reads

i ™

c = ’ (H + R · D)c.

™ (5.33)

Here D is the matrix representation of the non-adiabatic coupling vector

ˆ ˆ

operator D = ’i ∇R :

φn |∇R |φm .

Dnm = (5.34)

i

Note that D is purely imaginary. D is a vector in the multidimensional

space of the relevant nuclear coordinates R, and each vector element is an

operator (or matrix) in the Hilbert space of the basis functions.

In terms of the density matrix, the equation of motion for ρ for non-

stationary basis functions then is

i ™

[ρ, H + R · D].

ρ=

™ (5.35)

The shape of this equation is the same as in the case of time-independent

basis functions (5.29): the Hamiltonian is modi¬ed by a (small) term, which

can often be treated as a perturbation. The di¬erence is that in (5.29) the

perturbation comes from a, usually real, term in the potential energy, while

in (5.35) the perturbation is imaginary and proportional to the velocity

of the sources of the potential energy. Both perturbations may be present

122 Dynamics of mixed quantum/classical systems

simultaneously. We note that in the literature (see, e.g., Tully, 1990) the real

and antisymmetric (but non-Hermitian) matrix element (which is a vector

in R-space)

i

def

dnm = φn |∇R |φm = (5.36)

D nm

is often called the non-adiabatic coupling vector ; its use leads to the following

equation of motion for the density matrix:

i ™

™

ρ= [ρ, H] + R[ρ, d]. (5.37)

In the proof of (5.35), given below, we make use of the fact that D is a

Hermitian matrix (or operator):

D† = D. (5.38)

ˆ

This follows from the fact that D cannot change the value of φn |φm , which

equals δnm and is thus independent of R: