Reconstruction of the one-particle density matrix from expectation values in position and momentum space

Abstract

For the beryllium atom, it is demonstrated that coherent form factors F(k) can be insufficient for inferring the one-particle reduced density matrix (ODM). The description in terms of reciprocal form factors B(s) as the complementary momentum-space property is compared with the results for a least-squares fit to F(k) data. A virtually complete description of the true ODM may be obtained by using a combined data set, as can be shown by representing the ODM in spherically averaged form.

On the Possibility of Kinetic Energy Density Evaluation from the Experimental Electron-Density Distribution

“…Thus, K(r) and G(r) integrate to yield the same value of the total kinetic energy, T. In this sense, the local kinetic energy yielding the correct value of the total kinetic energy is only defined up to C * V2p(r), where C is an arbitrary constant. Although the problem of the reconstruction of the one-electron-density matrix p(r, r') from p(r) obtained from the diffraction experiment has been considered before (Aleksandrov, Tsirelson, Reznik & Ozerov, 1989;Tanaka, 1988;Schmider, Smith & Weyrich, 1992;Howard, Huke, Mallinson & Frampton, 1994), we are going to propose here an alternative less accurate but easier approach for the evaluation of the local kinetic energy density at the bond critical point of the multipolefitted electron density. A simple approximate way of directly relating the kinetic energy density to the electron density was introduced by the semiclassical Thomas-Fermi equation (March, 1957) with gradient quantum corrections (von Weizsacker, 1935;Kirzhnitz, 1957).…”

Atomic Orbitals from Compton Profiles

“…[28], Table 3. Expectation values of the total momentum moments <p 9 > for different wave functions. We report the Compton profile at the peak, J(0) = <p-1 >/2 and the kinetic energy <T> = <p 2 )/2, rather than the corresponding moments themselves.…”

“…The occupation numbers are determined separately for each functional evaluation (equivalent to a linear least-squares fit with linear constraints). The employed method is general and has been applied to test systems, where electron correlation yielded dramatic improvement in the data reproduction [8][9][10].…”

“…The application of this method to theoretical model-systems has shown that the inclusion of electron correlation is necessary to reproduce mixed sets of position and momentum density dependent data [8][9][10]. The experience gained on these systems indicates, on the other hand, that the idempotency condition may well be retained in cases where only data of one kind are used.…”

“…where r are functions of q including some nonlinear parameter(s) Gaussians [11,12], Lorentzians [13,14,12] and hydrogenic Lorentzians [15] have been employed as functional forms of t. Such fits allow the straightforward evaluation of the isotropic momentum distribution n(p), its moments <p 9 ) [16] and more complicated momentum-space quantities such as stopping powers [17]. However, since they do not have the form of an ODM, they do not permit the direct calculation of position-space expectation values, except through additional approximations in the framework of density-functional theory [12,18].…”