Single Determinant Reference Research Articles

The correlation factor approach: Combining density functional and wave function theory.

Several of the limitations of approximate exchange-correlation functionals within Kohn-Sham density functional theory can be eliminated by extending the single-determinant reference system to a multi-determinant one. Here, we employ the correlation factor ansatz to combine multi-configurational, self-consistent field (MCSCF) with approximate density functionals. In the proposed correlation factor approach, the exchange-correlation hole ρXC(r, u), a function of the reference point r and the electron-electron separation u, is written as a product of the correlation factor fC(r, u) and an exchange plus static-correlation hole ρXS(r, u), i.e., ρXC CFXS(r, u) = fC(r, u)ρXS(r, u). ρXS(r, u) is constructed to reproduce the exchange-correlation energy of an MCSCF reference wave function. The correlation factor fC(r, u) is designed to account for dynamic correlation effects that are absent in ρXS(r, u). The resulting approximation to the exchange-correlation energy, which we refer to as CFXStatic, is free of empirical parameters, and it combines the qualitatively correct description of the electronic structure obtainable with MCSCF with the advantages of approximate density functionals in accounting for dynamic correlation.

Clean and Convenient Tessellations for Number Counting Jastrow Factors.

We introduce a basis of counting functions that, by cleanly tessellating three-dimensional space, allows real space number counting Jastrow factors to be straightforwardly applied to general molecular situations. By exerting direct control over electron populations in local regions of space and encoding pairwise correlations between these populations, these Jastrow factors allow even very simple reference wave functions to adopt nodal surfaces well suited to many strongly correlated settings. Being trivially compatible with traditional Jastrow factors and diffusion Monte Carlo and having the same cubic per-sample cost scaling as a single determinant trial function, these Jastrow factors thus offer a powerful new route to the simultaneous capture of weak and strong electron correlation effects in a wide variety of molecular and materials settings. In multiple strongly correlated molecular examples, we show that even when paired with the simplest possible single determinant reference, these Jastrow factors allow quantum Monte Carlo to out-perform coupled cluster theory and approach the accuracy of traditional multireference methods.

Excited electronic states from a generalization of the extended Koopmans theorem

Inspired by the extended Koopmans theorem, we demonstrate that excited electronic states can efficiently and very accurately be computed from the ground-state density correlators. That this is possible in principle does not come as a surprise. However, what correlator order is needed is an important question. There are a number of methods that differ in the way in which higher-order correlators, which are very expensive to evaluate, are treated: the equation-of-motion approach, also known as the random phase approximation, the cumulant, and the Hermitian operator methods. Here it is shown that there exists, in fact, a close connection between the extended Koopmans theorem and the equation-of-motion approach. A dramatic improvement over the conventional linear-response calculations is numerically demonstrated for a paradigmatic molecular system and explained by comparing with the equation-of-motion approach for a single-determinant reference state and for the case of composite excitations. Our approach opens prospects for systematic improvements of the adiabatic approximation of the time-dependent density functional theory by exploiting properties of the correlated ground state.

Transition matrices and orbitals from reduced density matrix theory.

In this contribution, we report two different methodologies for characterizing the electronic structure reorganization occurring when a chromophore undergoes an electronic transition. For the first method, we start by setting the theoretical background necessary to the reinterpretation through simple tensor analysis of (i) the transition density matrix and (ii) the natural transition orbitals in the scope of reduced density matrix theory. This novel interpretation is made more clear thanks to a short compendium of the one-particle reduced density matrix theory in a Fock space. The formalism is further applied to two different classes of excited states calculation methods, both requiring a single-determinant reference, that express an excited state as a hole-particle mono-excited configurations expansion, to which particle-hole correlation is coupled (time-dependent Hartree-Fock/time-dependent density functional theory) or not (configuration interaction single/Tamm-Dancoff approximation). For the second methodology presented in this paper, we introduce a novel and complementary concept related to electronic transitions with the canonical transition density matrix and the canonical transition orbitals. Their expression actually reflects the electronic cloud polarisation in the orbital space with a decomposition based on the actual contribution of one-particle excitations from occupied canonical orbitals to virtual ones. This approach validates our novel interpretation of the transition density matrix elements in terms of the Euclidean norm of elementary transition vectors in a linear tensor space. A proper use of these new concepts leads to the conclusion that despite the different principles underlying their construction, they provide two equivalent excited states topological analyses. This connexion is evidenced through simple illustrations of (in)organic dyes electronic transitions analysis.

Double, Rydberg and charge transfer excitations from pairing matrix fluctuation and particle-particle random phase approximation

Double, Rydberg, and charge transfer (CT) excitations have been great challenges for time-dependent density functional theory (TDDFT). Starting from an (N ± 2)-electron single-determinant reference, we investigate excitations for the N-electron system through the pairing matrix fluctuation, which contains information on two-electron addition/removal processes. We adopt the particle-particle random phase approximation (pp-RPA) and the particle-particle Tamm-Dancoff approximation (pp-TDA) to approximate the pairing matrix fluctuation and then determine excitation energies by the differences of two-electron addition/removal energies. This approach captures all types of interesting excitations: single and double excitations are described accurately, Rydberg excitations are in good agreement with experimental data and CT excitations display correct 1/R dependence. Furthermore, the pp-RPA and the pp-TDA have a computational cost similar to TDDFT and consequently are promising for practical calculations.

Communication: Two-determinant mixing with a strong-correlation density functional

In recent papers [A. D. Becke, J. Chem. Phys. 138, 074109 (2013); ibid. 138, 161101 (2013)], a density functional for strong correlations in quantum chemistry was introduced. The functional is designed to capture molecular dissociation limits using symmetry-restricted orbitals. Here we demonstrate that the functional describes, with good accuracy, two-determinant multi-reference states. The examples of this work involve 50∕50 mixing of symmetry-equivalent Slater determinants at avoided crossings. We employ exactly-computed exchange and fractional spin-orbital occupancies. The connection with dissociated systems and single-determinant reference states is explained.

Benchmark studies of variational, unitary and extended coupled cluster methods

Comparative benchmark calculations are presented for coupled cluster theory in its standard formulation, as well as variational, extended, and unitary coupled cluster methods. The systems studied include HF, N(2), and CN, and with cluster operators that for the first time include up to quadruple excitations. In cases where static correlation effects are weak, the differences between the predictions of molecular properties from each theory are negligible. When, however, static correlation is strong, it is demonstrated that variational coupled cluster theory can be significantly more robust than the traditional ansatz and offers a starting point on which to base single-determinant reference methods that can be used beyond the normal domain of applicability. These conclusions hold at all levels of truncation of the cluster operator, with the variational approach showing significantly smaller errors.

O(3P) + C2H4 Potential Energy Surface: Study at the Multireference Level

The O((3)P) + C(2)H(4) reaction provides a crucial, initial understanding of hydrocarbon combustion. In this work, the lowest-lying triplet potential energy surface is extensively explored at the multiconfiguration self-consistent field (MCSCF) and MRMP2 levels with a preliminary surface crossing investigation; and in cases that additional dynamical correlation is necessary, MR-AQCC stationary points are also determined. In particular, a careful determination of the active space along the intrinsic reaction pathway is necessary; and in some cases, more than one active space must be explored for computational feasibility. The resulting triplet potential energy surface geometries mostly agree with geometries from methods using single determinant references. However, although the selected multireference methods lead to energetics that agree well, only qualitative agreement was found with the energetics from the single determinant reference methods. Challenges and areas of further exploration are discussed.

Quadratic canonical transformation theory and higher order density matrices

Canonical transformation (CT) theory provides a rigorously size-extensive description of dynamic correlation in multireference systems, with an accuracy superior to and cost scaling lower than complete active space second order perturbation theory. Here we expand our previous theory by investigating (i) a commutator approximation that is applied at quadratic, as opposed to linear, order in the effective Hamiltonian, and (ii) incorporation of the three-body reduced density matrix in the operator and density matrix decompositions. The quadratic commutator approximation improves CT's accuracy when used with a single-determinant reference, repairing the previous formal disadvantage of the single-reference linear CT theory relative to singles and doubles coupled cluster theory. Calculations on the BH and HF binding curves confirm this improvement. In multireference systems, the three-body reduced density matrix increases the overall accuracy of the CT theory. Tests on the H(2)O and N(2) binding curves yield results highly competitive with expensive state-of-the-art multireference methods, such as the multireference Davidson-corrected configuration interaction (MRCI+Q), averaged coupled pair functional, and averaged quadratic coupled cluster theories.

Approximate treatment of higher excitations in coupled-cluster theory. II. Extension to general single-determinant reference functions and improved approaches for the canonical Hartree–Fock case

The theory and implementation of approximate coupled-cluster (CC), in particular approximate CC singles, doubles, triples, and quadruples methods, are discussed for general single-determinant reference functions. While the extension of iterative approximate models to the non-Hartree-Fock case is straightforward, the generalization of perturbative approaches is not trivial. In contrast to the corresponding perturbative triples methods, there are additional terms required for non-Hartree-Fock reference functions, and there are several possibilities to derive approximations to these terms. As it turns out impossible to develop an approach that is consistent with the canonical Hartree-Fock-based theory, several new approximations have been implemented and their performance for total energies and heats of formation has been assessed. The numerical results show that the performance of the methods does not depend strongly on the approximations assumed. Furthermore, the new perturbative quadruples methods, when applied to canonical Hartree-Fock reference functions, outperform at least for absolute energies the existing ones without increasing the computational costs.

Improving upon CCSD(T): ΛCCSD(T). I. Potential energy surfaces

Despite the successes of spin-restricted coupled-cluster singles, doubles, and perturbative triples [CCSD(T)], it fails for systems away from equilibria, which might raise questions about transition states, e.g. In an attempt to remedy this failure, we implemented LambdaCCSD(T) in a general purpose form for open and closed shells and with arbitrary single determinant reference functions, and applied it to a wide variety of bond-breaking examples. LambdaCCSD(T) is shown to substantially improve the behavior of CCSD(T) at long bond lengths without significantly affecting the quality of the equilibrium results. Weighted average nonparallelity errors for HF, C(2), N(2), and H(2)O are reduced from 22 mhartree for CCSD(T) to 10 mhartree for LambdaCCSD(T). Using LambdaCCSD(T) with a Brueckner reference provides the best single reference coupled-cluster description of N(2)'s dissociation curve to date. Application of CCSD(T) and LambdaCCSD(T) to the concerted transition state of 1,3,5-trinitrohexahydro-1,3,5-triazine (RDX) indicates that this transition state is well described by either methods, and indicates that the activation barrier is too high for it to be a major pathway of decomposition.

Interactions in Diatomic Dimers Involving Closed-Shell Metals

Interaction energies of dimers containing alkaline earth (Be, Mg, and Ca) metals have been investigated using symmetry-adapted perturbation theory (SAPT) and supermolecular (SM) methods. Also, to enable broader comparisons, some calculations have been performed on the Zn dimer and on the He-Mg dimer. Although all of the investigated metallic atoms have closed electronic shells, the quasidegeneracy of the ground states of these atoms with the lowest-lying excited states leads to convergence problems in theories based on a single-determinant reference state. The main goal of the present work was to establish how the quality of the interaction energies computed using various electronic-structure methods changes across the range of atoms. We show that although the convergence problems become somewhat less severe with the increase of the atomic number, single-determinant-based methods do not provide reliable interaction energies for any of the investigated metallic dimers even at the level of the coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)]. However, interaction energies accurate to within a few percent can be obtained if CCSD(T) calculations in large basis sets are extrapolated to the complete basis set limit and followed by full configuration interaction (FCI) calculations with a frozen-core (FC) approximation. Since the systems considered contain only two valence electrons, FCI/FC calculations have been feasible for all of them except for Zn2, providing the best theoretical estimates of the binding energies to date. We found that a large part of the error of the SAPT results originates from limiting some exchange components to terms proportional to the squares of the intermonomer orbital overlap integrals. When the neglected terms were approximately accounted for, the accuracy improved significantly and became comparable to that of CCSD(T), allowing us to obtain for the first time a physical interpretation of the interaction energies in metallic dimers.

Dipole moments calculations of transition metal mononitrides: ScN, TiN, VN, and CrN: Limits of the CCSD(T) method

AbstractCoupled‐cluster calculations of selected molecular properties of ScN, TiN, VN, and CrN in their ground states are presented. It is shown that the single determinant reference Restricted Open Shell Hartree‐Fock Coupled Clusters Singles Doubles noniterative Triples (ROHF‐CCSD(T)) method provides reasonably accurate bond distances, harmonic frequencies, and most dipole moments of these molecules but fails to predict sufficiently accurate dipole moment of CrN. The reason for the low accuracy in obtaining the dipole moment of CrN is analyzed and is attributed to the presence of the series of large single excitation CC amplitudes. This affects the correct response of the molecular CC wavefunction to the external field. Quasirelativistic effects are considered using the Mass‐velocity‐Darwin correction. Changes of dipole moments due to relativity are stressed, and trends throughout the series of molecules are analyzed using their bonding characteristics. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Orbital functional theory of linear response and excitation

Orbital functional theory (OFT) is based on a rule that determines a single-determinant reference state Φ for any exact N-electron eigenstate Ψ. An OFT model postulates an explicit correlation energy functional Ec of occupied orbital functions {ϕi} and occupation numbers {ni}. The orbital Euler–Lagrange equations are analogous to Kohn–Sham equations, but do not in general contain local potential functions. Time-dependent Hartree–Fock theory is generalized in OFT to a formally exact linear response theory that includes electronic correlation. In the exchange-only limit, the theory reduces to the random-phase approximation of many-body theory. The formalism determines excitation energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

Towards multireference equivalents of the G2 and G3 methods

The effect of replacing the standard single-determinant reference wave functions in variants of G2 and G3 theory by multireference (MR) wave functions based on a full-valence complete active space has been investigated. Twelve methods of this type have been introduced and comparisons, based on a slightly reduced G2-1 test set, are made both internally and with the equivalent single-reference methods. We use CASPT2 as the standard MR-MP2 method and MRCl+Q as the higher correlation procedure in these calculations. We find that MR-G2(MP2,SVP), MR-G2(MP2), and MR-G3(MP2) perform comparably with their single-reference analogs, G2(MP2,SVP), G2(MP2), and G3(MP2), with mean absolute deviations (MADs) from the experimental data of 1.41, 1.54, and 1.23 kcal mol−1, compared with 1.60, 1.59, and 1.19 kcal mol−1, respectively. The additivity assumptions in the MR-Gn methods have been tested by carrying out MR-G2/MRCI+Q and MR-G3/MRCI+Q calculations, which correspond to large-basis-set MRCI+Q+ZPVE+HLC calculations. These give MADs of 1.84 and 1.58 kcal mol−1, respectively, i.e., the agreement with experiment is somewhat worse than that obtained with the MR-G2(MP2) and MR-G3(MP2) methods. In a third series of calculations, we have examined pure MP2 and MR-MP2 analogs of the G2 and G3 procedures by carrying out large-basis-set MP2 and CASPT2(+ZPVE+HLC) calculations. The resultant methods, which we denote G2/MP2, G3/MP2, MR-G2/MP2, and MR-G3/MP2, give MADs of 4.19, 3.36, 2.01, and 1.66 kcal mol−1, respectively. Finally, we have examined the effect of using MCQDPT2 in place of CASPT2 in five of our MR-Gn procedures, and find that there is a small but consistent deterioration in performance. Our calculations suggest that the MR-G3(MP2) and MR-G3/MP2 procedures may be useful in situations where a multireference approach is desirable.

Normal ordering and a Wick-like reduction theorem for fermions with respect to a multi-determinantal reference state

We introduce the notion of normal ordering and Wick-like expansion of a product of fermionic creation/annihilation operators with respect to a multi-determinantal reference state ψ 0. The new normal ordered products possess the following desirable properties: (a) their expectation values with respect to ψ 0 vanish, and (b) the normal product of N operators does not depend in a special way on N. The analogues of contractions, unlike in the case of a single determinant reference function, can have n creation and n annihilation operators with n ≥ 1. We prove a Wick-like reordering theorem for a product of creation/annihilation operators, which generates a sum of products in the new normal ordering with any number of contractions. The formula can be generalized to cover products of normal ordered products of operators as well.

Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method

The theory for analytic energy derivatives of excited electronic states described by the equation-of-motion coupled cluster (EOM-CC) method has been generalized to treat cases in which reference and final states differ in the number of electrons. While this work specializes to the sector of Fock space that corresponds to ionization of the reference, the approach can be trivially modified for electron attached final states. Unlike traditional coupled cluster methods that are based on single determinant reference functions, several electronic configurations are treated in a balanced way by EOM-CC. Therefore, this quantum chemical approach is appropriate for problems that involve important nondynamic electron correlation effects. Furthermore, a fully spin adapted treatment of doublet electronic states is guaranteed when a spin restricted closed shell reference state is used—a desirable feature that is not easily achieved in standard coupled cluster approaches. The efficient implementation of analytic gradients reported here allows this variant of EOM-CC theory to be routinely applied to multidimensional potential energy surfaces for the first time. Use of the method is illustrated by an investigation of the formyloxyl radical (HCOO), which suffers from notorious symmetry breaking effects.

Coupled-cluster methods with noniterative triple excitations for restricted open-shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients

A new, noniterative triples correction to the coupled-cluster singles and doubles (CCSD), method, for general single determinant reference functions is proposed and investigated numerically for various cases, including non-Hartree–Fock (non-HF) reference functions. It is correct through fourth-order of perturbation theory for non-HF references, and unlike other such methods, retains the usual invariance properties common to CC methods, while requiring only a single N7 step. In the canonical Hartree–Fock case, the method is equivalent to the usual CCSD(T) method, but now permits the use of restricted open-shell Hartree-Fock (ROHF) and quasirestricted Hartree–Fock (QRHF) reference determinants, along with many others. Comparisons with full configuration interaction (FCI) results are presented for CH2, CH2+, CH3, NH2, and SiH2. The paper also reports the derivation and initial computational implementation of analytical gradients for the ROHF-CCSD(T) method, which includes unrestricted Hartree–Fock (UHF) CCSD(T) and RHF-CCSD(T) as special cases. Applications of analytical gradients are presented for HOO, the CN radical, which is highly spin contaminated at the UHF level, and HCO, the latter with several large basis sets. With these developments of analytical gradients, these highly accurate generalized CCSD(T) methods can be widely applied.

Fock space multireference coupled-cluster theory for general single determinant reference functions

The technique of Fock space multireference coupled-cluster (FSMRCC) theory is applied for the first time to problems involving a high-spin open-shell ground state. Explicit spin–orbital equations applicable to any single determinant reference state are presented and some computational aspects of FSMRCC are discussed. The method is illustrated by two applications in which calculations are limited to single and double excitation operators (FSMRCCSD). First, several basis sets and choices of open-shell reference function are used to calculate selected ionization potentials of O2. The FSMRCCSD results obtained with a large generally contracted basis set are uniformly within 0.1 eV of experiment. In addition, FSMRCCSD is applied to a study of symmetry breaking in the 3A2 state of CO2, a classic multireference problem. The force constant for asymmetric distortion is shown to be predicted correctly as positive, unlike ordinary single-reference CCSD which predicts a double-minimum potential. The results of this paper suggest that the open-shell reference FSMRCC approach has wide applicability for the solution of chemical problems, particularly when significant nondynamic electron correlation effects are present.

Long bonds in silicon clusters: a failure of conventional Møller—Plesset perturbation theory?

On the basis of generalized valence bond calculations we propose a new type of structure for the Si 10 cluster. This structure is found to have large intra-pair correlation effects, suggesting the inadequacy of a single determinant reference wave function in conjunction with Møller—Plesset perturbation theory for determining the relative energetics of silicon clusters. The bonding in Si 10 is analyzed and the origin of the large intra-pair correlation effects is found to be four “long bonds” in the new Si 10 geometry.