Redfield tensor

The Redfield tensor S is defined in terms of stationary correlation functions of the system-bath coupling operator, V, evolving under the bath Hamiltonian, Thus the dynamics of the bath are retained in Eq. (9), the only assumptions being that the bath is in thermal equilibrium and that its dynamics are independent of the state of the system beyond some correlation time, t, short compared to the rate of change of cr. The tensor element R,, / can be written [26, 42]... 

An important property of the Redfield tensor is that when the bath is treated quantum mechanically, the physically required detailed-balance conditions are naturally satisfied, so that the subsystem relaxes to thermal... 

To evaluate the correlation functions in Eqs. (12) and (13), it is usual to complete the separation of the system and bath by decomposing the system-bath coupling into a sum of products of pure system and bath operators. This allows the correlation functions of the system-bath coupling to be replaced, without loss of generality, by correlation functions of bath operators alone, evolving under the uncoupled bath Hamiltonian. Moreover, as we have previously pointed out [39,40], this decomposition of the system-bath coupling make it possible to write the Redfield equation in a highly compact form, without explicit reference to the Redfield tensor at all. 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

We recently demonstrated that when the system-bath coupling is written in the sum-of-products form [Eq. (16)], a substantial amount of structure is imparted to the Redfield tensor, 3 , allowing the product Roto be rewritten without reference to the individual tensor elements at all [39]. Instead, is replaced by commutators of a and the system operators defined in Eqs. (16) and (18). The result is... 

This result demonstrates that the Redfield tensor can be applied to any operator without the explicit construction of the full tensor. The G and G are ordinary operators, so numerical evaluation of the right side of Eq. (20) in an W-dimensional basis involves only the storage and multiplication oi N y N matrices. The full Redfield tensor, in contrast, is an X operator in Liouville space, whose application to an x density matrix requires 0 N ) scalar multiplications, as opposed to the 0 N ) operations required to multiply NxN matrices. The ability to apply the Redfield tensor to the density matrix using Eqs. (20) or (23) therefore allows a significant savings in both computer time and memory, particularly as N becomes large. 

There are three important issues to consider in the numerical solution of the Redfield equation. The first is the evaluation of the Redfield tensor matrix elements I ,To obtain these matrix elements, it is necessary to have a representation of the system-bath coupling operator and of the bath Hamiltonian. Two fundamental types of models are used. First, the system-bath coupling can be described using stochastic fluctuation operators, without reference to a microscopic model. In this case, the correlation functions appearing in the formulas for parame-... 

One issue that arises when classical correlation functions are used is that they do not satisfy the detailed-balance relation Eq. (14) (because they are even with respect to i = 0) and hence cannot produce a Redfield tensor that lets the subsystem come to thermal equilibrium. Therefore, before being introduced into Eq. (18), the classical results must be modified to satisfy detailed balance. Unfortunately, there is no unique way to accomplish this, and a handful of different approaches are found in the literature. 

This relation is, itself, sufficient to satisfy the detailed-balance-condition and define a Redfield tensor that allows the system to come to thermal equilibrium. It has thus been used directly by some as an ad hoc correction for classical or stochastic correlation functions [38]. 

The strength of the bath coupling to each system variable is described by the coupling constants / and, because they enter at second order, the rate constant for the dissipation process arising from each term in Eq. (38) will be proportional to f I- The only important properties of the F t) are their autocorrelation and cross-correlation functions, (FJfi)F t)) and F (0)Fi,(t)), which enter the definition of the Redfield tensor in Eq. (18). These, like the classical correlation functions discussed earlier, do not satisfy the detailed-balance relation and must be corrected in the same way. It is convenient, but not necessary, that the variables be chosen to be independent, so that the cross-correlation functions vanish. 

The system-bath coupling operator [Eq. (108)] in the final effective-bath Hamiltonian [Eq. (101)] has five terms with five distinct bath operators each is either a sum of the bath positions, of the bath momenta, or an exponentiated sum of the bath momenta. For reference, we summarize here the autocorrelation and crosscorrelation functions of these three classes of operator from these, the various functions needed to define the Redfield tensor [Eq. (18)] are easily obtained. The Heisenberg operators here all evolve under the bath Hamiltonian,... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

This Hamiltonian must be put in the form of equation (Al) (See Appendix A ) for the Redfield theory to be applicable, and depending on the origin, different treatments of the perturbation is necessary. How the direct product is handled is determined by the correlation between the different parts. For simple liquids, the dipole-dipole tensor fluctuates on the picosecond to nanosecond time scale and it is thus not correlated with the nuclear spins. 

Under certain assumptions in equation (A3), the time dependence in the lattice interaction tensors is sufficient to describe the relaxation and derive expressions for the relaxation times. This is the basis of Redfield theory, in which first the Master equation is expanded... 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

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