X-ray molecular orbital analysis. I. Quantum mechanical and crystallographic framework

Abstract

Molecular orbitals were obtained by X-ray molecular orbital analysis (XMO). The initial molecular orbitals (MOs) of the refinement were calculated by the ab initio self-consistent field (SCF) MO method. Well tempered basis functions were selected since they do not produce cusps at the atomic positions on the residual density maps. X-ray structure factors calculated from the MOs were fitted to observed structure factors by the least-squares method, keeping the orthonormal relationship between MOs. However, the MO coefficients correlate severely with each other, since basis functions are composed of similar Gaussian-type orbitals. Therefore, a method of selecting variables which do not correlate severely with each other in the least-squares refinement was devised. MOs were refined together with the other crystallographic parameters, although the refinement with the atomic positional parameters requires a lot of calculation time. The XMO method was applied to diformohydrazide, (NHCHO), without using polarization functions, and the electron-density distributions, including the maxima on the covalent bonds, were represented well. Therefore, from the viewpoint of X-ray diffraction, it is concluded that the MOs averaged by thermal vibrations of the atoms were obtained successfully by XMO analysis. The method of XMO analysis, combined with X-ray atomic orbital (AO) analysis, in principle enables one to obtain MOs or AOs without phase factors from X-ray diffraction experiments on most compounds from organic to rare earth compounds.

Quantum Crystallography in the Last Decade: Developments and Outlooks

“…In this context, different attempts have been proposed over the years. Just to cite a few of them, we mention the pioneering works of Clinton, Massa and their coworkers since the early 1960s [22][23][24][25][26][27][28][29][30] or the more recent molecular orbital occupation number (MOON) approach [37,38] and Tanaka's X-ray atomic orbital (XAO) [39] and X-ray molecular orbital (XMO) methods [40]. Nevertheless, within this family of techniques, the strategies that emerged the most or witnessed a significant evolution in the last decade were essentially two: (i) the X-ray constrained/restrained wavefunction (XCW/XRW) method originally devised by Jayatilaka and (ii) the approaches proposed by Gillet and collaborators to refine (spin-resolved) one-particle reduced density matrices (1-RDMs).…”

X-ray molecular orbital analysis. II. Application to diformohydrazide, (NHCHO)2

“…X-ray molecular orbital (XMO) analysis was proposed in our previous paper (Tanaka, 2018, hereinafter referred to as I). It determines molecular orbitals (MOs) from X-ray structure factors using a least-squares method, though the starting MOs for the nonlinear least-squares refinement were calculated employing the Hartree-Fock-Roothaan (HFR) equation.…”

The Advent of Quantum Crystallography: Form and Structure Factors from Quantum Mechanics for Advanced Structure Refinement and Wavefunction Fitting

“…The pioneering studies in this field date back to the 1960s when Clinton, Massa and collaborators proposed strategies to extract N-representable one-electron density matrices from X-ray diffraction data [96][97][98][99][100]. These studies were the starting points for the development of other techniques to obtain "experimental" wavefunctions and density matrices, such as Tanaka's X-ray Atomic Orbital [40] (XAO) and X-ray Molecular Orbital [101] (XMO) strategies, the Molecular Orbital Occupation Number [102, 103] (MOON) method and all the approaches aiming at reconstructing the diagonal and the off-diagonal parts of the one-electron density matrices by simultaneously exploiting X-ray diffraction data, magnetic structure factors and inelastic…”