Coupled-cluster Equations Research Articles

Ab initio study of the electronic spectrum of peroxyacetyl nitrate.

A theoretical study of the ground and excited states of peroxyacetyl nitrate (PAN), CH3C(O)OONO2, has been carried out using high level ab initio molecular orbital methods. The ground state geometry and vibrational frequencies are calculated using the coupled-cluster method. The vertical excitation energies for the lowest three excited states are calculated using the complete active space self-consistent field method along with the multireference internally contracted configuration interaction method. These results are compared with vertical excitation energies calculated with the coupled cluster equation of motion method. The calculation provides relevant insight into the origin of PAN absorption in the UV wavelength region from 200 to 300 nm. The nature of the electron transitions for these excited states is discussed.

Optical properties of singly charged conjugated oligomers: a coupled-cluster equation of motion study.

We have implemented a coupled-cluster equation of motion approach combined with the intermediate neglect of differential overlap parametrization and applied it to study the excited states and optical absorptions in positively and negatively charged conjugated oligomers. The method is found to be both reliable and efficient. The theoretical results are in very good agreement with experiments and confirm that there appear two subgap absorption peaks upon polaron formation. Interestingly, the relative intensities of the polaron-induced subgap absorptions can be related to the extent of the lattice geometry relaxations.

Method of moments of coupled-cluster equations: a new formalism for designing accurate electronic structure methods for ground and excited states

The method of moments of coupled-cluster equations (MMCC), which provides a systematic way of improving the results of the standard coupled-cluster (CC) and equation-of-motion CC (EOMCC) calculations for the ground- and excited-state energies of atomic and molecular systems, is described. The MMCC theory and its generalized MMCC (GMMCC) extension that enables one to use the cluster operators resulting from the standard as well as nonstandard CC calculations, including those obtained with the extended CC (ECC) approaches, are based on rigorous mathematical relationships that define the many-body structure of the differences between the full configuration interaction (CI) and CC or EOMCC energies. These relationships can be used to design the noniterative corrections to the CC/EOMCC energies that work for chemical bond breaking and potential energy surfaces of excited electronic states, including excited states dominated by double excitations, where the standard single-reference CC/EOMCC methods fail. Several MMCC and GMMCC approximations are discussed, including the renormalized and completely renormalized CC/EOMCC methods for closed- and open-shell states, the quadratic MMCC approaches, the CI-corrected MMCC methods, and the GMMCC approaches for multiple bond breaking based on the ECC cluster amplitudes.

New coupled-cluster methods with singles, doubles, and noniterative triples for high accuracy calculations of excited electronic states.

The single-reference ab initio methods for high accuracy calculations of potential energy surfaces (PESs) of excited electronic states, termed the completely renormalized equation-of-motion coupled-cluster approaches with singles, doubles, and noniterative triples [CR-EOMCCSD(T)], are developed. In the CR-EOMCCSD(T) methods, which are based on the formalism of the method of moments of coupled-cluster equations, the suitably designed corrections due to triple excitations are added, in a state-selective manner, to the excited-state energies obtained in the standard equation-of-motion coupled-cluster calculations with singles and doubles (EOMCCSD). It is demonstrated that the CR-EOMCCSD(T) approaches, which can be regarded as the excited-state analogs of the ground-state CR-CCSD(T) theory, provide a highly accurate description of excited states dominated by double excitations, excited states displaying a manifestly multireference character, and PESs of excited states along bond breaking coordinates with the ease of the ground-state CCSD(T) or CR-CCSD(T) calculations. The performance of the CR-EOMCCSD(T) methods is illustrated by the results of calculations for the excited states of CH+, HF, N2, C2, and ozone.

Singular value decomposition approach for the approximate coupled-cluster method

We present a method for the approximate solution of the coupled-cluster (CC) equation, based upon the singular value decomposition (SVD). The key idea of this method is that we use SVD for the cluster amplitudes to exploit the physically important states in the CC equation. This method enables us to significantly reduce the computational requirements for CC calculations without losing the essence of the method. Relationships to the density matrix renormalization group theory and the local correlation methods are mentioned. We perform pilot calculations on some atoms and molecules to investigate the applicability of the method.

General-model-space state-universal coupled-cluster theory: Connectivity conditions and explicit equations

We present a new version of the state-universal (SU), multireference, coupled-cluster (CC) theory that is capable of handling completely general, incomplete model spaces. This is achieved by exploiting the concept of “locality” for the active molecular spin orbitals and by introducing the constraining conditions (C conditions) on cluster amplitudes that are associated with the internal excitations transforming one reference configuration into another one. These C conditions make it possible to represent the exact (i.e., full configuration interaction) wave function via the SU CC cluster ansatz based on an arbitrary model space. The C conditions are then taken into account together with the standard SU CC equations for the external amplitudes, thus enabling us to reach the exact result in the limit, while preserving the connectivity property and thus the size extensivity. We also present compact expressions for the matrix elements of the effective Hamiltonian as well as the explicit expressions for the most important coupling coefficients that are required at the single and double excitation level. All other expressions are the same as in the single reference CC formalism.

Method of moments of coupled-cluster equations: The quasivariational and quadratic approximations

The method of moments of coupled-cluster equations (MMCC) and the renormalized coupled-cluster (CC) approaches [see, e.g., K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)] are extended to potential energy surfaces involving multiple bond breaking by introducing the new quasivariational (QV) and quadratic (Q) MMCC approximations. The QMMCC approximations retain the single-reference and noniterative character of the renormalized CC methods, while allowing us to obtain the highly accurate description of multiple bond stretching or breaking. The discussion of the general QVMMCC and QMMCC theories is augmented by the results of test calculations for the double dissociation of H2O and triple bond breaking in N2.

Coupled-cluster approach for studying the optical properties of charged π-conjugated oligomers

We have developed a size-consistent correlated quantum-chemical approach, the coupled-cluster equation of motion (CC-EOM) method, to describe the charged excited states of conjugated materials. We apply this formalism to study a variety of conjugated oligomers: polyenes, oligophenylenevinylenes, oligothiophenes, and oligophenylenes. The transition energies and absorption intensities of the positively charged species are calculated. In all cases, two subgap absorption features are found to dominate the optical spectrum, which are the characteristic optical signatures for polarons in conjugated materials. The relative intensity of these two bands and the dependence on chain length and chemical structure are analyzed. Excellent agreement is found with experimental data.

Parallelization of four-component calculations. II. Symmetry-driven parallelization of the 4-Spinor CCSD algorithm.

Given the importance of the Coupled-cluster (CC) method as an efficient and accurate way to take electron correlation into account, we extend the parallelization technique in the second part of this series also to the 4-Spinor CCSD algorithm implemented in the Dirac-Fock packages DIRAC and MOLFDIR. The present implementation is based on the availability of the transformed molecular two-electron integrals on an external storage medium. The linearity of the CC equations in these two-electron integrals is used in a parallelization strategy that is based on distribution of the two largest integral classes that carry three or four virtual spinor indices. The corresponding partial contributions to the T(1) and T(2) amplitudes are calculated on each node and added using Message Passing Interface (MPI) library calls. Although we did not employ a master/slave principle, one specific node was assigned to also perform the remaining serial parts of the algorithm. In the critical sections considerable savings in storage requirements and computer time could be achieved, and this allows for computations on larger systems in the framework of four-component theory.

Multiple solutions of coupled-cluster equations for PPP model of [10]annulene

Multiple (real) solutions of the coupled-cluster (CC) equations (corresponding to the CCD, ACP and ACPQ methods) are studied for the Pariser–Parr–Pople (PPP) model of [10]annulene, C 10H 10. The long-range electrostatic interactions are represented either by the Mataga–Nishimoto potential, or Pople’s R −1 potential. The multiple solutions are obtained in a quasi-random manner, by generating a pool of starting amplitudes and applying a standard CC iterative procedure combined with Pulay’s DIIS method. Several unexpected features of these solutions are uncovered, including the switching between two CCD solutions when moving between the weakly and strongly correlated regime of the PPP model with Pople’s potential.

Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches

The recently developed new approach to the many-electron correlation problem in atoms and molecules, termed the method of moments of coupled-cluster (CC) equations (MMCC), is reviewed. The ground-state MMCC formalism and its extension to excited electronic states via the equation-of-motion coupled-cluster (EOMCC) approach are discussed. The main principle of all MMCC methods is that of the non-iterative energy corrections which, when added to the ground- and excited-state energies obtained in the standard CC calculations, such as CCSD or EOMCCSD, recover the exact, full configuration interaction (CI) energies. Three types of the MMCC approximations are reviewed in detail: (i) the CI-corrected MMCC methods, which can be applied to ground and excited states; (ii) the renormalized and completely renormalized CC methods for ground states; and (iii) the quasi-variational MMCC approaches for the ground-state problem, including the quadratic MMCC models. It is demonstrated that the MMCC formalism provides a new theoretical framework for designing 'black-box' CC approaches that lead to an excellent description of entire potential energy surfaces of ground- and excited-state molecular systems with an ease of use of the standard single-reference methods. The completely renormalized (CR) CCSD(T) and CCSD(TQ) methods and their quadratic and excited-state MMCC analogues remove the failing of the standard CCSD, CCSD(T), EOMCCSD and similar methods at larger internuclear separations and for states that normally require a genuine multireference description. All theoretical ideas are illustrated by numerical examples involving bond breaking, excited vibrational states, reactive potential energy surfaces and difficult cases of excited electronic states. The description of the existing and well-established variants of the MMCC theory, such as CR-CCSD(T), is augmented by the discussion of future prospects and potentially useful recent developments, including the extension of the black-box CR-CCSD(T) method to excited states.

A general state-selective multireference coupled-cluster algorithm

A state-selective multireference coupled-cluster algorithm is presented which is capable of describing single, double (or higher) excitations from an arbitrary complete model space. One of the active space determinants is chosen as a formal Fermi-vacuum and single, double (or higher) excitations from the other reference functions are considered as higher excitations from this determinant as it has been previously proposed by Oliphant and Adamowicz [J. Chem. Phys. 94, 1229 (1991)]. Coupled-cluster equations are generated in terms of antisymmetrized diagrams and restrictions are imposed on these diagrams to eliminate those cluster amplitudes which carry undesirable number of inactive indices. The corresponding algebraic expressions are factorized and contractions between cluster amplitudes and intermediates are evaluated by our recent string-based algorithm [J. Chem. Phys. 115, 2945 (2001)]. The method can be easily modified to solve multireference configuration interaction problems. Performance of the method is demonstrated by several test calculations on systems which require a multireference description. The problem related to the choice of the Fermi-vacuum has also been investigated.

The State-Universal Multi-Reference Coupled-Cluster Theory: An Overview of Some Recent Advances

Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.

Diagrammatic structure of the general coupled cluster equations

The general formula for the number of diagrammatic terms occurring in the Tn equation within a particular coupled cluster model is derived. Both the antisymmetrized and Goldstone diagrams are considered. In addition to the full coupled cluster equation approximate approaches are discussed, and for each the general formula for the number of terms is given. Analogous expressions are presented for the number of diagrammatic terms contributing to the elements of the transformed Hamiltonian [Hbar] = e−T HeT .

Use of a New Cluster Ansatz to Treat Strong Relaxation and Correlation Effects: A Direct Method for Energy Differences

We have presented in this paper a new cluster Ansatz for the wave operator for open-shell and/or quasidegenerate states, which takes care of strong relaxation and correlation effects in a compact and efficient manner. This Ansatz allows contraction among the various cluster operators via spectator orbitals, accompanied by suitable combinatorial factors. Since both the orbital and the correlation relaxations are treated on the same footing, it allows us to develop a very useful direct method for energy differences for open shell states relative to a closed-shell ground state, where the total charge for the two states may differ. We have discussed a new spin-free coupled cluster (CC) based direct method and illustrated its performance by evaluating electron affinity of a neutral doublet radical. We have also indicated how the scope of the theory can be extended to compute the state energies of simple open shell configurations as well. In that case, the CC equations terminate after the quartic power of cluster operators – exactly as in the closed-shell situation, which is not the case for the current methods.

Method of Moments of Coupled-Cluster Equations: Externally Corrected Approaches Employing Configuration Interaction Wave Functions

A new approach to the many-electron correlation problem, termed the method of moments of coupled-cluster equations (MMCC), is further developed and tested. The main idea of the MMCC theory is that of the noniterative energy corrections which, when added to the energies obtained in the standard coupled-cluster calculations, recover the exact (full configuration interaction) energy. The MMCC approximations require that a guess is provided for the electronic wave function of interest. The idea of using simple estimates of the wave function, provided by the inexpensive configuration interaction (CI) methods employing small sets of active orbitals to define higher–than–double excitations, is tested in this work. The CI-corrected MMCC methods are used to study the single bond breaking in HF and the simultaneous breaking of both O–H bonds in H2O.

Extension of the method of moments of coupled-cluster equations to excited states: The triples and quadruples corrections to the equation-of-motion coupled-cluster singles and doubles energies

The recently proposed extension of the method of moments of coupled-cluster equations (MMCC) to excited states via the equation-of-motion coupled-cluster (EOMCC) formalism [K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966 (2001)] is developed further. A new approximate variant of the excited-state MMCC theory, termed the MMCC(2,4) method, is proposed and tested. In the MMCC(2,4) method, relatively simple noniterative corrections due to triples and quadruples are added to the excited-state energies obtained in the standard EOMCCSD (EOMCC singles and doubles) calculations. The performance of the MMCC(2,4) approach is illustrated by the results of calculations for the excited states of N2, C2, and CH+. The MMCC(2,4) energies are compared with the results of the MMCC(2,3) calculations, in which noniterative corrections due to triples only are added to the EOMCCSD energies, and with the results of other EOMCC calculations, including various EOMCC triples schemes.

A compact spin-free cluster expansion formalism for simple open-shell configurations

A compact spin-free coupled cluster (CC) approach is presented for simple open-shell configurations such as doublets or h– p excited singlets/triplets, using a new cluster Ansaz for the wave-operator. Unlike the extant methods, the CC equations for state energies are at most quartic in the cluster amplitudes, analogous to the closed-shell theory. It can also be utilized for energy differences relative to a closed-shell singlet. The Ansaz allows contractions between cluster operators via spectator orbitals, taking care of relaxation and correlation effects very efficiently. Applications to IP and doublet energies demonstrate its efficacy for outer, inner and core ionizations.

Coupled-cluster approach for studying the singlet and triplet exciton formation rates in conjugated polymer LED’s

The coupled-cluster equation of motion approach is applied to describe positively and negatively charged states as well as exciton states in conjugated polymers. The formation rates for singlet and triplet excitons associated with intermolecular charge-transfer processes are calculated. It is found that the interchain bond-charge correlation has a strong influence on the singlet/triplet ratio, since the charge-transfer configuration contributes differently to the singlet and triplet excitons. In addition, we find that the range of electron interaction potential has a strong influence on the formation rates. The ratio between the electroluminescence and photoluminescence quantum yields can exceed the 25% spin-degeneracy statistical limit.

Linear scaling local correlation approach for solving the coupled cluster equations of large systems.

A linear scaling local correlation approach is proposed for approximately solving the coupled cluster doubles (CCD) equations of large systems in a basis of orthogonal localized molecular orbitals (LMOs). By restricting double excitations from spatially close occupied LMOs into their associated virtual LMOs, the number of significant excitation amplitudes scales only linearly with molecular size in large molecules. Significant amplitudes are obtained to a very good approximation by solving the CCD equations of various subsystems, each of which is made up of a cluster associated with the orbital indices of a subset of significant amplitudes and the local environmental domain of the cluster. The combined effect of these two approximations leads to a linear scaling algorithm for large systems. By using typical thresholds, which are designed to target an energy accuracy, our numerical calculations for a wide range of molecules using the 6-31G or 6-31G* basis set demonstrate that the present local correlation approach recovers more than 98.5% of the conventional CCD correlation energy.