Position-space representation

Suppose that (xi,X2,..., xn) is the position-space representation of the N-electron wavefunction. It is a function of the space-spin coordinates Xk = (Ofcj ctjfc) in which is the position vector of the kth electron and Gk is its spin coordinate. The position-space wavefunction is obtained by solving the usual position- or r-space Schrodinger equation by one of the many well-developed approximate methods [32-34]. 

In addition, one should contrast the appearance of the plots from Fig.(l) with our more familiar position space representation of these spa hybrids, as shown in Fig.(4). There, we always see two distinct maxima [6], the sharper of which is located right at the origin. 

Figure 4. The sp, sp2, and sp3 hybrid density functions in the 12-plane of position space. As is often the case, orbitals that are quite different from one another in momentum space, can appear very similar in the corresponding position space representation.

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

Those intrinsically relativistic systems require the framework of relativistic quantum chemistry, which is based on the Dirac equation. Its position-space representation for a single electron in the attractive Coulomb potential of a nucleus with charge Z reads... 

In this appendix we present the interaction part (at,a, y o, a,<) of the static particle-hole self energy discussed in Sec. VIC. We assume Coulomb interacting electrons with the usual position space representation (106) of the two-body interaction V. The expression for the interaction part of the static self particle-hole self energy can then be readily evaluated, either from the definitions of the extended states (1) or from Eq. (68) ... 

The main requirement in the determination of bond orders is to derive rules on how to measure the number of electrons shared between two atoms. For this purpose, a definition of an atom in a molecule is required, which, however, cannot be formulated in a unique and unambiguous way [169]. Quantum chemical calculations are typically performed in the Hilhert-space analysis, where atoms are defined by their basis orbitals. Such an analysis, however, strongly depends on both the atomic basis set chosen and the type of wave function used. The position-space representation, on the other hand, where atoms are defined as basins in three-dimensional physical space does not suffer from these insufficiencies. In this chapter, we present one option for a three-dimensional atomic decomposition scheme and the reader is referred to Refs. [170-173] for further examples. 

Hence, one of the operators must be a differential operator while the other must be a simple multiplicative operator of the same variable. The first choice is called the position-space representation, while the second is called the momentum-space representation. Of course, one may add constants to these definitions but they are chosen to be zero since they would represent arbitrary shifts. Further, we must require that all arbitrary functions of position and momentum vanish. 

Silvi B, Fourre I, Alikhani ME (2005) The topological analysis of the electron localization function. A key for a position space representation of chemical bonds. Monatshefte Fur Chemie 136 855-879... 

The set of function values constitutes the position space representation of function yj/. 

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,... 

The so-called real-space methods provide a viable alternative to the supercell approach for molecules and clusters. Real-space methods use only the position-space representation (position space is also known as real space ), which implies that molecules and clusters can be dealt with directly, without artificial supercells. The Laplacian operator V, exactly evaluated in momentum space (see equation 97), has to be approximated in real-space methods. The most popular approaches " use a finite-difference approximation for the Laplacian. For example, the second derivative with respect to xofa function y, z) can be approximated by the following finite difference. 

The calculation of e in momentum space is analogous to that in position space. Starting with the r-representation, and expressing the quantity F(r)(pi(r) as the inverse Fourier transform of [F(r) (pi(r)]T(p), one easily finds that ... 

The one-electron energy ej has the same expression in the p-representation as in the position space where the different contributions can be expressed as follows ... 

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

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... 

The technology for solving the Schrddinger equation is so much farther advanced in r space than in p space that it is most practical to obtain the momentum-space from its position-space counterpart The transformation theories of Dirac [118,119] and Jordan [120,121] provide the hnk between these representations ... 

Using a representation of the one-particle density matrix in position space ... 

In this section we have shown how a representation on a vector space determines a representation on the dual of the vector space. We will find the dual representation useful in Section 5,5, More generally, duality is an important theoretical concept in many mathematical settings. Physically, momentum space is dual to position space, so the name "momentum space in the physics literature often connotes duality. 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

In systematic SAR analysis, molecular structure and similarity need to be represented and related to each other in a measurable form. Just like any molecular similarity approach, SAR analysis critically depends on molecular representations and the way similarity is measured. The nature of the chemical space representation determines the positions of the molecules in space and thus ultimately the shape of the activity landscape. Hence, SARs may differ considerably when changing chemical space and molecular representations. In this context, it becomes clear that one must discriminate between SAR features that reflect the fundamental nature of the underlying molecular structures as opposed to SAR features that are merely an artifact of the chosen chemical space representation. Consequently, activity cliffs can be viewed as either fundamental or descriptor- and metrics-dependent. The latter occur as a consequence of an inappropriate molecular representation or similarity metrics and can be smoothed out by choosing a more suitable representation, e.g., by considering activity-relevant physicochemical properties. By contrast, activity cliffs fundamental to the underlying SARs cannot be circumvented by changing the reference space. In this situation, molecules that should be recognized as... 

Note that in (7.3) time is a continuous variable, while the position n is discrete. We may go into a continuous representation also in position space by substituting... 

A third requirement is less absolute but still provide a useful consistency check for models that reduce to simple Brownian motion in the absence of external potentials The dissipation should be invariant to translation (e.g. the resulting friction coefficient should not depend on position). Although it can be validated only in representations that depend explicitly on the position coordinate, it can be shown that Redfield-type time evolution described in such (position or phase space) representations indeed satisfies this requirement under the required conditions. 

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. 

The momentum-space representation also proves particularly convenient for comparisons of the electron distributions of systems with different nuclear frameworks. Difference density plots in r-space are complicated by the different sets of nuclear positions. Such complications are absent in p-space and, in the case of polyenes [23], for example, momentum-space concepts have proved useful for examining the effects of bond alternation on the electron density - an important characteristic of such systems and of doped polyacetylene. 

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