DENSITY MATRIX EQUATION FOR A BATHED SMALL SYSTEM AND ITS APPLICATION TO MOLECULAR MAGNETS - Advances in Chemical Physics - page 250

where the matrix elements M ( k ) are defined by Eq. (164). Similarly to Eq. (165),
for one of the parts of R (2)
αβ,α β

one obtains

e − E / ( k B T ) δ ε α − E V (2)

π

V (2)
γ ,α

ε γ + E −

α,γ

Z b

e − E ν k λ ,ν q λ ,... / ( k B T ) δ ε α − ω k

π

= ε γ + ω q −

Z b

ν k λ ,ν q λ ...

k λ, q λ

(2)

M ξξ ( k , q ) (2) M ζζ ( k , q )

×

αγ,ξξ γα ,ζζ

δ ε α − ω k

π

= ε γ + ω q −

k λ, q λ

M − ,ξ ( k ) M + ,ξ ( q )

M − ,ξ ( k ) M + ,ξ ( q )

1
2

(2)
αγ,ξξ

× +

2 M

,ζ ( q )

1

(2)
γα ,ζζ

×

,ζ ( k ) M ,ζ ( q )

+ M

,ζ ( k ) M

− + − +

δ ε α − ω k

π

= ε γ + ω q −

k λ, q λ

(2)
αγ,ξξ

(2)

¯

M − ,ξ ( k ) M + ,ξ ( q ) ¯ × M

,ζ ( k ) M

,ζ ( q )

(185)

−

αγ,ξξ

+

where

(2)

αγ,ξ ξ

1
2

(2)
αγ,ξξ =

(2)

¯

αγ,ξξ +

(186)

and M are averaged over the phonon populations, ν k λ → n k λ , see Eq. (166).

Substituting the expressions for M from Eq. (164), one obtains

2

δ ε α − ω k n k n q + 1 ¯

π

8 MN

(2)
αγ,ξξ

(2)
γα ,ζζ

¯

ε γ + ω q −

k λ, q λ

q

e q λ ξ q e q λ ζ × ×

× ×

[ k e k λ ] ξ [ k e k λ ] ζ

× (187)

ω k ω q