Hartree-Fock Self-consistent Field Research Articles

Geometric Optimization of Restricted-Open and Complete Active Space Self-Consistent Field Wave Functions.

We explore Riemannian optimization methods for Restricted-Open-shell Hartree-Fock (ROHF) and Complete Active Space Self-Consistent Field (CASSCF) methods. After showing that ROHF and CASSCF can be reformulated as optimization problems on so-called "flag manifolds", we review Riemannian optimization basics and their application to these specific problems. We compare these methods to traditional ones and find robust convergence properties without fine-tuning of numerical parameters. Our study suggests that Riemannian optimization is a valuable addition to orbital optimization for ROHF and CASSCF, warranting further investigation.

Molecular self-assembled helix peptide nanotubes based on some amino acid molecules and their dipeptides: molecular modeling studies.

The paper considers the features of the structure and dipole moments of several amino acids and their dipeptides which play an important role in the formation of the peptide nanotubes based on them. The influence of the features of their chirality (left L and right D) and the alpha-helix conformations of amino acids are taken into account. In particular, amino acids with aromatic rings, such as phenylalanine (Phe/F), and branched-chain amino acids (BCAAs)-leucine (Leu/L) and isoleucine (Ile/I)-as well as corresponding dipeptides (diphenylalanine (FF), dileucine (LL), and diisoleucine (II)) are considered. The main features and properties of these dipeptide structures and peptide nanotubes (PNTs), based on them, are investigated using computational molecular modeling and quantum-chemical semi-empirical calculations. Their polar, piezoelectric, and photoelectronic properties and features are studied in detail. The results of calculations of dipole moments and polarization, as well as piezoelectric coefficients and band gap width, for different types of helical peptide nanotubes are presented. The calculated values of the chirality indices of various nanotubes are given, depending on the chirality of the initial dipeptides-the results obtained are consistent with the law of changes in the type of chirality as the hierarchy of molecular structures becomes more complex. The influence of water molecules in the internal cavity of nanotubes on their physical properties is estimated. A comparison of the results of these calculations by various computational methods with the available experimental data is presented and discussed. The main tool for molecular modeling of all studied nanostructures in this work was the HyperChem 8.01 software package. The main approach used here is the Hartree-Fock (HF) self-consistent field (SCF) with various quantum-chemical semi-empirical methods (AM1, PM3, RM1) in the restricted Hartree-Fock (RHF) and in the unrestricted Hartree-Fock (UHF) approximations. Optimization of molecular systems and the search for their optimal geometry is carried out in this work using the Polak-Ribeire algorithm (conjugate gradient method), which determines the optimized geometry at the point of their minimum total energy. For such optimized structures, dipole moments D and electronic energy levels (such as EHOMO and ELUMO), as well as the band gap Eg = ELUMO - EHOMO, were then calculated. For each optimized molecular structure, the volume was calculated using the QSAR program implemented also in the HyperChem software package.

Improved ground state energies of electronegative atoms using Hartree–Fock self-consistent field approximation

Improved ground state energies of electronegative atoms using Hartree–Fock self-consistent field approximation

Kylin 1.0: An ab-initio density matrix renormalization group quantum chemistry program.

The accurate evaluation of electron correlations is highly necessary for the proper descriptions of the electronic structures in strongly correlated molecules, ranging from bond-dissociating molecules, polyradicals, to large conjugated molecules and transition metal complexes. For this purpose, in this paper, a new ab-initio quantum chemistry program Kylin 1.0 for electron correlation calculations at various quantum many-body levels, including configuration interaction (CI), perturbation theory (PT), and density matrix renormalization group (DMRG), is presented. Furthermore, fundamental quantum chemistry methods such as Hartree-Fock self-consistent field (HF-SCF) and the complete active space SCF (CASSCF) are also implemented. The Kylin 1.0 program possesses the following features: (1) a matrix product operator (MPO) formulation-based efficient DMRG implementation for describing static electron correlation within a large active space composed of more than 100 orbitals, supporting both and symmetries; (2) an efficient second-order DMRG-self-consistent field (SCF) implementation; (3) an externally contracted multi-reference CI (MRCI) and Epstein-Nesbet PT with DMRG reference wave functions for including the remaining dynamic electron correlation outside the large active spaces. In this paper, we introduce the capabilities and numerical benchmark examples of the Kylin 1.0 program.

Structural and spectroscopic study of neutral dioxides MO2 (M = B, Al, Ga, In, Tl)

Both ab initio Hartree-Fock self consistent field (HF-SCF) with Cluster Coupled (CCSD(T)) and Density Functional Theory with B3LYP functional, high level theories are adopted for structural study of different dioxides MO2 (M = B, Al, Ga, In and Tl). A geometry optimization is performed using the avtz, aug-cc-pwcvnz (nz = tz; qz; 5z), 6–311++G** and Lanl2DZ basis sets. Vibration frequencies are reported for optimized geometries. We also report calculated M-O and O-O bond lengths, O-M-O and M-O-O, bond angles for stable isomers. Our calculations indicate that there are five possible isomers with linear or folded shapes. Some discrepancies with previous results are noted, on the other hand, according to our CCSD(T) calculations, MO2 molecule (M = B, Al and Ga) is in ground state when it is linear and symmetric O-M-O (2Πg), then in a higher energy state with the cyclic isomer (2A2, or 2A1) and in more higher state when it is dissymmetric M-O-O (2A’, 2∑+), B3LYP and CCSD(T) results agree on energetic ordering of various isomeric forms for BO2, AlO2,InO2 and TlO2 systems, while there is disagreement for GaO2, molecule. HOMO-1, HOMO, LUMO and LUMO + 1 molecular orbitals for the most stable structures are given as well. NBO analysis is performed to study the superoxide 2A2 and its stability which evolves according to electropositivity of metal along the 13th column of periodic table.

Electron-impact cross-sections of atmospherically relevant amines from intermediate to 5000 eV energy range

The amines are major source of environment pollutants emitted in atmosphere from various anthropogenic sources. The non-thermal plasma (NTP)-based technology has proved successful in controlling the emitted amines reaching the atmosphere. The efficient NTP reactors rely on accurate electron–molecule collision data. The electron impact cross-sections are thus obtained for a few amines from ionisation threshold to 5000 eV using the single centre expansion (SCE) formalism. The molecular wave function of each target is obtained from the multicentre expansion of the Gaussian-type orbitals within a single determinant Hartree–Fock self-consistent field scheme. The expansion of wave function, density and potential is carried out at the centre of mass of the molecules. The interaction potential included to model the electron interaction in the target comprises static, correlation-polarisation and exchange types of potentials. The elastic cross-sections are obtained after solving the coupled scattering equations using Volterra integral form. The inelastic effects contributing to electron–molecule scattering are approximated by the ionisation cross-sections. The total cross-sections obtained after summing the elastic and ionisation cross-sections are in good agreement with the available data. We have also tried to explain the effect of polarisation potential on scattering cross-sections. A semiempirical formula based on the spatial extent of the molecule is proposed to estimate the cross-sections for the homologous series of amine molecules.

Phosphorus mononitride: A difficult case for theory

AbstractPhosphorus nitride (PN) is the simplest molecule formed solely by phosphorus and nitrogen. It represents an interesting model for materials, where phosphorus is directly attached to nitrogen. Nevertheless, both theoretical and experimental studies often provide an incomplete picture on the structural, electronic, and spectral properties of PN. Theoretical predictions often suffer from insufficient level of theory, incomplete basis set, or from neglecting several effects, for example, zero‐point vibrational correction (ZPVC). Therefore, we performed an extensive benchmark study on structural, electronic, and spectral properties of PN at the Hartree‐Fock, density functional theory (DFT), or even the coupled‐cluster levels. We paid special attention to the basis set effect. We tested three variants of Dunning's aug‐cc‐pVXZ basis sets with the size from double‐ζ to sextuple‐ζ, as well as Jensen's aug‐pc‐n, aug‐pcJ‐n, and aug‐pcSseg‐n basis sets, where n = 1‐4. Obtained energetics, PN distance, dipole moment, vibrational frequencies, and nuclear magnetic resonance (NMR) parameters were extrapolated to the complete basis set limit (CBS) using three‐ or two‐parameter formulas. The 31P NMR shieldings estimated with the aug‐cc‐pVXZ and aug‐cc‐pV(X + d)Z basis sets strongly depend on the basis set size providing scattered convergence patterns toward CBS. The Hartree‐Fock self‐consistent field (HF‐SCF) NMR parameters evinced similar behavior as the coupled‐cluster data. The only smooth convergence was achieved using the aug‐cc‐pCVXZ basis sets that include core‐valence effects. The KT3 functional underestimated the phosphorus CBS shieldings by about 12 ppm compared to coupled cluster with singles and doubles (CCSD) (T). Nevertheless, KT3 unambiguously surpasses the HF‐SCF and CCSD levels that provide 31P shieldings that are lower by about 150 ppm and 24 ppm compared to CCSD(T). The convergence of nitrogen shieldings was regular for all basis set hierarchies and all theoretical methods. Relativistic and vibrational effects on selected properties were also discussed.

A comparative study of electron-impact cross sections of C4F6 isomers from 15 to 5000 eV

Electron-impact differential, integral, and momentum transfer cross sections (CS) are computed for C4F6 isomers from 15 to 5000 eV by employing the Single Center Expansion formalism. The molecular wavefunctions of isomers are obtained using the multicenter expansion of Gaussian-type orbitals within the single determinant Hartree-Fock self-consistent field scheme. The electron-molecule interaction is modeled by summing the static, exchange, and correlation-polarization types of potentials. The exchange and correlation-polarization potentials account for the indistinguishability of incident electron and target electrons and the distortion of charge density of target by the impinging electron, respectively. The multipole expansion of the target at center of mass includes the dipole and higher order terms. The local description of potential permits us to rewrite the scattering radial equations in a simplified form. The electron impact ionization CS are obtained using the Binary-Encounter-Bethe model. The elastic and inelastic CS are summed incoherently to obtain total CS over a wide energy range. A good agreement is observed with the available data for different types of CS obtained.

Higher-term contributions in the many-body calculation of the compressibility and thermodynamic properties of solid neon

To investigate both compressibility and thermodynamic properties of solid face-centered cubic neon, the many-body potential energy, which is expanded as a sum of two- to five-body potentials, was calculated. The calculation used the ab initio Hartree–Fock self-consistent field method in combination with the many-body expansion method. The results indicate that the many-body expansion potential is an exchange convergent series, and the even many-body potential contributions to the cohesive energy are repulsive. The odd many-body potential contributions to the cohesive energy, on the other hand, are attractive. The absolute values of the many-body potential energy Un obey |Un| > |Un+1|. Both the many-body potential energy and the total potential energy tend to saturate with the increase in atomic numbers and neighboring shell numbers. When the atomic distance R exceeds 2.60 A, the interaction energy may be described by two-body interactions. For atomic distances between 1.80 and 2.60 A, a three-body contribution to the many-body expansion potential is required, while for distances between 1.60 and 1.80 A, four-body contributions need to be considered. The calculated isotherm is in good agreement with the obtained experimental results in the studied pressure range (0–237 GPa), which considers the four-body potential if the pressure reaches 240 GPa. Below 1.60 A, we have to consider the five-body potential to accurately match the experimental data (for the pressure up to 280 GPa). Overall, the inclusion of the high many-body interaction energy makes it possible to obtain the most accurate equation of the state for solid neon under ambient conditions and higher pressure.

Four-body Interaction and Equation of State for Solid Neon from Ab Initio Calculation

Using ab initio Hartree-Fock self-consistent field method combined with many-body expansion method, the investigation is based on the first-principles. We have considered two-, three- and four-body potential energies of face-centered cubic (fcc) solid neon of which the atomic distance ranges from 1.6 to 3.0 . By discussing the truncation and convergence of many-body potential of solid neon, we obtain the cohesive energy, the zero-point vibration energy and equation of state (EOS). The results show that, when the number of neighboring atoms increases, two-body, three-body, and four-body potential energy tend to a saturation value for a certain atomic distance ( ). The even many-body contributions to the cohesive energy, such as two-, four-body terms and so on, are positive, whereas the odd many-body contributions to the cohesive energy, such as three-, five-body terms and so on, are negative. The zero-point vibration energy of solid neon is only 6% of the total atomic interaction energy, but should not be neglected. Compared with the experimental data, the importance of the four-body interactions in compressed solid neon is emphasized. Only taking into account the two-body term, the pressure is overestimated, and our calculated results are in good agreement with the experimental values at the low-pressure regions (<15GPa). Adding three-body term up to 55GPa, considering the four-body term, it has a good consistency at the experimentally studied pressure range of 0~237GPa, and maybe helpful to accurately explain the phenomenon of the experiment above 237GPa when the higher many-body effects are considered.

Convergence of nuclear magnetic shieldings and one-bond 1 J(11 B1 H) indirect spin-spin coupling constants in small boron molecules.

Self-consistent field Hartree-Fock, density functional theory, and coupled-cluster calculations of the nuclear magnetic shielding constants of BH and BH3 molecules have been conducted to characterize the convergence of individual results obtained with correlation-consistent and polarization-consistent basis sets. The individual 11 B and 1 H NMR parameters were estimated in the complete basis set limit and compared with benchmark results. Only the KT3 density functional accurately reproduced 11 B shielding in BH molecule.

Rovibrational spectrum calculations of four electronic states in carbon monoxide molecule: Comparison of two effect correction methods

Accurate calculation of molecular energy is of great significance for studying molecular spectral properties. In this work, the potential energy curve and rovibrational spectrum (Gν) of the ground state X1∑+ and the excited states a3Π, a'3∑+ and A1Π of carbon monoxide molecule are calculated by the multi-reference configuration interaction method. In the calculation, the core-valence correlation correction (CV) effect and scalar relativistic (SR) effect are included.In order to obtain an accurate energy of molecule, two computational schemes are adopted. In the first scheme, i.e. (m MRCI+Q/CBS(TQ5)+CV+SR), the molecular orbital wavefunction is obtained from the Hartree-Fock self-consistent field method by using the basis set aug-cc-pVnZ. The wavefunction is first calculated by the state-averaged complete active space self-consistent field approach. Then the multi-reference configuration interaction method (MRCI) is adopted to calculate the dynamic correlation energy in the potential energy curve. Finally, we use the basis set cc-pCVQZ and aug-cc-pVQZ to calculate the CV effect and SR effect by the MRCI method. In the second scheme (aug-cc-pwCVnZ-DK (n=T, Q, 5)), the potential energy curves (PECs) of these four electronic states are calculated by the MRCI method whose basis set (aug-cc-pwCVnZ-DK) contains the CV effect and SR effect. Finally, in order to reduce the error caused by the basis set, we extrapolate the basis sets of the two computational schemes to the complete basis set. On the basis of the PECs plotted by the different methods, we obtain the spectroscopic parameters of the X1∑+, a3Π, a'3∑+ and A1Π states of the carbon monoxide by solving the internuclear Schrödinger equations through utilizing the numerical integration program “LEVEL”.In this paper, we calculate the SR effect and the CV effect by using different schemes, and the latter is indispensable for accurately calculating the molecular structure. For the lowest two electronic states, we consider the dependence of the two effects on the calculation of the Gaussian basis group (Method B), and find that the accuracy of the rovibrational spectrum is improved. It can be seen that these electronic states have higher requirements for electronic correlation calculation. For higher electronic states, the electron cloud distribution is relatively loose, and the electronic correlation obtained by a single Gaussian basis group can achieve the corresponding calculation accuracy. Of course, since the calculation of the rovibrational spectra is essentially only the relative energy, the offset effect of the electronic correlation effect of different electronic states is also included here in this paper.

Electronic Structures of LNA Phosphorothioate Oligonucleotides.

Important oligonucleotides in anti-sense research have been investigated in silico and experimentally. This involves quantum mechanical (QM) calculations and chromatography experiments on locked nucleic acid (LNA) phosphorothioate (PS) oligonucleotides. iso-potential electrostatic surfaces are essential in this study and have been calculated from the wave functions derived from the QM calculations that provide binding information and other properties of these molecules. The QM calculations give details of the electronic structures in terms of e.g., energy and bonding, which make them distinguish or differentiate between the individual PS diastereoisomers determined by the position of sulfur atoms. Rules are derived from the electronic calculations of these molecules and include the effects of the phosphorothioate chirality and formation of electrostatic potential surfaces. Physical and electrochemical descriptors of the PS oligonucleotides are compared to the experiments in which chiral states on these molecules can be distinguished. The calculations demonstrate that electronic structure, electrostatic potential, and topology are highly sensitive to single PS configuration changes and can give a lead to understanding the activity of the molecules.

Computing sextic centrifugal distortion constants by DFT: A benchmark analysis on halogenated compounds

This work presents a benchmark study on the calculation of the sextic centrifugal distortion constants employing cubic force fields computed by means of density functional theory (DFT). For a set of semi-rigid halogenated organic compounds several functionals (B2PLYP, B3LYP, B3PW91, M06, M06-2X, O3LYP, X3LYP, ωB97XD, CAM-B3LYP, LC-ωPBE, PBE0, B97-1 and B97-D) were used for computing the sextic centrifugal distortion constants. The effects related to the size of basis sets and the performances of hybrid approaches, where the harmonic data obtained at higher level of electronic correlation are coupled with cubic force constants yielded by DFT functionals, are presented and discussed. The predicted values were compared to both the available data published in the literature and those obtained by calculations carried out at increasing level of electronic correlation: Hartree-Fock Self Consistent Field (HF-SCF), second order Møller-Plesset perturbation theory (MP2), and coupled-cluster single and double (CCSD) level of theory. Different hybrid approaches, having the cubic force field computed at DFT level of theory coupled to harmonic data computed at increasing level of electronic correlation (up to CCSD level of theory augmented by a perturbational estimate of the effects of connected triple excitations, CCSD(T)) were considered. The obtained results demonstrate that they can represent reliable and computationally affordable methods to predict sextic centrifugal terms with an accuracy almost comparable to that yielded by the more expensive anharmonic force fields fully computed at MP2 and CCSD levels of theory. In view of their reduced computational cost, these hybrid approaches pave the route to the study of more complex systems.

Calculation of the electronic structure in the field of a homogeneously charged core of a large radius.

Calculation of the electronic density and potential distribution in the field of a homogeneously charged core of a large radius are reviewed. The Hartree-Fock self-consistent field method and the density-functional method are applied for the problem. These methods are compared with simple model where not interacting electrons are in spherical potential well with infinite border. The applicability of model with infinite potential well is discussed.

Accurate and efficient calculation of excitation energies with the active-space particle-particle random phase approximation.

An efficient method for calculating excitation energies based on the particle-particle random phase approximation (ppRPA) is presented. Neglecting the contributions from the high-lying virtual states and the low-lying core states leads to the significantly smaller active-space ppRPA matrix while keeping the error to within 0.05 eV from the corresponding full ppRPA excitation energies. The resulting computational cost is significantly reduced and becomes less than the construction of the non-local Fock exchange potential matrix in the self-consistent-field (SCF) procedure. With only a modest number of active orbitals, the original ppRPA singlet-triplet (ST) gaps as well as the low-lying single and double excitation energies can be accurately reproduced at much reduced computational costs, up to 100 times faster than the iterative Davidson diagonalization of the original full ppRPA matrix. For high-lying Rydberg excitations where the Davidson algorithm fails, the computational savings of active-space ppRPA with respect to the direct diagonalization is even more dramatic. The virtues of the underlying full ppRPA combined with the significantly lower computational cost of the active-space approach will significantly expand the applicability of the ppRPA method to calculate excitation energies at a cost of O(K4), with a prefactor much smaller than a single SCF Hartree-Fock (HF)/hybrid functional calculation, thus opening up new possibilities for the quantum mechanical study of excited state electronic structure of large systems.

Massless Majorana-Like Charged Carriers in Two-Dimensional Semimetals

The band structure of strongly correlated two-dimensional (2D) semimetal systems is found to be significantly affected by the spin-orbit coupling (SOC), resulting in SOC-induced Fermi surfaces. Dirac, Weyl and Majorana representations are used for the description of different semimetals, though the band structures of all these systems are very similar. We develop a theoretical approach to the band theory of two-dimensional semimetals within the Dirac–Hartree–Fock self-consistent field approximation. It reveals partially breaking symmetry of the Dirac cone affected by quasi-relativistic exchange interactions for 2D crystals with hexagonal symmetry. Fermi velocity becomes an operator within this approach, and elementary excitations have been calculated in the tight-binding approximation when taking into account the exchange interaction of π ( p z ) -electron with its three nearest π ( p z ) -electrons. These excitations are described by the massless Majorana equation instead of the Dirac one. The squared equation for this field is of the Klein–Gordon–Fock type. Such a feature of the band structure of 2D semimetals as the appearance of four pairs of nodes is shown to be described naturally within the developed formalism. Numerical simulation of band structure has been performed for the proposed 2D-model of graphene and a monolayer of Pb atoms.

Three-dimensional hole transport in nickel oxide by alloying with MgO or ZnO

It has been shown previously that the movement of a hole in nickel oxide is confined to two dimensions, along a single ferromagnetic plane. Such confinement may hamper hole transport when NiO is used as a p-type transparent conductor in various solar energy conversion technologies. Here, we use the small polaron model, along with unrestricted Hartree-Fock and complete active space self-consistent field calculations to show that forming substitutional MxNi1−xO alloys with M = Mg or Zn reduces the barrier for movement of a hole away from the ferromagnetic plane to which it is confined. Such reduction occurs for hole transfer alongside one or two M ions that have been substituted for Ni ions. Furthermore, the Mg and Zn ions do not trap holes on O sites in their vicinity, and NiO's transparency is preserved upon forming the alloys. Thus, forming MxNi1−xO alloys with M = Mg or Zn may enhance NiO's potential as a p-type transparent conducting oxide, by disrupting the two-dimensional confinement of holes in pure NiO.

Evaluating Shielding‐Based Ring‐Current Models by Using the Gauge‐Including Magnetically Induced Current Method

AbstractMagnetically induced current densities and integrated ring‐current strength susceptibilities have been calculated at the density functional theory (DFT) level for a test set consisting of 17 ring‐shaped molecules using the gauge‐including magnetically induced current (GIMIC) method. Reliable values for the ring‐current strengths have been obtained by performing numerical integration of the current‐density susceptibility passing a cut plane perpendicularly to the molecular ring. The current densities and ring current strengths were calculated at the DFT level using the B3LYP functional and def2‐TZVP basis sets. Current densities and ring‐current strengths have also been calculated at the Hartree‐Fock self‐consistent field (HF‐SCF) level using Dunning’s aug‐cc‐pVTZ basis sets, which allow a direct comparison with ring‐current strengths that have previously been estimated using ring‐current models based on magnetic shielding calculations. Current density calculations at both levels of theory show that the magnetic shielding based ring‐current models are not a very accurate means to estimate the magnetically induced ring current strengths, whereas they provide qualitatively the correct aromaticity trends for the studied molecules.

Scaling up Hartree–Fock calculations on Tianhe-2

This paper presents a new optimized and scalable code for Hartree–Fock self-consistent field iterations. Goals of the code design include scalability to large numbers of nodes, and the capability to simultaneously use CPUs and Intel Xeon Phi coprocessors. Issues we encountered as we optimized and scaled up the code on Tianhe-2 are described and addressed. A major issue is load balance, which is made challenging due to integral screening. We describe a general framework for finding a well-balanced static partitioning of the load in the presence of screening. Work stealing is used to polish the load balance. Performance results are shown on Stampede and Tianhe-2 supercomputers. Scalability is demonstrated on large simulations involving 2938 atoms and 27,394 basis functions, utilizing 8100 nodes of Tianhe-2.