Unrestricted Hartree-Fock Wave Function Research Articles

Particle-Breaking Unrestricted Hartree-Fock Theory for Open Molecular Systems.

We recently introduced the particle-breaking restricted Hartree-Fock (PBRHF) model, a mean-field approach to address the fractional charging of molecules when they interact with an electronic environment. In this paper, we present an extension of the model referred to as particle-breaking unrestricted Hartree-Fock (PBUHF). The unrestricted formulation contains odd-electron states necessary for a realistic description of fractional charging. Within the PBUHF parametrization, we use two-body operators as they yield convenient operator transformations. However, two-body operators can change only the particle number by two. Therefore, we include noninteracting zero-energy bath orbitals to generate a linear combination of even and odd electron states. Depending on whether the occupied or virtual orbitals of a molecule interact with the environment, the average number of electrons is either decreased or increased. Without interaction, PBUHF reduces to the unrestricted Hartree-Fock wave function.

A geminal theory based on the generalized electron pairing.

As in the hierarchy of the Hartree-Fock theory in terms of spin symmetry, the extension of spin functions in a theory of two-electron units or geminal, was developed in this study. A trial wave function is constructed as an antisymmetrized product of geminals, in which singlet and triplet two-electron functions are fully mixed. We present a variational optimization method for this generalized pairing wave function in the strong orthogonality condition. The present method is considered an extension of the antisymmetrized product of strongly orthogonal geminals or perfect pairing generalized valence bond methods, maintaining the compactness of the trial wave function. The obtained broken-symmetry solutions were similar to the unrestricted Hartree-Fock wave functions in terms of spin contamination while giving lower energy due to the inclusion of the electron correlation effect in geminals. The degeneracy of the obtained broken-symmetry solutions in the Sz space is reported for the tested four-electron systems.

Theoretical Investigation on C11H8 Bicyclic Carbene and Allene Isomers

Recent studies postulate that the presence of polycyclic aromatic hydrocarbons (PAHs) in the interstellar medium (ISM) could have been formed through resonance-stabilized arylcarbene intermediates. However, identifying most of these reactive intermediates is very challenging experimentally due to their metastability and other experimental constrains. Thus, computational studies that cover the thermodynamic versus kinetic stability of various possible structures would be beneficial for successfully identifying new molecules either in the laboratory and/or in the ISM. In this paper, more than four hundred C11H8 carbene isomers have been theoretically investigated employing density functional theory (DFT). Hybrid density functionals B3LYP and ωB97XD with 6-311 + G (d,p) basis set have been used for singlet electronic states, whereas, triplet spin states were optimized at the same level using an unrestricted Hartree-Fock wavefunction. Although the skeletal structures of C11H8 can be categorized into monocyclic, bicyclic, tricyclic, tetracyclic and acyclic isomers, bicyclic carbenes have shown better stability due to the presence of resonance stabilized azulenyl/naphthyl rings. In this category, some isomers (1-, 2-, 5- and 6-azulenylcarbenes and 1- and 2-naphthylcarbenes) have also been detected recently in the laboratory and simple aromatic carbenes such as cyclopropenylidene and its homologues are detected in the ISM. Thus, we have systematically investigated the energetic and spectroscopic properties of resonance stabilized 5-, 6-, 7- and 8-membered ring containing bicyclic isomers of C11H8 and the fingerprint regions of the infrared spectrum for each class of these bicyclic compounds.

Half-Projection of the Strongly Orthogonal Unrestricted Geminals’ Product Wave Function

Half-projection of a wave function built as a product of singlet-triplet mixed two-electron fragments (geminals) is explored. The condition of strong orthogonality is imposed between geminals, the initial Ansatz accommodating the unrestricted Hartree-Fock (UHF) wave function as a special case. The here explored geminal product is more general than UHF, it allowing for single covalent bond breaking in a spin pure manner. The strong correlation present in two geminals simultaneously is accompanied by spin contamination at the geminal product level. Variation after half-projection leads to an eigenvalue problem of an effective two-electron Hamiltonian, conserving the computational economy of the initial model. While the resulting expressions are shown to violate size consistency, quantification reveals a suppressed error with half-projection as compared to the full spin projected then varied scheme. Spin contamination is also found to be diminished compared to the initial, unprojected wave function.

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.

Geminal perturbation theory based on the unrestricted Hartree–Fock wavefunction

A perturbative correction exploiting natural orbitals and the pair function structure of the unrestricted Hartree-Fock (UHF) wavefunction is devised. The method offers a simple framework for describing multireference systems where static correlation is captured by UHF. The UHF wavefunction is built of two-electron fragments (geminals), involving both singlet and triplet (ms = 0) parts. At order zero of the perturbative treatment, configuration interaction coefficients of UHF geminals are relaxed. The zero order Hamiltonian is of the Dyall-type, including explicit two-electron interaction within geminals and leading to a formal 6th power scaling. Adopting an effective one-electron zero order Hamiltonian term for the subset of virtual orbitals reduces scaling of the correction step to 4th power. Formal properties of the proposed schemes are discussed. Energetic data and natural occupation numbers of illustrative test systems are used to assess the new approach. The cases where the wavefunction becomes essentially spin pure at the level of reference show good performance. Spin contamination remaining at order zero is found to undermine the perturbative correction.

Spin-projected generalized Hartree-Fock method as a polynomial of particle-hole excitations

The past several years have seen renewed interest in the use of symmetry-projected Hartree-Fock for the description of strong correlations. Unfortunately, these symmetry-projected mean-field methods do not adequately account for dynamic correlation. Presumably, this shortcoming could be addressed if one could combine symmetry-projected Hartree-Fock with a many-body method such as coupled cluster theory, but this is by no means straightforward because the two techniques are formulated in very different ways. However, we have recently shown that the singlet $S^2$-projected unrestricted Hartree-Fock wave function can in fact be written in a coupled cluster-like wave function: that is, the spin-projected unrestricted Hartree-Fock wave function can be written as a polynomial of a double-excitation operator acting on some closed-shell reference determinant. Here, we extend this result and show that the spin-projected generalized Hartree-Fock wave function (which has both $S^2$ and $S_z$ projection) is likewise a polynomial of low-order excitation operators acting on a closed-shell determinant, and provide a closed-form expression for the resulting polynomial coefficients. We include a few preliminary applications of the combination of this spin-projected Hartree-Fock and coupled cluster theory to the Hubbard Hamiltonian, and comment on generalizations of the methodology. Results here are not for production level, but a similarity transformed theory that combines the two offers the promise of being accurate for both weak and strong correlation, and particularly may offer significant improvements in the intermediate correlation regime where neither projected Hartree-Fock nor coupled cluster is particularly accurate.

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.

First UHF Implementation of the Incremental Scheme for Open-Shell Systems.

The incremental scheme makes it possible to compute CCSD(T) correlation energies to high accuracy for large systems. We present the first extension of this fully automated black-box approach to open-shell systems using an Unrestricted Hartree-Fock (UHF) wave function, extending the efficient domain-specific basis set approach to handle open-shell references. We test our approach on a set of organic and metal organic structures and molecular clusters and demonstrate standard deviations from canonical CCSD(T) values of only 1.35 kJ/mol using a triple ζ basis set. We find that the incremental scheme is significantly more cost-effective than the canonical implementation even for relatively small systems and that the ease of parallelization makes it possible to perform high-level calculations on large systems in a few hours on inexpensive computers. We show that the approximations that make our approach widely applicable are significantly smaller than both the basis set incompleteness error and the intrinsic error of the CCSD(T) method, and we further demonstrate that incremental energies can be reliably used in extrapolation schemes to obtain near complete basis set limit CCSD(T) reaction energies for large systems.

Spin contamination and noncollinearity in general complex Hartree–Fock wave functions

An expression for the square of the spin operator expectation value, $$\langle S^2 \rangle$$ , is obtained for a general complex Hartree–Fock wave function and decomposed into four contributions: the main one whose expression is formally identical to the restricted (open-shell) Hartree–Fock expression. A spin contamination one formally analogous to that found for spin unrestricted Hartree–Fock wave functions. A noncollinearity contribution related to the fact that the wave function is not an eigenfunction of the spin-S z operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered.

Singlet and triplet instability theorems.

A useful definition of orbital degeneracy—form-degeneracy—is introduced, which is distinct from the usual energy-degeneracy: Two canonical spatial orbitals are form-degenerate when the energy expectation value in the restricted Hartree-Fock (RHF) wave function is unaltered upon a two-electron excitation from one of these orbitals to the other. Form-degenerate orbitals tend to have isomorphic electron densities and occur in the highest-occupied and lowest-unoccupied molecular orbitals (HOMOs and LUMOs) of strongly correlated systems. Here, we present a mathematical proof of the existence of a triplet instability in a real or complex RHF wave function of a finite system in the space of real or complex unrestricted Hartree-Fock wave functions when HOMO and LUMO are energy- or form-degenerate. We also show that a singlet instability always exists in a real RHF wave function of a finite system in the space of complex RHF wave functions, when HOMO and LUMO are form-degenerate, but have nonidentical electron densities, or are energy-degenerate. These theorems provide Hartree-Fock-theory-based explanations of Hund's rule, a singlet instability in Jahn-Teller systems, biradicaloid electronic structures, and a triplet instability during some covalent bond breaking. They also suggest (but not guarantee) the spontaneous formation of a spin density wave (SDW) in a metallic solid. The stability theory underlying these theorems extended to a continuous orbital-energy spectrum proves the existence of an oscillating (nonspiral) SDW instability in one- and three-dimensional homogeneous electron gases, but only at low densities or for strong interactions.

The analysis of polarization effects on the interelectronic separations in the atoms and molecules of the G1 test set

AbstractWe have calculated the differences between kinetic, nuclear attraction, and two‐electron energies as well as the position intracules, P(u), for 56 molecules and 17 atoms contained in the G1 test set, using basis sets differing in their degree of polarization. The calculations were performed using an unrestricted Hartree–Fock wave function that was expanded using the 6‐31G, 6‐31G(d,p), 6‐31G(3d,3p), 6‐31G(3df,3pd), 6‐311G, 6‐311G(d,p), 6‐311G(3d,3p), 6‐311G(3df,3pd), 6‐311++G, 6‐311++G(d,p), 6‐311++G(3d,3p), and 6‐311++G(3df,3pd) basis sets. In addition to these Pople basis sets, the Dunning's cc‐pVDZ and cc‐pVTZ bases as well as the non‐polarized portions of these basis sets were used. We observe a consistent contraction of electron pairs (as measured by the distribution in P(u)) as the number of polarization functions is increased beyond zero in molecular systems. This contraction is related to an increase in the electron density in the internuclear bonding regions of molecules as the wave functions are polarized. We note that the increased flexibility in the polarized basis set allows for a more accurate description of the chemical bond and is energetically favorable due to an increased level of attraction to the nuclei (despite the obvious increase in electronic repulsion energy). Because of the increase in electron density in internal regions of the molecule (at the expense of electron density depletions in outer regions), the probability of observing electrons closer together (i.e., at smaller u) increases due to the relationship between P(u) and its Coulomb component, J(u). © 2012 Wiley Periodicals, Inc.

Electronic Excited States Calculated Using Generalized Spin-Orbital Functions Including Spin-Orbit Interactions

We examined an equation-of-motion coupled-cluster (EOM-CC) method, which is basically equivalent to the SAC-CI method, starting from the generalized unrestricted Hartree-Fock (GUHF) wave function incorporated with a spin-dependent relativistic Hamiltonian. Demonstrations of the present method were carried out for the spin-orbit (SO) splitting energies of 3P and 2P states of relatively light atoms (Be - Ar). Although we considered only the lowest-order (c-2) Breit-Pauli one- and two-electron SO terms, it was found that the SO splitting energies calculated by the present method agree well with experimental ones, and are almost equivalent to those obtained by some conventional higher-order relativistic and electron-correlated methods.

Correlated ab Initio Spin Densities for Larger Molecules: Orbital-Optimized Spin-Component-Scaled MP2 Method

The performance of the orbital-optimized MP2 method (OO-MP2) and its spin-component-scaled variant is investigated for the prediction of atomic and molecular hyperfine coupling constants (hfccs). The calculated hfccs are compared to experimental as well as to CCSD(T) reference results. The OO-MP2 isotropic hfccs for a series of small radicals are almost of CCSD quality but are obtained with iterative O (N⁵) effort. The dipolar hfccs are less well predicted by the OO-MP2 methods, whereas spin-component scaling improves the description of the hyperfine structure. The spin contamination in the OO-MP2 wave function is drastically reduced compared to the standard unrestricted Hartree-Fock wave function. The applicability of the OO-MP2 to fairly large systems is demonstrated for the solvated p-benzosemiquinone radical anion, where calculations with almost 2000 basis functions have been performed.

Use of unrestricted hartree-fock wave functions in RPA calculations

Simple RPA theory breaks down when there exists a UHF wave function of lower energy than the RHF ground-state wave function. A UHF wave function as well as its projected form are used to investigate the excitation energies and transition moments within an RPA calculation on a frequently investigated two-electron model for ethylene.

Intracule functional models. IV. Basis set effects

We have calculated position and dot intracules for a series of atomic and molecular systems, starting from an unrestricted Hartree-Fock wave function, expanded using the STO-3G, 6-31G, 6-311G, 6-311++G, 6-311++G(d,p), 6-311++G(3d,3p), and 6-311++G(3df,3pd) basis sets as well as the nonpolarized part of Dunning's cc-pV5Z basis. We find that the basis set effects on the intracules are small and that correlation energies from the dot intracule ansatz are remarkably insensitive to the basis set quality. Mean absolute errors in correlation energies across the G1 data set agree to within 2 mE(h) for all basis sets tested.

Ground state potential energy surfaces and bound states of M–He dimers (M=Cu,Ag,Au): A theoretical investigation

We present an ab initio investigation on the ground state interaction potentials [potential energy surface (PES)] between helium and the group 11 metal atoms: copper, silver, and gold. To the best of our knowledge, there are no previous theoretical PESs proposed for Cu-He and Au-He, and a single one for Ag-He [Z. J. Jakubek and M. Takami, Chem. Phys. Lett. 265, 653 (1997)], computed about 10 years ago at MP2 level and significantly improved by our study. To reach a high degree of accuracy in the determination of the three M-He potentials (M=Cu,Ag,Au), we performed extensive series of test computations to establish the appropriate basis set, the theoretical method, and the computational scheme for these systems. For each M-He dimer we computed the PES at the CCSD(T) level of theory, starting from the reference unrestricted Hartree-Fock wave function. We described the inner shells with relativistic small core pseudopotentials, and we adopted high quality basis sets for the valence electrons. We also performed CCSDT computations in a limited set of M-He internuclear distances, adopting a medium-sized basis set, such as to define for each dimer a CCSD(T) to CCSDT correction term and to improve further the quality of the CCSD(T) interaction potentials. The Cu-He complex has minimum interaction energy (E(min)) of -28.4 microhartree at the internuclear distance of 4.59 A (R(min)), and the short-range repulsive wall starts at 4.04 A (R(E=0)). Quite interestingly, the PES of Ag-He is more attractive (E(min)=-33.8 microhartree) but presents nearly the same R(min) and R(E=0) values, 4.60 and 4.04 A, respectively. The interaction potential for Au-He is markedly deeper and shifted at shorter distances as compared to the lighter complexes, with E(min)=-69.6 microhartree, R(min)=4.09 A and R(E=0)=3.60 A. As a first insight in the structure of M-He(n) aggregates, we determined the rovibrational structure of the three M-He dimers. The Cu-He and Ag-He potentials support just few rotational excitations, while the Au-He PES admits also a bound vibrational excitation.

Toward accurate thermochemical models for transition metals: G3Large basis sets for atoms Sc–Zn

An augmented valence triple-zeta basis set, referred to as G3Large, is reported for the first-row transition metal elements Sc through Zn. The basis set is constructed in a manner similar to the G3Large basis set developed previously for other elements (H-Ar, K, Ca, Ga-Kr) and used as a key component in Gaussian-3 theory. It is based on a contraction of a set of 15s13p5d Gaussian primitives to 8s7p3d, and also includes sets of f and g polarization functions, diffuse spd functions, and core df polarization functions. The basis set is evaluated with triples-augmented coupled cluster [CCSD(T)] and Brueckner orbital [BD(T)] methods for a small test set involving energies of atoms, atomic ions, and diatomic hydrides. It performs well for the low-lying s-->d excitation energies of atoms, atomic ionization energies, and the dissociation energies of the diatomic hydrides. The Brueckner orbital-based BD(T) method performs substantially better than Hartree-Fock-based CCSD(T) for molecules such as NiH, where the starting unrestricted Hartree-Fock wavefunction suffers from a high degree of spin contamination. Comparison with available data for geometries of transition metal hydrides also shows good agreement. A smaller basis set without core polarization functions, G3MP2Large, is also defined.

Triplet-triplet energy-transfer coupling: Theory and calculation

Triplet-triplet (TT) energy transfer requires two molecular fragments to exchange electrons that carry different spin and energy. In this paper, we analyze and report values of the electronic coupling strengths for TT energy transfer. Two different methods were proposed and tested: (1) Directly calculating the off-diagonal Hamiltonian matrix element. This direct coupling scheme was generalized from the one used for electron transfer coupling, where two spin-localized unrestricted Hartree-Fock wave functions are used as the zero-order reactant and product states, and the off-diagonal Hamiltonian matrix elements are calculated directly. (2) From energy gaps derived from configuration-interaction-singles (CIS) scheme. Both methods yielded very similar results for the systems tested. For TT coupling between a pair of face-to-face ethylene molecules, the exponential attenuation factor is 2.59 A(-1)(CIS6-311+G(**)), which is about twice as large as typical values for electron transfer. With a series of fully stacked polyene pairs, we found that the TT coupling magnitudes and attenuation rates are very similar irrespective of their molecular size. If the polyenes were partially stacked, TT couplings were much reduced, and they decay more rapidly with distance than those of full-stacked systems. Our results showed that the TT coupling arises mainly from the region of close contact between the donor and acceptor frontier orbitals, and the exponential decay of the coupling with separation depends on the details of the molecular contacts. With our calculated results, nanosecond or picosecond time scales for TT energy-transfer rates are possible.

Triplet instability in doublet systems.

Some problems associated with unrestricted wave functions for open-shell molecules are discussed in the contest of coupled-cluster calculation of molecular properties. Particular attention is given to a phenomenon akin to the "triplet instability" of closed-shell molecules, where the approximate spin pairing of a nominal pair of electrons in the unrestricted Hartree-Fock wave function begins to give way to significant spin polarization. This problem-which gives rise to pronounced spin contamination-is discussed from the point of view of orbital instability and occupation numbers of the charge density matrix. The onset, rather than the magnitude of the spin contamination is analyzed in detail for diatomics, especially heteronuclear cases where the transition to significant spin contamination does not occur discontinuously. It is shown that the qualitative description of this phenomenon satisfactorily explains anomalous results for NO and PO, although the magnitude of spin contamination in these molecules is significantly less than in other cases where anomalous results are not observed. It appears that calculations of equilibrium molecular properties using coupled-cluster methods based on unrestricted Hartree-Fock reference should be monitored carefully for any molecule containing a multiple bond, especially when the bonded pair of atoms appear in different rows of the periodic table.