Unrestricted Hartree-Fock Solutions Research Articles

Broken-symmetry self-consistent GW approach: Degree of spin contamination and evaluation of effective exchange couplings in solid antiferromagnets

We adopt a broken-symmetry strategy for evaluating effective magnetic constants J within the fully self-consistent GW method. To understand the degree of spin contamination present in broken-symmetry periodic solutions, we propose several extensive quantities demonstrating that the unrestricted self-consistent GW preserves the broken-symmetry character of the unrestricted Hartree-Fock solutions. The extracted J are close to the ones obtained from multireference wave-function calculations. In this paper, we establish a robust computational procedure for finding magnetic coupling constants from self-consistent GW calculations and apply it to solid antiferromagnetic nickel and manganese oxides.

A black-box, general purpose quadratic self-consistent field code with and without Cholesky decomposition of the two-electron integrals

We present the implementation of a quadratically convergent self-consistent field (QCSCF) algorithm based on an adaptive trust-radius optimisation scheme for restricted open-shell Hartree–Fock (ROHF), restricted Hartree–Fock (RHF), and unrestricted Hartree–Fock (UHF) references. The algorithm can exploit Cholesky decomposition (CD) of the two-electron integrals to allow calculations on larger systems. The most important feature of the QCSCF code lies in its black-box nature – probably the most important quality desired by a generic user. As shown for pilot applications, it does not require one to tune the self-consistent field (SCF) parameters (damping, Pulay's DIIS, and other similar techniques) in difficult-to-converge molecules. Also, it can be used to obtain a very tight convergence with extended basis sets – a situation often needed when computing high-order molecular properties – where the standard SCF algorithm starts to oscillate. Nevertheless, trouble may appear even with a QCSCF solver. In this respect, we discuss what can go wrong, focusing on the multiple UHF solutions of ortho-benzyne.

Symmetry in Multiple Self-Consistent-Field Solutions of Transition-Metal Complexes.

We use a method based on metadynamics to locate multiple low-energy Unrestricted Hartree-Fock (UHF) self-consistent-field (SCF) solutions of two model octahedral d1 and d2 transition-metal complexes, [MF6]3- (M = Ti, V). By giving a group-theoretical definition of symmetry breaking, we classify these solutions in the framework of representation theory and observe that a number of them break spin or spatial symmetry, if not both. These solutions seem unphysical at first, but we show that they can be used as bases for Non-Orthogonal Configuration Interaction (NOCI) to yield multideterminantal wave functions that have the right symmetry to be assigned to electronic terms. Furthermore, by examining the natural orbitals and occupation numbers of these NOCI wave functions, we gain insight into the amount of static correlation that they incorporate. We then investigate the behaviors of the most low-lying UHF and NOCI wave functions when the octahedral symmetry of the complexes is lowered and deduce that the symmetry-broken UHF solutions must first have their symmetry restored by NOCI before they can describe any vibronic stabilization effects dictated by the Jahn-Teller theorem.

Spin-orbit coupling of sp2 nanocarbons and magnetism of fullerene C60 in view of spin peculiarities of unrestricted Hartree–Fock solution

ABSTRACTThe paper presents the first application of unrestricted Hartree–Fock formalism to the consideration of spin-orbit coupling (SOC) of open-shell systems in general and sp2 nanocarbons in particular. The SOC parameters determination is exhibited on the example of stretched ethylene. g-Factor determination addresses fullerene C60 due to its peculiar “paradiamagnetic” behavior at low temperature characterized by a set of local spins with different g-factors. The approach application has allowed not only to suggest a reliable explanation of the factor variability but to supplement it by a quantitative agreement with experimental data.

Finding symmetry breaking Hartree-Fock solutions: The case of triplet instability.

Determining the lowest unrestricted Hartree-Fock (UHF) solution is often difficult in even-electron systems. We have developed a deterministic method for locating approximately the UHF minimum using the restricted Hartree-Fock triplet instability matrix. The current method is truncated to fourth order. The minimum energy solution for this model can be determined by solving a small linear system of equations. This solution gives a suitable starting point to determine the exact UHF solution. This should be useful for the black-box determination of active spaces spanned by the fractionally occupied charge natural orbitals of the ground-state UHF wavefunction. The results can be generalized to higher (6th and 8th) degree expansions (odd expansion orders vanish by symmetry), and to other types of instability, including complex instability. The results are illustrated by calculations on ozone, benzene, nitrobenzene, butadiene, hexatriene, octatetraene, dichromium, and nickel porphine. Further examples are given in the supplementary material.

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.

Spin- and charge-density waves in the Hartree–Fock ground state of the two-dimensional Hubbard model

The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree–Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.

On the nature of electron correlation in C60

The ground state restricted Hartree Fock (RHF) wave function of C(60) is found to be unstable with respect to spin symmetry breaking, and further minimization leads to a significantly spin contaminated unrestricted Hartree Fock (UHF) solution (<S(2)> = 7.5, 9.6 for singlet and triplet, respectively). The nature of the symmetry breaking in C(60) relative to the radicaloid fullerene, C(36), is assessed by energy lowering of the UHF solution, <S(2)>, and the unpaired electron number. We conclude that the high value of each of these measures in C(60) is not attributable to strong correlation behavior as is the case for C(36). Instead, their origin is from the collective effect of relatively weak, global correlations present in the π space of both fullerenes. Second order perturbation (MP2) calculations of the singlet triplet gap are significantly more accurate with RHF orbitals than UHF orbitals, while orbital optimized opposite spin second order correlation (O2) performs even better.

Ground state of two electrons on concentric spheres

We extend our analysis of two electrons on a sphere [Phys. Rev. A {\bf 79}, 062517 (2009); Phys. Rev. Lett. {\bf 103}, 123008 (2009)] to electrons on concentric spheres with different radii. The strengths and weaknesses of several electronic structure models are analyzed, ranging from the mean-field approximation (restricted and unrestricted Hartree-Fock solutions) to configuration interaction expansion, leading to near-exact wave functions and energies. The M{\o}ller-Plesset energy corrections (up to third-order) and the asymptotic expansion for the large-spheres regime are also considered. We also study the position intracules derived from approximate and exact wave functions. We find evidence for the existence of a long-range Coulomb hole in the large-spheres regime, and infer that unrestricted Hartree-Fock theory over-localizes the electrons.

A resonating broken-symmetry CI study of cationic states of phenalenyl dimeric compounds

We investigated several phenalenyl cationic dimer systems using ab initio methods. First, we examine the through-space cation dimer. The B3LYP method fails to describe the correct dissociation as in the case of usual cation dimers. On the other hand, the resonating broken symmetry configuration interaction (Res-BS CI) method based on unrestricted Hartree–Fock solutions yields the correct potential surface and correct charge distributions. Furthermore, we investigated though-bond interactions of the phenalenyl cationic dimer via acetylene and ethylene bridges. The results are considerably different from those of the though-space case, implying that these bridging units play important roles to determine charge and spin distributions. Judging from these results, different bridges-or-stacks lead to different hole behaviors, indicating the possibility of various electronic functionalities of hole-doped phenalenyl compounds. In particular, the acetylene bridge is expected to enhance the conductivity of the hole-doped phenalenyl compounds.

Determination of the Hubbard model parameters by using the unrestricted Hartree–Fock solutions, and improvement of their energies

AbstractA strategy is proposed to determine parameters in model Hamiltonians from the difference of the density matrices calculated by ab initio methods. As an example, the parameters of the two‐site Hubbard model are determined. The parameters t and U are explicitly expressed in terms of the energy difference and the difference of the expectation values of the spin operator 〈S2〉 between two unrestricted Hartree–Fock (UHF) solutions. By using this relation and the exact energy formulas of the model Hamiltonian the energies of the ground and excited states of the ab initio Hamiltonian are estimated. This energy estimation scheme is applied to H2 and Li2 molecules and the results are in reasonable agreement with those of the full configuration‐interaction method. The errors of our estimation are also discussed and evaluated by using a modified Hubbard model. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Three, four and five interacting electrons in a spherical box: an exact diagonalization studyextended

We study the problem of N electrons confined to a hard three-dimensional spherical box, forN = 3,4 and 5. Full-configuration interaction and Hartree–Fock calculations have been performed. Wereport energies, densities, second-order density matrices and exchange–correlation holes. In theN = 4 system, we observe a spin transition as the radiusR of the sphere is increased, changing the ground state from3P to5S. TheN = 3 and 5 systemsremain 2P and4S, respectively,for all R investigated. Pair-correlation functions and physical exchange–correlation holes have beencomputed over a regime in which the systems transform from liquid droplets to Wigner molecules.These functions show marked, yet characteristic, changes in this regime. By computing|Ec/E|,where Ec is the correlation energy defined with respect to the unrestricted Hartree–Fock solution, weshow that the maximum relative error in the UHF energies occurs in this crossover regime.These systems provide challenging benchmarks for post-Hartree–Fock and densityfunctional theory methods.

Correlation effects inSr2RuO4andCa2RuO4: Valence-band photoemission spectra and self-energy calculations

We have studied the electronic structure of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ and ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$ using x-ray photoemission spectroscopy (XPS) and subsequent model calculations. While the ${t}_{2g}$ band of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ has substantial spectral weight at the Fermi level, that of ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$ has no spectral weight at ${E}_{F}$ and shows a peak at $\ensuremath{-}1.8\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. In the valence-band XPS spectrum of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, a satellite structure of the ${t}_{2g}$ band is observed. In order to explain the spectral features, we have carried out band-structure calculations using the unrestricted Hartree-Fock (HF) approximation and found good agreement with the experimental result for ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$. In order to explain the satellite structure of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, we have performed second-order perturbation calculations of the self-energy corrections around the unrestricted HF solution of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$.

Obtaining the right orbitals is the first step to calculating accurate binding energies for Cu + ion

We investigate the previously reported poor performance of the G2 method for simple copper systems. We optimized CuH +, CuO +, and CuSi + using the HF, MP2, B3LYP, mPW1PW91, and MPW1K methods with three basis sets. We found multiple solutions to the Hartree–Fock equations, which are the cause for previously reported poor behavior of the G2 method. The orbitals of the lowest-energy unrestricted Hartree–Fock solution do not always generate the lowest-energy correlated wavefunction. Hybrid density functional theory methods are shown to be quite successful for obtaining orbitals and for predicting geometries and atomization energies.

Spin-contamination of coupled-cluster wave functions

The propensity of approximate solutions of the electronic Schrödinger equation to break spin-symmetry is directly related to the quality of the approximate wave function, and thus can be used as a diagnostic tool. The quasi-variational nature of the (valence) optimized orbitals coupled-cluster doubles methods, (V)OO-CCD, enables a discussion of the stability of coupled-cluster wave functions in terms of both spin-contamination and a corresponding energy lowering relative to the pure spin solutions. The spin-contamination of (V)OO-CCD models has been studied for bond-breaking processes and diradicals. The main findings are: (i) The OO-CCD method is stable for a relatively large range of nuclear distortions and is capable of eliminating even very large spin-contamination of the unrestricted Hartree–Fock solution given that the molecular electronic configuration remains essentially single-reference. When a spin-contaminated solution arises, the energy splitting rapidly becomes large and 〈Ŝ2〉 approaches the Hartree–Fock value; (ii) The VOO-CCD method, which is designed to approximate a multi-reference model, remains stable over broader ranges; however, for pure diradicals it becomes unstable. In these cases, spin-contamination is also very large, but the energy lowering for the spin-unrestricted solutions is negligible; (iii) Higher order corrections described by perturbation theory lead to smaller energy splittings between restricted and unrestricted (V)OO-CCD energies. However, in case of spin-contaminated (V)OO-CCD solutions, these corrections may lead to unphysical shapes of the potential energy surfaces. Thus, in order to quantitatively characterize the quality of the wave functions, both spin-contamination and energy lowering due to the breaking of spin-symmetry must be considered.

The density matrix renormalization group method: Application to the PPP model of a cyclic polyene chain

The density matrix renormalization group (DMRG) method introduced by White for the study of strongly interacting electron systems is reviewed; the method is variational and considers a system of localized electrons as the union of two adjacent fragments A,B. A density matrix ρ is introduced, whose eigenvectors corresponding to the largest eigenvalues are the most significant, the most probable states of A in the presence of B; these states are retained, while states corresponding to small eigenvalues of ρ are neglected. It is conjectured that the decreasing behavior of the eigenvalues is Gaussian. The DMRG method is tested on the Pariser-Parr-Pople Hamiltonian of a cyclic polyene (CH)N up to N=34. A Hilbert space of dimension 5.×1018 is explored. The ground state energy is 10−3 eV within the full CI value in the case N=18. The DMRG method compares favorably also with coupled cluster approximations. The unrestricted Hartree-Fock solution (which presents spin density waves) is briefly reviewed, and a comparison is made with the DMRG energy values. Finally, the spin-spin and density-density correlation functions are computed; the results suggest that the antiferromagnetic order of the exact solution does not extend up to large distances but exists locally. No charge density waves are present.

Electronic Structures of Ions ofm-Phenylenebis (Phenylmethylene). II - The Electron Correlation Effects -

m -Phenylenebis (phenylmethylene) ( m -PBPM) has the quintet ground state in the neutral state, due to the dynamic spin polarization (DSP) in the π system and exchange interaction between n and π electrons on carbenes. The experimental results by the ESR show that both monocation and monoanion of m -PBPM have the quartet ground state. We calculate the electronic structures of the ions with the unrestricted Hartree-Fock (UHF) and resonating Hartree-Fock (Res HF) methods. We use the model that includes only n and π electrons developed in a previous paper. The broken spatial and spin symmetries of the UHF solutions are restored by the Res HF calculations of electron correlation effects. In the UHF approximation, the doped charge is always localized on a carbene and the adjacent phenylenes as a spin density wave (SDW) polaron. The n spins on the right and left carbenes are parallel and antiparallel in the quartet and doublet states, respectively. Due to this fact, the stabilization by the quantum motion of th...

Self-energy correction to unrestricted Hartree-Fock solutions of lattice models for 3d transition-metal oxides.

In order to explain the excitation properties of $3d$ transition-metal oxides in a unified framework, we have performed second-order perturbation calculations of the self-energy corrections around the unrestricted Hartree-Fock solution of lattice models using the electronic-structure parameters deduced from photoemission spectroscopy. The self-energy modifies the magnitude of the band gap and causes substantial spectral weight transfer over a wide energy range both in insulating and metallic compounds of the Mott-Hubbard type as well as of the charge-transfer type, resulting in an improved agreement between theory and experiment.

LINEAR EXCITATIONS AROUND BROKEN-SYMMETRY MEAN-FIELD STATES: DOPING STATES IN THE COPPER OXIDES

Nonlinear doping states in Peierls–Hubbard models are investigated through response functions in magnetic, optical and vibronic channels. A random phase approximation is used for their analysis on the basis of totally unrestricted Hartree–Fock solutions, which in general have broken symmetries. The validity of the present method is first discussed through a comparison with known exact results for a two-site two-electron system. Then this method is applied to a two-dimensional three-band system with one additional hole relative to stoichiometry, for modeling the lightly doped copper oxides. New excitation modes are induced by doping and observed in the different channels of response functions. These doping-induced modes distinguish between various doping states and helpful for the identification of doping states in specific materials.

Vibronic, Optical, and Magnetic Excitations around Solitons in a One-Dimensional Two-Band Peierls-Hubbard Model

Nonlinear doping states in a charge-density-wave background and those in a spin-density- wave background are investigated through their response functions in vibronic, optical and magnetic channels in a one-dimensional two-band Peierls-Hubbard model. A random phase approximation is used for their analysis on the basis of totally unrestricted Hartree-Fock solutions. Doping-induced changes of excitation spectra in these different channels are distinctive to the nature of individual doping states