Momentum space density

Schulke, W., Stuz, G., Wohlert, F. and Kaprolat, A. (1996) Electron momentum-space densities ofLi metal a high-resolution Compton-scattering study, Phys. Rev., B54, 14381-14395. 

So, by measuring J(pz) for a variety of different q-directions, one can, in principle, reconstruct in three-dimension the momentum space density. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Zhou Z, Chu S (2009) A time-dependent momentum-space density functional theoretical approach for electron transport dynamics in molecular devices. Europhys Lett 88 17008... 

Cooper and Allan ° have used momentum density in several studies. A problem remains in obtaining the momentum space densities because most calculations are performed with position space wave functions. In a sense, working in momentum space is yet another way to reduce the overweighting of the core electron density. Most of the following discussions on, e.g., molecular alignment and quantum similarity indices, remain valid when we... 

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... 

Other algorithms have been developed for speeding up the uniform sampling of phase space points. For example, the efficient microcanonical sampling series of schemes exploits the possibility of sampling independently the spatial coordinates and momenta, simply by weighting the sampled geometries by their associated momentum space density. 

Inasmuch as the inscribed sphere corresponds to only 226 electrons per unit cube, it seems likely that the density of energy levels in momentum space has become small at 250.88, possibly small enough to provide a satisfactory explanation of the filled-zone properties. However, there exists the possibility that the Brillouin polyhedron is in fact completely filled by valence electrons. If there are 255.6 valence electrons per 52 atoms at the composition Cu6Zn8, and if the valence of copper is one greater than the valence of zinc, then it is possible to determine values of the metallic valences of these elements from the assumption that the Brillouin polyhedron is filled. These values are found to be 5.53 for copper and 4.53 for zinc. The accuracy of the determination of the metallic valences... 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

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 ... 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

Allan, N.L. and Cooper, D. Momentum-Space Electron Densities and Quantum Molecular Similarity. 173, 85-111 (1995). 

FIGURE 5.4 (See color insert following page 302.) Coordinate and momentum space charge densities of H2 molecule illustrating the hond directionality principle. Isosurfaces from 0.04 to 0.01 a.u. are plotted for the coordinate space charge density (a). Isosurfaces from 1.0 to 0.01 a.u. are plotted for the momentum space charge density (h). 

To derive some chemistry out of the momentum distribution, a bridge between the electron densities in position and momentum spaces needs to be built. The studies in momentum space are therefore still a breeding ground for further conceptual and developmental work. 

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... 

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