David Z. Goodson

Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: Application to anharmonic oscillators

The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Pade approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

Extrapolating the coupled-cluster sequence toward the full configuration-interaction limit

Extrapolation methods that accelerate the convergence of coupled-cluster energy sequences toward the full configuration–interaction (FCI) limit are developed and demonstrated for a variety of atoms and small molecules for which FCI energies are available, and the results are compared with those from Moller–Plesset (MP) perturbation theory. For the coupled-cluster sequence SCF, CCSD, CCSD(T), a method based on a continued-fraction formalism is found to be particularly successful. It yields sufficient improvement over conventional CCSD(T) that the results become competitive with, and often better than, results from the MP4-qλ method (MP4 summed with quadratic approximants and λ transformation). The sequence SCF, CCSD, CCSDT can be extrapolated with a quadratic approximant but the results are not appreciably more accurate than those from the CCSD(T) continued fraction. Singularity analysis of the MP perturbation series provides a criterion for estimating the accuracy the CCSD(T) continued fraction.

A linear algebraic method for exact computation of the coefficients of the 1/D expansion of the Schrödinger equation

The 1/D expansion, where D is the dimensionality of space, offers a promising new approach for obtaining highly accurate solutions to the Schrodinger equation for atoms and molecules. The method typically employs an asymptotic expansion calculated to rather large order. Computation of the expansion coefficients has been feasible for very small systems, but extending the existing computational techniques to systems with more than three degrees of freedom has proved difficult. We present a new algorithm that greatly facilitates this computation. It yields exact values for expansion coefficients, with less roundoff error than the best alternative method. Our algorithm is formulated completely in terms of tensor arithmetic, which makes it easier to extend to systems with more than three degrees of freedom and to excited states, simplifies the development of computer codes, simplifies memory management, and makes it well suited for implementation on parallel computer architectures. We formulate the algorithm f...

A summation procedure that improves the accuracy of the fourth-order Mo/ller–Plesset perturbation theory

A procedure is demonstrated for summing the Mo/ller–Plesset many-body perturbation expansion based on the ability of quadratic summation approximants to locate branch point singularities in the complex plane of the perturbation parameter. Accuracy comparable to that from CCSDT coupled-cluster calculations is obtained using fourth-order perturbation theory.

Summation approximants improve the accuracy of ab initio calculations of molecular potential-energy hypersurfaces

Abstract Full configuration–interaction calculations for BH, HF, and CH 3 are used as benchmarks for determining the accuracy of summation approximants for CCSD(T) coupled-cluster theory and fourth-order Moller–Plesset perturbation theory as function of bond distances. A continued-fraction approximant [the CCSD(T)-cf method] reliably improves CCSD(T). The MP4-qλ procedure, in which repartitioned perturbation theory is summed with a quadratic approximant, is much more accurate than conventional MP4 but more sensitive to bond stretching than is CCSD(T)-cf. Diagnostics for estimating the accuracies of the methods, based on singularity analysis of perturbation series, are developed and discussed.

ON THE USE OF ALGEBRAIC APPROXIMANTS TO SUM DIVERGENT SERIES FOR FERMI RESONANCES IN VIBRATIONAL SPECTROSCOPY

Pade summation of large-order perturbation theory can often yield highly accurate energy eigenvalues for molecular vibrations. However, for eigenstates involved in Fermi resonances the convergence of the Pade approximants can be very slow. This is because the energy is a multivalued function of the perturbation parameter while Pade approximants are single valued, and Fermi resonances occur when a branch point lies close to the physical value of the parameter. Algebraic approximants are multivalued generalizations of Pade approximants. Using the (200) state of H2S and the (400) state of H2O as examples of Fermi resonances, it is demonstrated here that algebraic approximants greatly improve the summation convergence.

Perturbation theory for coupled anharmonic oscillators

Perturbation theory is applied to a pair of coupled oscillators with cubic anharmonicity. Large-order perturbation theory is shown to be more efficient computationally than numerical diagonalization of the Hamiltonian. Quadratic Pade summation of the energy expansions yields convergent results for the real and the imaginary parts of resonance eigenvalues.

Dimensional perturbation theory for vibration–rotation spectra of linear triatomic molecules

A very efficient large-order perturbation theory is formulated for the nuclear motion of a linear triatomic molecule. All coupling between vibration and rotation is included. To demonstrate the method, all of the experimentally observed rotational energies, with values of J almost up to 100, for the ground and first excited vibrational states of CO2 and for the ground vibrational states of N2O and of OCS are calculated. The perturbation expansions reported here are rapidly convergent. The perturbation parameter is D−1/2, where D is the dimensionality of space. Increasing D is qualitatively similar to increasing the angular momentum quantum number J. Therefore, this approach is especially suited for states with high rotational excitation. The computational cost of the method scales only in proportion to JNv5/3, where Nv is the size of the vibrational basis set.

DIMENSIONAL PERTURBATION THEORY : AN EFFICIENT METHOD FOR COMPUTING VIBRATION-ROTATION SPECTRA

Abstract Large-order perturbation theory in D −1 2 , where D is the dimensionality of space, is shown to be an accurate and computationally efficient method for computing molecular vibration-rotation spectra. Results are presented for the ground vibrational state of CO 2 with J as high as 80. The rate of convergence of the perturbation expansion increases with J and the computational cost scales on JN v 5 3 , where N v is the number of vibrational basis functions. Therefore, this approach seems to be especially well suited for treating highly excited rotational states.

UNIFIED APPROACH TO MOLECULAR STRUCTURE AND MOLECULAR VIBRATIONS

First-order dimensional perturbation theory is used to construct a Hamiltonian for the H+2 molecule without the Born-Oppenheimer approximation. The physical model that emerges has the three particles undergoing harmonic oscillations about a bent symmetric configuration. Despite its simplicity, the theory yields correct results for the ground-state energy, for the equilibrium internuclear distance, and for vibrational frequencies. Although the standard dimensional continuation of the Schrodinger equation leads to dissociation at large D, this model remains stable due to a quadratic polynomial in 1/D that is included in the potential energy. This Hamiltonian is a suitable starting point for a large-order perturbation expansion in 1/D.