Momentum-space properties

Spin-coupled wavefunctions have proved to be very useful in studies of momentum-space properties " . Except for very simple systems, it is rather difficult to solve the Schrodinger equation directly in the momentum representation fortunately, the momentum-space wavefunction is also given by the Fourier transform of that in position space and this indirect approach proves to be much more tractable. The momentum-space formalism is particularly convenient for the interpretation of various scattering techniques such as Compton scattering and binary (e, 2e) spectroscopy. 

In the early 1940s, an investigation of chemical bonding from the momentum-space viewpoint was initiated by Coulson and Duncanson (Coulson, 1941a,b Duncanson, 1941, 1943 Coulson and Duncanson, 1941,1942 Duncanson and Coulson, 1941) based on the Fourier transformation of the position wave function. [They also gave a systematic analysis of the momentum distributions and the Compton profiles of atoms (Duncanson and Coulson, 1944, 1945, 1948).] They first clarified the momentum-space properties of the fundamental two-center MO and VB wave functions, which may be outlined as follows. 

This article provides an introduction to the momentum perspective of the electronic structure of atoms and molecules. After an explanation of the genesis of momentum-space wave functions, relationships among one-electron position and momentum densities, density matrices, and form factors are traced. General properties of the momentum density are highlighted and contrasted with properties of the number (or charge) density. An outline is given of the experimental measurement of momentum densities and their computation. Several illustrative computations of momentum-space properties are summarized. 

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]. 

P. E. Regier, J. Fisher, B. S. Sharm and A. J. Thakkar, Int. J. Quantum Chem., 28, 429 (1985). Gaussian vs. Slater Representations of d Orbitals An Information Theoretic Appraisal Based on Both Position and Momentum Space Properties. 

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. 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

The symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space. 

Tab.l Chemical systems (atoms, molecules or chain) treated in momentum space methods and properties. 

It follows from the above considerations that local-scaling transformations can be advanced in momentum space on an equal footing with those in position space. In particular, wavefunctions in momentum space can be transformed so as to generate new wavefunctions that have the property of belonging to an orbit . 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

This Wigner representation of the density pw q, p) proves particularly useful since it, satisfies a number of properties that are similar to the classical phase-space distribu tion pd(q, p). For example, if p = pure state, then fdppw = probability density in coor- dinate space. Similarly, integrating pw over q gives the probability density in ( momentum space. These features are shared by the classical density p p, q) in phase space. Note, however, that pw is not a probability density, as evidenced by if the fact that it can be negative, a reflection of quantum features of the dynamics, ) [165], 3... 

We keep the quantum number / in the notation for the state because it is necessary to keep track of the parity, which is a property of coordinate or momentum space (represented here by the orbital eigenstate) and has nothing to do with spin space. The coordinate-spin representation of (3.91) may be called a jj coupling function because of its use in states for systems of several electrons where the total angular momentum is obtained by vector addition of the angular momenta J of each electron. 

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