Many-electron Wave Function and Momentum Density

Inelastic x-ray scattering at large momentum transfer is an ideal probe of the ground state of electrons in condensed matter. The experimental determination of the electron momentum density (EMD) is based on the Impulse Approximation (IA). In general, the EMD even in simple metals cannot be well represented by the mean-field Independent Particle Model (IPM). In other words, the many-electron wave function is not well described by a single Slater determinant. Instead the momentum density has to be constructed from a correlated state with average occupancies in between 0 and 1. For the Homogeneous Electron Gas (HEG) considerable effort has been made to deduce the size of the discontinuity Z at the Fermi momentum. The difference between the interacting and free HEG momentum distributions for several electron densities yields the Lam-Platzman Correction (LPC) within the Density Functional Theory (DFT). Since the Quantum Monte Carlo (QMC) method applied to the HEG is used to determine DFT correlation potentials [1], a consistent treatment of the LPC should utilize the results of these same QMC simulations. The LPC gives a redistribution of the EMD from small momentum regions to larger momentum values. It describes some interactions between the electrons beyond the IPM. However, the LPC is spherically symmetric while the correction in real materials should be anisotropic. For more realistic corrections, one can perform QMC or GW calculations. These calculations are much more time consuming than the DFT computations. Moreover it was found that QMC Compton profiles are still to high at zero momentum as compared to experiments in materials ranging from lithium [2] to silicon [3]. A recent GW calculation for lithium [4] agrees with the QMC simulation. This indicates that both the QMC and the GW methods do not capture all the effects observed with the Compton scattering. These effects could include, in some extent, the failure of the IA or multiple scattering corrections. However, Bouchard and Lhuillier have shown that the momentum density is also extremely sensitive to the way the anti-symmetry is implemented in the many-fermion wave function [5]. The current QMC and GW calculations assume that fermionic correlation is not catastrophically modified by interactions and that an adiabatic path exists between the free electron gas and the interacting liquid. In the QMC method, this assumption leads to the fixed node approximation. The standard picture of the interacting gas appears to be substantially correct in the aluminum Compton profile [6], but in other materials some deviations from may occur as soon as the Fermi surface is not rotationally symmetric. Two recent examples are given by the experimental Compton profiles of beryllium [7] and copper [8]. It is therefore worthwhile to investigate other ways to implement anti-symmetry such as the Antisymmetrized Geminal Product (AGP) [9]. The AGP yields an orbital-dependent approach in which the momentum density is constructed using the natural orbitals, and the corresponding occupation numbers are obtained through a variational procedure. Sometime it is advantageous to introduce the Wannier orbitals via the natural orbitals [10] in order to study the degree of locality of bonding properties. The range of the Wannier orbitals is a fundamental ingredient for understanding the existence of insulators and metals and, in particular, the Metal-Insulator Transition (MIT). Electron localization in a MIT can be produced both by correlation effects and by disorder in an otherwise ideal crystal lattice. Actually, the path integral formalism [11] connects the theory of interacting electrons to that of disordered systems, where electrons are moving in randomly distributed external potentials. Thus, correlation and disorder can produce some similar effects in the EMD. This work is supported in part by the US Department of Energy contract W-