Liouville equation space techniques

Whenever it is difficult to obtain an analytical solution or in cases where the analytical solution is unnecessary, for a practical problem in finite Liouville space, the exact evolution of a density operator can always be obtained numerically. This can be achieved either by direct computation of the time-ordering evolution operator, or by iteration of the Liouville equation. A general physical picture or universal conclusions cannot be drawn from the numerical technique alone, but it is helpful for providing a better understanding of a complicated problem and indicates the direction for general conclusions. Moreover, even for a problem where the analytical solution is available, numerical calculation helps to provide a visual account. 

It is immediate to see from equation (A2) that whenever elements of P are small, since /2m is a small parameter, equations adiabatically decouple into one-dimensional problems for the effective potentialse (p). In turn, these problems can be analyzed by the Liouville-Green WKB technique, which requires special care whenever e = E (turning points) but this problem is to be considered as effectively solved by the method of comparison equations. It is important to realize that proper coordinate choices may lead to wide regions of p space where this decoupling is very effective in such a case, approximate quantum numbers can be assigned, and it is possible to compute semiclassically bound or resonance states and scattering properties. 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

There are several ways of proceeding from here to arrive at the master equation for the reduced density matrix. The Zwanzig projection operator technique in Liouville space or the Kubo cumulant expansion may be used and both methods have recently been applied to study optical dephasing in solids. 

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