Formulation of unrestricted and restricted Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) method, dealing with Bogoliubov orbitals, which consist of particle and hole part, can provide states with pair correlations associated with Cooper pairs. The dimension of HFB Fock matrices can be reduced by restrictions of spin states of Bogoliubov orbitals similarly to ordinary Hartree–Fock (HF) equations such as restricted HF (RHF), unrestricted HF (UHF), and generalized HF (GHF). However, there are few studies of moderate restricted HFB equations such as UHF‐based HFB equations. In this article, formulation and calculations of restricted HFB equations are described. The solutions of general and restricted HFB equations are compared. Pair correlations taking account of restricted and general HFB equations are discussed. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

RHF and two-configuration SCF calculations are inappropriate for conjugated diradicals

Restricted Hartee Fock (RHF) and two-configuration self-consistent field (TCSCF) calculations provide qualitatively correct molecular orbitals for the two open-shell electrons in diradicals. Nevertheless, these calculations fail to give correct relative energies and in some cases they even lead to incorrect geometries. Examples of these failures are given for both singlet and triplet states of some conjugated diradicals. In several cases these failures are related to the “doublet instability problem” in RHF calculations on radicals. It is argued that unrestricted Hartee-Fock (UHF) calculations on triplet states are more likely that RHF to provide accurate geometries.     

Hartree–Fock instabilities and electronic properties

Hartree–Fock instabilities are investigated for about 80 compounds, from acetylene to mivazerol (27 atoms) and a cluster of 18 water molecules, within a double ζ basis set. For most conjugated systems, the restricted Hartree–Fock wave function of the singlet fundamental state presents an external or so‐called triplet instability. This behavior is studied in relation with the electronic correlation, the vicinity of the triplet and singlet excited states, the electronic delocalization linked with resonance, the nature of eventual heteroatoms, and the size of the systems. The case of antiaromatic systems is different, because they may present a very large internal Hartree–Fock instability. Furthermore, the violation of Hund's rule, observed for these compounds, is put in relation with the fact that the high symmetry structure in its singlet state has no feature of a diradical‐like species. It appears that the triplet Hartree–Fock instability is directly related with the spin properties of nonnull orbital angular momentum electronic systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 483–504, 2000     

Spin-projected EHF method. IV. Comparison of potential curves given by different one-electron methods

Some aspects of the computer realization of the spin-projected extended Hartree–Fock (EHF ) method at the ab initio level are briefly discussed for the algorithm of solution developed in the previous papers of this series. Calculations have been performed for the BH molecule by using a small basis of contracted Gaussian lobes with the purpose of comparing the potential curves given by the different one-electron methods RHF , UHF , UHF with subsequent spin projection, and EHF . It is concluded that the UHF and, in particular, the EHF methods give a qualitatively correct shape of the potential curve; the RHF method shows the known incorrect dissociation behavior while the potential curve obtained by subsequent spin projection of the UHF wave function exhibits spurious extrema at intermediate internuclear separations.     

A molecular-orbital theoretical classification of reactions of singlet ground-state molecules

The reaction mechanisms of molecular systems in the ground singlet state are discriminated with the triplet stability—instability criterion of the restricted Hartree—Fock (RHF) solution for the entire system and with the phase continuity criterion of the highest occupied orbital involved. The criteria used are discussed in relation to the outcomes from the configuration-interaction approach.     

钴硫团簇ConSn+-1(n=2,3)的结构和稳定性

用ab initio分子轨道方法(RHF,UHF)和密度泛函(DFT)方法研究了团簇Co2S+,Co3S2+的各种可能的几何构型和电子结构,并计算了相应的较稳定构型的振动光谱,发现Co2S+和Co3S2+团簇最稳定结构均具有C,对称性.对团簇的成键作用机理进行了理论分析.     

Approximate spin-extended Hartree—Fock calculations

Projected-unrestricted Hartree—Fock wavefunctions can be improved by minimizing the energy with respect to variations in the occupation numbers of the charge-density natural orbitals. A series of pi-electron calculations are compared to the spin extended calculations of Mayer and Kertesz. Ab initio calculations on the lowest singlet and triplet states of ozone are compared to the GVB results of Hay et al.     

Addressing an instability in unrestricted density functional theory direct dynamics simulations

In Density Functional Theory (DFT) direct dynamics simulations with Unrestricted Hartree Fock (UHF) theory, triplet instability often emerges when numerically integrating a classical trajectory. A broken symmetry initial guess for the wave function is often used to obtain the unrestricted DFT potential energy surface (PES), but this is found to be often insufficient for direct dynamics simulations. An algorithm is described for obtaining smooth transitions between the open-shell and the closed-shell regions of the unrestricted PES, and thus stable trajectories, for direct dynamics simulations of dioxetane and its •O CH₂-CH₂ O• singlet diradical. © 2018 Wiley Periodicals, Inc.     

Unrestricted Hartree–Fock method for infinite systems with antiferromagnetic array: Analysis of antiferromagnetic state of trans-polyacetylene

An improved unrestricted Hartree–Fock (UHF) crystal orbital method for infinite one-dimensional systems having antiferromagnetic spin array is presented. This variational version of the UHF method is composed of two steps of self-consistent-field (SCF) calculations. The second SCF equation is to be solved using the canonical transformation between specific crystal orbitals around the Fermi level of the first conventional SCF solution. The energy gain and the forbidden band gap (correlation gap) associated with the hypothetical antiferromagnetic state of trans-polyacetylene are examined as an example. © 1993 John Wiley & Sons, Inc.     

Spin-unrestricted calculations in quantum chemistry

The unrestricted complete active space self-consistent field (UCASSCF ) function is defined, and a proof that a UCASSCF eigenfunction of the spin operator S ² is a CASSCF function is given. The spin-contamination for an unrestricted Hartree–Fock (UHF ) function is evaluated by using Araki angle operators, and the UHF function is then projected on the restricted open-shell Hartree–Fock (ROHF ) space. The present analysis has deep consequences since it implies that the only non-spin-contaminated UHF functions are the ROHF functions. This is illustrated in a calculation of the spin density of He. © 1993 John Wiley & Sons, Inc.     

The relationship between the unrestricted and projected Hartree–Fock methods in a simple three-electron model system

Various forms of the unrestricted Hartree–Fock (UHF ) scheme are studied for a simple three-electron model system, represented by the PPP (Pariser–Parr–Pople) π-electron model of the allyl radical. Both spin and space symmetry are violated in the UHF trial wave function, either individually or simultaneously. A comparison with the projected Hartree–Fock (PHF ) schemes studied earlier is made and the effect of the order in which various symmetries are broken in both UHF and PHF schemes is studied. The effectiveness of various schemes follows from a comparison of the correlation energy and the wave function is obtained by various UHF (or projected UHF ) and PHF schemes, in the whole range of the coupling constant, with the corresponding quantities given by the exact solution of the model. Finally, the implications of the stability of the restricted HF solutions for the behavior of various single- and multiparameter UHF and PHF schemes are briefly outlined and exemplified on the studied model.     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (I): The unrestricted approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent, two‐component relativistic calculations, termed the Kramers‐unrestricted HF (KUHF) method, is developed. The present KUHF method, which is formulated as a relativistic counterpart of nonrelativistic UHF, is based on quaternion algebra and partly uses time‐reversal symmetry. The fundamental characteristics of KUHF are discussed in this study. From numerical assessments, it was revealed that KUHF gives a corresponding solution to nonrelativistic UHF; furthermore, KUHF properly describes spin‐orbit interactions. In addition, KUHF can improve the self‐consistent field convergence behavior in spin‐dependent calculations, for example, for f‐block elements.     

Simple derivation of conditions for instability in the Hartree–Fock and projected Hartree–Fock schemes

The conditions for instability of solutions of Hartree–Fock and projected Hartree–Fock equations are derived in a form involving finite real symmetric matrices. These conditions are also expressed in terms of the Fock–Dirac density matrix, both at the spin–orbital and at the orbital level. The particular variations which give rise to the so-called singlet and triplet instabilities are described.     

Half-projected and projected Hartree-Fock calculations for singlet ground states. II. Lithium hydride

The half-projected Hartree–Fock function (HPHF ) for singlet states is defined as a linear combination of two Slater determinants which contains only spin eigenstates with even spin quantum numbers. The possible uses of such an approach for determining molecular properties are investigated computing the potential energy curve, binding energy, force constant, and dipole moment variation corresponding to the lithium hydride ground state. Full projected and restricted Hartree–Fock calculations (PHF and RHF ) are performed simultaneously for comparison purposes. It is found that the HPHF model yields very satisfactory results, very close to those of the PHF scheme. Both models predict properly the molecular behavior as a function of nuclear separation, whereas the RHF one fails. A discussion is given in terms of configuration equivalents. It is concluded that the HPHF scheme seems to be useful for determining molecular properties specially in the case of large systems in which the more sophisticated methods are unmanageable.     

Comparison of Magnetic Properties between Spin Singlet and Triplet Li<Subscript>3</Subscript>Al<Stack><Subscript>4</Subscript><Superscript>−</Superscript></Stack> Clusters

With a second-order Møller–Plesset perturbation theory and Hartree–Fock nuclear magnetic resonance calculations, we investigated the magnetic properties of spin singlet and triplet Li₃Al ₄ ^? clusters. The obtained gauge-independent atomic orbital magnetic shielding tensors confirm the paramagnetism of singlet Li₃Al ₄ ^? and diamagnetism of the triplet. The planar rings composed of four aluminum atoms make the magnetic properties of Li₃Al ₄ ^? clusters versatile. The localized molecular orbital, low symmetry of geometric conformation and narrow gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital are found to correlate with the paramagnetism of singlet Li₃Al ₄ ^? . The origin of the paramagnetism is explained. In triplet Li₃Al ₄ ^? , the two outmost orbitals are degenerate, causing a conversion from the paramagnetism to diamagnetism.     

The simulated ab initio molecular orbital technique. VI. Open-shell radicals in the spin restricted formalism

The simulated ab initio molecular orbital (SAMO ) method previously applied to RHF closed-shell and UHF open-shell situations has been applied to open-shell radicals, such as the butyl radical and the pentyl radical, within the RHF open-shell framework. The open-shell Hartree–Fock theory is developed such that a rapidly convergent iterative method for evaluating the SAMO wave-function can be employed. Results closely parallel those for the same systems using the UHF method and are of comparable accuracy to SAMO closed-shell results.     

Electron correlation in weakly coupled binuclear transition metal compounds: Cyclopentadienyl-allyl-cyclobutadiene-dinickel

The Hartree—Fock (HF) instabilities of the binuclear 3d complex cyclobutadiene-allyl-cyclobutadiene-dinickel (1) have been investigated by means of a semi-empirical INDO approach. It is demonstrated that the HF solution is unstable with respect to singlet, non-singlet and complex variations. The MO fluctuation leading to the singlet instability is predominantly localized in the region of Ni₂ and the four-membered ring. The lowest eigenvalue of the triplet instability corresponds to a spin decoupling between both Ni centers. Consequences arising from HF instabilities in binuclear transition metal compounds for the comparison between theoretical model calculations and experimental data (e.g. vertical ionization potentials, geometries, geometrical preferences, transport properties) are shortly discussed.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Theoretical evidence for the singlet diradical character of square planar nickel complexes containing two o-semiquinonato type ligands

Density functional theory and complete active space self-consistent field computations are applied to elucidate the singlet diradical character of square planar, diamagnetic nickel complexes that contain two bidentate ligands derived from o-catecholates, o-phenylenediamines, o-benzodithiolates, o-aminophenolates, and o-aminothiophenolates. In the density functional framework, the singlet diradical character is discussed within the broken symmetry formalism. The singlet-triplet energy gaps, the energy gained from symmetry breaking, the spin distribution in the lowest triplet state, and the form of the magnetic orbitals are applied as indicators for the singlet diradical character. Moreover, a new index for the diradical character is proposed that is based on symmetry breaking. All symmetry breaking criteria show that the complexes obtained from o-catecholates and o-benzodithiolates have the largest and the smallest singlet diradical character, respectively. The singlet diradical character should be intermediate for the complexes derived from o-phenylenediamines, o-aminophenolates, and o-aminothiophenolates. The diradical character of all complexes suggests the presence of Ni(II) central atoms. This is also indicated by the d-populations computed by means of the natural population analysis.     

Quantum chemistry of many-electron systems: high-priority aspects

The review generalizes the studies devoted to the development of a new quantum chemistry method representing an alternative to the Hartree–Fock approximation. Based on the hypothesis of prohibition of equipotential surfaces, which clarifies the physical sense of the Pauli exclusion principle, and taking account of the condition for antisymmetrical wave function of the triplet state (3S) of He atom, the Hartree–Fock approximation is inappropriate for a priori determination of the nodal surfaces of many-electron wave functions (MWFs) for the test systems traditionally used in quantum chemistry, namely, excited triplet state of H₂ molecule and the ground electronic states of Li atom and LiH molecule. The nodal surfaces of the wave functions corresponding to the minimum basis set of Slater orbitals in the Hartree–Fock approximation are constructed and analyzed. An alternative to the Hartree–Fock approximation is provided by the MWF quantum chemical method being developed by the authors. In the MWF method, the nodal surfaces for H₂(³Σ _u ^v ) and Li(²S) are specified a priori. Some aspects of geometric interpretation of the Pauli exclusion principle are discussed. Unlike the MWF method, the Hartree–Fock approximation is unsuitable for taking account of the dependence of the MWF nodal surfaces on the nuclear charges and on correlation effects related to the motion of electrons with antiparallel spins because such nodal surfaces are predefined by the mathematical properties of Slater determinants rather than by physically clear and more practically valuable algebraic products of electrostatic potential differences.