Unrestricted Hartree-Fock States Research Articles

An auxiliary-Field quantum Monte Carlo perspective on the ground state of the dense uniform electron gas: An investigation with Hartree-Fock trial wavefunctions

We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combination of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC) was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤ 2.0 and provided new benchmark data for future method development. We found that the UEG models with rs = 5.0 remain to be challenging for RHF+ph-AFQMC. Employing nonorthogonal configuration expansions or unrestricted HF states as trial wavefunctions was also found to be ineffective in the case of the 14-electron UEG model with rs = 5.0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.

Quantum mechanical calculation of the longitudinal elastic modulus and of deviations from Hooke’s law in polyethylene

The microscopic longitudinal elastic modulus, deviations from Hooke’s law, the maximum of the retractive force and tensile strength are calculated for single infinite polyethylene chains using a first principle crystal orbital method and including electron correlation effects by perturbation theory. The best theoretical value of the modulus is 334 GPa at the single-particle level and 276 GPa including correlation. Significant deviations from Hooke’s law start at small deformations and amount to 30%–40% at ε=0.10–0.15. The maximum of the retractive force is f=0.814 mdyn at ε=0.38 and f=0.631 mdyn at ε=0.31, using restricted or unrestricted Hartree–Fock states as zeroth-order wave functions for perturbation theory, respectively. The corresponding maxima of the microscopic tensile strength are at 44.6 and 34.2 GPa, respectively.

Theory of Electronic Structures and Lattice Distortions in Polyacetylene and Itinerant Peierls Systems. I: UHF Transfer Matrix Method Adapted to the Long Range Coulomb Interaction and UHF States in Regular Bond Alternated Lattice

We develop in this series a theory of itinerant Peierls systems that mainly aims to study polyacetylene. In order to investigate roles of the long range Coulomb interaction in electronic structures and lattice distortions in a Peierls system, we develop a transfer matrix technique adapted to the. unrestricted Hartree·Fock (UHF) approximation. The method makes it pos­ sible to calculate UHF states and equilibrium lattice geometries in a lattice with any aperiodic structure due to the presence of solitons. Applying the method, we obtain the UHF states in the case of regular lattice that may become the HF ground state. There are at least eight such UHF states which are distinguished by the long range orders in the spin and charge densities and the bond orders. The nearest neighbour exchange interaction is responsible to produce such plentiful UHF states. We calculate the bond alternation potential and the band gap in trans polyacetylene using paramet­ rizations of the Hamiltonian and the elastic potential that are able to well reproduce spectra and equilibrium geometries of small conjugated hydrocarbons. The bond alternation potential is very sensitive to the strength of the nearest neighbour Coulomb interaction. The Coulomb potential with fast damping prevents the bond alternation. The band gap is much larger than the energy 2 eV of the lowest absorption band as long as the Coulomb interaction in poly­ acetylene is similar to that in small conjugated hydrocarbons, suggesting the excitonic nature of the absorption band. We make also an illustrative calculation for an itinerant spin Peierls system with an antiferromagnetic spin order as well as a lattice dimerization.

Studies on the Electronic Structures of Hydrocarbon Diradicals by the Unrestricted Hartree-Fock Theory

Abstract Electronic structures of low-lying states of cyclobutadiene, trimethylenemethane (2), and tetramethyleneethane (3) are studied by the unrestricted Hartree-Fock (UHF) theory. Characteristics of electronic correlations, called dynamical spin polarization effect, in these molecules are well described by corresponding orbitals and spin density structures of low-lying UHF solutions. The spin polarization of σ electrons is found to be relatively small but not negligible for quantitative discussion. The two low-lying singlet states of 2 are described by two UHF solutions with different spin density structures. The fact that the singlet UHF state of 3 lies a little below the triplet UHF state is explained by the difference in the spin polarization of the bonding π electrons between the singlet and triplet states. Two rules are given which predict the spin structure of low-lying UHF states and the spin multiplicity of ground states of hydrocarbon diradicals. These rules are applied to 1,1,2,3,3-pentamethylenepropane and 1,3-dimethylenecyclobutadiene, which confirms their validity.

Spin dependent properties of perturbed wave functions: An analytic comparison of the exact, UHF, and spin-projected UHF states

A generally applicable diagrammatic representation for spin-annihilated wave functions is developed; the diagrammatic approach avoids the complex algebra usually associated with the application of spin-annihilation operators. The diagrammatic formulation is first applied to the perturbation expansion for the exact wave function to elucidate the diagrammatic origin of the (correct) spin eigenfunction properties of the exact perturbed state. Employing a previously derived perturbation expansion for the unrestricted Hartree–Fock (UHF) wave function, the diagrammatic spin-annihilation formalism is then used to analyze the effect of projecting unwanted spin states from the UHF wave function. Results obtained for the projected UHF state are compared to those appropriate to both the exact perturbed wave function and the unprojected UHF wave function. It is shown that results obtained by annihilating only the lowest unwanted spin multiplicity are expected, in a perturbation theory sense, to be very similar to those obtained by annihilating all unwanted spin states. However, it is further shown that additional terms are introduced into the UHF wave function by the spin projection procedure and that these terms are in general unrelated to the exact perturbative corrections to the UHF state. In particular, the wave function, energy, one and two-electron density matrices, and spin densities obtained for the projected state are all shown to be in error in the lowest order of perturbation theory; in contrast, the UHF state leads to the correct one-electron density matrix and spin densities in lowest order.

The Unrestricted Hartree-Fock Theory of Chemical Reactions. V: The Reaction of Three Hydrogens in Isosceles Triangular Conformations

A UHF theoretical analysis is carried out for the reaction of three hydrogens in iso­ sceles triangular conformations. It is shown that the UHF ground state of the H 8 system consists of· three different phases; an RHF configuration in a region of acute triangles, an ASDW configuration in a region of flat triangles and a TSDW configuration iri a region near equilateral triangles. The potentials of the UHF states are calculated and their interconnection relation is studied. A configurational analysis in terms of the RHF orbitals is carried out for the UHF states at equilateral triangles. It is shown that the TSDW state is a good approximation to ·the full CI ground state. The reason for the missing of the Jahn-Teller cusp in the UHF ground state potential is discussed. Importance of spin flipping instabilities in mechanistic characterization of radical reactions is pointed out. In previous papers,n-5> we have carried out unrestricted Hartree-Fock (UHF) theoretical analyses of the chemical reactions in singlet ground state. We consider, in this paper, the three hydrogen system with isosceles triangular conformations as one of the simplest model system for insertion reactions of free radicals. We show from an analysis of the H 3 system that instability of a ground state configu­ ration and appearance of a new ground state of broken symmetry character are phenomena which may occur not only in the reacting systems with a singlet ground state but also in the systems with a doublet ground state, i.e., in free radical reactions. The UHF ground state of the H 3 system undergoes phase transitions between a restricted Hartree-Fock (RHF), an axial spin density wave (ASDW) and a torsional spin density wave (TSDW) configurations, indicating the presence of a hierarchy of doublet radicals with electronic structures of increas­ ing complexities as in singlet radicals discussed previously.5> A configurational analysis is carried out for the ASDW and the TSDW states at equilateral triangles in order to clarify the nature of the electronic correlation effects incorporated into the UHF states. The reason for the missing of the Jahn-Teller cusp6> in the UHF ground state

Pi-Electron Theory of Acetylene. II. Unrestricted Hartree—Fock Theory

Unrestricted Hartree—Fock (UHF) theory was applied to a Pariser—Parr—Pople pi-electron Hamiltonian for acetylene. The symmetries that must be broken in order to obtain the best UHF ground state were examined as a function of the Hamiltonian parameters. Improved ground-state wavefunctions were obtained for this two-band system by projecting out the undesired symmetries. Ground-state energies and correlation functions of the symmetrized UHF states were compared to the exact diagonalization of the Hamiltonian. For the model used, electron correlation is very well represented by the symmetrized UHF states.