Complete Active Space SCF Method Research Articles

Intermolecular-Type Conical Intersections in Benzene Dimer.

The equilibrium and conical intersection geometries of the benzene dimer were computed in the framework of the conventional, linear-response time-dependent and spin-flipped time-dependent density functional theories (known as DFT, TDDFT and SF-TDDFT) as well as using the multiconfigurational complete active space self-consistent field (CASSCF) method considering the minimally augmented def2-TZVPP and the 6-31G(d,p) basis sets. It was found that the stacking distance between the benzene monomers decreases by about 0.5 Å in the first electronic excited state, due to the stronger intermolecular interaction energy, bringing the two monomers closer together. Intermolecular-type conical intersection (CI) geometries can be formed between the two benzene molecules, when (i) both monomer rings show planar deformation and (ii) weaker (approximately 1.6-1.8 Å long) C-C bonds are formed between the two monomers, with parallel and antiparallel orientation with respect to the monomer. These intermolecular-type CIs look energetically more favorable than dimeric CIs containing only one deformed monomer. The validity of the dimer-type CI geometries obtained by SF-TDDFT was confirmed by the CASSCF method. The nudged elastic band method used for finding the optimal relaxation path has confirmed both the accessibility of these intermolecular-type CIs and the possibility of the radiationless deactivation of the electronic excited states through these CI geometries. Although not as energetically favorable as the previous two CI geometries, there are other CI geometries characterized by the relative rotation of monomers at different angles around a vertical C-C axis.

Diffusion quantum Monte Carlo method on diradicals using single- and multi-determinant-Jastrow trial wavefunctions and different orbitals.

In this work, the diffusion quantum Monte Carlo (DMC) method is employed to calculate the energies of singlet and triplet states for a series of organic diradicals and diatomic diradicals with π2 configuration. Single-determinant-Jastrow (SDJ) trial wavefunctions for triplet states, two-determinant-Jastrow (2DJ) trial wavefunctions for the singlet states, and multi-determinant-Jastrow (MDJ) trial wavefunctions are employed in DMC calculations using restricted open-shell B3LYP (ROB3LYP) orbitals, complete-active-space self-consistent field (CASSCF) orbitals, state-average CASSCF orbitals, or frozen-CASSCF orbitals. Our results show that DMC energies using either SDJ/2DJ or MDJ with ROB3LYP orbitals are close to or lower than those with the other orbitals for organic diradicals, while they are not very sensitive to the employed orbitals for diatomic diradicals. Furthermore, using MDJ can reduce DMC energies to some extent for most of the investigated organic diradicals and some diatomic diradicals. The importance of MDJ on DMC energies can be estimated based on the percentage of main determinants in the CASCI wavefunction. On the other hand, singlet-triplet gaps can be calculated reasonably with DMC using MDJ with a mean absolute error of less than 2 kcal/mol with all these orbitals. CASCI wavefunctions using density functional theory orbitals are preferred in constructing MDJ trial wavefunctions in practical DMC calculations since it is easier to obtain such wavefunctions than CASSCF methods.

Theoretical investigation of CH-bond activation by photocatalytic excited SO2 and the effects of C-, N-, S-, and Se-doped TiO2.

The photocatalytic sulfoxidation on TiO2 discovered by Parrino et al. represents a new, interesting and lower energy route for the synthesis of sulfonic acids. Sulfonic acids are important precursors for dyes, detergents and drugs. In the commonly known industrial process, SO2 and a specific hydrocarbon are converted into sulfonic acids using high-energy UV light. In this reaction, SO2 is excited into a metastable triplet state (3SO2), which has the potential to activate a CH-bond of hydrocarbons and start a radical reaction cycle. By introducing TiO2 as a photocatalyst, it has been shown that visible light can be used for the synthesis. This offers the potential to be a cost-effective reaction approach for industrial use. However, experimental studies indicate that the initial excitation mechanism of SO2 on TiO2 is significantly different from the catalyst-free mechanism. Parrino et al. were able to reveal first evidence for the existence of a charge-transfer process from SO2 to the TiO2 surface by means of electrochemical experiments. First theoretical investigations from first principles were able to further substantiate the existence of a charge-transfer. However, to fully understand this mechanism, more accurate methods such as Time Dependent Density Functional Theory (TD-DFT) or ab initio multireference methods such as the Complete Active Space Self Consistent Field (CASSCF) method are required. Furthermore, after understanding the charge-transfer mechanism, the introduction of dopants into TiO2 can be investigated in order to possibly redshift the excitation energy. This might open the route to using lower energy light for the sulfoxidation of hydrocarbons on TiO2 as a new potential industrial reaction for the synthesis of sulfonic acids. In this work, we will study the initial step of the photocatalytic sulfoxidation of hydrocarbons using the TD-DFT and CASSCF methods by using a combined approach consisting of calculations with periodic boundary conditions and a newly constructed embedded cluster model. Furthermore, we will explore the effects of doping by introducing four heteroatoms (C, N, S, and Se) into the TiO2 surfaces anatase[101] and rutile[110] to find a possible enhancement of the photocatalytic reactivity by lowering the electronic excitation energy.

Redox states of dinitrogen coordinated to a molybdenum atom.

Chemical structures bearing a molybdenum atom have been suggested for the catalytic reduction of N2 at ambient conditions. Previous computational studies on gas-phase MoN and MoN2 species have focused only on neutral structures. Here, an ab initio electronic structure study on the redox states of small clusters composed of nitrogen and molybdenum is presented. The complete-active space self-consistent field method and its extension via second-order perturbative complement have been applied on [MoN]n and [MoN2]n species (n = 0, 1±, 2±). Three different coordination modes (end-on, side-on, and linear NMoN) have been considered for the triatomic [MoN2]n. Our results demonstrate that the reduced states of such systems lead to a greater degree of N2 activation, which can be the starting point of different reaction channels.

Multicomponent CASSCF Revisited: Large Active Spaces Are Needed for Qualitatively Accurate Protonic Densities.

Multicomponent methods seek to treat select nuclei, typically protons, fully quantum mechanically and equivalent to the electrons of a chemical system. In such methods, it is well-known that due to the neglect of electron-proton correlation, a Hartree-Fock (HF) description of the electron-proton interaction catastrophically fails leading to qualitatively incorrect protonic properties. In single-component quantum chemistry, the qualitative failure of HF is normally indicative of the need for multireference methods such as complete active space self-consistent field (CASSCF). While a multicomponent CASSCF method was implemented nearly 20 years ago, it is only able to perform calculations with very small active spaces (∼105 multicomponent configurations). Therefore, in order to extend the realm of applicability of the multicomponent CASSCF method, this study derives and implements a new two-step multicomponent CASSCF method that uses multicomponent heat-bath configuration interaction for the configuration interaction step, enabling calculations with very large active spaces (up to 16 electrons in 48 orbitals). We find that large electronic active spaces are needed to obtain qualitatively accurate protonic densities for the HCN and FHF- molecules. Additionally, the multicomponent CASSCF method implemented here should have further applications for double-well protonic potentials and systems that are inherently electronically multireference.

Optimal Orbital Selection for Full Configuration Interaction (OptOrbFCI): Pursuing the Basis Set Limit under a Budget.

Full configuration interaction (FCI) solvers are limited to small basis sets due to their expensive computational costs. An optimal orbital selection for FCI (OptOrbFCI) is proposed to boost the power of existing FCI solvers to pursue the basis set limit under a computational budget. The optimization problem coincides with that of the complete active space SCF method (CASSCF), while OptOrbFCI is algorithmically quite different. OptOrbFCI effectively finds an optimal rotation matrix via solving a constrained optimization problem directly to compress the orbitals of large basis sets to one with a manageable size, conducts FCI calculations only on rotated orbital sets, and produces a variational ground-state energy and its wave function. Coupled with coordinate descent full configuration interaction (CDFCI), we demonstrate the efficiency and accuracy of the method on the carbon dimer and nitrogen dimer under basis sets up to cc-pV5Z. We also benchmark the binding curve of the nitrogen dimer under the cc-pVQZ basis set with 28 selected orbitals, which provide consistently lower ground-state energies than the FCI results under the cc-pVDZ basis set. The dissociation energy in this case is found to be of higher accuracy.

Are Inorganic Single-Molecule Magnets a Possibility? A Theoretical Insight into Dysprosium Double-Deckers with Inorganic Ring Systems.

Computational studies of sandwich dysprosium double-decker complexes [Dy(L)2]+/3+ as candidates for single-ion magnets with several inorganic aromatic ring systems (P5-, N5-, B3N3H6, B3P3H6, B3S3H3) have been performed. The molecular structures were optimized with the TPSSh functional, and the ground state properties were investigated with the complete active space SCF method (CASSCF) complemented by the dynamic correlation dressed correction (DCD-CAS(2)) or NEVPT2. Besides the evaluation of the magnetic moment blocking barrier, the impact of the molecular vibration on the relaxation of magnetization was also inspected. We were able to make predictions about the performance of those molecules as single-molecule magnets, where estimated effective energy barrier, Ueff, values are as high as 1475 K in the case of [Dy(N5)2]-, which is the most anisotropic complex from our choice of studied compounds, making them a potentially very effective carbon-free alternative to organometallic double-decker dysprosocenium high-temperature single-molecule magnets.

Benchmarks for Electronically Excited States with CASSCF Methods.

The accuracy of three different complete active space (CAS) self-consistent field (CASSCF) methods is investigated for the electronically excited-state benchmark set of SchreiberM.; et al. J. Chem. Phys.2008, 128, 13411018397056. Comparison of the CASSCF linear response (LR) methods MC-RPA and MC-TDA and the state-averaged (SA) CASSCF method is made for 122 singlet excitation energies and 69 oscillator strengths. Of all CASSCF methods, when considering the complete test set, MC-RPA performs best for both excitation energies and oscillator strengths with a mean absolute error (MAE) of 0.74 eV and 51%, respectively. MC-TDA and SA-CASSCF show a similar accuracy for the excitation energies with a MAE of ∼1 eV with respect to more accurate coupled cluster (CC3) excitation energies. The opposite trend is observed for the subset of n → π* excitation energies for which SA-CASSCF exhibits the least deviations (MAE 0.65 eV). By looking at s-tetrazine in more detail, we conclude that better performance for the n → π* SA-CASSCF excitation energies can be attributed to a fortunate error compensation. For oscillator strengths, SA-CASSCF performs worst for the complete test set (MAE 100%) as well as for the subsets of n → π* (MAE 192%) and π → π* excitations (MAE 84.9%). In general, CASSCF gives the worst performance for excitation energies of all excited-state ab initio methods considered so far due to lacking the major part of dynamic electron correlation, though MC-RPA and TD-DFT (BP86) show similar performance. Among all LR-type methods, LR-CASSCF oscillator strengths are the ones with the least accuracy for the same reason. As state-specific orbital relaxation effects are accounted for in LR-CASSCF, oscillator strengths are significantly more accurate than those of MS-CASPT2. Our findings should encourage further developments of response theory-based multireference methods with higher accuracy and feasibility.

CASSCF linear response calculations for large open-shell molecules

The complete active space self-consistent-field (CASSCF) linear response method for the simulation of ultraviolet-visible (UV/Vis) absorption and electronic circular dichroism (ECD) spectra of large open-shell molecules is presented. By using a one-index transformed Hamiltonian, the computation of the most time-consuming intermediates can be pursued in an integral-direct fashion, which allows us to employ the efficient resolution-of-the-identity and overlap-fitted chain-of-spheres approximation. For the iterative diagonalization, pairs of Hermitian and anti-Hermitian trial vectors are used which facilitate, on the one hand, an efficient solution of the pair-structured generalized eigenvalue problem in the reduced space, and on the other hand, make the full multiconfigurational random phase approximation as efficient as the corresponding Tamm-Dancoff approximation. Electronic transitions are analyzed and characterized in the particle-hole picture by natural transition orbitals that are introduced for CASSCF linear response theory. For a small organic radical, we can show that the accuracy of simulated UV/Vis absorption spectra with the CASSCF linear response approach is significantly improved compared to the popular state-averaged CASSCF method. To demonstrate the efficiency of the implementation, the 50 lowest roots of a large Ni triazole complex with 231 atoms are computed for the simulated UV/Vis and ECD spectra.

Can large active-space CASSCF calculation make sense to the reaction analysis of iron complex? A benchmark study of methane oxidation reaction by FeO.

A methane oxidation reaction by FeO+ cation was theoretically investigated based on the density functional theory (DFT) and the complete active-space self-consistent field (CASSCF) method as well as the coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) to explore the active-space dependency to computational analyses in such strongly correlated reaction systems. A small active-space CASSCF(5e in 5o) calculation, which only includes five 3d orbitals of the Fe atom in the active-space, showed remarkable difference both in energy and geometry compared to those computed by the DFT and CCSD(T) methods. Interestingly, a large active-space CASSCF(17e in 17o) calculation, which includes almost all the valence orbitals gives a qualitative agreement with either the DFT or the CCSD(T) results in the first half part of the reaction, although it varies from them in the latter half part. Therefore, it is indicated that the active-space dependency is serious in some part of the reaction and the small active-space CASSCF might lead a wrong discussion. We further investigated the optimized geometry of the intermediate complex with the small and the large active-space CASSCF methods as well as the CCSD(T) method, and found that the CASSCF(5e in 5o)-optimized geometry is considerably different from the others. In consequence, a small active-space CASSCF/CASPT2 calculation does not really work for such a strongly correlated reaction system even qualitatively, and a sophisticated assessment using the large active-space CASSCF/CASPT2 method will be indispensable. © 2018 Wiley Periodicals, Inc.

Extended Koopmans’ Approximation for CASDFT Exchange-Correlation Functional

A new density functional theory approach based on a complete active space self-consistent field (CASSCF) reference function in Extended Koopmans’ approximation is discussed. Recently, the number of generalizations of density functional theory based on a multiconfigurational CASSCF reference function with exact exchange (CASDFT) was introduced. It was shown by one of the authors (Dr. Gusarov) that such a theory could be formulated by introducing a special form of exchange-correlation potential. To take into account an active space and to avoid double counting of correlation energy the dependence from on-top pair density P2(r) as a new variable was introduced. Unfortunately, this requires a deep review and reparametrization of existing functional expressions which lead to additional computational difficulties. The presented approach does not require introducing additional variables (like on-top pair density, P2(r)) and is based on Extended Koopmans’ theorem (EKT) approximation for multiconfigurational wave function within CASSCF method.

Potential Energy Curves, Transition Dipole Moments, and Franck-Condon Factors of the 12 Low-Lying States of BrO- Anion.

This work investigates the spectroscopic parameters, vibrational levels, and transition probabilities of 12 low-lying states, which are generated from the first dissociation limit, Br(2Pu) + O-(2Pu), of the BrO- anion. The 12 states are X1Σ+, 21Σ+, 11Σ-, 11Π, 21Π, 11Δ, a3Π, 13Σ+, 23Σ+, 13Σ-, 23Π, and 13Δ. The potential energy curves are calculated with the complete active-space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with Davidson modification. The dissociation energy D0 of X1Σ+ state is determined to be approximately 26876.44 cm-1, which agrees well with the experimental one of 26494.50 cm-1. Of these 12 states, the 21Σ+, 11Σ-, 21Π, 11Δ, 13Σ+, 23Σ+, 23Π, and 13Δ states are very weakly bound states, whose well depths are only several-hundred cm-1. The a3Π, 23Π, and 13Δ states are inverted and account for the spin-orbit coupling effect. No states are repulsive regardless of whether the spin-orbit coupling effect is included. The spectroscopic parameters and vibrational levels are determined. The transition dipole moments of 12-pair electronic states are calculated. Franck-Condon factors of a number of transitions of more than 20-pair electronic states are evaluated. The electronic transitions are discussed. The spin-orbit coupling effect on the spectroscopic parameters and vibrational properties is profound for all the states except for X1Σ+, a3Π, and 11Π. The spectroscopic parameters and transition probabilities obtained in this paper can provide some powerful guidelines for observing these states in a proper spectroscopy experiment, in particular the states that have very shallow potential wells.

Analytic Energy Gradients for Variational Two-Electron Reduced-Density-Matrix-Driven Complete Active Space Self-Consistent Field Theory.

Analytic energy gradients are presented for a variational two-electron reduced-density-matrix (2-RDM)-driven complete active space self-consistent field (CASSCF) method. The active-space 2-RDM is determined using a semidefinite programing (SDP) algorithm built upon an augmented Lagrangian formalism. Expressions for analytic gradients are simplified by the fact that the Lagrangian is stationary with respect to variations in both the primal and the dual solutions to the SDP problem. Orbital response contributions to the gradient are identical to those that arise in conventional CASSCF methods in which the electronic structure of the active space is described by a full configuration interaction (CI) wave function. We explore the relative performance of variational 2-RDM (v2RDM)- and CI-driven CASSCF for the equilibrium geometries of 20 small molecules. When enforcing two-particle N-representability conditions, full-valence v2RDM-CASSCF-optimized bond lengths display a mean unsigned error of 0.0060 Å and a maximum unsigned error of 0.0265 Å, relative to those obtained from full-valence CI-CASSCF. When enforcing partial three-particle N-representability conditions, the mean and maximum unsigned errors are reduced to only 0.0006 and 0.0054 Å, respectively. For these same molecules, full-valence v2RDM-CASSCF bond lengths computed in the cc-pVQZ basis set deviate from experimentally determined ones on average by 0.017 and 0.011 Å when enforcing two- and three-particle conditions, respectively, whereas CI-CASSCF displays an average deviation of 0.010 Å. The v2RDM-CASSCF approach with two-particle conditions is also applied to the equilibrium geometry of pentacene; optimized bond lengths deviate from those derived from experiment, on average, by 0.015 Å when using a cc-pVDZ basis set and a (22e,22o) active space.

Excited-state minima and emission energies of retinal chromophore analogues: Performance of CASSCF and CC2 methods as compared with CASPT2.

This study provides gas-phase S1 excited-state geometries along with emission and adiabatic energies for methylated/demethylated and ring-locked analogues of protonated Schiff base retinal models comprising system of five conjugated double bonds (PSB5), using second order multiconfiguration perturbation theory (CASPT2). CASPT2 results serve as reference data to assess the performance of CC2 (second-order approximate coupled cluster singles and doubles) and a commonly used CASSCF/CASPT2 protocol, that is, complete active space self-consistent field (CASSCF) geometry optimization followed by CASPT2 energy calculation. We find that the CASSCF methodology fails to locate planar S1 minimum energy structures for four out of five investigated planar models in contrast to CC2 and CASPT2 methods. However, for those which were found: one planar and two twisted minima, there is an excellent agreement between CASSCF and CASPT2 results in terms of geometrical parameters, one-electron properties, as well as emission and adiabatic energies. CC2 performs well for in-plane S1 minima and their spectroscopic and electronic properties. However, this picture deteriorates for twisted minima. As expected, the CC2 description of the S2 electronic state, with strong multireference and significant double excitation character, is very poor, exhibiting errors in transition energies exceeding 1 eV. They may be substantially diminished by recalculating transition energies with CASPT2 method. Our work shows that CASSCF/CASPT2 and CC2 shortcomings may influence gas-phase retinal analogues' excited state description in a dramatic way. © 2017 Wiley Periodicals, Inc.

Theoretical Investigation of Anion-Radical States of Edge-Oxidized Carbon Model Clusters.

A set of flat carbon clusters and ultrashort carbon nanotubes in different anion-radical states was investigated by density functional theory and complete-active space self-consistent field methods. It was found that carbon nanoparticles with pervasively oxidized edges are extremely strong oxidants, and their ground states in catalytic studies should be considered as negatively charged rather than neutral, as it is traditionally done. Negative charging renders initially diamagnetic/semiconducting types of carbon nanoparticles into half-metallic, which can also be achieved by doping with transition metals.

Ligand non-innocence and strong correlation in manganese superoxide dismutase mimics.

We examine the 1-electron reduction of manganese porphyrin complexes Mn(iii) porphyrin and Mn(iii) TDE-2-ImP5+, which have attracted recent interest due to their properties as superoxide dismutase mimics. We perform a series of electronic structure calculations using the variational 2-electron reduced density matrix (2-RDM) method with a large [30,30] active space that represents a wavefunction with 1019 variables, as well as the more traditional complete active space self-consistent field (CASSCF) method with a [14,14] active space. We show that the larger 2-RDM calculation, intractable with CASSCF, is required to capture the full effects of electron correlation in the molecule and predict the non-innocence of the porphyrin ligand during the reduction. The CASSCF method predicts single-reference systems exhibiting a metal-centered reduction, but the 2-RDM method predicts a strongly correlated system exhibiting a ligand-centered reduction. Based on these results, we find that the porphyrin ligand is reduced rather than the manganese, and suggest that the electron correlation plays a role in driving the ligand non-innocence.

¹H Chemical Shifts in Paramagnetic Co(II) Pyrazolylborate Complexes: A First-Principles Study.

We apply the theory of the nuclear magnetic resonance (NMR) chemical shift for paramagnetic systems to demanding cobalt(II) complexes. Paramagnetic NMR (pNMR) chemical shift results by density-functional theory (DFT) can be very far from the experimental values. Therefore, it is of interest to investigate the applicability of electron-correlated ab initio computational methods to achieve useful accuracy. Here, we use ab initio wave function based electronic structure methods to calculate the pNMR chemical shift within the theoretical framework established recently. We applied the N-electron valence-state perturbation theory (NEVPT2) on three Co(II) systems, where the active space of the underlying complete active space self-consistent field (CASSCF) wave function consists of seven electrons in the five metal 3d orbitals. These complexes have the S = 3/2 electronic ground state consisting of two doublets separated by zero-field splitting (ZFS). To calculate the hyperfine coupling tensor A, DFT was used, while the g- and ZFS-tensors were calculated using the ab initio CASSCF and NEVPT2 methods. These results were combined to obtain the total chemical shifts. The shifts obtained from these calculations are in generally good agreement with the experimental results, in some cases suggesting a reassignment of the signals. The accuracy of this mixed ab initio/DFT approach is very promising for further applications to demanding pNMR problems involving transition metals.

Configuration interaction singles natural orbitals: an orbital basis for an efficient and size intensive multireference description of electronic excited states.

Multireference quantum chemical methods, such as the complete active space self-consistent field (CASSCF) method, have long been the state of the art for computing regions of potential energy surfaces (PESs) where complex, multiconfigurational wavefunctions are required, such as near conical intersections. Herein, we present a computationally efficient alternative to the widely used CASSCF method based on a complete active space configuration interaction (CASCI) expansion built from the state-averaged natural orbitals of configuration interaction singles calculations (CISNOs). This CISNO-CASCI approach is shown to predict vertical excitation energies of molecules with closed-shell ground states similar to those predicted by state averaged (SA)-CASSCF in many cases and to provide an excellent reference for a perturbative treatment of dynamic electron correlation. Absolute energies computed at the CISNO-CASCI level are found to be variationally superior, on average, to other CASCI methods. Unlike SA-CASSCF, CISNO-CASCI provides vertical excitation energies which are both size intensive and size consistent, thus suggesting that CISNO-CASCI would be preferable to SA-CASSCF for the study of systems with multiple excitable centers. The fact that SA-CASSCF and some other CASCI methods do not provide a size intensive/consistent description of excited states is attributed to changes in the orbitals that occur upon introduction of non-interacting subsystems. Finally, CISNO-CASCI is found to provide a suitable description of the PES surrounding a biradicaloid conical intersection in ethylene.

Erratum to "Geometries and Vertical Excitation Energies in Retinal Analogues Resolved at the CASPT2 Level of Theory: Critical Assessment of the Performance of CASSCF, CC2, and DFT Methods".

A systematic investigation of structural properties and vertical excitation energies of a series of structurally modified 11-cis-retinal chromophores in vacuo was performed by means of multiconfigurational second-order perturbation theory (CASPT2). CASPT2-based geometries agree reasonably well with Moller–Plesset second-order perturbation theory (MP2), local second-order approximate coupled cluster singles and doubles (LCC2), and density functional theory (DFT) geometries, while the complete active space self-consistent field (CASSCF) method exaggerates dramatically the bond length pattern in the polyene chain. The quality of the resulting vertical excitation energies obtained by employing CASSCF, second-order approximate coupled cluster singles and doubles (CC2), LCC2, and time-dependent density functional theory (TD-DFT) approaches is assessed with respect to the CASPT2 data. We show that the commonly used CASSCF/CASPT2 approach works reasonably well in the case of vertical excitation energies of planar...

Gradient of molecular Hartree–Fock–Bogoliubov energy with a linear combination of atomic orbital quasiparticle wave functions

The analytical gradient for the atomic-orbital-based Hartree-Fock-Bogoliubov (HFB) energy functional, the modified form of which was proposed by Staroverov and Scuseria to account for the static electron correlation [J. Chem. Phys. 117, 11107 (2002)], is derived. Interestingly, the Pulay force for the HFB energy is expressed with the same formula as that for the Hartree-Fock method. The efficiency of the present HFB energy gradient is demonstrated in the geometry optimizations of conjugated and biradical systems. The geometries optimized by using the HFB method with the appropriate factor ζ, which controls the degree of static correlation included, are found to show good agreement with those obtained by using a complete active-space self-consistent field method, although they are significantly dependent on ζ.