Density matrices from generalized products

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

To generalize Eq. (10.81), consider a system with M excited states. If we can evaluate the product of the density matrix and the fluorescence operator for an ensemble of systems, we can use Eq. (10.14) to calculate the fluorescence from any given excited state. Equation (10.74) suggests that the operator F for the dipole strength of fluorescence with polarization if can be written symbolically as... 

The principle of minimum entropy production holds in the macroscopic description in which the entropy is considered to be a function of the diagonal density matrix, as seen from Klein and Meijer s theory. An attempt has been made by Callen to generalize this principle for the cases where the contribution of the off-diagonal elements of the density matrix to the entropy cannot be neglected. 

The matrices F(n) and aFM are easily calculated in the AO basis from the Roothaan-Bagus integrals. Conversely, the construction of Q(n> (and thus the total Fock matrix) requires MO integrals with one general and three active indices. The Q(n> matrix can be calculated as a matrix product if all density and integral elements for a given distribution vx are held in core at the same time. 

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