Momentum space density matrix

Finally, we consider density functional theory (DFT) computations of p-space properties. A naive way of calculating p-space properties is to use the Kohn-Sham orbitals obtained from a DFT computation to form a one-electron, r-space density matrix Fourier transform / according to Eq. (14), and proceed further. This approach is incorrect because the Kohn-Sham density matrix F is not the true one and, in fact, corresponds to a fictitious non-interacting system with the same p(r) as the true system. On the other hand, Hamel and coworkers [112] have shown that if the exact Kohn-Sham exchange potential is used, then the spherically averaged momentum densities of the Kohn-Sham orbitals should be very close to those of the Hartree-Fock orbitals. Of course, in practical computations the exact Kohn-Sham exchange potential is not used since it is generally not known. 

A third technique, which has been proposed by Randeria et al. (1995), relies on the approximate sum rule relating the spectral function to the momentum space density of states (DOS), n(k). As noted in their letter, n(k) = f A(k,a))/((o)da). To see how this relates to what is measured in an ARPES experiment, note that, within the sudden approximation, a valence-band ARPES spectrum may be approximated as a sum over bands of the product of a matrix element for each band with the corresponding single band spectral function I k,a)) = Mn k,hv)f (0)A k,a)). It is... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space ... 

By its size, this chapter fails to address the entire background of MQS and for more information, the reader is referred to several reviews that have been published on the topic. Also it could not address many related approaches, such as the density matrix similarity ideas of Ciosloswki and Fleischmann [79,80], the work of Leherte et al. [81-83] describing simplified alignment algorithms based on quantum similarity or the empirical procedure of Popelier et al. on using only a reduced number of points of the density function to express similarity [84-88]. It is worth noting that MQS is not restricted to the most commonly used electron density in position space. Many concepts and theoretical developments in the theory can be extended to momentum space where one deals with the three components of linear momentum... 

R. Benesch and V. H. Smith, Jr., Density matrix methods in X-ray scattering and momentum space calculations, in Wave Mechanics—the First Fifty Years, W. C. Price, S. S. Chissick, and T. Ravensdale, eds. (Butterworths, London, 1973), pp. 357-377. 

To the extent that our model holds true, one can use the sum of the expressions (9.39) in the case of the large strata of the atom on hand for the density matrix of the atom in momentum space. But the knowledge of the density matrix allows one — as f)irac especially has pointed out—to answer all questions about the atom, in particular the calculation of the atom form factors. 

R. Benesch and S. R. Singh, Chem. Phys. Lett., 10, 151 (1971). On the Relationship of the X-Ray Form Factor to the First-Order Density Matrix in Momentum Space. 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

According to Chaix et al. (1989), in the absence of external potentials ( A = V = 0) it has been argued that for stability we need a < 2/n and for instability we need a > 7r/log4. The stability result has been proved by Bach et al. (1999). To demonstrate the instability result a translationally invariant density matrix yc based on a rotation in momentum space has been employed (Chaix et al. 1989). But since the model is defined on R3, such a density matrix is either zero or has an infinite number of particles, it has either zero eneigy or infinite eneigy. This is because the state represented by the density matrix is the same in every unit volume. Therefore, the arguments presented by Chaix et al. (1989) could just show the instability within a unit volume. 

What is needed for a correct computation of momentum-space properties from DPT is an accurate functional for approximating the exact first-order reduced density matrix r f f ), or failing that, good functionals for each of the p-space properties of interest. Of course, a sufficiently good functional for (p ) would obviate the necessity of using Kohn-Sham orbitals and enable the formulation of an orbital-free DFT. Unfortunately, a kinetic energy functional sufficiently accurate for chemical purposes remains an elusive goal [118,119]. 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix formulation yields a consistent treatment of electronic populations and coherences, and the momentum changes associated with an electronic transition can be directly derived from the formalism without the need of ad hoc assumptions. Employing a Monte-Carlo sampling scheme of local classical trajectories, however, we have to face two major complications, that is, the representation of nonlocal phase-space operators and the sampling problem caused by rapidly varying phases. At the present time, the... 

EMD is defined as the diagonal element of the six-dimensional Fourier transform of the one-electron density matrix from coordinate to momentum space ... 

To get the matrix elements of Vxc, we proceed as follow. First, we get the position space representation of p, by applying the FFT to equation (91). This will give us the value of the density at all the grid points, pj. Second, we compute the exchange-correlation potential v c at all the grid points, [vxcj]-Third, we apply the ITT to to get the momentum space representation of Vxc,... 

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