Can X-ray constrained Hartree-Fock wavefunctions retrieve electron correlation?

Alessandro Genoni,Piero Macchi,Leonardo H R Dos Santos,Benjamin Meyer

doi:10.1107/s2052252516019217

Abstract

The X-ray constrained wavefunction (XC-WF) method proposed by Jayatilaka [Jayatilaka & Grimwood (2001) ▸, Acta Cryst. A57, 76-86] has attracted much attention because it represents a possible third way of theoretically studying the electronic structure of atoms and molecules, combining features of the more popular wavefunction- and DFT-based approaches. In its original formulation, the XC-WF technique extracts statistically plausible wavefunctions from experimental X-ray diffraction data of molecular crystals. A weight is used to constrain the pure Hartree-Fock solution to the observed X-ray structure factors. Despite the wavefunction being a single Slater determinant, it is generally assumed that its flexibility could guarantee the capture, better than any other experimental model, of electron correlation effects, absent in the Hartree-Fock Hamiltonian but present in the structure factors measured experimentally. However, although the approach has been known for long time, careful testing of this fundamental hypothesis is still missing. Since a formal demonstration is impossible, the validation can only be done heuristically and, to accomplish this task, X-ray constrained Hartree-Fock calculations have been performed using structure factor amplitudes computed at a very high correlation level (coupled cluster) for selected molecules in isolation, in order to avoid the perturbations due to intermolecular interactions. The results show that a single-determinant XC-WF is able to capture the electron correlation effects only partially. The largest amount of electron correlation is extracted when: (i) a large external weight is used (much larger than what has normally been used in XC-WF calculations using experimental data); and (ii) the high-order reflections, which carry less information on the electron correlation, are down-weighted (or even excluded), otherwise they would bias the fitting towards the unconstrained Hartree-Fock wavefunction.

Highlights

As is well known, X-ray scattering is the Fourier image of the dynamic electron-density distribution
In agreement with previous studies (Gatti et al, 1988; Boyd & Wang, 1989), the effect of the electron correlation on (r) is tiny and difficult to capture, especially for methods based on fitting procedures like the X-ray constrained wavefunction or the more traditional multipolar expansions
These indices allowed us to assess the global performances of the X-ray constrained wavefunctions and, together with the other more local indicators mentioned above, enabled us to analyse the following features: (i) The overall similarity between the charge distributions, by means of the real-space R (RSR) value, the Carbodistance and the root-mean-square deviation (RMSD) and mean absolute deviation (MAD) indexes; (ii) The displacement of electron density due to electron correlation, visualized by electron-density differences along representative chemical bonds, as well as by difference density maps and attachment and detachment densities; Figure 3 Comparison between the CCSD electron densities and the RHF, XC-WF/0.5, XC-WF/0.7, XC-WF/1.2 and XC-WF/2.0 charge distributions along some selected chemical bonds of the six investigated molecules

Summary

Introduction

X-ray scattering is the Fourier image of the dynamic electron-density distribution. It is well established that, by fitting suitable multipole models or through maximum likelihood estimations, one can reconstruct accurate electron-density maps from the X-ray diffraction of a crystal. There are three main levels of detail that one may observe: (i) the hybridization of the atomic orbitals due to chemical bonding; (ii) the electron polarization caused by inter- or intramolecular electrostatic interactions; and (iii) the tiny electron-density redistribution, which reflects the instantaneous electron–electron repulsions (electron correlation). 4, 136–146 research papers covalent bonds or to the localization of lone pairs. They can be proposed by Perdew & Schmidt (2001). Intra- or of the ladder to the does not always guarantee a intermolecular electrostatic interactions induce electron- systematic improvement in the calculations

Objectives

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Conclusion