1549380323-Statistical Mechanics Theory and Molecular Simulation

Frequency spectra 545

Writing theδ-function as an integral, eqn. (14.3.3) becomes

R(ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dt

∑

i,f

wiei(Ef−Ei− ̄hω)t/ ̄h

∣

∣

∣〈Ef|Vˆ|Ei〉

∣

∣

∣

2

=

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

∑

i,f

wi〈Ei|Vˆ|Ef〉〈Ef|Vˆ|Ei〉eiEft/ ̄he−iEit/ ̄h

=

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

∑

i,f

wi〈Ei|Vˆ|Ef〉〈Ef|ei

Hˆ 0 t/ ̄hˆ

Ve−i

Hˆ 0 t/ ̄h

|Ei〉. (14.3.4)

In the last line, we have used the fact that|Ei〉and|Ef〉are eigenstates ofHˆ 0 to
bring the two exponential factors into the angle brackets as the unperturbed propa-
gator exp(−iHˆ 0 t/ ̄h) and its conjugate exp(iHˆ 0 t/ ̄h). Note, however, that the operator

exp(iHˆ 0 t/ ̄h)Vˆexp(−iHˆ 0 t/ ̄h) =Vˆ(t) is just the representation of the operatorVˆin
the interaction picture (see eqn. (14.2.3)). Thus, the average transition rate can be
expressed as

R(ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

∑

i,f

wi〈Ei|Vˆ(0)|Ef〉〈Ef|Vˆ(t)|Ei〉, (14.3.5)

where theVˆ(0) is the operator in the interaction picture att= 0. Thus, both operators
in eqn. (14.3.5) are represented within the same quantum-mechanical picture. Note
that the sum over final states can now be performed using the completeness relation
∑

f

|Ef〉〈Ef|=Iˆ (14.3.6)

of the eigenstates ofHˆ 0. Eqn. (14.3.5) thus becomes

R(ω) =

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

∑

i

wi〈Ei|ˆV(0)Vˆ(t)|Ei〉

=

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt

1

Q(N,V,T)

Tr

(

e−β

Hˆ (^0) ˆ
V(0)ˆV(t)

)

=

1

̄h^2

|F(ω)|^2

∫∞

−∞

dte−iωt〈Vˆ(0)Vˆ(t)〉. (14.3.7)

The last line shows that the ensemble-averaged transition rate at frequencyωis just
the Fourier transform of the quantum time correlation function〈ˆV(0)Vˆ(t)〉(Berne,
1971).
In general, a quantum time correlation function of two operatorsAˆandBˆwith
respect to an unperturbed HamiltonianHˆ 0 is given by

CAB(t) =

Tr

[

Aˆ(0)Bˆ(t)e−βHˆ^0

]

Tr

[

e−βHˆ^0

]. (14.3.8)