Phase-Space Approach to the Density-Functional Calculation of Compton Profiles of Atoms and Molecules

Abstract

The phase-space distribution function corresponding to a ground-state density of a many-electron system proposed earlier is explored as a means for generation of momentum-space properties through density-functional theory. Excellent results are found for the spherically averaged Compton profiles for several atoms and the molecules H2 and N2, as dwell as the directional Compton profiles for N2, thereby providing both a useful scheme for computation of such profiles and confirrnation of the basic theory. The entropy-maximization procedure employed is discussed from the point of vie~of information theory, PACS numbers: 31.15.+ q, 31.20.Lr, 31,90.+ s, 32.80.Cy In connection with a thermodynamic transcription of the density-functional theory of electronic structure, ' there recently has been proposed a phase-space distribution function f(r, p) corresponding to a groundstate electron density p(r). A unique f(r, p) was obtained by the invoking of an entropy-maximization principle, in analogy to the classical case. The resulting distribution function is Maxwellian in nature with a local temperature T(r) and leads to various thermodynamic and fluid-theoretic equations for the electron cloud. " In the present work, this same f(r, p), and a simple generalization of it, are used to predict momentum-space properties of an atom or molecule: the spherically averaged and directional Compton profiles.Consider an N-electron system characterized by the ground-state density p(r). Identify a phase-space distribution function f(r, p) with this density, and assume it to yield the correct kinetic energy density r(r, p) as weil: p(r) ="' d p f(r, p), z d r p(r) =N; (I) t (r, p) = , ' J, d'p p'f (r,-p), f(r, p) = [2mkT(r)] '~'p(r)exp[ -p'/2kT(r)]. (5) Here P(r) = [kT(r)] ', where the local temperature T(r) is defined in analogy with the ideal-gas expression for kinetic energy by -' , p(r) kT(r) = r(r, p). (6) tained' by a maximization of the entropy defined as S = "d r s (r), s (r) = -k J~d 'p f (lnf' -1), (3) subject to the constraints of correct density [Eq. (1)] and correct kinetic energy density [Eq. (2)]. In Eq. (3), k is the Boltzmann constant. Introducing Lagrange multipliers o. (r) and P(r) for the two constraints"respectively, one obtains f(r, p) = exp [ -~( r) -p(r) p'/2], which, on evaluation of the Lagrange multipliers from Eqs. (1) and (2), becomes J d'r t(r, p) =Ek;". ( ) I x&p 8, . p;() If we presume the validity of f(r, p) as a phasespace distribution function, the spherically averaged The most appropriate distribution function is then ob-~m omentum density X(p) is given by X(p) = Jfd r f(r, p) = J d r [p(r)/2n] p(r)exp[ -p(r)p /2]. (7) and the spherically averaged Compton profile, within the impulse approximation, can be obtained fromIq I which is a basic equation for calculation of the averaged Compton profiles of atomic and molecular sysAlthough the term V2p does not contribute to the glotems. bal kinetic energy, it introduces quantum oscillations Now consider the form of the kinetic energy density in the kinet...

Phase-space explorations in time-dependent density functional theory

“…In fact the earlier interest in developing ways to associate phase-space distributions to a given ground-state coordinate-space density [49,48,50] was motivated by trying to get good momentum-space properties derived from the coordinate-space density. One may view the density-matrix developments as a rigorous approach to this problem; as discussed in Section 2, the exact one-body density matrix, or equivalently, the one-body phase-space density, directly yields all one-body information, i.e.…”

“…Such distributions are particularly useful in a semiclassical context, relating the quantum states to the underlying classical trajectories, and have been exploited extensively, for example in quantum chaos and quantum optics. Phase-space approaches have however been largely, although not entirely [48][49][50][51][52][53][54] neglected in density functional theories, where position plays a preferred role over momentum. Most recently [52][53][54], Gill and co-workers, have developed and tested models for the ground-state correlation energy based on various ''phasespace intracules" which are essentially different contractions of the two-body reduced Wigner function.…”

Density functional theory on phase space

“…This had been conjured by Ghosh, Berkowitz, and Parr [24,25] by means of heuristic "local thermodynamics" arguments. The GBP Ansatz was put to good use [26][27][28] in the calculation of Compton profiles, exchange energies, and corrections to the Thomas-Fermi-Dirac energy functional. As the GBP distribution cannot be a true Wigner quasiprobability, its success should partially be attributed to the intrinsic strength of the phase space formalism.…”

Time-dependent density functional theory as a thermodynamics

“…The analogy with the classical thermodynamics of fluids was developed by Ghosh and Berkowitz [3]. Exchange energies [4] and Compton profiles [5] predicted by the GBP theory are very good. This approach nicely takes the form of a thermodynamics with a local temperature.…”