Analysis of coupled cluster methods

Summary A procedure is developed that leads from CI to size-extensive CI (ECI) by stepwise cancellation of disconnected terms in the CI equations. The ECI methods thus obtained are identical with the corresponding coupled cluster (CC) methods with the exception of CISD and CISDT, which convert to size-extensive quadratic CI (QCISD) and ECISDT. The latter method has similar properties as CCSDT, but does not offer any significant time-savings as compared to CCSDT. Therefore, the idea of extending CI methods to size-extensive CI methods does not lead to a hierarchy of independent CC methods. However, the procedure of obtaining ECI methods lays the basis for deriving QCI methods that are truly size-extensive. This is accomplished by (a) deleting the first linear term of thep-fold CI excitation equations (p 3) since this term always represents a disconnected term and (b) including just the connected part of appropriate quadratic correction terms in all but the energy equation. In this way, size-extensive QCISDT and QCISDTQ are obtained and their properties are discussed in comparison with QCISD(T) and CCSDT.This paper is dedicated to Prof. Werner Kutzelnigg on the occasion of his sixtieth birthday     

Direct evaluation of one-electron properties in coupled cluster methods

Summary An analysis of a method for approximate calculations of expectation values for one-electron operators from available coupled cluster amplitudes is presented and illustrated numerically for the polarizability of the Be atom. The one-particle density matrix resulting from the present approach is accurate through the fourth order in the electron correlation perturbation. It has been found that, in order to obtain quantitative agreement between the energy derivative results and the approximate expectation value formalism, the third orderT ₁ T ₂⁽⁰⁾ wave function term must be included into the calculation of the one-particle density matrix. The present method is also considered as a promising tool for calculations of higher-order atomic and molecular properties from high level correlated wave functions.     

On expectation value calculations of one-electron properties using the coupled cluster wave functions

The ability of various approximate coupled cluster (CC) methods to provide accurate first-order one-electron properties calculated as expectation values is theoretically analysed and computationally examined for BH and CO. For actual calculations the infinite number of terms of the expectation value expansion (O=¦exp (T ⁺)O exp (T)¦_c) was truncated so that T ₁ T ₂, T ₃, and (1/2) T ₂T₂ clusters were retained on both sides of O. The role of individual clusters is carefully discussed. Inclusion of T ₁, is unavoidable, but if triples are essential in the energy evaluation, they may play an even more important role in the property expansion, as shown in the case of CO. It is shown that the CC wave function, which is exact to second order, effectively satisfies the Hellmann-Feynman theorem.     

Electron correlation in the interacting quantum atoms partition via coupled‐cluster lagrangian densities

The electronic energy partition established by the Interacting Quantum Atoms (IQA) approach is an important method of wavefunction analyses which has yielded valuable insights about different phenomena in physical chemistry. Most of the IQA applications have relied upon approximations, which do not include either dynamical correlation (DC) such as Hartree‐Fock (HF) or external DC like CASSCF theory. Recently, DC was included in the IQA method by means of HF/Coupled‐Cluster (CC) transition densities (Chávez‐Calvillo et al., Comput. Theory Chem. 2015, 1053, 90). Despite the potential utility of this approach, it has a few drawbacks, for example, it is not consistent with the calculation of CC properties different from the total electronic energy. To improve this situation, we have implemented the IQA energy partition based on CC Lagrangian one‐ and two‐electron orbital density matrices. The development presented in this article is tested and illustrated with the H₂, LiH, H₂O, H₂S, N₂, and CO molecules for which the IQA results obtained under the consideration of (i) the CC Lagrangian, (ii) HF/CC transition densities, and (iii) HF are critically analyzed and compared. Additionally, the effect of the DC in the different components of the electronic energy in the formation of the T‐shaped (H₂)₂ van der Waals cluster and the bimolecular nucleophilic substitution between F^– and CH₃F is examined. We anticipate that the approach put forward in this article will provide new understandings on subjects in physical chemistry wherein DC plays a crucial role like molecular interactions along with chemical bonding and reactivity. © 2016 Wiley Periodicals, Inc.     

Reciprocal adjustment of approximate coupled cluster and configuration interaction approaches

The linear version of the externally corrected coupled cluster method with singles and doubles (ecLCCSD), the recently proposed coupled cluster corrections to the multireference configuration interaction (ccMRCI) energies, and the so‐called self‐consistent, size‐consistent [(SC)²] approaches, which are designed to correct for the dynamic correlation effects and the size inconsistency of the MRCI energies, are analyzed and compared using several illustrative examples, including the dissociation of a triple‐zeta (TZ) model of the N₂ molecule. It is emphasized that the exponential cluster ansatz for the wave function is the basis of all these approaches, and appropriate cluster analysis of the MRCI wave function is the key step for both ecLCCSD and ccMRCI. The contributions from the orthogonal complement of the MRCI space, which can be generated by relying on such a cluster analysis, are responsible for a substantial part of the missing correlation energy. The ecLCCSD approach seems to represent a particularly attractive alternative to other highly accurate methods for the calculation of the ground‐state energy in the presence of quasidegeneracy, both due to its efficiency and affordability. It may in fact be regarded as a simple alternative to the iterative reduced multireference (RMR) CCSD method. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 693–703, 2000     

Theoretical study of neutral and protonated triple bonded molecules formed between C, N, Si, P, B and Al

A theoretical study of the possible protonation sites of simple molecules formed by C, N, Si, P, B and Al that present a triple bond between those atoms has been carried out. The calculations performed include MP2 and CCSD(T) methods with the aug-cc-pVTZ basis set. The nature of the protonated species has been analyzed with the Atoms In Molecules methodology. To Serafin, a free spirit and a good friend. Contribution to the Serafin Fraga Memorial Issue.     

Use of a unitary wavefunction in the calculation of static electronic properties

A unitary coupled cluster method is advocated in this paper for the calculation of static properties. Corresponding to the perturbed Hamiltonian H() including the relevant static property, a suitable unitary wavefunction is envisaged. It is shown that a specific nonvariational model of calculating various order static properties utilising this unitary ansatz results in simplifications compared to the previous Coupled Cluster Theories using only hole-particle excitation parameters formulated for this purpose.NCL Communications No. 3533     

Coupled cluster approaches with an approximate account of triexcitations and the optimized inner projection technique

Summary A general orthogonally spin-adapted formalism for coupled cluster (CC) approaches, with an approximate account of triexcited configurations, and for optimized inner projection (OIP) technique is described. Modifying the linear part of the CC equations for pair clusters (CCD) we obtain the orthogonally spin-adapted, non-iterative version of the CCDT-1 method of Bartlett et al. [J. Chem. Phys. 80, 4371 (1984), 81, 5906 (1984), 82, 5761 (1985)]. Similar modification of an approximate coupled pair theory corrected for connected quadruply excited clusters (ACPQ) yields a new approach called ACPTQ. Both the CCDT-1 and ACPTQ methods can be formulated in terms of effective interaction matrix elements between the orthogonally spin-adapted biexcited singlet configurations. The same matrix elements also appear in the orthogonally spin-adapted form of the CCD + T(CCD) perturbative estimate of triply excited contributions due to Raghavachari [J. Chem. Phys. 82, 4607 (1985)] and Urban et al. [J. Chem. Phys. 83, 4041 (1985)], and in the OIP method when applied to the Pariser-Parr-Pople (PPP) model Hamiltonians. We use the diagrammatic approach based on the graphical methods of spin algebras to derive the explicit form of these interaction matrix elements. Finally, the relationship between different diagrammatic spin-adaptation procedures and their relative advantages are discussed in detail.Also affiliated with the Department of Chemistry, and Guelph-Waterloo Center for Graduate Work in Chemistry, Waterloo Campus, University of Waterloo, Waterloo, Ontario, Canada. Killam Research Fellow 1987–89     

A relativistic 4-component general-order multi-reference coupled cluster method: initial implementation and application to HBr

We present the initial implementation of a determinant-based general-order coupled cluster method which fully accounts for relativistic effects within the four-component framework. The method opens the way for the treatment of multi-reference problems through a state-selective expansion of the model space. The evaluation of the coupled cluster vector function is carried out via relativistic configuration interaction expansions. The implementation is based on a large-scale configuration interaction technique, which may efficiently treat long determinant expansions of more than 10⁸ terms. We demonstrate the capabilities of the new method in calculations of complete potential energy curves of the HBr molecule. The inclusion of spin–orbit interaction and higher excitations than coupled cluster double excitations, either by multi-reference model spaces or the inclusion of full iterative triple excitations, lead to highly accurate results for spectral constants of HBr. An erratum to this article can be found at     

Calculating the first-order kinetics of three coupled,reversible processes

Calculation of the concentration-time profile in four consecutive, well-mixed compartments that are connected by diffusional transport is a frequently occurring problem for chemists and engineers. Mathematically this is equivalent to many other problems such as the concentration profiles of a parent compound and its three consecutive reaction products resulting from reversible first-order kinetics. Here we present an analytical solution to this problem implemented in a Microsoft Excel spreadsheet (available for download) and we discuss various examples of how this simple-to-use tool can be applied to very different scenarios from various fields of science.     

On the calculation of expectation values and transition matrix elements by coupled cluster method

Summary Finite order expressions are derived for expectation values and transition matrix elements within the framework of the coupled cluster method.     

A variational coupled cluster theory for closed shells using a propagator modification procedure

A general partial summation method for including arbitrary classes of diagrams to all orders in the coupled cluster based size consistent energy functional for closed shell states is developed. Since the various reduced density matrices which appear in the energy functional are essentially the time-independent analogues of the corresponding many body Green functions, it is possible to derive Dyson-like equations for these quantities. By expanding the associated proper self energy parts in terms of the T-amplitudes, one can carry out partial summations in the reduced density matrices and thus in energy. At a higher level, higher order terms in a proper self energy can also be generated by renormalizing the internal propagators in it, and considering only the irreducible self-energy terms.     

Dispersion coefficients for first hyperpolarizabilities using coupled cluster quadratic response theory

The frequency dependence of third-order properties can in the normal dispersion region be expanded in a Taylor series in the frequency arguments. The dispersion coefficients thus obtained provide an efficient way of expressing the dispersion of frequency-dependent properties and are transferable between different optical processes. We derive analytic expressions for the dispersion coefficients of third-order properties in coupled cluster quadratic response theory and report an implementation for the three coupled cluster models CCS, CC2, and CCSD. Calculations are performed for the first hyperpolarizability of the NH₃ molecule. The convergence of the dispersion expansion with the order of the coefficients is examined and we find good convergence up to about half the frequency at which the first pole in the hyperpolarizability occurs. Padé approximants improve the convergence dramatically and extend the application range of the dispersion expansion to frequencies close to the first pole. The sensitivity of the dispersion coefficients on the dynamic correlation treatment and on the choice of the one-electron basis set is investigated. The results demonstrate that, contrary to presumptions in the literature, the dispersion coefficients are sensitive to basis set effects and correlation treatment similar to the static hyperpolarizabilities. Received: 26 March 1998 / Accepted: 21 July 1998 / Published online: 19 October 1998     

Static electric properties of LiH: explicitly correlated coupled cluster calculations

Explicitly correlated MBPT-R12 and coupled cluster [up to CCSD(T)-R12] methods have been used in calculations of various (vibrationless) electrical properties for the LiH molecule, including the dipole and quadrupole moments, dipole and quadrupole polarizability tensors, dipole hyperpolarizability tensors, and the second dipole hyperpolarizability tensors. Generally, with extension of the basis set the R12 method did not lead to faster convergence for the calculated properties towards the basis limit. Nevertheless, R12 calculations serve as useful indicators to judge the reliability of the results, and substantially help in determining the accuracy. Results obtained with the 11s8p6d5f/9s8p6d5f basis and CCSD(T)-R12 calculated within this work should be close to the basis set limit. Received: 8 June 1998 / Accepted: 23 July 1998 / Published online: 7 October 1998     

A variational method to calculate static electronic properties

Using a coupled cluster form of the wave function, a variational method is formulated for calculation of static properties of any order. Corresponding to an appropriate perturbed hamiltonian H() including the relevant static property, a size consistent functional is set up. In a hierarchical fashion, properties of different orders may be found out using a variational method.     

Method of moments approach and coupled cluster theory

Summary The single reference coupled cluster (CC) approach to the many-electron correlation problem is examined from the viewpoint of the method of moments (MM). This yields generally an inconsistent (overcomplete) set of equations for cluster amplitudes, which can be solved either in the least squares sense or by selective projection process restricting the number of equations to that of the unknowns. These resulting generalized MM-CC equations always contain the standard CC equations as a special case. Since, in the MM-CC formalism, the Schrödinger equation will be approximately satisfied on a subspace spanned by non-canonical configurations, this procedure may be helpful in extending the standard single reference CC theory to quasi-degenerate situations. To examine the potential usefulness of this idea, we explore the linear version of the CC approach for systems with a quasi-degenerate reference, in which case the standard linear theory is plagued with singularities due to the intruder states. Implications of this analysis for the structure of the wavefunction are also briefly discussed.Killam Research Fellow 1987–89