Parallel algorithms for coupled-cluster methods

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Abstract

Coupled-cluster (CC) methods are now widely used in quantum chemistry to calculate the electron correlation energy and many other properties of atoms and molecules. In this paper we outline the basics of the theory, discuss some computational aspects, and review work that has been done toward developing and implementing algorithms for CC methods on parallel computers.

Introduction

Coupled-cluster (CC) theory [1], [2], [3], [4] is widely used in ab initio quantum chemistry today. Its wide use is a result of its well-documented accuracy, its availability in several software packages, and efficient algorithms enabling calculations on small and medium-sized molecules to be performed at “moderate” cost. As with all advances in computational chemistry, the increasing use and applicability of CC methods over the last 20 years would not have been possible without major advances in hardware.

Although great advances have been made, it is nevertheless true that CC calculations place great demands on computer resources. This is an inevitable consequence of how their cost scales as the number of electrons and the size of the basis set increase. Vector processing certainly has enabled larger CC calculations to be done than would otherwise have been possible, and effective use of parallel processing computers could provide even greater benefits. It should be realized, however, that parallel algorithms for CC methods are significantly more complex, and the programming effort correspondingly greater, than for Hartree–Fock/self-consistent field (SCF) or density-functional theory (DFT) methods. Parallel SCF calculations were first performed and used in large-scale applications in the mid-1980s [5], [6], [7], while the development of parallel CC programs began in the 1990s and is still at a comparatively early stage.

In this paper, we review work that has been done in the area of parallel algorithm development and implementation for CC methods. We also indicate possible future lines of development in this active and challenging field.

In the next section, we outline the basics of CC theory. After that we take a more detailed look at the equations, emphasizing some of the key computational aspects, particularly those that are crucial to parallel processing. Next, we review work that has been done. Finally, we look at possible future developments.

Section snippets

Basics of CC theory

In the standard CC theory, the wavefunction is given by an exponential excitation operator acting on a Slater determinant of orthonormal spin orbitalsΨCC=eT|O〉.The operator T is the sum of single, double, triple, … excitation operators.T1=a,itiaa+i,T2=1/4a,b,i,jtijaba+ib+jand so on. The coefficients tia, tijab,… are called cluster amplitudes. The operator a+i excites an electron from orbital i to orbital a (a single excitation); a+ib+j creates a double excitation, exciting electrons from

Practical CC methods and some computational aspects

As is the situation for FCI, the computational demands of complete CC theory render it impractical except for very small systems. All computationally feasible CC methods are necessarily approximations to the complete CC theory. In all of these methods, the first approximation is the truncation of the cluster operator T to an excitation level (much) less than the number of electrons in the system. Fortunately, a very large percentage of the correlation energy is obtained with these

Parallel processing

Prior to discussing efforts at developing and implementing algorithms for parallel CC methods, we shall mention briefly efforts that have been made for other methods of including electron correlation.

The simplest such method for including electron correlation is second-order many-body perturbation theory MBPT(2). If the Møller–Plessett partitioning is used, this is termed MP2. A large number of parallel implementations of the MP2 energy evaluation this have been reported [16], [17], [18], [19],

Conclusions

Looking back at the decade of the 1990s, one can see that tremendous advances have been made in the computational implementation of CC methods. As far as single-processor machines are concerned, algorithms based on the MO basis set formalism are very well developed. Several groups have vectorized codes that take advantage of point-group symmetry (more precisely D2h and its subgroups), and a very large number of applications have been reported. There is a sense that we are reaching the limits of

Acknowledgements

Several people are thanked for providing references to their work and/or useful comments thereon, which helped in the preparation of this article: Dr. Timothy Lee, Dr. Alistair Rendell, Dr. Peter Taylor and Dr. Hans-Joachim Werner. Dr. Ming-Ju Huang is thanked for helping with word processing.

References (40)

  • G.E. Scuseria et al.

    Chem. Phys. Lett.

    (1988)
  • K. Raghavachari et al.

    Chem. Phys. Lett.

    (1989)
  • R. Wiest et al.

    Comput. Phys. Commun.

    (1991)
  • D.E. Bernholdt et al.

    Chem. Phys. Lett.

    (1996)
  • R. Kobayashi et al.

    Chem. Phys. Lett.

    (1997)
  • A.P. Rendell et al.

    Chem. Phys. Lett.

    (1992)
  • C. Hampel et al.

    Chem. Phys. Lett.

    (1992)
  • H. Koch et al.

    Chem. Phys. Lett.

    (1994)
  • A.P. Rendell et al.

    Chem. Phys. Lett.

    (1991)
  • J. Cizek

    J. Chem. Phys.

    (1966)
  • M. Urban, I. Cernusak, V. Kello, J. Noga, in: S. Wilson (Ed.), Method in Computational Chemistry, Plenum Press, New...
  • R.J. Bartlett

    J. Phys. Chem.

    (1989)
  • J. Paldus, Methods in Computational Molecular Physics, NATO ASI,...
  • E. Clementi

    J. Phys. Chem.

    (1985)
  • M. Dupuis et al.

    Theoret. Chim. Acta

    (1987)
  • H.O. Villar et al.

    J. Chem. Phys.

    (1988)
  • G.D. Purvis III et al.

    J. Chem. Phys.

    (1982)
  • J. Noga et al.

    J. Chem. Phys.

    (1987)
  • N. Oliphant et al.

    J. Chem. Phys.

    (1991)
  • S.A. Kucharski et al.

    J. Chem. Phys.

    (1992)
  • Cited by (0)

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