Unrestricted Hartree-Fock Research Articles

Restricted and unrestricted non-Hermitian Hartree-Fock: Theory, practical considerations, and applications to metastable molecular anions.

This work describes the implementation and applications of non-Hermitian self-consistent field (NH-SCF) theory with complex basis functions for the ab initio computation of positions and widths of shape resonances in molecules. We utilize both the restricted open-shell and the previously unexplored spin-unrestricted variants to compute Siegert energies of several anionic shape resonances in small diatomic and polyatomic molecules including carbon tetrafluoride which has been the subject of several recent experimental studies. The computation of general molecular properties from a non-Hermitian wavefunction is discussed, and a density-based analysis is applied to the (2)B1 shape resonance in formaldehyde. Spin-unrestricted NH-SCF is used to compute a complex potential energy surface for the carbon monoxide anion which correctly describes dissociation.

Theoretical Investigation of the Reaction Paths of the Aluminum Cluster Cation with Water Molecule in the Gas Phase: A Facile Route for Dihydrogen Release.

The gas-phase reaction of the Al6(+) cation with a water molecule is investigated computationally by coupled cluster and density functional theories. Several low-energy paths of the mechanism for dihydrogen production from H2O by the positively charged aluminum cluster are identified. This reaction involves the initial formation of the association complex, exothermic by 25 kcal/mol, followed by the water dissociation and H2 elimination major steps, yielding the Al6O(+) product oxide with either the nonplanar or planar structure. The H2O dissociation on Al6(+) is the rate-determining step. Of the paths probed, the one kinetically most preferred leads from the O-H bond dissociation transition state lying below the separated reactants to the immediate HAl6OH(+) intermediate of the "open" type and involves further the more compact intermediate from which H2 is eliminated. The other reaction paths explored involve the activation enthalpy (at 0 K) for the rate-determining step of less than 2 kcal/mol relative to the Al6(+) + H2O. Natural population analysis based charges indicate that forming of H2 along the elimination coordinate is facilitated by the interaction of the hydridic and protic hydrogens. For the kinetically most favorable route detected, the coupled cluster singles and doubles with perturbative triples (CCSD(T)) relative energies calculated with the unrestricted and restricted HF references are in a good agreement. This investigation is relevant specifically to the recent mass spectrometric study of the reactivity of Aln(+) with water by Arakawa et al., and it provides a mechanistic insight into the formation of the observed AlnO(+) product oxide with n = 6.

Selection of active spaces for multiconfigurational wavefunctions.

The efficient and accurate description of the electronic structure of strongly correlated systems is still a largely unsolved problem. The usual procedures start with a multiconfigurational (usually a Complete Active Space, CAS) wavefunction which accounts for static correlation and add dynamical correlation by perturbation theory, configuration interaction, or coupled cluster expansion. This procedure requires the correct selection of the active space. Intuitive methods are unreliable for complex systems. The inexpensive black-box unrestricted natural orbital (UNO) criterion postulates that the Unrestricted Hartree-Fock (UHF) charge natural orbitals with fractional occupancy (e.g., between 0.02 and 1.98) constitute the active space. UNOs generally approximate the CAS orbitals so well that the orbital optimization in CAS Self-Consistent Field (CASSCF) may be omitted, resulting in the inexpensive UNO-CAS method. A rigorous testing of the UNO criterion requires comparison with approximate full configuration interaction wavefunctions. This became feasible with the advent of Density Matrix Renormalization Group (DMRG) methods which can approximate highly correlated wavefunctions at affordable cost. We have compared active orbital occupancies in UNO-CAS and CASSCF calculations with DMRG in a number of strongly correlated molecules: compounds of electronegative atoms (F2, ozone, and NO2), polyenes, aromatic molecules (naphthalene, azulene, anthracene, and nitrobenzene), radicals (phenoxy and benzyl), diradicals (o-, m-, and p-benzyne), and transition metal compounds (nickel-acetylene and Cr2). The UNO criterion works well in these cases. Other symmetry breaking solutions, with the possible exception of spatial symmetry, do not appear to be essential to generate the correct active space. In the case of multiple UHF solutions, the natural orbitals of the average UHF density should be used. The problems of the UNO criterion and their potential solutions are discussed: finding the UHF solutions, discontinuities on potential energy surfaces, and inclusion of dynamical electron correlation and generalization to excited states.

Multiorbital kinetic effects on charge ordering of frustrated electrons on the triangular lattice

The role of the multiorbital effects on the emergence of frustrated electronic orders on the triangular lattice at half filling is investigated through an extended spinless fermion Hubbard model. By using two complementary approaches, unrestricted Hartree-Fock and exact diagonalizations, we unravel a very rich phase diagram controlled by the strength of both local and off-site Coulomb interactions and by the interorbital hopping anisotropy ratio $t'/t$. Three robust unconventional electronic phases, a pinball liquid, an inverse pinball liquid, and a large-unit-cell $\sqrt{12} \times \sqrt{12}$ droplet phase, are found to be generic in the triangular geometry, being controlled by the band structure parameters. The latter are also stabilized in the isotropic limit of our microscopic model, which recovers the standard SU(2) spinful extended single-band Hubbard model.

Spin contamination analogy, Kramers pairs symmetry and spin density representations at the 2-component unrestricted Hartree–Fock level of theory

“Kramers pairs symmetry breaking” is evaluated at the 2-component (2c) Kramers unrestricted and/or general complex Hartree–Fock (GCHF) level of theory, and its analogy with “spin contamination” at the 1-component (1c) unrestricted Hartree–Fock (UHF) level of theory is emphasized. The GCHF “Kramers pairs symmetry breaking” evaluation is using the square of overlaps between the set of occupied spinorbitals with the projected set of Kramers pairs. In the same fashion, overlaps between α and β orbitals are used in the evaluation of “spin contamination” at the UHF level of theory. In this manner, UHF S^2 expectation value is made formally extended to the GCHF case. The directly evaluated GCHF expectation value of the S^2 operator is considered for completeness. It is found that the 2c GCHF Kramers pairs symmetry breaking has a very similar extent in comparison to the 1c UHF spin contamination. Thus higher excited states contributions to the 1c and 2c unrestricted wave functions of open shell systems have almost the same extent and physical consequences. Moreover, it is formally shown that a single determinant wave function in the restricted open shell Kramers case has the expectation value of K^2 operator equal to the negative number of open shell electrons, while the eigenvalue of K^2 for the series of simple systems (H, He, He*-triplet, Li and Li*-quartet) are found to be equal to minus the square of the number of open shell electrons. The concept of unpaired electron density is extended to the GCHF regime and compared to UHF and restricted open shell Hartree–Fock spin density. The “collinear” and “noncollinear” analogs of spin density at the GCHF level of theory are considered as well. Spin contamination and/or Kramers pairs symmetry breaking, spin populations and spin densities are considered for H2O+, Cl, HCl+, phenoxyl radical (C6H5O) as well as for Cu, Cu2+, Fe and the [OsCl5(1H-pyrazole)]− anion. The 1c and 2c unpaired electron density representation is found useful for visualization and/or representation of spin density.

A simple way to test for collinearity in spin symmetry broken wave functions: general theory and application to generalized Hartree Fock.

We introduce a necessary and sufficient condition for an arbitrary wavefunction to be collinear, i.e., its spin is quantized along some axis. It may be used to obtain a cheap and simple computational procedure to test for collinearity in electronic structure theory calculations. We adapt the procedure for Generalized Hartree Fock (GHF), and use it to study two dissociation pathways in CO2. For these dissociation processes, the GHF wave functions transform from low-spin Unrestricted Hartree Fock (UHF) type states to noncollinear GHF states and on to high-spin UHF type states, phenomena that are succinctly illustrated by the constituents of the collinearity test. This complements earlier GHF work on this molecule.

Assessment of Orbital-Optimized MP2.5 for Thermochemistry and Kinetics: Dramatic Failures of Standard Perturbation Theory Approaches for Aromatic Bond Dissociation Energies and Barrier Heights of Radical Reactions.

An assessment of orbital-optimized MP2.5 (OMP2.5) [ Bozkaya, U.; Sherrill, C. D. J. Chem. Phys. 2014, 141, 204105 ] for thermochemistry and kinetics is presented. The OMP2.5 method is applied to closed- and open-shell reaction energies, barrier heights, and aromatic bond dissociation energies. The performance of OMP2.5 is compared with that of the MP2, OMP2, MP2.5, MP3, OMP3, CCSD, and CCSD(T) methods. For most of the test sets, the OMP2.5 method performs better than MP2.5 and CCSD, and provides accurate results. For barrier heights of radical reactions and aromatic bond dissociation energies OMP2.5-MP2.5, OMP2-MP2, and OMP3-MP3 differences become obvious. Especially, for aromatic bond dissociation energies, standard perturbation theory (MP) approaches dramatically fail, providing mean absolute errors (MAEs) of 22.5 (MP2), 17.7 (MP2.5), and 12.8 (MP3) kcal mol(-1), while the MAE values of the orbital-optimized counterparts are 2.7, 2.4, and 2.4 kcal mol(-1), respectively. Hence, there are 5-8-folds reductions in errors when optimized orbitals are employed. Our results demonstrate that standard MP approaches dramatically fail when the reference wave function suffers from the spin-contamination problem. On the other hand, the OMP2.5 method can reduce spin-contamination in the unrestricted Hartree-Fock (UHF) initial guess orbitals. For overall evaluation, we conclude that the OMP2.5 method is very helpful not only for challenging open-shell systems and transition-states but also for closed-shell molecules. Hence, one may prefer OMP2.5 over MP2.5 and CCSD as an O(N(6)) method, where N is the number of basis functions, for thermochemistry and kinetics. The cost of the OMP2.5 method is comparable with that of CCSD for energy computations. However, for analytic gradient computations, the OMP2.5 method is only half as expensive as CCSD.

Insights into the structure of many-electron wave functions of Mott-insulating antiferromagnets: The three-band Hubbard model in full configuration interaction quantum Monte Carlo

We investigate the ground-state wave function of a prototypical strongly correlated system, a three-band $(p\ensuremath{-}d)$ Hubbard model of cuprates, using full configuration interaction quantum Monte Carlo. We show that the configuration interaction description of the exact ground state wave function is profoundly affected by the choice of single-particle representation, in a counterintuitive manner. Thus a broken-symmetry unrestricted Hartree-Fock basis, which at a single configuration level produces a qualitatively correct description of the antiferromagnet, results in a highly entangled exact solution consisting of high particle-hole excitations of the reference. This wave function is found to be very difficult to approximate using subspace diagonalizations. Conversely, a restricted Hartree-Fock basis, which yields at a single configuration level a qualitatively incorrect paramagnetic metal, results in a relatively rapidly converging configuration interaction expansion. Convergence can be further accelerated by adopting a natural orbital representation. Our results suggest that with the correct single-particle basis, such strongly correlated systems may be described by relatively compact wave functions.

Sign reversal of the Hall response in a crystalline superconductor

We consider the Hall conductivity due to the motion of a vortex in a lattice-model of a clean superconductor, using a combination of general arguments, unrestricted Hartree-Fock calculations, and exact diagonalization. In the weak coupling limit, $k_F \xi \gg 1$, the sign of the Hall response of the superconducting state is the same as that of the normal (non-superconducting) state. For intermediate and strong coupling, however, ($k_F \xi \sim 1$) we find that the sign of the Hall response in the superconducting state can be opposite to that of the normal state. In addition, we find that the sign reversal of the Hall response is correlated with a discontinuous change in the density profile at the vortex core. Implications for experiments in the cuprate superconductors are discussed.

Restricted Hartree Fock using complex-valued orbitals: a long-known but neglected tool in electronic structure theory.

Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H2 insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O2, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.

New XDM-corrected potential energy surfaces for Ar-NO(X(2)Π): a comparison with CCSD(T) calculations and experiments.

We report new potential energy surfaces for the ground state Ar-NO(X(2)Π) van der Waals system calculated using the unrestricted Hartree-Fock (UHF) method with the addition of the Becke-Roussel correlation functional and exchange-hole dipole moment dispersion correction (XDM). We compare UHFBR-XDM surfaces and those previously reported by Alexander from coupled cluster CCSD(T) calculations [J. Chem. Phys. 111, 7426 (1999)]. The bound states of Ar-NO have been investigated with these new UHFBR-XDM surfaces, including relative energy-level spacing, adiabatic bender states and wave functions, and spectroscopic data. These results have been found to be in good agreement with calculations based on the CCSD(T) PESs. These new PESs are used to investigate the inelastic scattering of NO(X) by Ar. Full close-coupling integral cross sections at collision energies of 442 cm(-1), 1774 cm(-1) and differential cross sections at collision energy of 530 cm(-1) were determined for transitions out of the lowest NO(X) rotational level (j = ω = 1/2,f). These cross sections are in good agreement with those calculated with CCSD(T) and accordingly in good agreement with the most recent initial and final state resolved experimental data. The UHFBR-XDM scheme yields high-quality potential surfaces with computational cost comparable to the Hartree-Fock method and our results may serve as a benchmark for application of this scheme to collisions between larger molecules.

Magnetic hyperfine interactions at the 51V nucleus in the pyrochlore Lu2V2O7

This is an ab initio calculation of the parameters of the magnetic hyperfine interactions at the 51V nucleus, in the Lu2V2O7 pyrochlore, in addition to a comparative analysis of the calculated values versus known experimental data. It is shown that the unrestricted Hartree-Fock method is more suitable when it comes to describing the magnetic properties at ion nuclei, in comparison to the density functional theory with hybrid functionals B3LYP and PBE0.

Обобщенный метод Хартри-Фока и его версии: от атомов и молекул до полимеров

There are considered the different versions of the generalized Hartree-Fock method: extended methods G1–GF, unrestricted HF method with and without projection on pure spin state, alternant MO methods (single- and multiparametrical), spin-optimized GI method. There are also discussed the properties of solutions of these methods and their applications to electronic shell calculations of atoms, molecules, and carbopolymers (polyenes, polyacetelenes, graphite, cumulenes).

Model Selection of NiGa2S4 Triangular Lattice by Bayesian Inference

We propose a novel method for extracting effective classical spin Hamiltonians from mean-field type electronic structural calculations by means of the Bayesian inference. We apply the method for a NiS2 triangular lattice in NiGa2S4, where the ground state is a spin disordered state. Starting from unrestricted Hartree–Fock calculations for the spin configurations of 16 Ni sites, we estimated that not only the strongest superexchange interaction between the third nearest neighbor-sites but also those between the nearest and the second nearest neighbor-sites should be considered for the effective classical spin Hamiltonian for NiGa2S4.

Qualitative breakdown of the unrestricted Hartree-Fock energy.

The stretching of closed-shell molecules is a qualitative problem for restricted Hartree-Fock that is usually circumvented by the use of unrestricted Hartree-Fock (UHF). UHF is well known to break the spin symmetry at the Coulson-Fischer point, leading to a discontinuous derivative in the potential energy surface and incorrect spin density. However, this is generally not considered as a major drawback. In this work, we present a set of two electron molecules which magnify the problem of symmetry breaking and lead to drastically incorrect potential energy surfaces with UHF. These molecules also fail with unrestricted density-functional calculations where a functional such as B3LYP gives both symmetry breaking and an unphysically low energy due to the delocalization error. The implications for density functional theory are also discussed.

Density‐matrix theory of quantum dynamics under a strong external field switched on nonadiabatically

Quantum time‐evolution equations for the density matrix are formulated in the unrestricted Hartree–Fock approximation, with an emphasis on the nonperturbative effect due to a sudden or gradual onset of a strong external field. Numerical simulations are performed for ideal Fermi gas around a square‐well potential which is switched on dynamically. When the switching is fast enough, an oscillatory motion of the particle is induced by a nonadiabatic transition at the Landau–Zener crossing point, which is most clearly seen in a small‐size system. When the switching is sufficiently slow, the simulation corroborates the adiabatic theorem. It is shown that the Anderson's infrared catastrophe in a metal is strongly enhanced by the nonperturbative effect. The Keldysh formula of atomic multiphoton ionization can also be derived from the nonperturbative term in the density‐matrix equation, indicating a wide applicability of the present theory. © 2014 Wiley Periodicals, Inc.

Physics of metal-corrrelated barrier with chemical modulation-metal heterostructure

Barrier planes described by the Ionic Hubbard model sandwiched between metallic planes on both sides are studied using unrestricted Hartree Fock. For zero onsite correlation, the presence of the metallic interface generates an additional gap in the energy spectrum away from half filling, if the chemically modulated potential in the barrier planes exceed a critical value. There is an insulator–metal–insulator transition as we tune onsite correlation for fixed strength of chemical modulation. The metallic behaviour penetrates the barrier planes due to proximity effect. The variation of the gaps for varying numbers of intermediate planes is also determined.

On the impact of intermolecular interactions between the quaternary ammonium ions on interlayer spacing of quat-intercalated montmorillonite: A molecular mechanics and ab-initio study

The work comprises the nature of intermolecular interactions in the organoclays modified with quaternary ammonium ions, the SOMASIF ME100 organofilized with N-octadecyltrimethylammonium ions and the commercially available Cloisite 15A. The potential energy scan was carried out for two large organoclay models with consistent-valence force field and a few others for comparison. Then, the two-body model of the interacting ions in their most and less preferable orientations was build for both systems and the more detailed, electrically neutral model of the two ions adsorbed on montmorillonite was constructed in order to investigate the non-additive interactions. The most of the work was devoted to study ab-initio calculations performed on the Unrestricted Hartree–Fock level of theory, but the electron correlation effects were also considered. The calculated interaction energy was partitioned into exchange, delocalization and electrostatic terms and the latter was distributed into its multipolar expansion. The specific intermolecular order occurring for the most and the least energetically preferred arrangements was discussed in detail with respect to the intermolecular interactions with the special emphasis on the structures underlying the occurrence of three distinctive phases in organoclay samples, for which the X-ray data was present in the literature. The basis-set dependency of the variational-perturbation scheme adopted to decompose the calculated Hartree–Fock interaction energies was examined with the use of cc-pVDZ and 6-31G basis sets. The comparison proved high reliability of the method used, even for the less extended 3-21G basis. The results revealed the presence of the relationship between the sum of two- and three-body delocalization energy terms and the inverse induced dipole moment, calculated for three body system as well as the independent functional relations between the second order electron correlation correction and Hartree–Fock delocalization and inverse Heitler–London exchange energy terms for the system of two N-octadecyltrimethylammonium cations. The equations proposed may help to predict the values of the electron correlation correction and the stabilizing delocalization energy term from the calculations at Hartree–Fock level of theory as well as from the measured module of the induced dipole moment for molecules.

Semi-canonical MOs in the varied open shell method and related approaches

The self-consistency schemes previously developed in J. Struct. Chem., 30, N 5, p. 3 (1986) are shown to belong to the class of semi-canonical orbital schemes. In the derivation they explicitly involve only a part of canonical Koopmans orbitals in the iteration process. In particular, within the restricted Hartree-Fock method for a non-degenerate case a semi-canonical variational scheme based on the matrix elements of the unrestricted Hartree-Fock method is studied. Errors present in the above mentioned work in the description of the scheme of a varied open shell are corrected. A possible generalization for the case of Roothaan type degeneration is proposed and specific examples of low-spin and high-spin states with a violated Aufbau filling principle are considered.

Phase diagram of the13-filled extended Hubbard model on the kagome lattice

We study the phase diagram of the extended Hubbard model on the kagome lattice at 1/3 filling. By combining a configuration interaction approach to an unrestricted Hartree-Fock, we construct an effective Hamiltonian which takes the correlations back on top of the mean-field solution. We obtain a rich phase diagram with, in particular, the presence of two original phases. The first one consists of local polarized droplets of metal standing on the hexagons of the lattice and an enlarged kagome charge order, inversely polarized, on the remaining sites. The second, obeying a local ice-rule type constraint on the triangles of the kagome lattice, is driven by an antiferromagnetic coupling of spins and is constituted of disconnected six-spin singlet rings. The nature and stability of these phases at large interactions are studied via trial wave functions and perturbation theory.