Density matrix nondiagonal

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

Transitions between superbarrier states (a > N, see Fig. 4.4) involve nondiagonal in cr elements of the density matrix which give, at AU > T, a small contribution to the spectral function for frequencies of the order a>h + 0cr - 0spectral function at frequencies near co, we can neglect this contribution. 

A measure for the electronic coherence of the wave function are the nondiagonal elements of the electronic density matrix, which, for example, in the diabatic representation are given by (k k )... 

To bypass these problems, we initially introduce broadening in energy, replacing the diagonal density matrix [Eq. (3.85)] by the nondiagonal... 

At high temperatures when the number of populated vibronic states is large enough, one can neglect the nondiagonality of the density matrix (13) on nuclear coordinates. This results in... 

Rydberg orbitals. The contribution to the total density matrix from unoccupied orbitals is generally small however, antibonds play an important role since they represent unused valence-shell capacity. The energy of a molecule can be decomposed into two components associated with covalent and noncovalent structures the former corresponds to the ideal Lewis contributions, and the latter is associated with non-Lewis contributions. The nondiagonal elements of the Fock matrix in the NBO basis are interpreted as the stabilizing interaction between occupied orbitals of the formal Lewis structure and unoccupied ones. Since the corrections to the energy of the Lewis-type picture are usually small, they can be approximated by second-order perturbation theory, Eq. (78),... 

This shows that the spectral function s(co), characteristic of the macroscopic pulse S(t), determines the energy domain in which nondiagonal density matrix elements p - are important, i.e., in which quantum-mechanical coherence exists between the spectral components. In particular, maximal coherence is obtained in the previously mentioned minimum uncertainty situation, where S(t) = <5(t) and s(- The other extreme situation corresponds to experiments without time resolution, where the time function S(t) is a constant, with S(x) = (2k) and s((u) = (w). For this case,... 

Broad-band coherent excitation, when A > de, and A g > Se. This is the case where extensive molecular coherence is initially prepared by the exciting field, as represented by the nondiagonal density matrix elements (137b). 

If the phases 0 of the atomic wave function (2.124) are randomly distributed for the different atoms of the ensemble, the nondiagonal elements of the density matrix (2.125) average to zero and the incoherently excited system is therefore described by the diagonal matrix... 

The nondiagonal elements of the density matrix are therefore proportional to the expectation value of the dipole moment. 

In this case the target-projectile symmetry of the collision system is removed and the peaks in the spectra are due to different He states and their interferences. The analysis of such spectra makes it possible to determine relative phases of excitation amplitudes with different L or, equivalently, nondiagonal density matrix elements L M p LM). To some extent this is analogous to the more familiar case of observing radiation from hydrogen in an electric field, where states with different L were excited coherently. Also in that case it is not possible to distinguish transitions from different L states, and the resulting implications have, for example, been discussed by Blum and Kleinpoppen. ... 

In these expressions, the diagonal and nondiagonal matrix elements of the excitation density operator in th representation of the molecular... 

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