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.     

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

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

Constrained active space unrestricted mean-field methods for controlling spin-contamination

We have recently proposed a novel approach for obtaining high-spin restricted open-shell Hartree-Fock wave functions by imposing constraints on the unrestricted Hartree-Fock (UHF) method [T. Tsuchimochi and G. E. Scuseria, J. Chem. Phys. 133, 141102 (2010)]. We here extend these ideas to the case where the constraints are released in an active space but imposed elsewhere. If the active space is properly chosen, our constrained UHF (CUHF) method greatly benefits from a controlled broken-symmetry effect while avoiding the massive spin contamination of traditional UHF. We also revisit and apply Lo?wdin's projection operator to CUHF and obtain multireference wave functions with moderate computational cost. We report singlet-triplet energy splittings showing that our constrained scheme outperforms fully unrestricted methods. This constrained approach can be readily used in spin density functional theory with similar favorable effects.     

Electron affinities of p-benzoquinone, p-benzoquinone imine and p-benzoquinone diimine, and spin densities of their p-benzosemiq

Restricted and unrestricted (U) Hartree–Fock (HF), second-order Møller–Plesset perturbation (MP2), density functional (DF), hybrid HF/DF and semiempirical (half-electron (HE) method) models have been used to calculate adiabatic electron affinities (EAad values) of p- benzoquinone (I), p-benzoquinone imine (VI) and p-benzoquinone diimine (XI), as well as expectation values (S2) and spin density distributions in the radical anions of I, VI and XI. The AM1/AM1-HE and ab initio calculated structures are found to be in accord with each other. The ROHF/6-31G(d) method gave the poorest EAad result. The UHF and UMP2 wave functions were found to be substantially spin contaminated (for the radicals) and the accuracies of the EAad values calculated were also poor. The use of molecular energies obtained after spin annihilation did not lead to significant improvement of the UHF and UMP2 results. In contrast to the ROHF, UHF and UMP2 results, the DF(USVWN, UBVWN, UBLYP) and hybrid HF/DF(UB3LYP) methods, as well as the AM1-HE, gave much better results. The calculated EAad values decreased, as predicted by most of the models, in the order EAad(I) > EAad(VI) > EAad(XI). The differences in the EAs, EAad(I) – EAad(VI) and EAad(I) – EAad(XI), were consistently predicted to be about 8–9 and 17–18 kcal/mol, respectively, by the DF, B3LYP and AM1-HE models. The performance of the PM3 and SAM1 models was not as good as the AM1 model. Of all the methods tested, the B3LYP/6-311G(d,p) model is concluded to give the most accurate quantitative trend (I(42.6) > VI(33.1) > XI(23.7)) in EAad. The predicted trend in EA can satisfactorily be rationalized by the calculated LUMO orbital energies, atomic charges and spin density distributions. Analysis of the spin density data predicts that phenoxyl- and anilino-type radical anions predominate in the p-benzosemiquinones of I and XI, respectively, while both phenoxyl- and anilino-type radicals contribute to the structure of the p-benzosemiquinone of VI, with the anilino-type predominating.     

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 ( = 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, , 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.     

Ab initio spectroscopic characterization of the HNNO and ONHN radicals

A composite coupled cluster methodology is used with systematic sequences of correlation consistent basis sets to accurately determine the structure, vibrational frequencies, and isotopic shifts for trans-HNNO ((2)A'), cis-HNNO ((2)A'), and ONHN ((2)A'). Anharmonic corrections to the vibrational frequencies and rotational constants are obtained using density functional theory. With basis sets larger than double-zeta, large differences between restricted open-shell Hartree-Fock (ROHF)-based and unrestricted Hartree-Fock (UHF)-based coupled cluster harmonic frequencies are calculated, with the UHF-based ones judged to be more reliable based on an analysis of the orbital hessian eigenvalues. The final calculated anharmonic vibrational band origins are generally in good agreement with the experimental values measured in rare gas matrices. The calculation of the vibrational band origins of the isovalent NO(2) molecule at similar levels of theory exhibits an agreement with experiment to within a few wavenumbers. In the latter case, however, a ROHF treatment was required since the UHF approach failed to provide realistic frequencies for the antisymmetric stretching mode. The heat of formation at 0 K of trans-HNNO is calculated to be 50.5 ± 0.5 kcal∕mol using a very similar composite coupled cluster methodology as in the structure and harmonic frequency determinations.     

Koopmans' theorem in the ROHF method: canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.     

Electronic properties of the low‐lying spin states of dimethylnitrosamine coordinated to Fe(III) heme models: An ab initio study

The unusual O‐coordination mode of nitrosamines to Fe(III) heme models has been observed in the bis(dimethylnitrosamine)(meso‐tetraphenylporphyrinate)iron(III) cation. For the first time, this latter as well as the simpler bis(dimethylnitrosamine)(porphinate)iron(III) heme model cations have been studied through ab initio methods. The sextet, quartet, and doublet spin states of both cations have been studied through single‐point calculations based on the experimental (X‐ray) geometry. Their energies, charges, and spin densities have been analyzed. The obtained results (at the UHF/cc‐pVDZ and ROHF/cc‐pVDZ levels) indicate that the peripheral benzene rings are of secondary importance for the coordination of dimethylnitrosamine to the Fe(III) porphyrin core. The obtained energy ordering is sextet < quartet < doublet, at all computational levels. The UHF, ROHF, and UMP2 results indicate an excess of alpha spin density around the Fe atom, a low covalency for the Fe? O bond and a substantial charge transfer to the Fe atom. Our best estimates [obtained at ROMP2 level with the mixed cc‐pVDZ/cc‐pVTZ‐DK(Fe) basis set] for the energy differences (in eV) between the three spin states considered are 0.929 for the sextet‐quartet gap and 0.812 for the quartet‐doublet gap, which indicate that the spin crossover (at room temperature) is very unlikely. These results represent the substantial decrease in the uncorrelated values. The implications of spin contaminations at the UHF and UMP2 levels for subsequent geometry optimizations to be performed in the smaller cation have also been discussed. © 2013 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-Extended Hartree–Fock calculation for hydrocarbon π-radicals

An iterative method of solving the Mestechkin spin-extended Hartree–Fock equations is proposed. On this basis projected and spin-extended values of energies and spin densities for some hydrocarbon π-radicals are calculated. Relations among UHF , ASA , AA , projected, and spin-extended energies are discussed.     

Ab initio study of the cyclooctatetraenyl radical

Ab initio calculations have been carried out on the 1,3,5,7- and 1,2,4,7-tetraene configurations of the cyclooctatetraenyl radical at UHF, ROHF, MCSCF, ROCISD, QCISD, and CCSD(T) levels of theory with 6-311G(d,p) and cc-pVDZ basis sets. Although spin contamination is present, the ROCISD calculations support the energies obtained from less intensive, UHF-based coupled cluster calculations over the energies obtained from MCSCF analysis of the pi-electron orbitals. The 1,3,5,7-form is a local minimum at the coupled cluster levels, higher in energy than the resonance-stabilized 1,2,4,7-form by 10-13 kJ/mol, but bounded by a barrier of less than 0.5 kJ/mol. The isomerization surface connecting these two structures is described and results reported from integration of the vibrational Schr?dinger equation on that surface. Excited vibrational states at energies just above the isomerization barrier are dominated by the character of the 1,3,5,7-tetraenyl radical, which suggests that chemistry involving this intermediate at typical combustion temperatures may branch at this juncture.     

Symmetry breaking in the cyclic C(3)C(2)H radical

We have employed high-level coupled cluster methods including connected triple excitations to study the possibility of symmetry-breaking in the (2)B(2) ground state of the c-C(3)C(2)H radical. Specifically, we find that spin-restricted open-shell Hartree-Fock (ROHF) reference orbitals yield a C(2v) structure, whereas spin-unrestricted Hartree-Fock (UHF) and Brueckner orbitals lead to a symmetry-broken C(s) minimum-energy geometry. Equation-of-motion coupled cluster singles and doubles method for ionized states yields a C(s) structure with a double-zeta basis set, but not with a triple-zeta basis set. Through a detailed analysis of the orbital instability/near-instability behavior of each type of Hartree-Fock reference, we have determined that the UHF reference wave function is more reliable than the ROHF reference in this case and that the Born-Oppenheimer potential surface for c-C(3)C(2)H exhibits a symmetry broken C(s) global minimum. This result is supported by excited-state computations, which indicate that a second-order (pseudo) Jahn-Teller interaction is responsible for the symmetry-breaking.     

Ab initio study of possible intermediates in the no(4Π) catalyzed geometric isomerizations of olefins

The geometries of acyclic and three-membered ring (nitroxide) H₄C₂NO radicals in their ground ²Π electronic states have been optimized completely at ab initio UHF and ROHF theoretical levels with the STO-3G and the 6-31G** basis sets. The optimizations favour the cyclic nitroxide structure energetically. However ΔE(acyclic - cyclic) at the UHF and ROHF/6-31G** levels are only 3.2 and 1.9 kcal mol^-1, respectively. Incomplete MP2/6-311G** optimizations support these results. The zero-point energy computed at the ROHF/6-31G** level for the nitroxide radical is 2.5 kcal mol^-1 higher than that for the acyclic structure, thus reversing the relative energies by 0.6 kcal mol^-1. The energies of the two radical structures, relative to the sum of those for ethylene and NO, are very close to literature values of the activation energies for the thermal, NO catalyzed geometrical isomerizations of olefins. Thus cyclic nitroxide intermediates may play a role not only in the Hg 6(³P₁) photosensitized, but also in the thermal, NO-catalyzed geometric isomerizations of olefins. Paper dedicated to Professor Otto P. Strausz; presented in part at the 75th Canadian Chemical Congress and Exhibition, Edmonton, May 31 – June 4, 1992.     

Basis set generation for the SCF calculation

A method for basis set generation for SCF calculations is proposed. Using SCF orbitals and orbital energies obtained in the extended basis set the Fock operator can be expressed as its spectral resolution. The sum of differences between occupied orbital energies and corresponding eigenvalues obtained by the diagonalization of this operator in the new smaller basis set is a criterion of the quality of this new set. The present method consists of the minimization of this sum by changing the parameters that determine the new basis functions. An example of the optimization of the different Gaussian basis sets for the LiH molecule is described.     

Hyperfine interactions and lattice distortion of the F center in KCl, NaCl and LiCl crystals

Detailed embedded-cluster quantum-chemistry (ROHF, UHF and UMP2) calculations of hyperfine interactions for the F center in KCl, NaCl and LiCl are presented. Our values for the isotropic hyperfine interaction show a good general agreement with experiments. We show that a careful treatment of the lattice environment, local lattice distortion, spin polarization and correlation are important for this good agreement.     

Bond length alternation in cyclic polyenes. III. Alternant molecular orbital method

The bond length alternation problem in cyclic polyene models as described by the Pariser–Parr–Pople Hamiltonian and an empirical quasiharmonic π-core potential is investigated using the one-parameter alternant molecular orbital (AMO) method. It is shown that in contrast to the unrestricted Hartree–Fock (UHF) results, which lead to symmetric equidistant structures, the one-parameter AMO results yield bond length alternating structures similar to those obtained with the restricted HF approach. The correlation energy recovered by the AMO method is examined for the symmetric polyenes in the whole range of coupling constants for both the Pariser–Parr–Pople and Hubbard Hamiltonians and compared with exact full configuration interaction (FCI) results. For the first member of the cyclic polyene series we also compared the FCI and AMO correlation energies for different nuclear framework distortions. This comparison indicates that in contrast to the UHF results the fraction of the correlation energy recovered by the AMO approach is very uniform over the range of nuclear distortions considered. The AMO results thus strongly indicate the dimerization in the polyenic chains.     

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.     

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.