Large-order perturbation theory

Perturbation theory is a venerable tool of theoretical chemistry and physics. First-order and second-order perturbation theories are widely used for simple qualitative descriptions. Higher-order terms can, in principle, systematically improve the description and yield solutions of arbitrary accuracy. I have had a long-term interest in large-order perturbation theories of the Schroedinger equation for atoms and molecules. The perturbation series are often in practice only slowly convergent or even divergent, but the accuracy can be significantly improved with summation approximants that model the underlying functional form of the solution. The perturbation series itself can be used as the starting point for a functional analysis in the complex plane of the perturbation parameter. Suitable approximants can be used to characterize the singularities responsible for the poor convergence.

One focus of my work has been on semiclassical perturbation theories, in particular, dimensional perturbation theories [1-5]. These theories start with a collective picture of many-particle dynamics, in contrast to the more conventional many-body perturbation theories, which are based on an independent-particle approximation. Although more difficult to formulate, the semiclassical theories are in some ways easier to analyze. For this reason, they have been useful as models for developing general mathematical techniques for summation and singularity analysis. Specific techniques we have used include algebraic approximants [6], Pade-Borel approximants [7], and renormalization methods [5,7,8].

Recently, we have been using these techniques with more mainstream methods of quantum chemistry, including the Moeller-Plesset many-body perturbation theory [9,10] and the coupled-cluster theory [11] of the electronic Schroedinger equation.

[1] M. Dunn, T. C. Germann, D. Z. Goodson, C. A. Traynor, J. D. Morgan III, D. K. Watson, and D. R. Herschbach, "A Linear Algebraic Method for Exact Computation of the Coefficients of the 1/D Expansion of the Schroedinger Equation," J. Chem. Phys. 101, 5987-6004 (1994).

[2] New Methods in Quantum Theory, edited by C. A. Tsipis et al. (Kluwer, Dordrecht, 1996), pp. 1-98, 149-182.

[3] A. A. Suvernev and D. Z. Goodson, " Dimensional Perturbation Theory: An Efficient Method for Computing Vibration-Rotation Spectra," Chem. Phys. Lett. 269, 177-182 (1997).

[4] D. Z. Goodson, "Self-Consistent-Field Perturbation Theory for the Schroedinger Equation," Phys. Rev. A 55 4155-4163 (1997).

[5] D. K. Watson and D. Z. Goodson, "Dimensional Perturbation Theory for Weakly Bound Systems," Phys. Rev. A 51, R5-R8 (1995).

[6] A. V. Sergeev and D. Z. Goodson, "Summation of Asymptotic Expansions of Multiple-Valued Functions Using Algebraic Approximants: Application to Anharmonic Oscillators," J. Phys. A: Math. Gen. 31, 4301-4317 (1998).

[7] M. O. Elout, D. Z. Goodson, C. D. Elliston, S.-W. Huang, A. V. Sergeev, and D. K. Watson, "Improving the Convergence and Estimating the Accuracy of Summation Approximants of 1/D Expansions for Coulombic Systems," J. Math. Phys. 39, 5112-5122 (1998).

[8] S.-W. Huang, D. Z. Goodson, M. Lopez-Cabrera, and T. C. Germann, "Large-Order Dimensional Perturbation Theory for Diatomic Molecules within the Born-Oppenheimer Approximation," Physical Review A, 58, 250-257 (1998).

[9] D. Z. Goodson, "A Summation Procedure that Improves the Accuracy of Fourth-Order Moeller-Plesset Perturbation Theory," Journal of Chemical Physics 113, 6461-6464 (2000). [ abstract, postscript, or pdf file]

[10] D. Z. Goodson, "Convergent Summation of Moeller-Plesset Perturbation Theory," Journal of Chemical Physics 112, 4901-4909 (2000). [ abstract]

[11] D. Z. Goodson, "Extrapolating the Coupled-Cluster Sequence Toward the Full Configuration-Interaction Limit," Journal of Chemical Physics, in press. [ abstract]