R space

In one of the earliest DFT models, the Thomas-Fermi theory, the kinetic energy of an atom or a molecule is approximated using the above type of treatment on a local level. That is, for each volume element in r space, one... 

The fact that the set of hi is, in principle, complete in r-space allows the full (electronic and nuclear) wavefunction h to have its r-dependence expanded in terms of the hp... 

In all the equations above we have omitted the dependencies on r the symbol denotes convolution in r-space. The symmetry of the correlation functions implies that Similarly to the total pair corre-... 

According to a factor group analysis of R (space group I4cm with Z = 4), vibration modes can be presented in the center of the Brillouin zone (wave... 

As has been described in Ref. 70, this approach can reasonably account for membrane electroporation, reversible and irreversible. On the other hand, a theory of the processes leading to formation of the initial (hydrophobic) pores has not yet been developed. Existing approaches to the description of the probability of pore formation, in addition to the barrier parameters F, y, and some others (accounting, e.g., for the possible dependence of r on r), also involve parameters such as the diffusion constant in r-space, Dp, or the attempt rate density, Vq. These parameters are hard to establish from first principles. For instance, the rate of critical pore appearance, v, is described in Ref. 75 through an Arrhenius equation ... 

The number of each residue occupying the P and PPII regions of (0,t/r)-space is shown. 

Data reduction of EXAFS spectra was performed using WinXAS [14], The normalized spectra were analyzed over the Arrange of 2.5 to 10 A1. A square-weighted degree 7 spline was used to remove the background of the x(k) function. Finally, the data in -space were converted to R-space using a Bessel window to obtain the radial distribution function. 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

The phase space (r space) of the system is the Euclidean space spanned by the 2n rectangular Cartesian coordinates qL and pt. Every possible mechanical state of the system is represented by exactly one point in phase space (and conversely each point in phase space represents exactly one mechanical state). 

As the temperature is increased the double peak structure of hoo(s) becomes less and less pronounced, and by 150 °C there is only a single peak near s 2.5 A-1. In direct (R) space, this change corresponds to the loss of correlation between molecular centers in the region R > 4 A, shown by the decrease in amplitude of the oscillations of hoo(R) as T incerases. Despite this dramatic change for R > 4 A, local tetrahedrality remains the dominant feature of the structure of the liquid. 

A and an r-space interval of 2 A, application of the Nyqvist theorem limits the free parameters to 14. Finally, the chemical feasibility of the fit should be examined. If the number of free parameters is not limited, it is possible to fit any EXAFS spectrum to a high level of apparent precision, and it is this observation that has given EXAFS a poor reputation in the past. 

In order to restria attention to a smgle shell of scatterers, one selects a limited range of the R-space data for back-transformation to k-space, as illustrated m Figure 3B,C. In Ae ideA case, this procedure allows one to anAyze each shell separately, AAough in practice many shells cannot be adequately separated by Fourier tering (9). 

Figure 4.13 shows R-space plots (not corrected for phase shift) for the 1-con-nected and 3-connected catalysts just described. We are fortunate that the V=0,... 

All spectra were Fourier-transformed at fc = 30-160nm and fitted in an R space of R = 0.10-0.32nm. 

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]. 

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 ... 

In short, and are related by a 3A-dimensional, norm-preserving, Fourier transform. If the r-space wavefunction is constructed from one-electron functions, then there is an isomorphism [2] between and <1>. In particular, if the wavefunction tk can be written in terms of spin-orbitals / (x) as a single Slater determinant... 

If the r-space wavefunction is a linear combination of Slater determinants constructed from a set of spin-orbitals /. , then its p-space counterpart is the... 

If we are interested only in properties that can be expressed in terms of q-electron operators, then it is sufficient to work with the th-order reduced-density matrix rather than the A -electron wavefunction [122-126]. In this section, we consider links between the r- and p-space representations of reduced-density matrices. In particular, we show that if we need the th-order density matrix in p space, then it can be obtained from its counterpart in r space without reference to the /-electron wavefunction in p space. 

The r-space, ( th-order, reduced-density matrix [123] is defined by... 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

In this chapter, the primary focus is on the one-electron case. Dropping the q = I labels, we can write the r-space, first-order, density matrix as... 

If electronic spin is not a focus of attention, then the spin-traced versions of these density matrices can be used. The r-space, spin-traced, first-order, reduced-density matrix is... 

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