Perturbation theory relaxed density matrix

Spin-spin relaxation is handled by second order perturbation theory [ Abr 1 ] of the density matrix equation of motion (2.2.62) in the rotating frame (RCF) [Abrl],... 

The relationship between T and and the J s can be derived through time dependent perturbation theory as was done originally by Bloembergen, Purcell, and Pound or by the use of the density matrix which turns out to be equivalent. See the general references in Appendix A, especially Chapter 5 of Slichter. For our case of two spins on a rotating molecule, the relaxation rates are given by... 

The BPP expression was derived under the assumptions that the dipolar interactions formed a perturbation on the Zeeman levels and that the time dependent part of the dipolar interaction could be treated by time dependent perturbation theory or equivalently the density matrix approach to determine the relaxation expression. This does not rule out a BPP type expression for relaxation in the rotating frame. Specifically, a BPP type approach can be used to derive the following expression for T due to rotational motion under basically... 

Contributions from terms like Cs pyare smaU and ignored in the summation. Starting from the von Neumann-Liouville equation, a reduced density matrix for the system mode is derived using the time-dependent perturbation theory after tracing over the bath degrees of freedom. The commonly employed Markov approximation is avoided and no assumption of a separation in timescales between system and bath mode relaxation is invoked in this theory [84]. The ultimate result for the evolution of the ground state vibrational population is... 

In this subsection, we employ standard time-dependent perturbation theory with respect to the laser-matter interaction i/mt(i) in the wave-function formalism. Time-dependent perturbation theory may alternatively be formulated in the density-matrix formalism. The latter is more general in that it allows for the inclusion of finite temperature effects and a phenomenological description of relaxation phenomena. In this chapter, we are concerned with the description of the dynamics at conical intersections in a fully microscopic manner, and temperature effects play a minor role. The wave-function formalism is therefore appropriate for our purposes. Writing the time-dependent wave function as... 

On less formal grounds a Markovian type of equation was also obtained by Redfield [167]. Redheld used perturbation theory up to second order and derived a Master equation for the evolution of the density matrix Oaa y where a, a denote quantum states of the molecule with energy defined by ha. The simple equation for the relaxation of the density matrix was obtained by neglecting the ensemble-averaged first-order terms. Thus we have [167]... 

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