DENSITY MATRIX EQUATION FOR A BATHED SMALL SYSTEM AND ITS APPLICATION TO MOLECULAR MAGNETS - Advances in Chemical Physics - page 251

Now, using Eqs. (168)-(170), one can simplify this result to

δ ω α γ + ω k n k n q + 1

π
(8 MN ) 2

ω q −

k λ, q λ

2
3

2 k 2
ω k

q 2
ω q

(2)
αγ,ξξ

(2)
γα ,ζζ

¯ ¯

× δ ξζ δ ξ ζ

γα ,ξξ
kq

k 2
ω k

q 2
ω q

δ ω α γ + ω k n k n q + 1

(188)

π
(12 MN ) 2

(2)
αγ,ξξ

(2)

¯ ¯

= ω q −

Recalling Eq. (53) one obtains

(12 MN ) 2
kq

πD 2 k 2

ω k

q 2
ω q

n k n q +

1

R (2)
αβ,α β

=

αα ,γγ δ ω α γ +

ω k δ β β Q (2)

× − ω q −

γ

δ αα
γ

β β,γγ δ ω β γ +

ω k Q (2) − ω q −

ω k (189)

αβ,α β δ ω ββ +

ω k + δ ω αα + Q (2) + ω q − ω q −

where

D 2
ξξ

1

Q (2) (2)

αα ,ξξ

(2)
β β,ξξ

¯ ¯

αβ,α β ≡

(190)

Which is similar to Eq. (173). Since Raman processes can become important only
at high temperatures, one can drop spin transition frequencies in the energy δ -
functions. Next, one can replace summation by integration with the help of Eq.
(174) and introduce the characteristic Raman rate

D 2
24 2 π 3

(2) dω k ω k n k ( n k +

= 1) (191)

2 t

Then R (2)

can be written in the form

αβ,α β

(2) δ αα

γ

R (2) Q (2) Q (2) 2 Q (2)

αβ,α β αβ,α β =

−

αα ,γγ δ β β − β β,γγ +

(192)

γ