Hartree-Fock Density Research Articles

Steering Orbital Optimization out of Local Minima and Saddle Points Toward Lower Energy.

The general procedure underlying Hartree-Fock and Kohn-Sham density functional theory calculations consists in optimizing orbitals for a self-consistent solution of the Roothaan-Hall equations in an iterative process. It is often ignored that multiple self-consistent solutions can exist, several of which may correspond to minima of the energy functional. In addition to the difficulty sometimes encountered to converge the calculation to a self-consistent solution, one must ensure that the correct self-consistent solution was found, typically the one with the lowest electronic energy. Convergence to an unwanted solution is in general not trivial to detect and will deliver incorrect energy and molecular properties and accordingly a misleading description of chemical reactivity. Wrong conclusions based on incorrect self-consistent field convergence are particularly cumbersome in automated calculations met in high-throughput virtual screening, structure optimizations, ab initio molecular dynamics, and in real-time explorations of chemical reactivity, where the vast amount of data can hardly be manually inspected. Here, we introduce a fast and automated approach to detect and cure incorrect orbital convergence, which is especially suited for electronic structure calculations on sequences of molecular structures. Our approach consists of a randomized perturbation of the converged electron density (matrix) intended to push orbital convergence to solutions that correspond to another stationary point (of potentially lower electronic energy) in the variational parameter space of an electronic wave function approximation.

Nucleon localization and fragment formation in nuclear fission

An electron localization measure was originally introduced to characterize chemical bond structures in molecules. Recently, a nucleon localization based on Hartree-Fock densities has been introduced to investigate $\alpha$-cluster structures in light nuclei. Compared to the local nucleonic densities, the nucleon localization function has been shown to be an excellent indicator of shell effects and cluster correlations. Using the spatial nucleon localization measure, we investigate the emergence of fragments in fissioning heavy nuclei. To illustrate basic concepts of nucleon localization, we employ the self-consistent energy density functional method with a quantified energy density functional optimized for fission studies. We study the particle densities and spatial nucleon localization distributions along the fission pathways of $^{264}$Fm, $^{232}$Th and $^{240}$Pu. We demonstrate that the fission fragments are formed fairly early in the evolution, well before scission. We illustrate the usefulness of the localization measure by showing how the hyperdeformed state of $^{232}$Th can be understood in terms of a quasimolecular state made of $^{132}$Sn and $^{100}$Zr fragments. Compared to nucleonic distributions, the nucleon localization function more effectively quantifies nucleonic clustering: its characteristic oscillating pattern, traced back to shell effects, is a clear fingerprint of cluster/fragment configurations. This is of particular interest for studies of fragment formation and fragment identification in fissioning nuclei.

Unphysical divergences in response theory.

Transition densities between excited states are key for nonlinear theoretical spectroscopy and multi-state non-adiabatic molecular dynamics (NAMD) simulations. In the framework of response theory, these transition densities are accessible from poles of the quadratic response function. It was shown recently that the thus obtained transition densities within time-dependent Hartree-Fock (TDHF) and adiabatic time-dependent density functional theory (TDDFT) exhibit unphysical divergences when the difference in excitation energy of the two states of interest matches another excitation energy. This unphysical behavior is a consequence of spurious poles in the quadratic response function. We show that the incorrect pole structure of the quadratic response is not limited to TDHF and adiabatic TDDFT, but is also present in many other approximate many-electron response functions, including those from coupled cluster and multiconfigurational self-consistent field response theory. The divergences appear in regions of the potential energy surface where the ground state is perfectly well behaved, and they are frequently encountered in NAMD simulations of photochemical reactions. The origin of the divergences is traced to an incorrect instantaneous time-dependence of the effective Hamiltonian. The implications for computations of frequency-dependent response properties are considerable and call into question the validity of conventional approximate many-electron response theories beyond linear response.

Computation of indirect nuclear spin-spin couplings with reduced complexity in pure and hybrid density functional approximations.

We present a (sub)linear-scaling algorithm to determine indirect nuclear spin-spin coupling constants at the Hartree-Fock and Kohn-Sham density functional levels of theory. Employing efficient integral algorithms and sparse algebra routines, an overall (sub)linear scaling behavior can be obtained for systems with a non-vanishing HOMO-LUMO gap. Calculations on systems with over 1000 atoms and 20 000 basis functions illustrate the performance and accuracy of our reference implementation. Specifically, we demonstrate that linear algebra dominates the runtime of conventional algorithms for 10 000 basis functions and above. Attainable speedups of our method exceed 6 × in total runtime and 10 × in the linear algebra steps for the tested systems. Furthermore, a convergence study of spin-spin couplings of an aminopyrazole peptide upon inclusion of the water environment is presented: using the new method it is shown that large solvent spheres are necessary to converge spin-spin coupling values.

Multiferroic nature of intrinsic point defects inBiFeO3: A hybrid Hartree-Fock density functional study

To achieve a fundamental understanding of the multiferroic behavior and electronic properties of intrinsic vacancies in $\mathrm{BiFe}{\mathrm{O}}_{3}$, here we performed first-principles calculations based on hybrid Hartree-Fock density functional theories, which can accurately describe defect electronic structures. Oxygen vacancies, which behave as deep donors with high concentrations under oxygen-poor conditions, reduce the magnetic moments at neighboring Fe ions in the neutral state, while charged oxygen vacancies induce additional ferroelectric polarizations. Cation vacancies, on the other hand, are likely to form under oxygen-rich conditions and result in multiferroic properties distinct from those induced by oxygen vacancies. Bi vacancies act as triple-shallow acceptors and strongly suppress spontaneous polarization regardless of charge states, while Fe vacancies locally interfere with both electric and spin polarization and are thus regarded as multiferroic singular points in $\mathrm{BiFe}{\mathrm{O}}_{3}$. A rich variety of the multiferroic behavior of vacancies can be systematically understood from the localized/delocalized features of defect states, and the different formation conditions for vacancies provide a strategy to tailor the multiferroic properties of $\mathrm{BiFe}{\mathrm{O}}_{3}$ through control of the concentration and charge states of vacancies.

Strong-field absorption and emission of radiation in two-electron systems calculated with time-dependent natural orbitals

Recently introduced time-dependent renormalized-natural-orbital theory (TDRNOT) is based on the equations of motion for the so-called natural orbitals, i.e., the eigenfunctions of the one-body reduced density matrix. Exact TDRNOT can be formulated for any time-dependent two-electron system in either spin configuration. In this paper, the method is tested against high-order harmonic generation (HHG) and Fano profiles in absorption spectra with the help of a numerically exactly solvable one-dimensional model He atom, starting from the spin-singlet ground state. Such benchmarks are challenging because Fano profiles originate from transitions involving autoionizing states, and HHG is a strong-field phenomenon well beyond linear response. TDRNOT with just one natural orbital per spin in the helium spin-singlet case is equivalent to time-dependent Hartree-Fock or time-dependent density functional theory (TDDFT) in exact exchange-only approximation. It is not unexpected that TDDFT fails in reproducing Fano profiles due to the lack of doubly excited, autoionizing states. HHG spectra, on the other hand, are widely believed to be well-captured by TDDFT. However, HHG spectra of helium may display a second plateau that originates from simultaneous HHG in He$^+$ and neutral He. It is found that already TDRNOT with two natural orbitals per spin is sufficient to capture this effect as well as the Fano profiles on a qualitative level. With more natural orbitals (6--8 per spin) quantitative agreement can be reached. Errors due to the truncation to a finite number of orbitals are identified.

Improved DFT Potential Energy Surfaces via Improved Densities.

Density-corrected DFT is a method that cures several failures of self-consistent semilocal DFT calculations by using a more accurate density instead. A novel procedure employs the Hartree-Fock density to bonds that are more severely stretched than ever before. This substantially increases the range of accurate potential energy surfaces obtainable from semilocal DFT for many heteronuclear molecules. We show that this works for both neutral and charged molecules. We explain why and explore more difficult cases, for example, CH(+), where density-corrected DFT results are even better than sophisticated methods like CCSD. We give a simple criterion for when DC-DFT should be more accurate than self-consistent DFT that can be applied for most cases.

Second-order Møller–Plesset perturbation (MP2) theory at finite temperature: relation with Surján’s density matrix MP2 and its application to linear-scaling divide-and-conquer method

In 2005, Surjan showed two explicit formulas for evaluating the second-order Moller–Plesset perturbation (MP2) energy as a functional of the Hartree–Fock density matrix D (Chem Phys Lett 406:318, 2005 ), which are referred to as the Δ E MP2[D] functionals. In this paper, we present the finite-temperature (FT) MP2 energy functionals of the FT Hartree–Fock density matrix. There are also two formulas for the FT-MP2, namely the conventional and renormalized ones; the latter of which has recently been formulated by Hirata and He (J Chem Phys 138:204112, 2013 ). We proved that there exists one-to-one correspondence between the formulas of two FT-MP2 and the Δ E MP2[D] functionals. This fact can explain the different behavior of two Δ E MP2[D] functionals when an approximate Hartree–Fock density matrix is applied, which was previously investigated by Kobayashi and Nakai (Chem Phys Lett 420:250, 2006 ). We also applied the FT-MP2 formalisms to the linear-scaling divide-and-conquer method for improving the accuracy with tiny addition of the computational efforts.

Positive semidefinite tensor factorizations of the two-electron integral matrix for low-scaling ab initio electronic structure.

Tensor factorization of the 2-electron integral matrix is a well-known technique for reducing the computational scaling of ab initio electronic structure methods toward that of Hartree-Fock and density functional theories. The simplest factorization that maintains the positive semidefinite character of the 2-electron integral matrix is the Cholesky factorization. In this paper, we introduce a family of positive semidefinite factorizations that generalize the Cholesky factorization. Using an implementation of the factorization within the parametric 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)], we study several inorganic molecules, alkane chains, and potential energy curves and find that this generalized factorization retains the accuracy and size extensivity of the Cholesky factorization, even in the presence of multi-reference correlation. The generalized family of positive semidefinite factorizations has potential applications to low-scaling ab initio electronic structure methods that treat electron correlation with a computational cost approaching that of the Hartree-Fock method or density functional theory.

The electronic properties of impurities (N, C, F, Cl, and S) in Ag3PO4: A hybrid functional method study.

The transition energies and formation energies of N, C, F, Cl, and S as substitutional dopants in Ag3PO4 are studied using first-principles calculations based on the hybrid Hartree-Fock density functional, which correctly reproduces the band gap and thus provides the accurate defect states. Our results show that NO and CO act as deep acceptors, FO, ClO, and SP act as shallow donors. NO and CO have high formation energies under O-poor condition therefore they are not suitable for p-type doping Ag3PO4. Though FO, ClO, and SP have shallow transition energies, they have high formation energies, thus FO, ClO, and SP may be compensated by the intrinsic defects (such as Ag vacancy) and they are not possible lead to n-type conductivity in Ag3PO4.

A correction for the Hartree-Fock density of states for jellium without screening.

We revisit the Hartree-Fock (HF) calculation for the uniform electron gas, or jellium model, whose predictions--divergent derivative of the energy dispersion relation and vanishing density of states (DOS) at the Fermi level--are in qualitative disagreement with experimental evidence for simple metals. Currently, this qualitative failure is attributed to the lack of screening in the HF equations. Employing Slater's hyper-Hartree-Fock (HHF) equations, derived variationally, to study the ground state and the excited states of jellium, we find that the divergent derivative of the energy dispersion relation and the zero in the DOS are still present, but shifted from the Fermi wavevector and energy of jellium to the boundary between the set of variationally optimised and unoptimised HHF orbitals. The location of this boundary is not fixed, but it can be chosen to lie at arbitrarily high values of wavevector and energy, well clear from the Fermi level of jellium. We conclude that, rather than the lack of screening in the HF equations, the well-known qualitative failure of the ground-state HF approximation is an artifact of its nonlocal exchange operator. Other similar artifacts of the HF nonlocal exchange operator, not associated with the lack of electronic correlation, are known in the literature.

Molecular Structure, Vibrational Assignments and Non-Linear Optical Properties of 4,4’ Dimethylaminocyanobiphenyl (DMACB) by DFT and <i>ab Initio</i> HF Calculations

In the present study, structural properties of 4,4’ dimethylaminocyanobiphenyl (DMACB) have been studied extensively by using ab initio Hartree Fock (HF) and density functional theory (DFT) employing B3LYP/B3PW91 exchange correlation levels of theory. The vibrational frequencies of DMACB in the ground state have been calculated by using Hartree Fock level and density functional method (B3LYP/B3PW91) with 6-31G(d, p), basis set. Nonlinear optical (NLO) behavior of the examined molecule is investigated by the determination of the electric dipole moment μ, the polarizability α, and the hyperpolarizability β by using the B3LYP/B3PW91 methods.

Mixed pentele-chalcogen cationic chains from aluminum and gallium halide melts.

The reactions of tellurium or selenium with bismuth or antimony in chloridogallate and iodidoaluminate melts in the presence of group 15 trihalides as weak oxidants yielded the compounds (Sb2Te2)[GaCl4] (1), (Sb2Te2)I[AlI4] (2), (Bi2Te2)Cl[GaCl4] (3a), (Bi2Se2)Cl[GaCl4] (3b), (Sb3Te4)[GaCl4] (4), and (SbTe4)[Ga2Cl7] (5). In the crystal structures one-dimensional polymeric cations (Sb2Te2(+))n (1), (Sb2Te2(2+))n (2), (Bi2Te2(2+))n (3a), (Bi2Se2(2+))n (3b), (Sb3Te4(+))n (4), and (SbTe4(+))n (5) are present. The polymeric cationic strands in 2, 3a, 3b, and 4 consist of pentele/chalcogen dumbbells, which are connected to ladder-shaped bands. The strands in 1 and 5 consist of condensed rings that involve four-membered Sb2Te2 rings for 1, and five-membered SbTe4 rings for 5. The counteranions are the weakly coordinating [GaCl4](-), [AlI4](-), and [Ga2Cl7](-) in addition to Cl(-) and I(-) anions, which are coordinated to the atoms of the cations. The crystal structures of 1-4 are characterized by a statistical disorder in the anions with alternatively occupied positions for the Al and Ga atoms. For 4 superstructure reflections appear in the diffractions patterns, indicating a partial order. A correct assignment of the Sb and Te positions in the cation of 5 was achieved by periodic quantum-chemical calculations, which were performed via a Hartree-Fock density functional theory hybrid method. A clear preference of the 4-fold coordinated site was obtained for Sb.

Spectroscopic studies and quantum chemical investigations of (3,4-dimethoxybenzylidene) propanedinitrile

The Fourier Transform Infrared (FTIR), Ultra-Violet Visible (UV–Vis) spectroscopy and Thermogravimetric (TG) analysis of (3,4-dimethoxybenzylidene) propanedinitrile have been carried out and investigated using quantum chemical calculations. The molecular geometry, harmonic vibrational frequencies, Mulliken charges, natural atomic charges and thermodynamic properties in the ground state have been investigated by using Hartree Fock Theory (HF) and Density Functional Theory (DFT) using B3LYP functional with 6-311G(d,p) basis set. Both HF and DFT methods yield good agreement with the experimental data. Vibrational modes are assigned with the help of Vibrational Energy Distribution Analysis (VEDA) program. UV–Visible spectrum was recorded in the spectral range of 190–800nm and the results are compared with the calculated values using TD-DFT approach. Stability of the molecule arising from hyperconjugative interactions, charge delocalization have been analyzed using natural bond orbital (NBO) analysis. The results obtained from the studies of Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) are used to calculate molecular parameters like ionization potential, electron affinity, global hardness, electron chemical potential and global electrophilicity.

MPW1PW91‐Calculated Structures, IR Spectra, and Frontier Orbitals of Benzoylmethoxythiacalix[4]arene Complexed with Alkali Metal Ions

The molecular structures of four conformers [cone (CONE), partial cone (PACO), 1,2‐alternate (12A), and 1,3‐alternate (13A)] of benzoylmethoxythiacalix[4]arene 1 and the corresponding alkali metal ion complexes were optimized by using a mPW1PW91/6‐31G(d,p) (hybrid Hartree–Fock density functional) calculation method. The 13A conformer was the most stable among the various conformers of 1 because of less steric hindrance. The total electronic and Gibbs free energies, complexation energies, and the gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) frontier orbitals of the complexes of 1 with the sodium and potassium ions were analyzed. Among the ditopic complexes, the 13A‐type 1•2Na+ complex exhibited the strongest binding efficiency. The binding efficiency of the CONE‐type 1•Na+ complex was much stronger than that of the 12A‐type 1•Na+ complex. However, the 1•K+ complexes were relatively insensitive to the conformation of host 1. The calculated binding preference of the various conformers is consistent with the empirical solution data. The infrared spectra of the conformers of 1 and their exo complexes with the alkali metal ions were calculated by mPW1PW91 and analyzed.

Multi-level coupled cluster theory.

We present a general formalism where different levels of coupled cluster theory can be applied to different parts of the molecular system. The system is partitioned into subsystems by Cholesky decomposition of the one-electron Hartree-Fock density matrix. In this way the system can be divided across chemical bonds without discontinuities arising. The coupled cluster wave function is defined in terms of cluster operators for each part and these are determined from a set of coupled equations. The total wave function fulfills the Pauli-principle across all borders and levels of electron correlation. We develop the associated response theory for this multi-level coupled cluster theory and present proof of principle applications. The formalism is an essential tool in order to obtain size-intensive complexity in the calculation of local molecular properties.

First-principles study of nitrogen doping and oxygen vacancy in cubic PbTiO3

The structural and electrical properties of the intrinsic PbTiO3, N-doped PbTiO3 and N-VO co-doping PbTiO3 have been studied by using the first-principles calculations based on the hybrid Hartree–Fock density functionals. In addition, Hall effect measurement experiment is performed, both the experimental and theoretical results are compared. In N-doped PbTiO3 system, the N impurity states occur at the top of valance band, and Fermi energy level intersects with valance bands. The band gap becomes narrowed, the N-doped PbTiO3 shows a typical p-type semiconductor characteristic. When oxygen vacancy and N impurity coexist in PbTiO3, the conduction bands move downward and the acceptors are found to be fully compensated.

Error accumulations in adhesive energies of dihydrogen molecular chains: performances of the XYG3 type of doubly hybrid density functionals.

We have systematically analyzed the error accumulations in the adhesive energies for a series of hydrogen molecular chains calculated by various kinds of density functional theory (DFT) methods. In particular, we have focused on some representative doubly hybrid (DH) functionals of either the B2PLYP type (B2PLYP, B2PLYP-D, and B2GP-PLYP) or the XYG3 type (XYG3, XYGJ-OS, and xDH-PBE0). The hydrogen molecular chain models have recently been proposed by Zheng et al. (J. Chem. Phys. 2012, 137, 214106) to identify the delocalization errors (DEs) in thermodynamic properties. From the perspective of DEs, it is shown here that the XYG3 type of DH functionals yield good performance on the calculated adhesive energies due to the minimizing effects of DEs, highlighting the underlying physics for the successes or failures of the approximate functionals. Examination was also extended to HF-DFT, where DFT energies are evaluated with the Hartree-Fock (HF) densities.

Fuzzy electron density fragments in macromolecular quantum chemistry, combinatorial quantum chemistry, functional group analysis, and shape-activity relations.

Conspectus Just as complete molecules have no boundaries and have "fuzzy" electron density clouds approaching zero density exponentially at large distances from the nearest nucleus, a physically justified choice for electron density fragments exhibits similar behavior. Whereas fuzzy electron densities, just as any fuzzy object, such as a thicker cloud on a foggy day, do not lend themselves to easy visualization, one may partially overcome this by using isocontours. Whereas a faithful representation of the complete fuzzy density would need infinitely many such isocontours, nevertheless, by choosing a selected few, one can still obtain a limited pictorial representation. Clearly, such images are of limited value, and one better relies on more complete mathematical representations, using, for example, density matrices of fuzzy fragment densities. A fuzzy density fragmentation can be obtained in an exactly additive way, using the output from any of the common quantum chemical computational techniques, such as Hartree-Fock, MP2, and various density functional approaches. Such "fuzzy" electron density fragments properly represented have proven to be useful in a rather wide range of applications, for example, (a) using them as additive building blocks leading to efficient linear scaling macromolecular quantum chemistry computational techniques, (b) the study of quantum chemical functional groups, (c) using approximate fuzzy fragment information as allowed by the holographic electron density theorem, (d) the study of correlations between local shape and activity, including through-bond and through-space components of interactions between parts of molecules and relations between local molecular shape and substituent effects, (e) using them as tools of density matrix extrapolation in conformational changes, (f) physically valid averaging and statistical distribution of several local electron densities of common stoichiometry, useful in electron density databank mining, for example, in medicinal drug design, and (g) tools for combinatorial quantum chemistry approaches using fuzzy fragment databanks and rapid construction of a large number of approximate electron densities for large sets of related molecules, relevant in theoretical molecular and nanostructure design.

Identifying the Hamiltonian structure in linear response theory.

We present a unifying framework for linear response eigenvalue equations that encompasses both variational Hartree-Fock and Kohn-Sham density functional theory as well as non-variational coupled-cluster theory. The joint description is rooted in the so-called Hamiltonian structure of the response kernel matrices, whose properties permit an immediate identification of the well-known paired eigenvalue spectrum describing a molecule in the isolated state. Recognizing the Hamiltonian structure underlying the equations further enables a generalization to the case of a polarizable-embedded molecule treated in variational and, in particular, in non-variational theories.