1549380323-Statistical Mechanics Theory and Molecular Simulation

Quantum linear response theory 555

By decomposing the susceptibility into its real and imaginary parts, we can relate it
directly to the energy spectrumQ(ω). In the limitǫ→ 0 +, we obtain

χVV(ω) = lim

ǫ→ 0 +

∫∞

0

dτe−ǫτΦVV(τ)e−iωτ

= lim

ǫ→ 0 +

∫∞

0

dτe−ǫτΦVV(τ) [cosωτ−isinωτ]

≡Re [χVV(ω)]−iIm [χVV(ω)], (14.5.18)

where

Re [χVV(ω)] = lim

ǫ→ 0 +

∫∞

0

dτe−ǫτΦVV(τ) cosωτ

Im [χVV(ω)] = lim

ǫ→ 0 +

∫∞

0

dτe−ǫτΦVV(τ) sinωτ. (14.5.19)

An important property of the susceptibilityχ(z) is its analyticity in the complexz-
plane. For any analytic function, the real and imaginary parts are not independent
but satisfy a set of relations known as theKramers–Kr ̈onig relations. Letχ′VV(ω) and
χ′′VV(ω) denote the real and imaginary parts ofχVV(ω), respectively, so thatχVV(ω) =
χ′VV(ω) +iχ′′VV(ω). The real and imaginary parts are related by

χ′VV(ω) =

1

π

P

∫∞

−∞

d ̃ω

ωχ ̃ ′′VV( ̃ω)

ω ̃^2 −ω^2

χ′′VV(ω) =−

ω

π

P

∫∞

−∞

d ̃ω

χ′VV( ̃ω)

̃ω^2 −ω^2

. (14.5.20)

Here, P indicates that the principal value of the integral is to be taken. The Kramers–
Kr ̈onig relations can be expressed equivalently as

χ′VV(ω) =

1

π

P

∫∞

−∞

d ̃ω

χ′′VV( ̃ω)

ω ̃−ω

χ′′VV(ω) =−

1

π

P

∫∞

−∞

d ̃ω

χ′VV( ̃ω)

ω ̃−ω

, (14.5.21)

which are known asHilberttransforms. We alluded to the use of these relations in
Section 14.4 where we presented infrared spectra for water and aqueous solutions.
We will now show that the frequency spectrum of eqn. (14.3.22) canbe related
to the imaginary partχ′′VV(ω) of the susceptibility. The spectrum of eqn. (14.3.22)

is given in terms of the anticommutator ofVˆ(0) andVˆ(t), while the susceptibility is
derived from the after-effect function, which involves a commutator betweenVˆ(0) and
Vˆ(t). Recall, however, that the frequency spectrum is defined as

Q(ω) = ̄hω[R(ω)−R(−ω)]. (14.5.22)