Density matrix singular

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. [Pg.53]

Equation (3.32a) implies normalization, and Eq. (3.32b) contains the essential probabilistic interpretation of the projections onto the momenta or coordinates. The last condition, a natural consequence of the definitions of the density matrix and the Wigner-Weyl transform, explicitly eliminates the singular distributions allowed in Eq. (3.31d). That is, although the completeness of the quantum PijXp, q) basis permits the construction of 8 function distributions, they make, unlike classical mechanics, no natural appearance in quantum mechanics wherein eigenfunctions of LQ are square integrable and such singular distributions are explicitly excluded in Eq. (3.32d). [Pg.408]

To extract the linearly independent excitations, we shall have to use the so-called singular value decomposition of the valence density matrices generated by the creation/annihilation operators with valence-labels which axe present in the particular excitation operator in T. To illustrate this aspect, let us take an example. For any excitation operator containing the destruction of a pair of active orbitals from V o the overlap matrix of all such excited functions factorize, due to our new Wick s theorem, into antisymmetric products of one-body densities with non-valence labels and a two-particle density matrix ... [Pg.184]

The Cl of the adiabatic PESs is a common phenomenon in molecules [11-13], The singular nonadiabatic coupling (NAC) associated with Cl is the origin of ultrafast non-Born-Oppenheimer transitions. For a number of years, the effects of Cl on IC (or other nonadiabatic processes) have been much discussed and numerous PESs with CIs have been obtained [11, 12] for qualitative discussion. Actual numerical calculations of IC rates are still missing. In this chapter, we shall calculate IC rate with 2-dependent nonadiabatic coupling for the pyrazine molecule as an example to show how to deal with the IC process with the effect of CL Recently, Suzuki et al. have researched the nn state lifetimes for pyrazine in the fs time-resolved pump-probe experiments [13]. The population and coherence dynamics are often involved in such fs photophysical processes. The density matrix method is ideal to describe these types of ultrafast processes and fs time-resolved pump-probe experiments [14-19]. [Pg.80]

Secondly, we note that the density matrix may become singular. is a Hermitian, positive semidefinite matrix and - as mentioned above - its eigenvalues, called natural weights, characterize the importance of the corresponding natural orbital. A zero eigenvalue occurs if there is a natural orbital (i.e., a linear combination of the single-particle functions) that does not contribute to the MCTDH wave function. Its time evolution may thus be modified by replacing with p( )... [Pg.3014]

Show how the method of Werner et al. (p. 273) may be applied in simple closed-shell SCF theory in order to obtain a rapidly convergent procedure. How would you implement the method of Section 8.3 (singular-value decomposition) in an actual calculation [Hint Start from an energy expression (e.g. (5.3.18)) identify the density-matrix elements (e.g. P = 2b , r, s occupied, = 0, otherwise) substitute in (8.4.12) and (8.4.13), and find the various elements of B.]... [Pg.284]

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... [Pg.162]

In contrast to (143), this is a diagonal matrix independent of the spatial variables. Hence, in exactly the same way as with the zero-point oscillations of energy density, the vacuum fluctuations of polarization in empty space has the global nature, while those in the presence of the singular point manifest certain spatial inhomogeneity. [Pg.462]

We assume of course H > 0 (else v = 0 according to Remark (ii) to Section 9.2). On the other hand, we can assume H < I see (9.2.33). But then rankA = H < I hence A is not of full row rank and the random variable v does not have a probability density as introduced in Appendix E. The covariance matrix of random variable v is thus (positive semidefinite but) singular (not regular). If e is Gaussian, the adjustment vector v has a degenerate Gaussian distribution we shall not examine its properties. We have anyway the mean... [Pg.313]

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