P- space density

The p-space density matrices form a hierarchy related through the analog of Eq. (5.7). 

Setting = 1 in Eq. (5.10) shows that the first-order r- and p-space density matrices (whether spin-traced or not) are related by a six-dimensional Fourier transform [127,129] ... 

Then, instead of performing the six-dimensional integral in Eq. (5.19) all at once, we perform successive three-dimensional integrals over s and R. The first step takes us to W R,P), the Wigner representation [130,131] of the density matrix, and the second step to the p-space density matrix, n(P — p/2 P + p/2). The reverse transformation of Eq. (5.20) can also be performed stepwise over P and p to obtain A( , p), the Moyal mixed representation [132], and then the r-space representation V R— s/2 R + s/2). These steps are shown schematically in Figure 5.2. 

Figure 5.2. The relationships among the r-space density matrix F, the p-space density matrix n, the Wigner representation W, and the Moyal representation A. Two-headed arrows with a T beside them signify reversible, three-dimensional, Fourier transformations.

Although the r- and p-space representations of wavefunctions and density matrices are related by Fourier transformation, Eqs. (5.19) and (5.20) show that the densities are not so related. This is easily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The operations of Fourier transformation and taking the absolute value squared do not commute, and so the p-space density is not the Fourier transform of its r-space counterpart. In this section, we examine exactly what the Fourier transforms of these densities are. 

Figure 5.3. Connections among r- and p-space densities, density matrices, and form factors. Two-headed arrows signify reversible transformations single-barbed arrows signify irreversible transformations. A Fourier transform is denoted by JF.

How is the form factor related to the p-space density matrix Substitution of Eq. (5.20) into Eq. (5.31) and integration over r andp yields [127,129]... 

The reciprocal nature of r- and p-space densities is seen clearly from Eqs. (55) and... 

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

Compound Syngony Cell parameters, A Space Density p, g/cm3 Reference... 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

Fig. 3.7. Schematic comparison of the multicanonical and iterative transition-matrix methods. The light gray box indicates a single iteration of several thousand or more individual Monte Carlo steps. The acceptance criterion includes the configuration-space density of the original ensemble, p, and is presented for symmetric moves such as single-particle displacements...

Consider a gas whose phase density in T space is represented by a microcanonical ensemble. Let it consist of molecules with //-spaces pi with probability distributions gt. Denote the element of extension in pi by fa. Since energy exchanges may occur between the molecules, pi cannot be represented by a microcanonical distribution. There must be a finite density corresponding to points of the ensemble that do not satisfy the requirement of constant energy. Nevertheless, the simultaneous probability that molecule 1 be within element di of its p-space, molecule 2 within dfa of its //-space, etc., equals the probability that the whole gas be in the element... 

Microscopic or molecular-a collection of reacting molecules sufficiently large to constitute a point in space, characterized, at any given instant, by a single value for each of c T, pressure (P), and density (p) for a fluid, the term element of fluid is used to describe the collection ... 

We consider a rigid system of / mechanical degrees of freedom in thermal contact with a solvent. As in the discussion of equilibria, p q,p) is the phase space density and /( ) is the reduced distribution for the coordinates alone. Following BCAH, we also define a conditional average (A)p of an arbitrary dynamical variable A with respect to the rapid fluctuations of the momenta and solvent forces, at fixed values of the coordinates q, as... 

Evolution of Probability Densities The phase space density p q,p) evolves according to the Liouville equation... 

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. 

B( ) is variously called the reciprocal form factor, the p-space form factor, and the internally folded density. B(s) is the basis of a method for reconstructing momentum densities from experimental data [145,146], and it is useful for the r-space analysis of Compton profiles [147-151]. The B(s) function probably first arose in an examination of the connection between form factors and the electron momentum density [129]. The B f) function has been rediscovered by Howard et al. [152]. 

This equation connects the large-p behavior of the momentum density with the small-r behavior of the electron density and small-w behavior of the intracule density. Hence, Eq. (5.49) is a quantitative manifestation of the reciprocal nature of r and p space. 

Thus there is an isomorphism between the first-order, r-space density matrix F and its p-space counterpart II, just as there is an isomorphism between a r-space wavefunction built from a one-particle basis set and the corresponding p-space wavefunction as described in Section II. 

Another vivid illustration of the reciprocity of densities in r and p spaces is provided by Figure 5.4, which shows the radial electron density... 

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