Hartree-Fock Electron Densities Research Articles

Unconventional Error Cancellation Explains the Success of Hartree-Fock Density Functional Theory for Barrier Heights.

Energy barriers, which control the rates of chemical reactions, are seriously underestimated by computationally efficient semilocal approximations for the exchange-correlation energy. The accuracy of a semilocal density functional approximation is strongly boosted for reaction barrier heights by evaluating that approximation non-self-consistently on Hartree-Fock electron densities, which has been known for ∼30 years. The conventional explanation is that the Hartree-Fock theory yields the more accurate density. This work presents a benchmark Kohn-Sham inversion of accurate coupled-cluster densities for the reaction H2 + F → HHF → H + HF and finds a strong, understandable cancellation between positive (excessively overcorrected) density-driven and large negative functional-driven errors (expected from stretched radical bonds in the transition state) within this Hartree-Fock density functional theory. This confirms earlier conclusions (Kaplan, A. D., et al. J. Chem. Theory Comput. 2023, 19, 532-543) based on 76 barrier heights and three less reliable, but less expensive, fully nonlocal density functional proxies for the exact density.

Density-Corrected Density Functional Theory for Molecular Properties.

Density-corrected (DC) density functional theory (DFT) has been proposed to overcome difficulties related to the self-interaction error. The procedure uses the Hartree-Fock electron density (matrix) non-self-consistently in conjunction with an approximate functional. DC-DFT has so far mainly been tested for total energy differences, whereas other types of molecular properties have not been evaluated systematically. This work focuses on the performance of DC-DFT for molecular properties, namely, dipole moments, static polarizabilities, and electric field gradients (EFGs) at atomic nuclei. Accurate reference data were generated from coupled-cluster theory to assess the performance of DC and self-consistent DFT calculations for twelve molecules, including diatomics with transition metals. DC-DFT does no harm in dipole moment calculations, but it negatively impacts the polarizability in at least one case. DC-DFT performs well for EFGs, even for the difficult case of CuCl.

On the relativistic impulse approximation for the calculation of Compton scattering cross sections and photon interaction coefficients used in kV dosimetry

We calculate differential and integrated cross sections for the Compton interaction as well as mass attenuation (), mass energy-transfer (), and mass energy-absorption () coefficients, within the relativistic impulse approximation (RIA) using Compton profiles (CPs) obtained from unrestricted Hartree–Fock electron densities. We investigate the impact of using molecular as opposed to atomic CPs on dosimetric photon interaction coefficients for air, water and graphite, and compare our cross sections to the simpler Waller–Hartree (WH) and Klein–Nishina (KN) formalisms. We find that differences in and resulting from the choice of CPs within the RIA are small relative to the differences between the RIA, WH, and KN calculations. Surprisingly, although the WH binding corrections seem accurate when considering , there are significant discrepancies between the WH and RIA results when we look at . The WH theory can differ substantially from the predictions of KN and the RIA in the tens of keV range (e.g. 6%–10% at 20 keV), when Compton scattering becomes the dominant interaction mechanism. For lower energies, the disagreement further grows to about one order of magnitude at 1 keV. However, since the photoelectric effect transfers more energy than the Compton interaction in the tens of keV range and below, the differences in the total values resulting from the choice of Compton models (KN, WH, or RIA) are not larger than 0.4%, and the differences between WH and the other two theories are no longer prominent.

Uniform electron gas limit of an exact expression for the Kohn–Sham exchange-correlation potential

Previously, we derived an exact formula for the Kohn–Sham exchange-correlation potential corresponding, in the basis-set limit, to the Hartree–Fock electron density of a given system. This formula expresses the potential in terms of the occupied Hartree–Fock and Kohn–Sham orbitals and orbital energies. Here, we show that, when applied to the Hartree–Fock description of a uniform electron gas, the formula correctly reduces to the exchange-only local density approximation.

The exact Fermi potential yielding the Hartree–Fock electron density from orbital‐free density functional theory

AbstractThe exact expression for the Fermi potential yielding the Hartree–Fock electron density within an orbital‐free density functional formalism is derived. The Fermi potential, which is defined as that part of the potential that depends on the particles’ nature, is in this context given as the sum of the Pauli potential and the exchange potential. The exact exchange potential for an orbital‐free density functional formalism is shown to be the Slater potential.

About the difference between density functionals defined by energy criterion and density functionals defined by density criterion: Exchange functionals

The difference between density functionals defined by energy criterion and density functionals defined by density criterion is studied for the exchange functional. It is shown that Slater potentials are exact exchange potentials in the sense that they yield the Hartree–Fock electron density if all operators are given by local expressions. © 2016 Wiley Periodicals, Inc.

Accurate and Efficient Approximation to the Optimized Effective Potential for Exchange

We devise an efficient practical method for computing the Kohn-Sham exchange-correlation potential corresponding to a Hartree-Fock electron density. This potential is almost indistinguishable from the exact-exchange optimized effective potential (OEP) and, when used as an approximation to the OEP, is vastly better than all existing models. Using our method one can obtain unambiguous, nearly exact OEPs for any reasonable finite one-electron basis set at the same low cost as the Krieger-Li-Iafrate and Becke-Johnson potentials. For all practical purposes, this solves the long-standing problem of black-box construction of OEPs in exact-exchange calculations.

Comprehensive study of some well-known molecular numerical integration methods

In a previous study, we compared, both in terms of accuracy and efficiency, the performance of some of the well-known grids, which use the Becke partitioning scheme for molecular numerical integration. We concluded, based on the number of electrons only, that the MultiExp grid performed well compared with the grids proposed by Becke, Gill et al., and Treutler and Ahlrichs. In this work, we re-examine the performance of the same set of grids in addition to the SG-0 grid and a benchmark grid. These grids are evaluated by integrating the Hartree–Fock electron density to calculate the number of electrons, dipole moment, potential energy, and Coulomb repulsion energy. Our results show that, except for the large benchmark grid, none of these grids were completely satisfactory.

The use of dipole lattice sums to estimate electric fields and dipole moment enhancement in molecular crystals

We describe the use of dipole lattice sums to estimate the electric field experienced by a molecule in a crystal, and apply it iteratively to ab initio calculations on a number of molecules of interest for nonlinear optical applications. The results demonstrate that it represents an efficient and reliable route to estimating molecular dipole moments in crystals. Typical dipole moment enhancements are in the range of 10–40%, agreeing well with the range resulting from partitioning of crystal Hartree–Fock electron densities, but much less than has been suggested in some previous work, and in particular from charge density analysis of X-ray diffraction data.

Electric field‐derived point charges to mimic the electrostatics in molecular crystals

Because of the way the electrostatic potential is defined in a crystal, it is not possible to determine potential-derived charges for atoms in a crystal. To overcome this limitation, we present a novel method for determining atomic charges for a molecule in a crystal based on a fit to the electric field at points on a surface around the molecule. Examples of fits to the electric field at points on a Hirshfeld surface, using crystal Hartree-Fock electron densities computed with a DZP basis set are presented for several organic molecular crystals. The field-derived charges for common functional groups are transferable, and reflect chemical functionality as well as the subtle effects of intermolecular interactions. The charges also yield an excellent approximation to the electric field surrounding a molecule in a crystal for use in cluster calculations on molecules in solids.

Dipole and quadrupole moments of molecules in crystals: A novel approach based on integration over Hirshfeld surfaces

Elegant expressions are derived for the computation of dipole and quadrupole moments of molecules using the electrostatic potential and electric field evaluated on an oriented molecular surface. These expressions are implemented for Hirshfeld surfaces, applied to various molecular crystals, and compared with the results from the quantum theory of atoms in molecules. The effect of intermolecular interactions is also explored by examining the differences between electrostatic moments derived from a periodic Hartree-Fock electron density and an electron density resulting from a superposition of noninteracting molecules. The enhancement of the dipole moment for hydrogen bonded molecular crystals is typically 30%-40% and shown to be largely independent of the partitioning scheme. Dipole moments calculated from Hirshfeld surfaces systematically underestimate those from zero-flux surfaces, a result attributed to the translation of the Hirshfeld surface relative to the zero-flux surfaces for these molecules. For acetylene and benzene, the differences between a crystal calculation and the sum of noninteracting molecules are small, and both partitioning schemes yield quadrupole and second moment results in close agreement.

Differential cross sections for elastic scattering of electrons by atoms and solids

Differential cross sections (DCSs) for elastic scattering of electrons by neutral atoms are extensively used in studies of electron transport in solids and liquids. A new NIST database has recently been released with DCSs calculated from a relativistic Dirac partial-wave analysis in which the potentials were obtained from Dirac–Hartree–Fock electron densities computed self-consistently for free atoms. We have compared calculated DCSs with measured DCSs for argon for electron energies between 50 eV and 3 keV, and found good agreement for electron energies above about 1 keV but with increasing deviations as the energy is reduced. These deviations are due to the neglect of absorption and polarizability effects in the calculations. Nevertheless, DCSs for neutral atoms have been successfully used in simulations of elastic backscattering of electrons by solid surfaces with energies down to 300 eV as well as for many other applications. It is suggested that this success might be due at least partially to the smaller absorption correction for the DCSs in solids on account of the smaller total inelastic scattering cross sections than for the corresponding free atoms.

CALCULATING MEAN EXCITATION ENERGY FROM SHANNON INFORMATION ENTROPY FOR K–Xe ATOMS

The calculation of the Shannon entropy of Hartree–Fock electron density is reported for the K–Xe atoms in the fourth and fifth rows of the periodic table in the ground state. In local plasma approximation, the Shannon entropy calculated is used directly to estimate mean excitation energies. To our knowledge, the mean excitation energies of K–Xe atoms are presented here for the first time.

The role of the local-multiplicative Kohn–Sham potential on the description of occupied and unoccupied orbitals

The optimum local-multiplicative exchange potential was found using as input the Hartree–Fock electron density, for the molecular systems: H2, LiH, HF, NH3, CH4, H2O, N2, CO, F2, C2H2 and C2H4. The Zhao and Parr method was used to obtain the local-multiplicative potential where the kinetic energy is minimized using a constrained-search formulation of density functional theory. Two orbital sets were compared, those obtained with the nonlocal Hartree–Fock potential and those obtained with the local-multiplicative potential, both sets yielding the same electron density. As expected, the highest occupied molecular orbital (HOMO) energy was similar in both orbital sets. In contrast, the virtual orbital energies, and in particular the lowest unoccupied molecular orbital (LUMO), exhibited considerable differences. The Hartree–Fock LUMO energy goes to zero in a complete basis set limit and to nearly zero with reasonably large basis sets (e.g., augmented triple zeta) with sufficient diffuse functions added. The LUMO provided by the local-multiplicative potential using the same large basis set goes to a bounded energy not equal to zero. The nonlocal Hartree–Fock potential generates a large gap between the HOMO and LUMO energies; this difference is equal to the negative of the HOMO energy at the complete basis set limit. Contrary to this behavior, the gap obtained with the local-multiplicative potential is a reasonable approximation to the lowest experimental vertical excitation energy. For some of the molecules tested, the ordering of the orbitals corresponding to the HF and local-multiplicative potential are different.

Molecular surfaces from the promolecule: A comparison with Hartree–Fock ab initio electron density surfaces

A new molecular surface, the promolecule electron density isosurface arising from the superposition of spherical atomic electron density functions, is compared and contrasted with the Hartree–Fock ab initio electron density isosurface, the fused-sphere van der Waals (CPK) surface, and the smooth Connolly surface. From application to a number of small to medium-sized molecules, including several amino acids, the promolecule surface is shown to be very similar to the Hartree–Fock electron density isosurface but, in contrast, is trivial to calculate. The promolecule electron density surface constructed with a contracted hydrogen atom provides remarkably reliable, and consistent, estimates of ab initio surface areas (typically within 0.5%) and volumes (routinely overestimated by less than 4%). Differences between ab initio and promolecule surfaces are explored visually by mapping the deformation density on the promolecule surface. To highlight the usefulness of the promolecule surface, a promolecule 0.002 au isosurface of the small protein crambin is shown. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 933–942, 2000

Molecular surfaces from the promolecule: A comparison with Hartree-Fockab initio electron density surfaces

A new molecular surface, the promolecule electron density isosurface arising from the superposition of spherical atomic electron density functions, is compared and contrasted with the Hartree–Fock ab initio electron density isosurface, the fused-sphere van der Waals (CPK) surface, and the smooth Connolly surface. From application to a number of small to medium-sized molecules, including several amino acids, the promolecule surface is shown to be very similar to the Hartree–Fock electron density isosurface but, in contrast, is trivial to calculate. The promolecule electron density surface constructed with a contracted hydrogen atom provides remarkably reliable, and consistent, estimates of ab initio surface areas (typically within 0.5%) and volumes (routinely overestimated by less than 4%). Differences between ab initio and promolecule surfaces are explored visually by mapping the deformation density on the promolecule surface. To highlight the usefulness of the promolecule surface, a promolecule 0.002 au isosurface of the small protein crambin is shown. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 933–942, 2000

Influence of intermolecular interactions on multipole-refined electron densities

This work examines the effect of intermolecular interactions on molecular properties derived from simulated X-ray diffraction data. Model X-ray data are computed from a superposition of ab initio molecular electron densities in the crystal, as well as from periodic crystal Hartree-Fock electron densities, for the hydrogen-bonded systems ice VIII, formamide and urea, as well as the weakly bound acetylene. The effects of intermolecular interactions on the electron density are illustrated at both infinite and finite data resolution, and it is concluded that multipole models are capable of quantitative retrieval of the interaction density, despite the known shortcomings of the radial functions in the model. Multipole refinement reveals considerable enhancement of the molecular dipole moment for hydrogen-bonded crystals, and negligible change in molecular second moments. Electric field gradients at H nuclei are significantly reduced in magnitude upon hydrogen bonding, and this change is also faithfully represented by the rigid pseudoatom model.

Critical points in a crystal and procrystal

The critical points in the model electron density distributions of LiF, NaF, NaCl, and MgO crystals, constructed from accurate X-ray diffraction data, are determined. For LiF and MgO they are compared with those obtained from a Hartree–Fock electron density calculation. Both experiment and theory show the same type of critical points on the bond lines. The topological features in areas between structural units, where the electron density is low and near-uniform, turn out to be model dependent and cannot be established well with the data available. Topological analysis of procrystals (hypothetical systems consisting of spherical atoms or ions placed on the same sites as atoms in real crystal) show that (3, −1) critical points, usually connected with bonding interaction, are observed on interatomic lines in these nonbonded systems as well.

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li+ to Cs+ and 43 anions from H− to I− in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions rniexp(−ζir). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54.

Analytical Hartree-Fock electron densities for atoms He through Lr

The Hartree–Fock electron density has an important property that it is identical to the exact density to first order in the perturbation theory. For the neutral atoms from He (Z=2) to Lr (Z=103) in their ground state, we report an accurate analytical approximation F(r) to the spherically averaged electron density ρ(r) obtained by the numerical Hartree–Fock method. The present density function F(r) is expressed by a linear combination of reasonable number (not more than 30) of basis functions rni exp(−ζir), and has the following properties: (i) F(r) is nonnegative, (ii) F(r) is normalized, (iii) F(r) reproduces the Hartree–Fock moments (k=−2 to +6), (iv) F(0) is equal to ρ(0), (v) F′(0) satisfies the cusp condition, and (vi) F(r) has the correct exponential decay in the long-range asymptotic region.