One-particle density

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

The third equation is the kinetic equation, which describes the evolution of the one-particle density matrix p(r, r, E) of the electron in the process of multiple elastic and inelastic scattering in a solid... 

In general the spectral one-particle density matrix p(r, rE) describes the mutual coherence of the wave field of high-energy electrons at the points r and r. For the simplest case of time-independent interaction potential the diagonal elements of... 

This is the desired result which may be substituted into the scattering amplitude formula (6). The resulting scattering formula is the same as found by other authors [5], except that in this work SI units are used. The contributions to the Fourier component of magnetic field density are seen to be the physically distinct (i) linear current JL and (ii) the magnetisation density Ms associated with the spin density. A concrete picture of the physical system has been established, in contrast to other derivations which are heavily biased toward operator representations [5]. We note in passing that the treatment here could be easily extended to inelastic scattering if transition one particle density matrices (x x ) were used in Equations (12)—(14). 

As we ourselves kept plugging along, the quantum chemical community largely was negative about DFT, even antagonistic. Their house journal International Journal of Quantum Chemistry, in 1980, published a pointed criticism of it [3] There seems to be a misguided belief that a one-particle density can determine the exact N-body ground state. In 1982, Mel Levy and John Perdew replied with a... 

For a 1, the Hamiltonian Hf is independent of i. For any other value of a, the adiabatic Hamiltonian depends on i and we have different Hamiltonians for different excited states. Thus the noninteracting Hamiltonian (a 0) is different for different excited states. If there are several external potentials V =0 leading to the same density we select that potential for which the one-particle density matrix is closest to the interacting one-particle density matrix. 

FIGURE 20.3 One-particle density for four noninteracting fermions in a one-dimensional box. The dots on the abscissa show different positions in which the density has the same value. 

Let us now look at the one particle density and compare it with the pair density x2(tx). We have to examine now the two possibilities, both electrons with the same spin or with different spin. In Figure 20.4, we have the pair density for the... 

Here we will present the formulae needed for calculating the reduced one-particle density matrices from the floating correlated Gaussians used in this work. The first-order density matrix for wave function T (ri,r2,..., r ) for particle 1 is defined as... 

Thus the density matrix elements take on the familiar form of a Gaussian with a shifted center. We can now find the one-particle density matrix element by... 

The constraint of a finite volume requires that the one-particle density distribution satisfy the condition... 

In the Hohenberg-Kohn formulation, the problem of the functional iV-representability has not been adequately treated, as it has been assumed that the 2-matrix IV-representability condition in density matrix theory only implies an N-representability condition on the one-particle density [21]. Because the latter can be trivially imposed [26, 27], the real problem has been effectively avoided. 

Through extensive analyses of accurate one-particle densities, Bader et al. [89,88] have been able to determine the topological properties that characterize these objects. Consider, for example, a one-electron density corresponding to the ground state of a molecule - whose nuclei are fixed by the vectors i... 

It is clear that arbitrary one-particle densities of a molecular system need not have the same topology. In fact, only those belonging to the same structural region will share this property. To make these concepts clearer, consider two nuclear configurations X and Y belonging to the nuclear configuration space in the context of the Born-Oppenheimer approximation. The corresponding one-electrons densities are p r X) and p(r T), respectively. Consider the... 

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