Spin-unrestricted Hartree-Fock Research Articles

A combined first- and second-order optimization method for improving convergence of Hartree–Fock and Kohn–Sham calculations

In this work, we investigate the optimization of Hartree-Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected "second-order domain," and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree-Fock (RHF, UHF), configuration-averaged Hartree-Fock (CAHF), as well as Kohn-Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

Utilizing Essential Symmetry Breaking in Auxiliary-Field Quantum Monte Carlo: Application to the Spin Gaps of the C36 Fullerene and an Iron Porphyrin Model Complex.

We present three distinct examples where phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) can be reliably performed with a single-determinant trial wave function with essential symmetry breaking. Essential symmetry breaking was first introduced by Lee and Head-Gordon [ Phys. Chem. Chem. Phys. 2019, 21, 4763-4778, 10.1039/C8CP07613H]. We utilized essential complex and time-reversal symmetry breaking with ph-AFQMC to compute the triplet-singlet energy gap in the TS12 set. We found statistically better performance of ph-AFQMC with complex-restricted orbitals than with spin-unrestricted orbitals. We then showed the utilization of essential spin symmetry breaking when computing the singlet-triplet gap of a known biradicaloid, C36. ph-AFQMC with spin-unrestricted Hartree-Fock (ph-AFQMC+UHF) fails catastrophically even with spin-projection and predicts no biradicaloid character. With approximate Brueckner orbitals obtained from regularized orbital-optimized second-order Møller-Plesset perturbation theory (κ-OOMP2), ph-AFQMC quantitatively captures strong biradicaloid character of C36. Lastly, we applied ph-AFQMC to the computation of the quintet-triplet gap in a model iron porphyrin complex where brute-force methods with a small active space fail to capture the triplet ground state. We show unambiguously that neither triplet nor quintet is strongly correlated using UHF, κ-OOMP2, and coupled-cluster with singles and doubles (CCSD) performed on UHF and κ-OOMP2 orbitals. There is no essential symmetry breaking in this problem. By virtue of this, we were able to perform UHF+ph-AFQMC reliably with a cc-pVTZ basis set and predicted a triplet ground state for this model geometry. The largest ph-AFQMC in this work correlated 186 electrons in 956 orbitals. Our work highlights the utility, scalability, and accuracy of ph-AFQMC with a single-determinant trial wave function with essential symmetry breaking for systems mainly dominated by dynamical correlation with little static correlation.

The use of low-lying excited states of zethrene and its homologs in singlet fission within Pariser-Parr-Pople model Hamiltonian: A Density Matrix Renormalization Group study

The low-lying excited states of zethrene, its homologs, and conjugated molecules like bis-phenaleno-indenofluorene, diaryl pentaleno diacenaphthylene and pentalenodiphenalene have been calculated using the Density Matrix Renormalization Group (DMRG) method within model Pariser-Parr-Pople(PPP) Hamiltonian. DMRG calculation shows that the energy ordering of low-lying singlet and triplet excited states of most of these molecules with 4nπ electrons satisfy two essential energetic conditions related to singlet fission (SF), a multiexcitonic process where one singlet exciton converts into two triplet excitons generating more number of charge carriers with longer lifetime enhancing the photoelectric conversion efficiency(PCE) in solar cell applications. The lowest singlet and triplet excited states calculated in DMRG are in very good agreement with experimental results in the literature. Such low-lying excitations and intermediate diradical characters of these molecules calculated through projected spin unrestricted Hartree–Fock (PUHF) method establish the fact that these molecules are potential candidates for designing SF materials.

Singlet-Triplet Energy Gaps in Binuclear Copper Complexes and Organic Diradicals by Approximate Spin Projected Spin-unrestricted Coupled Cluster Method

The performance of the approximate spin projected (AP) coupled cluster singles and doubles (CCSD) methods based on the spin-unrestricted Hartree-Fock and Brueckner determinants, namely AP-UCCSD and AP-UBD, is benchmarked for singlet-triplet energy gaps in various open-shell species. The test set includes the TS12 set, organic diradicals, and two binuclear copper complexes. We demonstrate that chemical accuracy (1 kcal/mol) can be reached for small open-shell molecules. These methods also provide highly accurate results for larger systems, in good agreement with the established equation of motion spin-flip CCSD (EOM-SF-CCSD) results.

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.

Finite-temperature coupled-cluster, many-body perturbation, and restricted and unrestricted Hartree-Fock study on one-dimensional solids: Luttinger liquids, Peierls transitions, and spin- and charge-density waves.

One-dimensional (1D) solids exhibit a number of striking electronic structures including charge-density wave (CDW) and spin-density wave (SDW). Also, the Peierls theorem states that at zero temperature, a 1D system predicted by simple band theory to be a metal will spontaneously dimerize and open a finite fundamental bandgap, while at higher temperatures, it will assume the equidistant geometry with zero bandgap (a Peierls transition). We computationally study these unique electronic structures and transition in polyyne and all-trans polyacetylene using finite-temperature generalizations of ab initio spin-unrestricted Hartree-Fock (UHF) and spin-restricted coupled-cluster doubles (CCD) theories, extending upon previous work [He et al., J. Chem. Phys. 140, 024702 (2014)] that is based on spin-restricted Hartree-Fock (RHF) and second-order many-body perturbation (MP2) theories. Unlike RHF, UHF can predict SDW as well as CDW and metallic states, and unlike MP2, CCD does not diverge even if the underlying RHF reference wave function is metallic. UHF predicts a gapped SDW state with no dimerization at low temperatures, which gradually becomes metallic as the temperature is raised. CCD, meanwhile, confirms that electron correlation lowers the Peierls transition temperature. Furthermore, we show that the results from all theories for both polymers are subject to a unified interpretation in terms of the UHF solutions to the Hubbard-Peierls model using different values of the electron-electron interaction strength, U/t, in its Hamiltonian. The CCD wave function is shown to encompass the form of the exact solution of the Tomonaga-Luttinger model and is thus expected to describe accurately the electronic structure of Luttinger liquids.

Структура и свойства малых кластеров нитридов переходных 3d-элементов

The results of calculations of the parameters of the equilibrium structures of small clusters of transition 3d-element nitrides are presented. The calculations are performed by the spin-unrestricted Hartree-Fock method. The properties of the most stable structures, i.e., those with the lowest total energies among all those obtained, are discussed.

Spin-Unrestricted Multi-Configuration Time-Dependent Hartree Fock Theory

Based on spin-unrestricted hartree fock theory, we present the spin unrestricted multi-configuration time dependent hartree fock theory (UMCTDHF) to describe the electron correlation dynamics of systems interacting with laser field. The positive spin orbitals and the negative spin orbitals are propagated in their own subspace respectively. The spin orbital in the spin-down subspace acts with that in the spin-up subspace by the reduced density matrix and mean field operator. The ground energy is acquired by propagating the trial wave function in the imaginary time by using spin-restricted MCTDHF (RMCTDHF) and UMCTDHF respectively. Then the ionization probabilities and the electrons energies are calculated by using RMCTDHF and UMCTDHF when the laser field is present. The ionization probability calculated with UMCTDHF agrees with the previous theoretical reports very well. The UMCTDHF method is accurate and applicable for open shell system beyond the capability of the RMCTDHF method.

The Hartree-Fock method with orthogonality restrictions for doubly excited and ionized states

It is shown that the doubly excited and ionized states with vacancies in the K electronic shells can be considered within the framework of the same scheme on the basis of the asymptotic projection method proposed earlier. In contrast to the conventional methods, the electron detachment effect is achieved by imposing the requirement that the orbitals of the excited (hole) state be orthogonal to the orbitals of removed electrons. The solution of equations that determine the orbitals is implemented in terms of the conventional spin-unrestricted Hartree-Fock method with additional conditions that ensure the spin purity and provide a unique choice of the zero-order approximation in constructing the many-particle perturbation theory for taking into account the correlation effects. The performance of the method is demonstrated by calculating the excitation energies and ionization potentials corresponding to the removal of electrons from the inner K-shell.

Diradical Character Based Design for Singlet Fission of Condensed-Ring Systems with 4nπ Electrons

By applying the diradical character based molecular design guideline for singlet fission (SF), we investigate the feasibility of efficient SF in condensed-ring π-conjugated molecules with 4nπ electrons (n = 4, 5, ...), i.e., antiaromatic polycyclic hydrocarbons composed of five- and six-membered rings. The multiple diradical character (yi), which takes a value between 0 (closed shell) and 1 (pure open shell), is defined as the occupation number of the lowest unoccupied natural orbital (LUNO) + i (i = 0, 1, ...) calculated using the approximately spin-projected spin-unrestricted Hartree–Fock method. The excitation energies are also evaluated using the tuned long-range corrected time-dependent density functional theory method with the Tamm–Dancoff approximation to examine the energy level matching conditions for SF: (i) 2E(T1) – E(S1) ∼ 0 or ≤ 0 and (ii) 2E(T1) – E(T2) < 0, where S1, T1 and T2 represent singlet first, triplet first, and triplet second excited states, respectively. It turns out that the ener...

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.

Reply to the ‘Comment on “Dynamic factors in the reactions between the magic cluster Al13− and HCl/HI”: A wavefunction instability problem.’ by Young-Kyu Han, Phys. Chem. Chem. Phys. 2012, DOI: 10.1039/C2CP23908F

The reaction between a metal cluster Al13− and an inorganic molecule HI involves charge transfer and density functional calculations has deficiency to estimate the activation barrier due to the self-interaction error. Coupled-cluster calculation with single and double excitation (CCSD) based on spin-unrestricted Hartree-Fock wavefunction can improve the calculated barrier and the choice of basis set is important for the calculation. Energy difference can be as large as 4.9 kcal mol−1 for the reaction by using aug-cc-pvdz and 6-31++G** basis sets respectively.

Symmetry and broken symmetry in molecular orbital description of unstable molecules IV: comparison between single- and multi-reference computational results for antiaromtic molecules

First principle calculations of the effective exchange integrals (J) in the Heisenberg model for diradical species are presented for both symmetry-adapted multi-reference (MR) and single-reference broken-symmetry (BS) methods. The Mukherjee-type state-specific MR coupled cluster singles and doubles (MkCCSD) method with several different reference orbitals including BS natural orbitals is used to calculate the singlet–triplet energy gaps (S–T energy gap or 2J) and diradical characters for the antiaromatic molecules [a cyclopropenyl anion (CPA), b cyclobutadiene (CBD), and c cyclopentadienyl cation (CPC)], the cyclobutadiene derivatives with polar substitutents [d aminocyclobutadiene (ACBD), e formylcyclobutadiene (FCBD), and f 1-amino-2-formyl-cyclobutadiene (AFCBD)] and finally the cyclobutadine derivatives with radical substitutents [g 1,2-bis(methylene)cyclobutadiene (1,2-BMCBD) and h 1,3-bis(methylene)cyclobutadiene (1,3-BMCBD)]. For the BS methods, the spin-unrestricted Hartree–Fock based CCSD (UHF-CCSD), the CCD with the spin-unrestricted Brueckner determinant (UBD), and BS density functional theory (UDFT) computations are performed. Comparison between MkCCSD and the UHF-CCSD results indicates that spin-contamination of UHF-CCSD solutions still remains. In comparison with UHF-CCSD, the UBD results show that spin-contamination involved in BS solutions is greatly suppressed. To eliminate the spin contamination, an approximate spin-projection (AP) scheme is applied to the BS solutions. The AP procedure with the use of the expectation value of the total-spin operator corresponding to UHF-CCSD and UBD results yields good agreement with the MkCCSD results. As for the AP correction of the UDFT methods, three different computational schemes for predicting the expectation value of the total-spin operator are examined. Systematic comparisons between these methods are presented for the S–T energy gaps (2J). Implications of the present computational results have been discussed in relation to the design of magnetic oligomers and polymers.

Potential Energy Curve for Ring-Opening Reactions: Comparison Between Broken-Symmetry and Multireference Coupled Cluster Methods

The spin-unrestricted Hartree-Fock (UHF)-based coupled cluster singles and doubles (UHF-CCSD) and Mukherjee's state-specific multireference CCSD (MkCCSD) methods are applied to four ring-opening reactions. The spin-restricted Hartree-Fock (RHF)-based CCSD (RHF-CCSD) calculations are also performed for comparison. In the case of the UHF-CCSD method, an approximate spin-projection (AP) method is applied to the broken-symmetry (BS) singlet solution to remove the spin contamination effect. For potential energy curves (PECs) of all reactions presented in this study, the results of RHF-CCSD and UHF-CCSD are substantially different from those of MkCCSD, while the results after the AP method (AP-UCCSD) reproduce the MkCCSD results well. It strongly suggests that the spin contamination effect should be removed by the AP correction even at the UHF-CCSD level to predict reliable energetics of these reactions.

Structure and properties of small clusters of transition 3d-element oxides

The results of calculations of the parameters of the equilibrium structures of small clusters of transition 3d-element oxides are presented. The calculations are performed by the spin-unrestricted Hartree-Fock method. The properties of the most stable structures, i.e., those with the lowest total energies among all those obtained, are discussed.

Divide-and-conquer self-consistent field calculation for open-shell systems: Implementation and application

In this Letter, the divide-and-conquer (DC) linear-scaling self-consistent field method is extended to the spin-unrestricted Hartree–Fock (UHF) method or Kohn–Sham density functional theory (UDFT) for treating large open-shell systems. Although the DC method is one of the fragmentation-based linear-scaling schemes, the present DC-UHF/UDFT framework can avoid specifying the number of up- and down-spin electrons in each fragment by introducing up- and down-spin Fermi levels. Test calculations for oligothiophenes demonstrate the high efficiency and accuracy of the DC-UHF/UDFT method even for spin-delocalized systems.

MkMRCC, APUCC and APUBD approaches to 1,n-didehydropolyene diradicals: the nature of through-bond exchange interactions

Mukherjee-type (Mk) state specific (SS) multi-reference (MR) coupled-cluster (CC) calculations of 1,n-didehydropolyene diradicals were carried out to elucidate singlet-triplet energy gaps via through-bond coupling between terminal radicals. Spin-unrestricted Hartree–Fock (UHF) based coupled-cluster (CC) computations of these diradicals were also performed. Comparison between symmetry-adapted MkMRCC and broken-symmetry (BS) UHF-CC computational results indicated that spin-contamination error of UHF-CC solutions was left at the SD level, although it had been thought that this error was negligible for the CC scheme in general. In order to eliminate the spin contamination error, approximate spin-projection (AP) scheme was applied for UCC, and the AP procedure indeed eliminated the error to yield good agreement with MRCC in energy. The CCD with spin-unrestricted Brueckner's orbital (UB) was also employed for these polyene diradicals, showing that large spin-contamination errors at UHF solutions are dramatically improved, and therefore AP scheme for UBD removed easily the rest of spin-contaminations. Pure- and hybrid-density functional theory (DFT) calculations of the species were also performed. Three different computational schemes for total spin angular momentums were examined for the AP correction of the hybrid DFT. The AP DFT calculations yielded the singlet-triplet energy gaps that were in good agreement with those of MRCC, AP UHF-CC and AP UB-CC. Chemical indices such as the diradical character were calculated with all these methods. Implications of the present computational results are discussed in relation to previous RMRCC calculations of diradical species and BS calculations of large exchange coupled systems.

Symmetry breaking in the cyclic C3C2H 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.

Resonating coupled‐cluster CI approach to ion‐radical systems: Comparison with the unrestricted coupled‐cluster approach

AbstractWe implemented the resonating configuration interaction (Res‐CI) approach with spin‐unrestricted coupled‐cluster (UCC) solutions as basis for ion‐radical systems. In previous studies (Nishihara et al., Int J Quantum Chem 2008, 108, 2966; Nishihara et al., J Phys: Condens Matt 2009, 21, 064227), we showed that it is possible to satisfy an important condition proposed by Perdew et al. using Res‐CI based on two complementary spin‐unrestricted Hartree–Fock solutions. In this study, we extended the method to Res‐CI calculations based on UCC solutions (Res‐CC CI) for simple ion‐radical systems, which is to our knowledge, a new type of ab initio method. We examined the asymmetric effects on whether the hole (or the excess electron) localizes at one site or delocalizes over two sites. The computational results of Res‐CC CI are compared with those of UCC, and the implications are discussed. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Intuitive local picture for spin polarization of chemical bonds

AbstractThe spin polarization of chemical bonds near radical centers is investigated by an analytical spin unrestricted Hartree–Fock model with numerical examples. The centroid analysis of localized molecular orbitals is also introduced to obtain an intuitive local picture for the spin polarization. The alternation of spin alignments in molecules are discussed with orbital symmetries and Hund's rule through chemical bonds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010