Open-shell Coupled-cluster Theory Research Articles

Analysis of different sets of spin-adapted substitution operators in open-shell coupled cluster theory

In spin-adapted open-shell coupled cluster (CC) theory, the choice of spin-free spatial substitution operators is generally not unique. Due to an increasing linear dependence of the cluster operator (with increasing substitution level), the options to span identical linear spaces increase rapidly. In this work, several sets of non-orthogonal as well as orthogonal spin-adapted substitution operators are generated and used in consecutive configuration interaction (CI) and CC calculations. All (full) operator sets were generated to span the same linear space. The results are analysed in terms of the produced wave function quality and the amount of recovered correlation energy w.r.t. full CI (FCI). In particular, the influence of different amounts of spectators, orthogonality as well as spin incompleteness was investigated. It was found that CC calculations involving fewer spectators lead to more accurate results in general. Here, correlation energy differences of up to 0.32% for minimal to maximal spectating sets were obtained. As expected, all conducted calculations led to identical results for non-orthogonal and orthogonal operator sets. Spin completeness was found to be of great importance. Spin-incomplete cluster operators led to significant errors in both the correlation energies and the FCI overlap.

Inclusion of orbital relaxation and correlation through the unitary group adapted open shell coupled cluster theory using non-relativistic and scalar relativistic Hamiltonians to study the core ionization potential of molecules containing light to medium-heavy elements.

The orbital relaxation attendant on ionization is particularly important for the core electron ionization potential (core IP) of molecules. The Unitary Group Adapted State Universal Coupled Cluster (UGA-SUMRCC) theory, recently formulated and implemented by Sen et al. [J. Chem. Phys. 137, 074104 (2012)], is very effective in capturing orbital relaxation accompanying ionization or excitation of both the core and the valence electrons [S. Sen et al., Mol. Phys. 111, 2625 (2013); A. Shee et al., J. Chem. Theory Comput. 9, 2573 (2013)] while preserving the spin-symmetry of the target states and using the neutral closed-shell spatial orbitals of the ground state. Our Ansatz invokes a normal-ordered exponential representation of spin-free cluster-operators. The orbital relaxation induced by a specific set of cluster operators in our Ansatz is good enough to eliminate the need for different sets of orbitals for the ground and the core-ionized states. We call the single configuration state function (CSF) limit of this theory the Unitary Group Adapted Open-Shell Coupled Cluster (UGA-OSCC) theory. The aim of this paper is to comprehensively explore the efficacy of our Ansatz to describe orbital relaxation, using both theoretical analysis and numerical performance. Whenever warranted, we also make appropriate comparisons with other coupled-cluster theories. A physically motivated truncation of the chains of spin-free T-operators is also made possible by the normal-ordering, and the operational resemblance to single reference coupled-cluster theory allows easy implementation. Our test case is the prediction of the 1s core IP of molecules containing a single light- to medium-heavy nucleus and thus, in addition to demonstrating the orbital relaxation, we have addressed the scalar relativistic effects on the accuracy of the IPs by using a hierarchy of spin-free Hamiltonians in conjunction with our theory. Additionally, the contribution of the spin-free component of the two-electron Gaunt term, not usually taken into consideration, has been estimated at the Self-Consistent Field (ΔSCF) level and is found to become increasingly important and eventually quite prominent for molecules with third period atoms and below. The accuracies of the IPs computed using UGA-OSCC are found to be of the same order as the Coupled Cluster Singles Doubles (ΔCCSD) values while being free from spin contamination. Since the UGA-OSCC uses a common set of orbitals for the ground state and the ion, it obviates the need of two N5 AO to MO transformation in contrast to the ΔCCSD method.

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.

Gas-phase detection of the FCO2 radical by millimeter wave and high resolution infrared spectroscopy assisted by ab initio calculations

Low pressure pyrolysis at 600 K of bis(fluoroformyl) peroxide, FC(O)OOC(O)F, yields the fluorocarboxyl radical, FCO2, in a concentration high enough to allow its detection by millimeter wave and infrared spectroscopy. The radical was first identified from its high resolution infrared spectrum obtained using a Fourier transform infrared interferometer. Observation and identification of its millimeter wave (MMW) spectrum were made possible by reliable ab initio calculations at the level of open-shell coupled cluster theory using large basis sets. The excellent agreement between the experimental and theoretical results confirms the structure of the FCO2 radical and the efficiency of the synthesis. The analysis of the MMW spectrum has given a set of ground state parameters including rotational, quartic centrifugal distortion, fine and hyperfine constants.

Theoretical studies of the X̃ 2Π and à 2Σ+ states of He⋅SH and Ne⋅SH complexes

The two-dimensional potential energy surfaces for the X̃ 2Π and à 2Σ+ states of the He⋅SH and Ne⋅SH complexes have been calculated using the restricted open-shell coupled cluster theory [RCCSD(T)] and the triple-zeta augmented correlation consistent polarized basis sets with an additional (3s3p2d2f1g) set of bond functions. In the case of the à 2Σ+ state of Ne⋅SH the entire surface has also been developed using the quadruple-zeta basis set with bond functions as exploratory calculations demonstrated significant differences between the RCCSD(T) results obtained with the triple- and quadruple-zeta basis sets. These potentials are somewhat shallower and less anisotropic in comparison to the surfaces for the related He⋅OH and Ne⋅OH complexes. In contrast to He⋅OH and Ne⋅OH, we find that the linear Rg–SH (Rg=He, Ne) configurations are in all but one case lower in energy than the Rg–HS geometries. Variational calculations of the bound rotation-vibration states have been performed using Hamiltonians that included the RCCSD(T) potentials. The calculated ground-vibrational-state dissociation energy, D0, the frequency of the intermolecular stretching vibration, and the rotational constant are in very good agreement with the available experimental results for the X̃ 2Π state of both Ne⋅SH and Ne⋅SD. The energies of rotation-vibration levels for the Ne⋅SH and Ne⋅SD complexes in the à 2Σ+ state calculated using the triple- or quadruple-zeta potentials differ significantly, but agreement with the experimental rovibrational transition frequencies and rotational constants is very good regardless of which potential is used.

A new diagnostic for open-shell coupled-cluster theory

We present a new diagnostic for open-shell coupled-cluster theory, readily computed from the single substitution amplitudes in the CCSD wavefunction. The new diagnostic, D 1(ROCCSD), is designed to be comparable to the previously proposed D 1(CCSD) diagnostic. Unlike other approaches, the D 1 diagnostics are independent of system size and have the same invariance properties as the energy with respect to orbital rotations. Calibration of the D 1(ROCCSD) diagnostic on 34 molecular systems indicates that for values of D 1(ROCCSD) of 0.025 or below the quality of the CCSD results are, in general, excellent, whereas values larger than 0.025 signal inadequacies in the CCSD approach.

Spin-restricted open-shell coupled-cluster theory for excited states

Using a linear-response approach, the recently introduced spin-restricted coupled-cluster (SR-CC) theory is extended to the treatment of excited states of high-spin open-shell molecules. Explicit equations are given within the usual singles and doubles approximation and our implementation (within an existing spin–orbital code) is described. It is shown that in SR-CC theory, due to spin constraints, the spin-expectation value for the excited states calculated as corresponding energy derivatives always corresponds to the exact value. In addition, the SR-CC singles and doubles (SR-CCSD) approach is extended to include also the so-called pseudotriple excitations (best described as double excitations with an additional spin–flip in one open-shell orbital) which are important for the description of so-called low-spin excited states. Exploratory calculations for a few diatomic systems (BeH, OH, NO, CN, and CO+) show that problems due to spin contamination in the unrestricted Hartree–Fock (UHF) CCSD treatment of excited states are rectified by using a restricted open-shell Hartree–Fock (ROHF) reference, as it is done in the SR-CC approach. While SR-CCSD performs well for high-spin excited states, the closely related partially spin-adapted (PSA) CC approach is shown to be inferior and errors in the computed excitation energies are generally larger than the typical accuracy of about 0.2 eV in CCSD excited state treatments. So-called low-spin states (e.g., the 2 2B1 state of NH2) are shown to require inclusion of pseudotriple excitations for even a qualitatively correct description. If they are included, ROHF-CC, SR-CC, and PSA-CC give essentially identical results.

Relativistic coupled-cluster static dipole polarizabilities of the alkali metals from Li to element 119

Static dipole polarizabilities ${\ensuremath{\alpha}}_{D}$ of the first main group elements from Li down to superheavy element 119 were calculated using a relativistic Douglas-Kroll Hamiltonian within an open-shell coupled-cluster theory. Spin-orbit effects were investigated using a fully relativistic four-component Dirac-Hartree-Fock scheme. Our final recommended values for the dipole polarizabilities from Li to Cs should be more accurate than currently available experimental data. Relativistic effects in ${\ensuremath{\alpha}}_{D}$ are roughly $\ensuremath{\sim}{Z}^{2}$ and we establish a clear relationship for relativistic and electron correlation effects between the dipole polarizability and the ionization potential of the neutral atom. Spin-orbit effects become non-negligible for Fr and element 119. For the latter, relativistic effects clearly dominate over electron correlation resulting in a very small ${\ensuremath{\alpha}}_{D}$ value comparable to that of sodium.

Unitary group based open-shell coupled cluster theory: Application to van der Waals interactions of high-spin systems

The performance of the unitary group approach (UGA) based coupled cluster singles and doubles (CCSD) method in application to van der Waals interactions involving high-spin open-shell systems is examined. The tested approach is fully spin-adapted in the sense that any intermediate quantity appearing in the formulation of the theory is free from spin contamination contributions. Interaction energies are computed within the supermolecular approach and corrected for the basis set superposition error. Several methods of solving UGA CCSD equations are used with the emphasis on iterative processes based on the Hamiltonian partitionings employed in the spin-restricted many-body perturbation theories. Test calculations are performed for the ground states of HeLi, H2Li, and for the excited a 3Σu+ state of Li2. The UGA CCSD interaction energies are compared with those computed using the spin-unrestricted and valence universal coupled cluster methods, spin-restricted and spin-unrestricted many-body perturbation expansions, and the full configuration interaction method. The obtained results show that the UGA CCSD method performs very well in applications to weakly bound open-shell systems, giving results as good or better than other open-shell coupled cluster approaches.

Spin-restricted open-shell coupled-cluster theory

Spin-restricted CC theory is suggested as a new approach for the treatment of high-spin open-shell systems in CC theory. Spin constraints are imposed on the wave function in the sense that the projected spin eigenvalue equations are fulfilled within the (truncated) excitation space. These constraints allow a reduction in the number of independent amplitudes, thus decreasing the computational cost when implemented efficiently. The approach ensures that the spin expectation value always corresponds to the exact value, though the wave function is (for truncated CC treatments) not rigorously spin-adapted. For the specific case of high-spin doublets, detailed equations are derived for amplitudes, energy and first derivatives of the energy within the computationally most useful singles and doubles approximation. Numerical examples demonstrate the excellent performance of the spin-restricted CC approach relative to the more traditional spin-orbital based CC treatments, suggesting that it might be an attractive alternative for the treatment of high-spin open-shell systems.

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.

Spin adapted restricted open shell coupled cluster theory. Linear version

We describe the linear version of the spin adapted coupled cluster theory with all single and double excitations for open shell systems in restricted Hartree–Fock formalism. The method is based on the partitioning of the Hamiltonian developed by Hubač and Čársky [I. Hubač and P. Čársky, Phys. Rev. A 22, 2392 (1980)] in restricted Roothaan open shell formalism. Correlation energy of some simple molecular systems is calculated.

Cluster relations for multireference coupled-cluster theories: A model study

The validity of various coupled-cluster conditions, that were recently formulated for both the valence-universal open-shell coupled-cluster theory based on Lindgren’s exponential ansatz and the state-universal theory employing the Jeziorski–Monkhorst ansatz, are examined for two model systems consisting of two and four slightly stretched, interacting hydrogen molecules. These cluster conditions ascertain the possibility to represent four-electron terms in the wave operator through appropriate products of two-electron terms and thus justify the truncation of the open-shell cluster operators employed to at most two-electron terms. A single parameter that determines the geometry of the studied model systems makes it possible to continuously vary the extent of quasidegeneracy between the two lowest energy states over a wide range. It is shown that the cluster conditions are relatively well satisfied only in the strongly quasidegenerate region, except for a few very small amplitudes whose effect on the resulting energies should be insignificant. To assess the validity of the often made assumption of negligibility of one-electron amplitudes, the generalized cluster conditions involving these terms were also examined. The role played by these additional terms was shown to be small. Finally, in the valence-universal case, the role of orbital choice was also investigated by performing cluster analysis with both system and core spin orbitals. The effect of using different orbital alternatives was again found to be small.

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.

The automated solution of second quantization equations with applications to the coupled cluster approach

Theoretical methods in chemistry frequently involve the tedious solution of complex algebraic equations. Then the solutions, sometimes still quite complex, are usually hand-coded by a programmer into an efficient computer language. During this procedure it is all too easy to make an error which will go undetected. A better approach would be to introduce the computer at an even earlier stage in the development of the theory by programming it to first solve the set of equations and then compile the solution into an efficient computer language. In this research a program has been written in the C programming language which can efficiently compute the quasivacuum expectation value of a product of creation and annihilation operators and scalar arrays. The terms in the resulting expressions are then transformed into a canonical form so that all equivalent terms can be combined. Finally, the equations are compiled into a simple representation which can be rapidly interpreted by a Fortran program. This symbol manipulator has been applied to open-shell coupled cluster theory. Two coupled cluster methods using high-spin open-shell references are presented. In one of these methods, the cluster operator contains the unitary group generators, and products thereof, which generate all single and double excitations with respect to the reference. The other uses a simplified cluster operator which generates equations that must be spin-projected. These methods are compared to other descriptions of electron correlation for the CH2 singlet-triplet splitting and the NH2 potential energy surface.

Application of open-shell coupled cluster theory to the ground state of GaAs

Theoretical calculations at the coupled cluster level of theory including all single, double and perturbative triple excitations, CCSD(T), are carried out for the3Σ− ground state of GaAs. Employing a (7s5p3d1f) basis set, the theoretical predictions forre (2.560 A), ω e (217 cm−1),De (1.84 eV), and IP (7.80 eV), are in good agreement with recent experimental results. The importance of includingf-type polarization functions in the basis set and the effect of correlating 3d electrons are discussed in detail.

Applications of open-shell coupled cluster theory using an eigenvalue-independent partitioning technique: Approximate inclusion of triples in IP calculations

Using our eigenvalue-independent partitioning (EIP) approach for the calculation of open-shell coupled cluster (CC) energy differences, we have computed the ionization potentials of HF and H 2O using basis sets with and without polarization functions. Our results include the three-body cluster operator for the ionized states at the lowest order of approximation. It is found that a CCSD calculation for the ground state, followed by a CCSD calculation for the ionized states - with additional inclusion of triples using the converged CCSD amplitudes - produces results that are accurate up to third order and recovers the relaxation and differential correlation energies consequent on ionization in a balanced and compact manner.