Transition reduced density matrix

Density matrices of the state functions provide a compact graphical representation of important microscopic features for second order nonlinear optical processes. The transition moment y is expressed in terms of the transition density matrix p jji(r,r ) by nn " /j, ptr Pjj t(r,r )dr and the dipole moment difference Ay by the difference density function p - p between the excited and ground state functions = -e / r( p -p )dr where p is the first order reduced density matrix. 

The formulas above give the gradient and the Hessian in terms of matrix elements of the excitation operators. They can be evaluated in terms of one-and two-electron integrals, and first and second order reduced density matrices, by inserting the Hamiltonian (3 24) into equations (4 9), (4 11), and (4 13)-(4 15). Note that transition density matrices and are needed for the evaluation of the Cl coupling matrix (4 15). 

Many apparent anomalies in the spectra produced by double resonance experiments can be resolved by a density matrix treatment. Thus in A- X experiments with a reduced amplitude of the irradiated field the normal effect of the NOE upon the intensities of the transitions may be severely modified and both emission and absorption may be observed. (58) It turns out that the overall behaviour depends upon... 

The oscillating factors exp [—i ( — ) t] are characteristic for the "unitary-type" dynamics caused by the commutator part —i [H, p] of the master Eq. (24) for the reduced (or relevant) density matrix p. These factors have the absolute value 1 and do not affect the numerical value of the transition rate. 

When V 0 transitions between L and R can take place, and their populations evolve in time. Defining the total L and R populations by our goal is to characterize the kinetics of the L R process. This is a reduced description because we are not interested in the dynamics of individual level /) and r), only in the overall dynamics associated with transitions between the L and R species. Note that reduction can be done on different levels, and the present focus is on Pl and Pr and the transitions between them. This reduction is not done by limiting attention to a small physical subsystem, but by focusing on a subset of density-matrix elements or, rather, their combinations. 

A comprehensive discussion of this important derivation can be found in Refs. 38 and 39. The nuclear contributions to the TM leading to vibrational fine structure in the spectra are discussed in the following section. Alternatively, the electronic part of the TM is obtained from the first-order reduced transition density matrix y as... 

These quantities are much simpler to calculate, and fewer in number, than those of the type (8.6.12). The terms that depend on the /i-parameters are calculated exactly as in (8.6.4), the only difference being that the ket (or the bra) contains 4> instead of V when such matrix elements are reduced, in terms of transition density matrices, the density-matrix elements in (8.2.3) are simply replaced by partial sums such as... 

It is seen from (60)-(61) that there are two alternative ways to calculate the density variation i) through the transition density and matrix elements of Qfc-operator and ii) through the ground state density. The second way is the most simple. It becomes possible because, in atomic clusters, Vres has no T-odd Ffc-operators and thus the commutator of Qk with the full Hamiltonian is reduced to the commutator with the kinetic energy term only ... 

The case of responses is more involved in the sense that matrix elements of the second operator in the commutator are transition densities which are generally complex. However, the first operator in the commutator still has real (for T-even A) or image (for T-odd A) matrix elements and so the averages can be finally reduced to... 

The three terms in brackets represent contributions from ED, MD and EQ transitions, respectively, and the matrix elements in terms of reduced matrix elements, 3-j and 6-j symbols are given elsewhere for MD [65, 112] and EQ [113, 114] transitions. The dimensionless factors 77, rf, 77" [115] in Eq. (19a) correct the vacuum linestrengths to those appropriate for the crystal, by correcting the polarizability of the medium and/or the density of radiative modes. The ZPL (i.e. electronic origin) of a transition refers to the 0-0 transition, i.e. from the v=0 vibrational level of one state to the v =0 level of the other in the case of absorption. 

---