Open-shell Coupled-cluster Research Articles

Reappraisal of the normal ordered Jeziorski-Monkhorst ansatz in the UGA-OSCC theory for a study of IP, EA and EE

The Unitary Group Adapted Open Shell Coupled Cluster (UGA-OSCC) theory, which is the single reference CSF limit of the UGA-State Universal Multireference CC (UGA-SUMRCC) formulation by Mukherjee et al. [Sen, Shee, and Mukherjee, J. Chem. Phys. 137, 074104 (2012)], was proposed and successfully applied via a normal ordered Jeziorski-Monkhorst (J-M)-like CC ansatz for the wave operator. This generates a spin-adapted, size extensive energy. The computed energy differences of spectroscopic interest such as IP, EA and EE were consequently size intensive. Here we revisit the formulation by suggesting a rigorous version which completely bypasses the use of a sufficiency condition posited in our earlier formulation via an exact factorisation of the wave operator in the Bloch equation. This brings in certain connected composites involving contractions among the cluster operators in the resultant equations. This formulation is flexible enough to accommodate truncation of these composites while remaining rigorously size-extensive for each such approximant. We test the effect of these new terms in the Bloch equation for this formulation by evaluating IP, EA and EE for a test set of molecules, which shifts the respective results closer in the order of 0.1 to 0.001 eV towards the benchmark EOM-CCSDt or FCI or experimental values.

Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12.

We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with additions of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3*A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calculations on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calculations are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol-1, and this maximum error is reduced to below 0.2 kcal mol-1 when large local domains are used. For larger systems, results using different thresholds for the local approximations are compared to demonstrate that 1 kcal mol-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approximations are only a fraction of 1 kcal mol-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calculations, the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calculations. The only exception is the iterative solution of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approximation. For most systems, even the semicanonical approximation (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for molecules with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster.

Accurate Prediction of Hyperfine Coupling Tensors for Main Group Elements Using a Unitary Group Based Rigorously Spin-Adapted Coupled-Cluster Theory.

We present the development of a perturbative triples correction scheme for the previously reported unitary group based spin-adapted combinatoric open-shell coupled-cluster (CC) singles and doubles (COS-CCSD) approach and report on the applications of the newly developed method, termed "COS-CCSD(T)", to the calculation of hyperfine coupling (HFC) tensors for radicals consisting of hydrogen, second- and third-row elements. The COS-CCSD(T) method involves a single noniterative step with [Formula: see text] scaling of the computational cost for the calculation of triples corrections to the energy. The key feature of this development is the use of spatial semicanonical orbitals generated from standard restricted open-shell Hartree-Fock (ROHF) orbitals, which allows the unperturbed Hamiltonian operator to be defined in terms of a diagonal spin-free Fock operator. The HFC tensors are computed as a first-order property via implementation of an analytic derivative scheme. The required one-particle spin density matrix is computed by using one- and two-particle spin-free density matrices that are obtained from the analytic derivative implementation, in this way avoiding the use of any spin-dependent operator and maintaining spin adaptation of the CC wavefunction. Benchmark calculations of HFC tensors for a set of 21 radicals indicate reasonably good agreement of the COS-CCSD(T) results with experiment and a consistent improvement over the COS-CCSD method. We demonstrate that the accuracies of the isotropic hyperfine coupling constants obtained in unrestricted HF (UHF) reference based spin-orbital CCSD(T) calculations deteriorate when spin contamination in the UHF wavefunction is large, and the results may even become qualitatively incorrect when spin polarization is the driving mechanism. Within a similar noniterative perturbative treatment of triple excitations, the spin-adapted COS-CCSD(T) approach produces accurate results, thus ensuring cost-effectiveness together with reliability.

Investigation of challenging spin systems using Monte Carlo configuration interaction and the density matrix renormalization group.

We investigate if a range of challenging spin systems can be described sufficiently well using Monte Carlo configuration interaction (MCCI) and the density matrix renormalization group (DMRG) in a way that heads toward a more "black box" approach. Experimental results and other computational methods are used for comparison. The gap between the lowest doublet and quartet state of methylidyne (CH) is first considered. We then look at a range of first-row transition metal monocarbonyls: MCO when M is titanium, vanadium, chromium, or manganese. For these MCO systems we also employ partially spin restricted open-shell coupled-cluster (RCCSD). We finally investigate the high-spin low-lying states of the iron dimer, its cation and its anion. The multireference character of these molecules is also considered. We find that these systems can be computationally challenging with close low-lying states and often multireference character. For this more straightforward application and for the basis sets considered, we generally find qualitative agreement between DMRG and MCCI. © 2017 Wiley Periodicals, Inc.

Benchmark calculations on the nuclear quadrupole-coupling parameters for open-shell molecules using non-relativistic and scalar-relativistic coupled-cluster methods.

Quantum-chemical computations of nuclear quadrupole-coupling parameters for 24 open-shell states of small molecules based on non-relativistic and spin-free exact two-component (SFX2C) relativistic equation-of-motion coupled-cluster (EOM-CC) as well as spin-orbital-based restricted open-shell Hartree-Fock coupled-cluster (ROHF-CC) methods are reported. Relativistic effects, the performance of the EOM-CC and ROHF-CC methods for treating electron correlation, as well as basis-set convergence have been carefully analyzed. Consideration of relativistic effects is necessary for accurate calculations on systems containing third-row (K-Kr) and heavier elements, as expected, and the SFX2C approach is shown to be a useful cost-effective option here. Further, it is demonstrated that the EOM-CC methods constitute flexible and accurate alternatives to the ROHF-CC methods in the calculations of nuclear quadrupole-coupling parameters for open-shell states.

Communication: spin densities within a unitary group based spin-adapted open-shell coupled-cluster theory: analytic evaluation of isotropic hyperfine-coupling constants for the combinatoric open-shell coupled-cluster scheme.

We report analytical calculations of isotropic hyperfine-coupling constants in radicals using a spin-adapted open-shell coupled-cluster theory, namely, the unitary group based combinatoric open-shell coupled-cluster (COSCC) approach within the singles and doubles approximation. A scheme for the evaluation of the one-particle spin-density matrix required in these calculations is outlined within the spin-free formulation of the COSCC approach. In this scheme, the one-particle spin-density matrix for an open-shell state with spin S and MS = + S is expressed in terms of the one- and two-particle spin-free (charge) density matrices obtained from the Lagrangian formulation that is used for calculating the analytic first derivatives of the energy. Benchmark calculations are presented for NO, NCO, CH2CN, and two conjugated π-radicals, viz., allyl and 1-pyrrolyl in order to demonstrate the performance of the proposed scheme.

Analytic first derivatives for a spin-adapted open-shell coupled cluster theory: evaluation of first-order electrical properties.

An analytic scheme is presented for the evaluation of first derivatives of the energy for a unitary group based spin-adapted coupled cluster (CC) theory, namely, the combinatoric open-shell CC (COSCC) approach within the singles and doubles approximation. The widely used Lagrange multiplier approach is employed for the derivation of an analytical expression for the first derivative of the energy, which in combination with the well-established density-matrix formulation, is used for the computation of first-order electrical properties. Derivations of the spin-adapted lambda equations for determining the Lagrange multipliers and the expressions for the spin-free effective density matrices for the COSCC approach are presented. Orbital-relaxation effects due to the electric-field perturbation are treated via the Z-vector technique. We present calculations of the dipole moments for a number of doublet radicals in their ground states using restricted open-shell Hartree-Fock (ROHF) and quasi-restricted HF (QRHF) orbitals in order to demonstrate the applicability of our analytic scheme for computing energy derivatives. We also report calculations of the chlorine electric-field gradients and nuclear quadrupole-coupling constants for the CCl, CH2Cl, ClO2, and SiCl radicals.

An explicitly spin-free compact open-shell coupled cluster theory using a multireference combinatoric exponential ansatz: Formal development and pilot applications

In this paper, we present a comprehensive account of an explicitly spin-free compact state-universal multireference coupled cluster (CC) formalism for computing the state energies of simple open-shell systems, e.g., doublets and biradicals, where the target open-shell states can be described by a few configuration state functions spanning a model space. The cluster operators in this formalism are defined in terms of the spin-free unitary generators with respect to the common closed-shell component of all model functions (core) as vacuum. The spin-free cluster operators are either closed-shell-like n hole-n particle excitations (denoted by T(mu)) or involve excitations from the doubly occupied (nonvalence) orbitals to the singly occupied (valence) orbitals (denoted by S(e)(mu)). In addition, there are cluster operators with exchange spectator scatterings involving the valence orbitals (denoted by S(re)(mu)). We propose a new multireference cluster expansion ansatz for the wave operator with the above generally noncommuting cluster operators which essentially has the same physical content as the Jeziorski-Monkhorst ansatz with the commuting cluster operators defined in the spin-orbital basis. The T(mu) operators in our ansatz are taken to commute with all other operators, while the S(e)(mu) and S(re)(mu) operators are allowed to contract among themselves through the spectator valence orbitals. An important innovation of this ansatz is the choice of an appropriate automorphic factor accompanying each contracted composite of cluster operators in order to ensure that each distinct excitation generated by this composite appears only once in the wave operator. The resulting CC equations consist of two types of terms: a "direct" term and a "normalization" term containing the effective Hamiltonian operator. It is emphasized that the direct term is almost quartic in the cluster amplitudes, barring only a handful of terms and termination of the normalization term depends on the valence rank of the effective Hamiltonian operator and the excitation rank of the cluster operators at which the theory is truncated. Illustrative applications are presented by computing the state energies of neutral doublet radicals and doublet molecular cations and ionization energies of neutral molecules and comparing our results with the other open-shell CC theories, benchmark full CI results (when available) in the same basis, and the experimental results. Highly encouraging results show the efficacy of the method.

Spin-restricted Brueckner orbitals for coupled-cluster wavefunctions

The theory of spin-restricted Brueckner orbitals for high-spin open-shell coupled-cluster wavefunctions is presented. The orbitals are based on single-excitation amplitudes constructed using the symmetric spin–orbital basis coupled-cluster method of Jayatilaka and Lee. It is shown how this approach may be easily implemented within existing open-shell coupled-cluster programs. The method’s performance is compared to conventional spin-unrestricted Brueckner orbitals for C̃ 2A2 NO2 and X̃ 2A2′ NO3, for which instabilities in the Hartree–Fock reference determinant cause serious difficulties for highly correlated wavefunctions.

Method of intermediate hamiltonians via eigenvalue-independent partitioning: application to theoretical spectroscopy

We develop and apply in this paper a coupled cluster (CC)-based intermediate hamiltonian method that is suitable for describing both the lower- and higher-lying excited/ionized states relative to a closed-shell ground state. Generation of the main roots corresponding to the lower-lying states is attempted via an open-shell CC expansion. This expansion dresses the hamiltonian appropriately to incorporate the effect of the virtual space Q in a size-extensive manner. We have shown that by recasting the CC equations in a pseudo-eigenvalue equation form, we may also generate the higher-lying states approximately. The space in which the dressed matrix works is the union of the starting model space (which now becomes the “main” model space, P m ) and the space reached by the action of the first power of the cluster operator on the main model space functions (they span the intermediate space P i ). Using the wave operator in the Fock space, the same kind of dressing is maintained for both the main and the intermediate functions via the use of the same transformed hamiltonian for both these types. This dressing thus incorporates the same decoupling of the Q space from both the P m and P i spaces. The P i , roots, however, miss the “extra” dressing which should arise because of extra hole-particle vacancies in the P i -space functions; to this extent, they are distorted. The pseudo-eigenvalue form of our working equations bypasses the difficult problem of intruder states in a straightforward and obvious way. Applications to compute the main and satellite Auger spectra of H 2O produce encouraging results, indicating the viability of the method.

Interaction of the Fe+ cation with heavy noble gas atoms

The diatomic iron–noble gas complexes FeAr+, FeKr+, FeXe+, and FeRn+ have been theoretically investigated by means of quantum-chemical calculations including an extensive treatment of electron correlation. Potential energy curves and spectroscopic constants for the lowest 4Δ and 6Δ states are derived from an open-shell coupled-cluster approach and the relative energies of all seven low-lying electronic states are evaluated by the multireference configuration interaction method. While in FeAr+ the lowest quartet and sextet states are found to be energetically almost degenerate, the heavier Fe+–noble gas molecules are predicted to exhibit 4Φ ground states. From a qualitative point of view bonding in these species is shown to be electrostatic in origin with intrinsically higher interaction energies for the quartet as compared to the sextet states. For calibration purposes, also an accurate calculation of the 4F(4s03d7)–6D(4s13d6) energy difference in the atomic Fe+ cation is provided.

Valence-specific open-shell coupled-cluster approach using a common vacuum. An application to doublet electronic states

We propose a simple valence-specific formulation of open-shell coupled-cluster theory using a single-exponential Ansätz for the wave operator. Formally resembling the well-known Fock space coupled-cluster approach, our method is easy to implement and free from spin contamination, but avoids the necessity to generate the cluster amplitudes hierarchically, starting from the zero-valence problem. Illustrative applications to the low-lying doublet electronic states of the F atom and the OH molecule are reported.

Computation of ionization potentials using the unitary group based open-shell coupled-cluster theory

The unitary group based open-shell state specific (SS) coupled-cluster (CC) theory is used to compute energies and ionization potentials for several electronic states of methylene, using a double-zeta plus polarization basis set model. The results are compared with the exact full configuration interaction (FCI) solutions as well as with various limited CI results. It is found that the open-shell SS CC theory yields similar results to those obtained by CI involving up to and including triple excitations. The error in calculated ionization potentials never exceeds half an electron volt.

Automation of the implementation of spin-adapted open-shell coupled-cluster theories relying on the unitary group formalism

A new implementation of the orthogonally spin-adapted open-shell (OS) coupled-cluster (CC) formalism that is based on the unitary group approach to many-electron correlation problem is described. Although the emphasis is on the so-called state specific single-reference but multiconfigurational OS CC approach, the developed algorithms as well as the actual codes are also amenable to multireference CC applications of the state-universal type. A special attention is given to simple OS doublets and OS singlet and triplet cases, the former being applicable to the ground states of radicals and the latter to the excited states of closed shell systems. The encoding of the underlying formalism is fully automated and is based on a convenient decomposition of the Hamiltonian into the effective zero-, one-, and two-orbital contributions as well as on the general strategy that focuses on the excitation operator driven evaluation of individual absolute, linear, quadratic, etc., coupled cluster coefficients, rather than on the standard molecular (spin) orbital driven algorithms. In this way unnecessary duplications are avoided and efficient codes are developed both for the general formula generation and final executable modules. A thorough testing of this procedure on a number of model cases is described and several illustrative applications at the ab initio level are provided.

An open‐shell coupled‐cluster response method for static properties

AbstractThis paper deals with a multideterminantal model space‐based coupled‐cluster response approach suitable for the calculation of static properties of quasi‐degenerate systems. The firstorder static property of one‐valence‐hole/particle system using this approach has been specifically discussed.

Aspects of separability in the coupled cluster based direct methods for energy differences

In this paper we have discussed in detail the aspects of separability of the energy differences obtained from coupled cluster based “direct” methods such as the open-shell Coupled Cluster (CC) theory and the Coupled Cluster based Linear Response Theory (CC-LRT). It has been emphasized that, unlike the state energiesper se, the energy differences have a semi-local character in that, in the asymptotic limit of non-interacting subsystemsA, B, C, etc., they are separable as ΔE A , ΔE B , ΔE A + ΔE B , etc. depending on the subsystems excited. We classify the direct many-body methods into two categories: core-extensive and core-valence extensive. In the former, we only implicitly subtract the ground state energy computed in a size-extensive manner; the energy differences are not chosen to be valence-extensive (separable) in the semi-local sense. The core-valence extensive theories, on the other hand, are fully extensive — i.e., with respect to both core and valence interactions, and hence display the semi-local separability. Generic structures of the wave-operators for core-extensive and core-valence extensive theories are discussed. CC-LRT is shown to be core-extensive after a transcription to an equivalent wave-operator based form. The emergence of valence disconnected diagrams for two and higher valence problems are indicated. The open-shell CC theory is shown to be core-valence extensive and hence fully connected. For one valence problems, the CC theory and the CC-LRT are shown to be equivalent. The equations for the cluster amplitudes in the Bloch equation are quadratic, admitting of multiple solutions. It is shown that the cluster amplitudes for the main peaks, in principle obtainable as a series inV from the zeroth order roots of the model space, are connected, and hence the energy differences are fully extensive. It is remarkable that the satellite energies obtained from the alternative solutions of the CC equations are not valence-extensive, indicating the necessity of a formal power series structure inV of the cluster amplitudes for the valence-extensivity. The alternative solutions are not obtainable as a power series inV. The CC-LRT is shown to have an effective hamiltonian structure respecting “downward reducibility”. A unitary version of CC-LRT (UCC-LRT) is proposed, which satisfy both upward and downward reducibility. UCC-LRT is shown to lead to the recent propagator theory known as the Algebraic Diagrammatic Construction. It is shown that both the main and the satellite peaks from UCC-LRT for the one valence problems are core-valence extensive owing to the hermitized nature of the underlying operator to be diagonalized.

A comparative study of core-extensive and core—valence-extensive coupled-cluster theories for energy differences: Excitation energies

We have investigated the relative computational efficacies of the open-shell coupled-cluster (OSCC) theory and the CC-based linear response theory (CC-LRT) for computing excitation energies directly. It is emphasized that the core—valence-extensive nature of OSCC and the core-extensive nature of CC-LRT stem essentially from the differences in the cluster structures of the wave operators. Results obtained from CH +, H 2O and N 2 indicate that both methods perform well for low-lying excited states. The model space used in our OSCC method consists of h—p determinants, and consequently, the excited states dominated by the 2h—2p determinants are poorly described. In contrast, these states are well described in CC-LRT, since the working space in this case has both h—p and 2h—2p determinants. We comment on the relative ease of implementation of the two methods.

The eigenvalue-independent partitioning technique in Fock space: An alternative route to open-shell coupled-cluster theory for incomplete model spaces

Our recent open-shell coupled-cluster (CC) theory for incomplete model spaces, having valence holes and valence particles, is cast in an alternative form having computational advantages. An eigenvalue-independent partitioning technique in Fock space converts the CC equations for each m-hole, n-particle model space of dimension N (m,n) to a non-Hermitian eigenproblem of larger dimension, whose N (m,n) roots are the energies of interest. This alternative strategy works even in the presence of intruders, as is demonstrated by applying it to the main and satellite peaks of the IP and Auger spectra of HF and to the excitation energies of N 2.

Intruder states and incomplete model spaces in multireference coupled-cluster theory: The 2p2 states of Be.

The open-shell coupled-cluster (CC) method is applied to the excited 2${p}^{2}$ states of Be. With the 2${s}^{2}$ and 2${p}^{2}$ configurations included in the model (P) space, the CC equations prove very difficult to converge. When they do converge, very large (>5) excitation amplitudes are observed, and the second $^{1}\mathrm{S}$ corresponds to the intruder 2s3s configuration, rather than the desired 2${p}^{2}$. The inclusion of 2s3s (but not 3${s}^{2}$) in the model space, which thereby becomes incomplete, improves convergence significantly, and gives energies in very good agreement with values known from other sources.

On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces

In this paper we study the following aspects of the open-shell coupled-cluster (CC) theories: (a) we examine the current theoretical status regarding the existence or non-existence of a linked-cluster theorem, ensuring the connectedness of the cluster amplitudes and the effective Hamiltonian; (b) we lay down the necessary and sufficient conditions for connectivity for a general (incomplete) model space, involving valence particles as well as valence holes; and (c) critically re-assess the earlier theoretical works from a comprehensive and unifying view point.