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Molecular Nanomagnets$

Dante Gatteschi, Roberta Sessoli, and Jacques Villain

Print publication date: 2006

Print ISBN-13: 9780198567530

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567530.001.0001

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Molecular Nanomagnets
Oxford University Press

In this appendix, it is shown that all eigenvalues of the ‘master matrix’

Θ j i = γ j i δ j i i γ i
are real and negative, except that which corresponds to equilibrium and is equal to 0. The indices i, j denote the eigenstates of the (time+independent) Hamiltonian and the Θ matrix appears in equation (5.12) which can be written as
d d t p ( t ) = Θ p ( t ) .
The transition probabilities γji are real and positive and satisfy the principle of detailed balance
γ j i = γ i j exp [ β ( ε j ε i ) ] .

It is appropriate to introduce the real, symmetric, and therefore Hermitian matrix

Γ ij = γ ij exp [ β ( ε j ε i ) / 2 ] = Γ ij .
Let v i be an eigenvectorof Θ with the eigenvalue λ:
j i γ j i v j v i i γ i = λ v i .
Substituting (D.4) one obtains
j i Γ j i u j α i u i = λ u i
u i = v i exp [ βε i / 2 ]
α i = i γ i = i γ i exp [ β ( ε i ε ) / 2 ] .

Thus, any eigenvalue λ of Θ is also an eigenvalue of the matrix Γji − δji αi, which is real, symmetric, and therefore Hermitian. This implies that λ is real. (p.330) To show that these eigenvalues are non+positive, it is sufficient to show that the quadratic form

Φ = Σ ij Γ ji x i x j Σ i α i x i 2
cannot be positive. Indeed, it is easily shown that (D.9) reads
Φ = ( 1 / 2 ) Σ ij Γ ji { x i exp [ β ( ε i ε j ) / 4 ] x j exp [ ( ε j ε i ) / 4 ] } . 2
The only non-vanishing contributions are those of coefficients Γij with ij which are positive equal to zero. Therefore, expression (D.10) is negative except for xi = exp(βε i/2), which corresponds to thermal equilibrium.

The desired property is thus proven.