Momentum-space approach

In this contribution our purpose is to review the principles and the results of the momentum space approach for quantum chemistry calculations of molecules and polymers. To avoid unnecessary complications, but without loss of generality, we shall consider in details the case of closed-shell systems. 

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

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. 

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... 

The counterpart wavefunction in momentum-space, 4>(yi,y2 is a function of momentum-spin coordinates % = (jpk, k) in which pk is the linear momentum of the feth electron. There are three approaches to obtaining the momentum-space wavefunction, two direct and one indirect. The wavefunction can be obtained directly by solving either a differential or an integral equation in momentum- or p space. It can also be obtained indirectly by transformation of the position-space wavefunction. 

In the case of semiconductors, the idea of electron tunneling has been used by Zener [42] to describe the so-called interband tunneling. Such tunneling represents one of the possible mechanisms of semiconductor breakdown. To understand the nature of interband tunneling, we shall first follow Ziman [43] and consider the one-dimensional motion of an electron in a separate band under the influence of an electric field. If we use the scheme of repeated bands, then the electron motion in momentum space is an up and down motion along the OABC periodic curve (Fig. 16). In the coordinate space, the electron, starting from point O, accelerates then slows down as it approaches point A here, the direction of the motion is changed to the... 

Another approach to the problem of rare gas scattering is to replace the spatial wavefunctions of Eq. (11.4) with their Fourier transforms, the momentum space wavefunctions. These wavefunctions represent the velocity distributions of the electron in the Rydberg states. Proceeding along these lines, we rewrite Eq. (11.4)... 

Pair transfer interaction between the states of an electron system components can cause the gauge symmetry breaking realized in superconductivity. This circumstance forms the basis of the two-band model of superconductivity known already during a considerable time [1,2], The basic advantage of such approaches consists in the possibility to reach pairing by a repulsive interband interaction which operates in a considerable volume of the momentum space. An electronic energy scale is... 

By forming translational wave packets, molecules can henceforth be localized in momentum space. This process may happen under non stationary conditions. This possibility results from the separation approach used here. 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

This PWE renormalization method was also called noncovariant contrary to the covariant approach described above where the covariant procedure in 4-dimensional momentum space was used to separate out and cancel the divergent terms. In principle, the noncovariant procedure should not lead to any differences provided that both bound -term and counterterm are described in the same way. Such a difference may arise only if the counterterm, unlike the bound -term is written in covariant form [12]. 

Calculation of the infinite lattice sums remains the most difficult step in ab initio polymer calculations. They can be evaluated in configuration space as well as in momentum space (or the two procedures can be also combined). There is not enough experience accumulated in the literature to decide which approach... 

D.L. Cooper, and N.L. Allan, Molecular Similarity and Momentum Space, R. Carbo, Ed. Molecular Similarity and Reactivity From Quantum Chemical to Phenomenological Approaches Kluwer Academic Publ. Dordrecht, The Netherlands, 1995, pp 31-55. 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for in momentum space. For... 

V. Aquilanti, S. Cavalli, C. Coletti, D. Di Domenico, and G. Grossi, Hyperspherical harmonics as sturmian orbitals in momentum space a systematic approach to the few-body coulomb problem. Int. Rev. in Phys. Chem., 20 673-709, 2001. 

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. 

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... 

Momentum-space similarity and dissimilarity indices tend to be especially useful in situations for which the bonding topology is less important than the long-range valence electron density. In addition to model systems, the approach has now been applied successfully to a number of real problems, such as... 

In summary, our unconventional approach to molecular similarity, based on indices derived from momentum-space electron densities, appears to show considerable promise for a wide range of applications. Work on a wide range of applications, both biological and non-biological, is currently in progress and the results will be reported in due course. 

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