DENSITY MATRIX EQUATION FOR A BATHED SMALL SYSTEM AND ITS APPLICATION TO MOLECULAR MAGNETS - Advances in Chemical Physics - page 256

one obtains

√ 2 C α |

ψ m

1

α C

|

χ α =

− |

ψ

− S − α

θ
2 =−

θ
2

α
√ 2

=− ( C | ψ + αC α | ψ m ) α | χ (213)

− α − S − α

Thus in Eq. (211) the scalar products are

θ
2

θ
2

χ α χ β =−

α χ β =−

α χ αδ (214)

− − α,β

and

χ α χ γ ρ γβ + χ γ χ β =− αρ

θ
2

βρ α, − β

ρ αγ + (215)

− α,β

γ

The density operator in the initial state typically is

ρ (0) = | ψ ψ | (216)

− S − S

so that the density matrix in the diagonal basis is given by

1
2 C α C β

ρ αβ (0) = α | ρ (0) | β = α | ψ ψ | β = (217)

− S − S

where Eq. (209) with m =− S was used. In particular,

1
2 (1

1
2 (1

ρ (0) = + cos θ ) , ρ (0) = − cos θ )

++ −−

1
2 sin θ

ρ (0) = ρ (0) = (218)

+− −+

2. Ground-Ground-State Resonance

The results obtained above already allow us to consider the dynamics at the ground-
state resonance, m =

S. In this case, the relaxation terms in Eq. (211) contain only

( ω 0 ) ω 0 , so that the secular approximation is applicable. Dropping nonsecular

terms in Eq. (211) one obtains

θ
2 +

d
dt ρ

++ =−

( ρ

+− +

ρ ) R ρ

++ +

R ρ

−+ ++ , ++ ++ , −− −−

θ
2 −

d
dt ρ

+− =−

( ρ

−− −

ρ ) iω 0 ρ

+− +

R ρ (219)

++ +− , +− +−