Restricted Hartree-Fock Orbitals Research Articles

Electronic structure of the S1 state in methylcobalamin: Insight from CASSCF/MC‐XQDPT2, EOM‐CCSD, and TD‐DFT calculations

The methylcobalamin cofactor (MeCbl), which is one of the biologically active forms of vitamin B12, has been the subject of many spectroscopic and theoretical investigations. Traditionally, the lowest-energy part of the photoabsorption spectrum of MeCbl (the so-called α/β band) has been interpreted as an S0→S1 electronic transition dominated by π→π* excitations associated with the C=C stretching of the corrin ring. However, a more quantitative band-shape analysis of the α/β spectral region, along with circular dichroism (CD), magnetic CD, and resonance Raman data, has revealed the presence of a second electronic transition that involves the Co-C(Me) bond weakening. Conversely, the lowest-energy excitations based on transient absorption spectroscopy measurements have been interpreted as metal-to-ligand charge transfer (MLCT) transitions. To resolve the existing controversy about the interpretation of the S1 state of MeCbl, calculations have been performed using two independent ab initio wavefunction-based methods. These include the modified variant of the second-order multiconfigurational quasi-degenerate perturbation theory (MC-XQDPT2), using complete active space self-consistent field orbitals, and the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) approach using restricted Hartree-Fock orbitals. It is shown that both ab initio methods provide a consistent description of the S1 state as having an MLCT character. In addition, the performance of different types of functionals, including hybrid (B3LYP, MPW1PW91, TPSSh), generalized-gradient-approximation-type (GGA-type) (BP86, BLYP, MPWPW91), meta-GGA (TPSS), and range-separated (CAM-B3LYP, LC-BLYP) approaches, has been examined and the results of the corresponding time-dependent density functional theory calculations have been benchmarked against the MC-XQDPT2 and EOM-CCSD data. The hybrid functionals support the interpretation in which the S1 state represents a π→π* transition localized on corrin, while pure GGA, meta-GGA, and LC-BLYP functionals produce results consistent with the MLCT assignment.

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.

The coupled cluster approach with a hybrid treatment of connected triple excitations based on the restricted Hartree–Fock reference

A generalization of the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples [denoted as CCSD(T)-h] [Shen et al., J. Chem. Phys. 132, 114115 (2010)] to the restricted Hartree-Fock (RHF) reference is presented. In this approach, active (or pseudoactive) RHF orbitals are constructed automatically by performing unitary transformations of canonical RHF orbitals so that they spatially mimic the natural orbitals of the unrestricted Hartree-Fock reference. The present RHF-based CCSD(T)-h approach has been applied to study the potential energy surfaces in several typical bond breaking processes and the singlet-triplet gaps in a diradical (HFH)(-1). For all systems under study, the overall performance of CCSD(T)-h is very close to that of the corresponding CCSD(T) (CC singles, doubles, and triples), and much better than that of CCSD(T) (CC singles, doubles, and perturbative triples).

Analytic gradients for Mukherjee’s multireference coupled-cluster method using two-configurational self-consistent-field orbitals

Analytic gradients for the state-specific multireference coupled-cluster method suggested by Mahapatra et al. [Mol. Phys. 94, 157 (1998)] (Mk-MRCC) are reported within the singles and doubles approximation using two-configurational self-consistent field (TCSCF) orbitals. The present implementation extends our previous work on Mk-MRCC gradients [E. Prochnow et al., J. Chem. Phys. 131, 064109 (2009)] which is based on restricted Hartree-Fock orbitals and consequently the main focus of the present paper is on the treatment of orbital relaxation at the TCSCF level using coupled-perturbed TCSCF theory. Geometry optimizations on m-arynes and nitrenes are presented to illustrate the influence of the orbitals on the computed equilibrium structures. The results are compared to those obtained at the single-reference coupled-cluster singles and doubles and at the Mk-MRCC singles and doubles level of theory when using restricted Hartree-Fock orbitals.

Hybrid correlation models based on active-space partitioning: Seeking accurate O(N5) ab initio methods for bond breaking

Moller-Plesset second-order (MP2) perturbation theory remains the least expensive standard ab initio method that includes electron correlation, scaling as O(N5) with the number of molecular orbitals N. Unfortunately, when restricted Hartree-Fock orbitals are employed, the potential energy curves calculated with this method are of little use at large interatomic separations because of the divergent behavior of MP2 in these regions. In our previous study [J. Chem. Phys. 122, 234110 (2005)] we combined the MP2 method with the singles and doubles coupled cluster (CCSD) method to produce a hybrid method that retains the computational scaling of MP2 and improves dramatically the shape of the MP2 curves. In this work we expand the hybrid methodology to several other schemes. We investigate a new, improved MP2-CCSD method as well as a few other O(N5) methods related to the Epstein-Nesbet pair correlation theory. Nonparallelity errors across the dissociation curve as well as several spectroscopic constants are computed for BH, HF, H2O, CH+, CH4, and Li2 molecules with the 6-31G* basis set and compared with the corresponding full configuration interaction results. We show that among the O(N5) methods considered, our new hybrid MP2-CCSD method is the most accurate and significantly outperforms MP2 not only at large interatomic separations, but also near equilibrium geometries.

Analytic energy gradients with frozen molecular orbitals in coupled-cluster and many-body perturbation theory methods: Systematic study of the magnitude and trends of the effects of frozen molecular orbitals

Analytic coupled-cluster (CC) and many-body perturbation theory (MBPT) energy gradient methods with restricted Hartree–Fock (RHF), unrestricted Hartree–Fock (UHF), restricted open-shell Hartree–Fock (ROHF), and quasi-RHF(QRHF) reference functions are extended to permit dopping core and excited orbitals. By using the canonical property of the semicanonical ROHF orbitals and the RHF orbitals from which the QRHF reference function is constructed, it is shown that a general procedure can be established not only for RHF and UHF, but also for ROHF and QRHF reference functions. The basic theory and implementation are reported. To provide a systematic study of the trends and magnitudes of the effects of dropped molecular orbitals (MOs) on the structures, harmonic frequencies, and ir intensities, we study HCN, C2H2, CO2, HO2, and C2H4 at increasing levels of correlation and basis sets. The effects of the dropped MOs with the largest basis sets are about 0.003 Å and 0.1° in structures and about 1% on harmonic frequencies and ir intensities. The magnitude and the direction of the drop-MO effect tend to be almost constant from MBPT(2) to CCSD(T) methods. The two isomers of S3 are studied by the drop-MO-method, yielding very accurate results.

Accurate electrical and spectroscopic properties ofX 1?+ BeO from coupled-cluster methods

This paper reports a series of coupled-cluster (CC) calculations through CCSDT on the theoretically challenging ground state of the BeO molecule. Along with CC methods, quadratic configuration interaction (QCI) approximations to CC theory have been used (QCISD and QCISD(T)), which show several dramatic failings. Equilibrium electrical properties (μ, α xx , and α zz ) and basic spectroscopic properties (r e, θe,D e, and infrared intensity (I)) have been computed. Basis set and electron correlation effects are analyzed in order to arrive at accurate values of the dipole moment and polarizability, which are not known experimentally. For the dipole moment, we obtain a value of 6.25 D, with an uncertainty of about 0.1 D. For α xx and α zz , we suggest respective values of 32 and 36 atomic units (a.u.) and error bars of about 1 and 2 a.u. With extended basis sets, the spectroscopic propertiesr e, θe, andD e are reproduced to high accuracy, which is the first time this has been achieved for this species byab initio methods. At the highest calculation levels,I is predicted to be very small. AlthoughI has not been measured, some support for this prediction comes from a recent infrared study of BeO-rare gas complexes. The QCI methods are shown to be much more sensitive to basis set, and even with large basis sets yield values of α zz andI which differ from CC results by an order of magnitude and three orders of magnitude, respectively. These differences doubtless arise from the importance of single excitations (T 1) for this molecule, as several terms involvingT 1 are neglected in the QCISD approximation compared with CCSD. We also report CC calculations with Brueckner orbitals, which yield results similar to those obtained with restricted Hartree-Fock orbitals.

Hybrids and bond formation: Excursions in momentum space

Carbon sp, sp 2 and sp 3 hybrids, formed by combining restricted Hartree-Fock orbitals, are described in position space and momentum space. A modern valence bond study of the 3 Δ state of ScH is used to illustrate the process of bond formation for a transition metal atom in a low oxidation state. Additional insight into the parentage of the various orbitals is provided by the momentum space representation.

Charge current waves in a one-dimensional Coulomb gas

As a first step toward a solution of the general Hartree–Fock equations for a one-dimensional Coulomb system we construct charge current waves by pairing restricted Hartree–Fock orbitals with functions from their orthogonal complement. The degree of pairing is determined by the solution of the corresponding gap equation. Special attention is paid to the screening problem.

Analytic configuration interaction (CI) gradient techniques for potential energy hypersurfaces. A method for open-shell molecular wave functions

A formalism is presented for the determination of analytic energy first derivatives for the most common types of open-shell correlated wave functions. These are the cases for which the electronic structure is qualitatively understood in terms of a set of molecular orbitals which are eigenfunctions of Roothaan’s restricted Hartree–Fock (RHF) operator. Using these RHF orbitals, configuration interaction (CI) wave functions of broad generality are included in the formalism. The method has been implemented in conjunction with the loop-driven graphical unitary group approach. Application to the vibrational frequencies of methylene suggests that the triplet state has 0.5 kcal more zero-point vibrational energy than does the lowest singlet state.

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

Spin Polarization and Spin Delocalization in Unrestricted Hartree–Fock Method

A method to separate the spin density calculated with the unrestricted Hartree–Fock method into components due to spin-polarization and spin-delocalization mechanisms is presented, and applied to some doublet and triplet radicals. The results are examined by means of the UHF natural orbitals and of the open-shell restricted Hartree–Fock orbitals, and the validity of the method is confirmed.

Even-parity states of 43Sc

Restricted Hartree-Fock orbitals are determined for four-particle-one-hole (4p1h) states in 43Sc, and the model Hamiltonian is diagonalized within the subspace of states obtained by angular momentum projection from the lowest k π = 3 2 + and 1 2 + configurations. It is found that to account for the observed even-parity spectrum in terms of 4p1h states, it is necessary to use a value for the d 3 2 s 1 2 single-hole gap which is less than half the value deduced from the 39K or 39Ca spectra. The main features of electromagnetic transition data are then quite well reproduced.