Unrestricted Hartree-Fock Theory Research Articles

Maximum Ionization in Restricted and Unrestricted Hartree-Fock Theory

In this paper, we investigate the maximum number of electrons that can be bound to a system of nuclei modelled by Hartree-Fock theory. We consider both the Restricted and Unrestricted Hartree-Fock models. We are taking a non-existence approach (necessary but not sufficient), in other words we are finding an upper bound on the maximum number of electrons. In giving a detailed account of the proof of Lieb’s bound [Theorem 1, Phys. Rev. A 29 (1984), 3018] for the Hartree-Fock models we establish several new auxiliary results, furthermore we propose a condition that, if satisfied, will give an improved upper bound on the maximum number of electrons within the Restricted Hartree-Fock model. For two-electron atoms we show that the latter condition holds.

Interplay of mass imbalance and frustration in correlated band insulators

We report the emergence or broadening of exotic magnetic metallic phases upon explicit breaking of $SU(2)$ symmetry by introduction of mass imbalance in a variant of Hubbard model, known as the ionic Hubbard model in the presence of frustration at half-filling on a square lattice. The ionic Hubbard model has in addition to hopping($\sim t$) and onsite coulomb repulsion ($\sim U$), a staggered ionic potential ($\sim \Delta$) which breaks translational symmetry of the underlying lattice. In the low to intermediate ranges of $U$ and $\Delta$, we use unrestricted Hartree-Fock theory to construct the phase diagram in the $U-\Delta$ plane for a fixed mass imbalance($\sim\eta$). Where as in the limit where both $U,\Delta$ are comparable and much larger than the first $(\sim t_{\sigma})$ and second $(\sim t_{\sigma}')$ neighbor hopping amplitudes , we employ the technique of generalized Gutzwiller approximation and subsequently use renormalized mean field theory to construct the phase diagram for varying values of hopping asymmetry. In both cases, starting from a correlated band insulator with weak antiferromagnetic spin density wave order, if we tune $U/\Delta$, we observe the opening of novel magnetically ordered metallic phases such as spin imbalanced ferromagnetic metal, ferrimagnetic metal and antiferromagnetic half metal. We also study singlet superconductivity in the d-wave and extended s-wave channels in the strong coupling limit of this model in the presence of mass imbalance.

Accurate real-time evolution of electron densities and ground-state properties from generalized Kohn-Sham theory

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation (xc) potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalised by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.

Singlet open-shell diradical nature and redox properties of conjugated carbonyls: a quantum chemical study

In the present work, a set of 90 conjugated carbonyls based on cyclic hydrocarbons with different ring sizes are studied by means of theoretical methods of quantum chemistry in order to analyze their singlet open-shell diradical character and its influence in different molecular properties. In particular, we have investigated the molecular structures, singlet–triplet gaps, ionization energies (IE), electron affinities (EA), first (E 1) and second (E 2) redox potentials and coordination of the reduced species to lithium cation. The diradical character is estimated using the diradical index from the spin-projected unrestricted Hartree–Fock theory (y PUHF). Non-negligible diradical character is accounted for most of the studied molecules, and trends may be devised in relation to their molecular properties. Thus, in those groups where several or all molecules show certain diradical character, this feature accomplishes larger EAs, E 1 and E 2. In this study, 35 carbonyls display E 1 values greater than 3 V, and the largest calculated value is 4.22 V. Regarding the IEs, no clear relationship may be established, although in some cases with large values of y PUHF small IEs are recorded. Finally, the binding energy to Li+ mainly depends on the relative position of the carbonyls, and the lithiation process is favored when the carbonyl moieties are adjacent or close in the molecule. These results highlight the importance of the diradical character on conjugated carbonyls and its effect in molecular properties such as redox potentials, a characteristic that may be exploited both theoretically and experimentally to provide with a deeper insight on the application of these diradicals as organic cathode materials.

Importance of Three-Body Interactions in Molecular Dynamics Simulations of Water Demonstrated with the Fragment Molecular Orbital Method.

The analytic first derivative with respect to nuclear coordinates is formulated and implemented in the framework of the three-body fragment molecular orbital (FMO) method. The gradient has been derived and implemented for restricted second-order Møller-Plesset perturbation theory, as well as for both restricted and unrestricted Hartree-Fock and density functional theory. The importance of the three-body fully analytic gradient is illustrated through the failure of the two-body FMO method during molecular dynamics simulations of a small water cluster. The parallel implementation of the fragment molecular orbital method, its parallel efficiency, and its scalability on the Blue Gene/Q architecture up to 262,144 CPU cores are also discussed.

Communication: Comparing ab initio methods of obtaining effective U parameters for closed-shell materials

The density functional theory (DFT)+U method is an efficient and effective way to calculate the ground-state properties of strongly correlated transition metal compounds, with the effective U parameters typically determined empirically. Two ab initio methods have been developed to compute the U parameter based on either constrained DFT (CDFT) or unrestricted Hartree-Fock (UHF) theory. Previous studies have demonstrated the success of both methods in typical open-shell materials such as FeO and NiO. In this Communication we report numerical instability issues that arise for the CDFT method when applied to closed-shell transition metals, by using ZnO and Cu2O as examples. By contrast, the UHF method behaves much more robustly for both closed- and open-shell materials, making it more suitable for treating closed-shell transition metals, as well as main group elements.

Computational strategy for graphene: Insight from odd electrons correlation

AbstractThe correlation of odd electrons in graphene turns out to be significant so that the species should be attributed to correlated ones. This finding profoundly influences the computational strategy addressing it to multireference computational schemes. Owing to serious problems related to the schemes realization, a compromise can be suggested using single‐determinant approaches based on either Hartree–Fock or density‐functional theory in the form of unrestricted open‐shell presentation. Both computational schemes enable to fix the electron correlation, whereas only the Hartree–Fock theory suggests a set of quantities to be calculated that can quantitatively characterize the electron correlation and be used for a quantitative description of such graphene properties as magnetism, chemical reactivity, and mechanical response. The article presents concepts and algorithms of the unrestricted Hartree–Fock theory applied for the consideration of magnetic properties of nanographenes, their chemical modification by the example of stepwise hydrogenation, as well as a possible governing the electron correlation by the carbon skeleton deformation. © 2012 Wiley Periodicals, Inc.

Hole transport in pure and doped hematite

Hematite (α-Fe2O3) is a promising candidate for use in photovoltaic (PV) and photoelectrochemical devices. Its poor conductivity is one major drawback. Doping hematite either p-type or n-type greatly enhances its measured conductivity and is required for potential p-n junctions in PVs. Here, we study hole transport in pure and doped hematite using an electrostatically embedded cluster model with ab initio quantum mechanics (unrestricted Hartree-Fock theory). Consistent with previous work, the model suggests that hole hopping is via oxygen anions for pure hematite. The activation energy for hole mobility is predicted to be at least 0.1 eV higher than the activation energy for electron mobility, consistent with the trend observed in experiments. We examine four dopants—magnesium(II), nickel(II), copper(II), and manganese(II/III) in direct cation substitution sites—to gain insight into the mechanism by which conductivity is improved. The activation energies are used to assess qualitative effects of different dopants. The hole carriers are predicted to be attracted to O anions near the dopants. The magnitude of the trapping effect is similar among the four dopants in their +2 oxidation states. The multivalent character of Mn doping facilitates local hole transport around Mn centers via a low-barrier O-Mn-O pathway, which suggests that higher hole mobility can be achieved with increasing Mn doping concentration, especially when a network of these low-barrier pathways is produced. Our results suggest that the experimentally observed conductivity increase in Mg-, Ni-, and Cu-doped p-type hematite is mostly due to an increase in hole carriers rather than improved mobility, and that Mg-, Ni-, and Cu-doping perform similarly, while the conductivity of Mn-doped hematite might be significantly improved in the high doping concentration limit.

Unrestricted Prescriptions for Open-Shell Singlet Diradicals: Using Economical Ab Initio and Density Functional Theory to Calculate Singlet–Triplet Gaps and Bond Dissociation Curves

Here we present and test several computational prescriptions for calculating singlet-triplet (ST) gap energies and bond dissociation curves for open-shell singlet diradicals using economical unrestricted single reference type calculations. For ST gap energies from Slipchenko and Krylov's atom and molecule test set (C, O, Si, NH, NF, OH(+), O(2), CH(2), and NH(2)(+)) spin unrestricted Hartree-Fock and MP2 energies result in errors greater than 15 kcal/mol. However, spin-projected (SP) Hartree-Fock theory in combination with spin-component-scaled (SCS) or scaled-opposite-spin (SOS) second-order perturbation theory gives ST gap energies with a mean unsigned error (MUE) of less than 2 kcal/mol. Density functionals generally give poor results for unrestricted energies and only the ωB97X-D, the M06, and the M06-2X functionals provide reasonable accuracy after spin-projection with MUE values of 4.7, 4.3, and 3.0 kcal/mol, respectively, with the 6-311++G(2d,2p) basis set. We also present a new one parameter hybrid density functional, diradical-1 (DR-1), based on Adamo and Barone's modified PW exchange functional with the PW91 correlation functional. This DR-1 method gives a mean error (ME) of 0.0 kcal/mol and a MUE value of 1.3 kcal/mol for ST gap energies. As another test of unrestricted methods the bond dissociation curves for methane (CH(4)) and hydrofluoric acid (H-F) were calculated with the M06-2X, DR-1, and ωB97X-D density functionals. All three of these functionals give reasonable results for the methane C-H bond but result in errors greater than 50 kcal/mol for the H-F bond dissociation. Spin-projection is found to significantly degrade bond dissociation curves past ~2.2 Å. Although unrestricted Hartree-Fock theory provides a very poor description of H-F bond dissociation, unrestricted SCS-MP2 and SOS-MP2 methods give accurate results.

Half-metallicity and spin-contamination of the electronic ground state of graphene nanoribbons and related systems: An impossible compromise?

An analysis using the formalism of crystalline orbitals for extended systems with periodicity in one dimension demonstrates that any antiferromagnetic and half-metallic spin-polarization of the edge states in n-acenes, and more generally in zigzag graphene nanoislands and nanoribbons of finite width, would imply a spin contamination <linear span>S(2)<linear span> that increases proportionally to system size, in sharp and clear contradiction with the implications of Lieb's theorem for compensated bipartite lattices and the expected value for a singlet (S = 0) electronic ground state. Verifications on naphthalene, larger n-acenes (n = 3-10) and rectangular nanographene islands of increasing size, as well as a comparison using unrestricted Hartree-Fock theory along with basis sets of improving quality against various many-body treatments demonstrate altogether that antiferromagnetism and half-metallicity in extended graphene nanoribbons will be quenched by an exact treatment of electron correlation, at the confines of non-relativistic many-body quantum mechanics. Indeed, for singlet states, symmetry-breakings in spin-densities are necessarily the outcome of a too approximate treatment of static and dynamic electron correlation in single-determinantal approaches, such as unrestricted Hartree-Fock or Density Functional Theory. In this context, such as the size-extensive spin-contamination to which it relates, half-metallicity is thus nothing else than a methodological artefact.

Fragmentation patterns of core-ionized thymine and 5-bromouracil

Photofragmentation of thymine and 5-bromouracil into cation and neutral fragments following the core ionization by soft x-rays using photoelectron-photoion-photoion coincidence technique has been studied. The fragment ion mass spectra were recorded in coincidence with the C 1s photoelectron spectra. In the case of thymine, deuterated samples were used to identify fragments. Deuteration or bromination allowed us to study not only the main fragmentation channels of these pyrimidine bases, but also to investigate if replacement of an exocyclic functional group affects molecular fragmentation. We found that the dominant fragmentation channels involve only one starting geometry, and the base ring and other bond cleavages, leading to the detected fragments, are essentially identical between thymine and 5-bromouracil. In addition, the relative intensities of the strongest fragmentation channels were determined and compared with calculated appearance energies using ab initio unrestricted Hartree-Fock theory.

Constrained-pairing mean-field theory. IV. Inclusion of corresponding pair constraints and connection to unrestricted Hartree–Fock theory

Our previously developed constrained-pairing mean-field theory (CPMFT) is shown to map onto an unrestricted Hartree-Fock (UHF) type method if one imposes a corresponding pair constraint to the correlation problem that forces occupation numbers to occur in pairs adding to one. In this new version, CPMFT has all the advantages of standard independent particle models (orbitals and orbital energies, to mention a few), yet unlike UHF, it can dissociate polyatomic molecules to the correct ground-state restricted open-shell Hartree-Fock atoms or fragments.

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.

Orbitals That Are Unrestricted in Active Pairs for Generalized Valence Bond Coupled Cluster Methods

Spin unrestriction is typically defined as a free variation of the molecular orbitals of different spins in order to lower the molecular energy. When applied to approximate active space electron correlation methods, such as generalized valence bond coupled cluster (GVB-CC) models, this approach often leads to undesirable artifacts. We therefore present an alternative unrestricted in active pairs (UAP) procedure for spin polarization in GVB-CC methods, which resembles the corresponding orbitals of unrestricted Hartree-Fock theory. The UAP procedure permits spin polarization only within the two orbital subspaces describing each electron pair (consisting of one nominally occupied and one virtual orbital). This great reduction to just a linear number of degrees of freedom associated with spin polarization eliminates many of the undesirable artifacts associated with unconstrained spin unrestriction. The UAP procedure is tested on a variety of potential curves for bond breaking and the properties of small radicals as well as larger polyenyl radicals, allyl, pentadienyl, and heptatrienyl.

Quantum Monte Carlo Studies of Transition Metal Oxides

Electronic structure of the manganese and iron oxide solids is studied by the quantum Monte Carlo (QMC) methods. The trial wavefunctions are built using orbitals from unrestricted Hartree-Fock and Density Functional Theory, and the electron-electron correlation is recovered by the fixed-node QMC. The correlation effects are significant and QMC estimations of the gap and cohesion show a very good agreement with experiment. Comparison with hybrid functional results points out the importance of the exact exchange for improvement of the Density Functional description of transition metal oxide systems. The QMC methodology have enabled us to calculate the FeO equation of state as well as the transition pressure between the low- and high-pressure FeO structures.

Ground state of two electrons on a sphere

We have performed a comprehensive study of the singlet ground state of two electrons on the surface of a sphere of radius $R$. We have used electronic structure models ranging from restricted and unrestricted Hartree-Fock theory to explicitly correlated treatments, the last of which lead to near-exact wavefunctions and energies for any value of $R$. M{\o}ller-Plesset energy corrections (up to fifth-order) are also considered, as well as the asymptotic solution in the large-$R$ regime.

Unrestricted hartree-fock SCF calculations on some hypothetical molecules with triplet ground states

Self consistent field /SCF/ calculations of dipyridine glyoxal and bianthrone photoproduct molecules with triplet ground state, using unrestricted Hartree-Fock theory

Rotationally invariant ab initio evaluation of Coulomb and exchange parameters for DFT+U calculations

Conventional density functional theory (DFT) fails for strongly correlated electron systems due to large intra-atomic self-interaction errors. The DFT+U method provides a means of overcoming these errors through the use of a parametrized potential that employs an exact treatment of quantum mechanical exchange interactions. The parameters that enter into this potential correspond to the spherically averaged intra-atomic Coulomb (U) and exchange (J) interactions. Recently, we developed an ab initio approach for evaluating these parameters on the basis of unrestricted Hartree-Fock (UHF) theory, which has the advantage of being free of self-interaction errors and does not require experimental input [Mosey and Carter, Phys. Rev. B 76, 155123 (2007)]. In this work, we build on that method to develop a more robust and convenient ab initio approach for evaluating U and J. The new technique employs a relationship between U and J and the Coulomb and exchange integrals evaluated using the entire set of UHF molecular orbitals (MOs) for the system. Employing the entire set of UHF MOs renders the method rotationally invariant and eliminates the difficulty in selecting unambiguously the MOs that correspond to localized states. These aspects overcome two significant deficiencies of our earlier method. The new technique is used to evaluate U and J for Cr(2)O(3), FeO, and Fe(2)O(3). The resulting values of U-J are close to empirical estimates of this quantity for each of these materials and are also similar to results of constrained DFT calculations. DFT+U calculations using the ab initio parameters yield results that are in good agreement with experiment. As such, this method offers a means of performing accurate and fully predictive DFT+U calculations of strongly correlated electron materials.

Unrestricted Hartree-Fock theory of Wigner crystals

We demonstrate that unrestricted Hartree-Fock theory applied to electrons in a uniform potential has stable Wigner crystal solutions for ${r}_{s}&gt;~1.44$ in two dimensions and ${r}_{s}&gt;~4.5$ in three dimensions. The correlation energies of the Wigner crystal phases are considerably smaller than those of the fluid phases at the same density.

Electronic structure of dangling bonds in amorphous silicon studied via a density-matrix functional method

A structural model of hydrogenated amorphous silicon containing an isolated dangling bond is used to investigate the effects of electron interactions on the electronic level splittings, localization of charge and spin, and fluctuations in charge and spin. These properties are calculated with a recently developed density-matrix correlation-energy functional applied to a generalized Anderson Hamiltonian, consisting of tight-binding one-electron terms parametrizing hydrogenated amorphous silicon plus a local interaction term. The energy level splittings approach an asymptotic value for large values of the electron-interaction parameter U, and for physically relevant values of U are in the range 0.3-0.5 eV. The electron spin is highly localized on the central orbital of the dangling bond while the charge is spread over a larger region surrounding the dangling bond site. These results are consistent with known experimental data and previous density-functional calculations. The spin fluctuations are quite different from those obtained with unrestricted Hartree-Fock theory.