Trace operators

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

Before deriving equations that determine the RDMCs, we ought to clarify precisely which are the RDMCs of interest. It is clear, from Eqs. (25a) and (25b), that Ai and A2 contain the same information as D2 and can therefore be used to calculate expectation values (IT), where W is any symmetric two-electron operator of the form given in Eq. (1). Whereas the 2-RDM contains all of the information available from the 1-RDM, and affords the value of (IT) with no additional information, the 2-RDMC in general does not determine the 1-RDM [43, 65], so both Ai and A2 must be determined independently in order to calculate (IT). More generally, Ai,...,A are all independent quantities, whereas the RDMs Dj,..., D are related by the partial trace operation. The u-RDM determines all of the lower-order RDMs and lower-order RDMCs, but... 

Here, DF represents the matrix product of the density matrix Dj j and the matrix representation Fij = <( )ilF l(f>j> of the F operator, both taken in the (f>j basis, and Tr represents the matrix trace operation. 

The quantum version of the partition function is obtained by replacing the phase space integral and the classical Boltzmann distribution with the trace operation of the quantum Boltzmann operator, giving the usual expression... 

Aeeording to Eq. (10), (x 0(Xc,Pc) x") is aphase spaee path integral representation for the operator 27t/iexp —pA, where all the paths run from x to x", but their eentroids are eonstrained to the values of Xc and po. Integration over the diagonal element, whieh corresponds to the trace operation, leads to the usual definition of the phase space centroid density multiplied by 2nH. In this review and in Refs. 9,10 this multiplicative factor is included in the definition of the centroid distribution function, pc (xc, pc). Equation (6) thus becomes equivalent to... 

For an arbitrary canonical density operator, the phase space centroid distribution fimction is imiquely defined. However, this function does not directly contain any dynamical information from the quantum ensemble because such information has been lost in the course of the trace operation. The lost information may be recovered by associating to each value of the centroid distribution function the following normalized operator ... 

At last, the closeness relation may be equated to unity, and then the trace operation over the basis ( )) may be made implicit to give... 

Even though we do not invoke the full machinery of tensor analysis (Butkov, 1968), it is useful to keep the distinction between contravariant and covariant components clear. To avoid conflicting notation we do not use upper and lower indices to denote contravariant and covariant indices. Instead, we will use the suffix ao (lower case letters) on tensors whose indices are all contravariant, and AO (capital letters) on tensors whose indices are all covariant. No special suffix is used in the MO basis. For example, using the two- and four-index trace operators the energy is... 

In terms of two- and four-index trace operations the expectation value may be written more compactly as... 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

This follows from the fact that this relation has to hold for the diagonal representation and from the fact that the Trace operation does not depend on the representation used. 

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