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The Journal of Chemical Physics

Physical and mathematical content of coupled–cluster equations: Correspondence between coupled–cluster and configuration–interaction solutions

K. Jankowski, Karol Kowalski

Published Feb 22, 1999• 29 citations
Publisher

Abstract

To gain more insight into the physical and mathematical content of the equations of the coupled–cluster (CC) method, comprehensive numerical studies have been performed for various geometries of the H4 model which belongs to the simplest and best understood among the realistic many–electron model systems. These studies are for the first time based on the knowledge of the complete sets of geometrically isolated solutions of the relevant equations that are obtained when using a special version of the homotopy methods. The equations of the CC method including two–electron excitations (CCD) both in the spin–orbital and spin–symmetry–adapted versions are considered. To establish the correspondence of the solutions attained with those of the configuration interaction (CID) method, we have for the first time solved the unabridged characteristic equations (CE) of T. P. Živković and H. J. Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The complete sets of solutions to the spin–orbital and spin–symmetry–adapted versions of the CCD equations and CEs consist of 20 and 12 solutions, respectively. Their structures turned out to be unexpectedly simple, which makes it possible to understand the physical and mathematical significance of the individual solutions. Moreover, this simplicity rises hopes that the most serious irregularities previously found for low dimensional simulations of the CEs can be avoided for realistic systems. The present results confirm most of the findings of previous research. Attention is paid to studies of the impact of using mixed–symmetry cluster operators on the structure of solutions to the CC equations. The specific part played by the components of the cluster amplitudes corresponding to other spin–symmetry than that of the reference function has been demonstrated for the first time.

DOI
10.1063/1.478262

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Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.

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To gain more insight into the physical and mathematical content of the equations of the coupled–cluster (CC) method, comprehensive numerical studies have been performed for various geometries of the H4 model which belongs to the simplest and best understood among the realistic many–electron model systems. These studies are for the first time based on the knowledge of the complete sets of geometrically isolated solutions of the relevant equations that are obtained when using a special version of the homotopy methods. The equations of the CC method including two–electron excitations (CCD) both in the spin–orbital and spin–symmetry–adapted versions are considered. To establish the correspondence of the solutions attained with those of the configuration interaction (CID) method, we have for the first time solved the unabridged characteristic equations (CE) of T. P. Živković and H. J. Monkhorst [J. Math. Phys. 19, 1007 (1978)]. The complete sets of solutions to the spin–orbital and spin–symmetry–adapted versions of the CCD equations and CEs consist of 20 and 12 solutions, respectively. Their structures turned out to be unexpectedly simple, which makes it possible to understand the physical and mathematical significance of the individual solutions. Moreover, this simplicity rises hopes that the most serious irregularities previously found for low dimensional simulations of the CEs can be avoided for realistic systems. The present results confirm most of the findings of previous research. Attention is paid to studies of the impact of using mixed–symmetry cluster operators on the structure of solutions to the CC equations. The specific part played by the components of the cluster amplitudes corresponding to other spin–symmetry than that of the reference function has been demonstrated for the first time.
To discuss the usefulness of the entries of the complete set of solutions to approximate coupled-cluster (CC) equations for the description of the states of many-electron systems, one has to be able to establish a correspondence between the states and solutions. We have explored four methods of finding links between the states (represented for the model by the full configuration interaction wave functions and energies) and solutions to the CC equations. Numerical studies for the P4 model, which belongs to the simplest realistic many-electron model systems, show that the states-solutions correspondence pattern strongly depends on the method employed. Only for a few states all methods considered have assigned the same solutions. Among these states are those that are energetically the closest to the reference determinants. For the remaining states the assignment is not unique and the accuracy of description depends on the physical features considered. The article is concluded with a comparison of the present results and results of the most recent studies on the structure of the complete sets of solutions to the CC equations with the findings and intuitions being the content of the work by Živković and Monkhorst (J Math Phys 1978, 19, 1007). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 483–496, 1999
Abstract. Coupled cluster theory produced arguably the most widely used high-accuracy computational quantum chemistry methods. Despite the approach’s overall great computational success, its mathematical understanding is so far limited to results within the realm of functional analysis. The coupled cluster amplitudes, which are the targeted objects in coupled cluster theory, correspond to solutions to the coupled cluster equations, which is a system of polynomial equations of at most degree four. The high dimensionality of the electronic Schrödinger equation and the nonlinearity of the coupled cluster ansatz have so far stalled a formal analysis of this polynomial system. In this article, we present algebraic investigations that shed light on the coupled cluster equations and the root structure of this ansatz. This is of importance for the a posteriori evaluation of coupled cluster calculations. To that end, we investigate the root structure by means of Newton polytopes. We derive a general v-description, which is subsequently turned into an h-description for explicit examples. This perspective reveals an apparent connection between Pauli’s exclusion principle and the geometrical structure of the Newton polytopes. We also propose an alternative characterization of the coupled cluster equations projected onto singles and doubles as cubic polynomials on an algebraic variety with certain sparsity patterns. Moreover, we provide numerical simulations of two computationally tractable systems, namely, the two electrons in four spin-orbitals system and the three electrons in six spin-orbitals system. These simulations provide novel insight into the root structure of the coupled cluster solutions when the coupled cluster ansatz is truncated.
In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H4 molecule and the continuous deformation of the square H4 molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree–Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.
The viability of treating higher excitations in coupled-cluster theory is discussed. An algorithm is presented for solving coupled-cluster (CC) equations which can handle any excitation. Our method combines the formalism of diagrammatic many-body perturbation theory and string-based configuration interaction (CI). CC equations are explicitly put down in terms of antisymmetrized diagrams and a general method is proposed for the factorization of the corresponding algebraic expressions. Contractions between cluster amplitudes and intermediates are evaluated by a string-based algorithm. In contrast to our previous developments [J. Chem. Phys. 113, 1359 (2000)] the operation count of this new method scales roughly as the (2n+2)nd power of the basis set size where n is the highest excitation in the cluster operator. As a by-product we get a completely new CI formalism which is effective for solving both truncated and full CI problems. Generalization for approximate CC models as well as multireference cases is also discussed.
The Coupled-Cluster (CC) method is one of the most popular and efficient correlation methods in quantum chemistry. \nEspecially the CCSD(T) approximation, which includes single and double excitations by means of the application of the cluster operator \nto a reference determinant and triple excitations via a perturbative treatment, has become a standard tool \nin quantum chemical applications. However, the method is restricted to relatively small system sizes due to its unfavourable scaling (N^7, where N is the number of basis functions applied). \nFor the full treatment of triple excitations the scaling advances to N^8 and every further excitation level increases the exponent by two. \n \nThe goal of this work is to reduce the calculation time for closed shell systems at least by a factor growing with the excitation level for arbitrary truncation levels. \nThis is done by restricting the spatial parts of the spin orbitals and thus treat pairs of spin orbitals on the same footing. The restrictions can be easily constructed \nfor the CCSD model by spin integration. \nThe derivation of the restrictions arising in higher excited case will be done employing \nthe spatial orbital excitation operators E. \n \nIn the first part of this work an algorithm is derived that is capable of the derivation of the energy and amplitude equations for arbitrary excitation levels. In the second part \nan implementation of this algorithm is presented.
Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.