Correlation function quantum motion

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

In many cases, in order to compute the dynamics of condensed phase systems, one invokes a basis representation for the quantum degrees of freedom in the system. Typically, one computes the dynamics of these systems in order to obtain quantities of interest, such as an average value, A(t) = Tr [Ap(t)], or a correlation function, as will be discussed below. Since such averages are basis independent one may project Eq. (8) onto any convenient basis. This is in principle a nice feature, and one that is often exploited to aid in calculations. However, it is important to note that the basis onto which one chooses to project the QCLE has important implications on how one goes about solving the resulting equations of motion. Ultimately the time-dependent average value of an observable is expressed as a trace over quantum subsystem... 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

We now turn to quantum Brownian motion as described by the generalized Langevin equation (22), with the symmetrized correlation function of the random force as given by Eq. (20). 

Rather than carrying out a linear response derivation to obtain correlation function expressions for transport coefficients based on the quantum-classical equations of motion, in this section we show how transport coefficients can be obtained by a different route. We take as a starting point the quantum mechanical expression for a transport coefficient and consider a limit where the dynamics is approximated by quantum-classical dynamics [17,18]. The advantage of this approach is that the full quantum equilibrium structure can... 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

---