Density matrix interaction representation

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

We now regard Eq. (8-233) as analogous to Schrodinger s equation, and proceed to carry out the transformation to the interaction representation described in Chapter 7, Section 7.7. We define the transformed density matrix R and the transformed potential U by... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

G. Gidofalvi and D. A. Mazziotti, Boson correlation energies via variational minimization with the two-particle reduced density matrix exact V-representability conditions for harmonic interactions. Phys. Rev. A 69, 042511 (2004). 

The interaction between the matter and the light beam is weak and I compute the state TOt using perturbation theory based on the complete set of exact states /> , with energies ha> of the chiral medium in the absence of the light beam, noting that the information yielded by the experiment can then be related to the optically active medium alone. The density matrix, , for the medium in the absence of the light beam can be given a spectral representation in terms of this complete set of states, by virtue of the spectral theorem,... 

The macromolecular density matrix constructed from the fragment density matrices within the ADMA framework represents the same level of accuracy as the electron densities obtained with the MEDLA and ALDA methods. The effects of interactions between local fragment representations are determined to the same level of accuracy within the ADMA, the MEDLA, and the ALDA approaches. The ADMA direct density matrix technique allows small readjustments of nuclear geometries, in a manner similar to the ALDA technique however, within the ADMA framework, the geometry readjustment can be carried out directly on the macromolecule. 

As a matter of fact, it must be described by a density matrix p which is the partial trace over the environmental degrees of freedom of the total density matrix ptot of the closed compound system quantum computer + environment p = Tren ) [ptot] p generally evolves non-unitarily according to the operator-sum representation, the matrix [/ obtained after the interaction of the computer with its environment can be written under the form... 

The density matrix of the quasi-stationary state in the representation of the interaction Ho looks as follows ... 

Following McGurk et al.l, we transform to the density matrix in the interaction representation defined by... 

Upon transforming to an interaction representation (i.e., a reference frame rotating at frequency w) in which the density matrix is defined by Equation 7, and invoking the rotating wave approximation which consists of dropping all high-frequency motions with respect to u . Equation 25 becomes... 

The definition of the density matrix given in Eq. (10.12), in which in c and include the factors exp(—iH t/h) and e p —iHm t/h) along with the coefficients C (0 and C (0. uses what is called the Schrodinger representation. An alternative formulation called the interaction representatiOTi is= C (i)C)),(<). In the interaction representation, the factors e p —iH, t/h) and exp —iHmmt/fi) must be introduced separately in order to obtain the complete time-dependence of the system. Both representations are widely used, and the choice is mostly a matter of personal preference. We will use the Schrodinger representation. 

The first generally available implementation of COSMO was the MOPAC implementation in 1993. In this the matrix inversion algorithm has been used. The nonzero elements of the semiempirical density matrix are used for the density representation Q and the interaction of the density with the screening charges is expressed by the corresponding atom centered multipoles. This implementation follows the theory outlined above, closely. Several other semiempirical Implementations of COSMO have appeared meanwhile (MNDO, AMPAC, ZINDO) which closely follow the MOPAC/COSMO concept have appeared since (see MNDO). 

The preceding derivation of an expression for l/Tj, equation (29), was for a simple case with two eigenstates and only one interaction V. A more general case will involve further simultaneous equations, and relaxation times that depend on more than one integral of type Such cases can often be made more tractable by replacing expression (19) with a density matrix, whose elements are the averaged products C C . In this representation the wave equation converts to an equation involving this matrix. Its solution was first studied by Kubo and Tomita, and then developed by Redfield and many others. Their approach is essential in complex... 

---