Density matrix residual

The averaged density matrix becomes p (Rslow, qa. 0 = RJ> qj> 0) r, Rfast and the corresponding Liouville equation, in the hypothesis of no residual dynamic effect of averaging with respect to subpicosecond processes, can be simply written as in Equation (2.30) with H (Rslow, qa ) instead of H ( rj, Rt, qa ). 

This formula indicates that the residue of G(z) at an eigenvalue pole is just the density matrix, in the coordinate representation, summed over all eigenfunctions with this eigenvalue. This residue can be extracted conveniently by using Dirac s formula after displacing each eigenvalue into the lower complex plane,... 

All of the information concerning excited state populations in an ensemble of such dimers residues in the four elements p, of a time-dependent density matrix p where the indices k, I are 1 or 2 in the site representation, and -I- or - in the delocalized representation. The initial values of the density matrix elements in the delocalized representation, immediately after electric dipole-allowed absorption of a photon with polarization C are given by ... 

The chemist s idealized Lewis structure picture describes the N/2 electron pairs as localized in one-center ( lone pair ) or two-center ( bond ) regions of the molecule. The natural bond orbital (NBO) algorithm leads to an optimal set of one- and two-center orbitals that are in close correspondence with this picture. In effect, the algorithm searches the density matrix for the set of N/2 localized Lewis-type lone pair and bond orbitals of near-double occupancy that best describe the given wave-function, with the residual weakly occupied non-Lewis-type... 

Start from the result = (a ala a,.) in (S.4.20) and use an argument parallel to that which leads to (13.3.10) to express the 2-electron density matrix in terms of an appropriate polarization propagator. [Hint How should you choose the operators in (13.2.10) in order to obtain poles whose residues (numerators) will yield the desired elements Use the anticommutation rules and remember that n)(n is the unit operator.]... 

Figure 17 Convergence of purification transformation for different starting values (left). Purification of the density matrix after a typical geometry optimization step within D-QCSCF (see the following section for a definition) calculation (right), the logarithmic value of the norm of the residual (log P — P ) is plotted.

At each stage of the refinement of a new set of parameters, the hat matrix diagonal elements were calculated in order to detect the influential observations following the criterium of Velleman and Welsh [8,9]. The inspection of the residues of such reflections revealed those which are aberrant but progressively, these aberrations disappeared when the pseudo-atoms model was used (introduction of multipoler coefficients). This fact confirms that the determination of the phases in acentric structures is improved by sophisticated models like the multipole density model. 

So, MCBA builds a covariance matrix of the residuals around the inner model and from this matrix it obtains a probability density function as bayesian analysis does, taking into account that the dimensionality of the inner space correspondingly reduces the rank of the covariance matrix from which a minor must be extracted. 

Strutinsky procedure. Because asymmetric shapes is so shallow, it is worthwhile to deal with octupole correlation effects using a microscopic, two-body interaction treatment of the octupole-octupole residual interaction [CHA80]. The pairing force matrix elements, G. come from a density dependent delta interaction. This set of matrix elements [CHA77] explains many features of the actinides at low and... 

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