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WIREs Comput Mol Sci
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Resummation methods

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Abstract Resummation methods can significantly improve the accuracy of ab initio electronic structure computations without increasing the computational cost. For perturbation theories, resummation methods can be designed by constructing approximants to model the known singularity structure of the theory in the complex plane of the perturbation parameter. Quadratic approximants for the fourth‐order Møller–Plesset perturbation theory (MP4) greatly improve the accuracy for the ground‐state energy and provide information about singularity positions that can be used to select an optimal summation method. The Coupled cluster theories CCSD (coupled clusters with single and double excitations), CCSDT (with triple excitations), CCSDTQ (with quadruple excitations), and CCSD(T) (with a triples correction from perturbation theory) can be resummed using approximants that model the empirically observed convergence patterns of the Hartree–Fock (HF), CCSD, CCSD(T) and HF, CCSD, CCSDT, CCSDTQ sequences. Coupling‐constant perturbation theories of molecular vibration and of atoms in external fields, and semiclassical perturbation theories also benefit from appropriate approximants. © 2011 John Wiley & Sons, Ltd. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

Typical convergence patterns for Møller–Plesset perturbation theory, given by the model functions fa and fb of Eq. (4).

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A flowchart for choosing an appropriate method for an N7 ab initio computation.

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Effect of the bilinear mapping on a class α|α system. The mapping parameter λ ranges from 0 to −1.5 as indicated in increments of −0.5. The squares show z = 0 and 1, the fixed points of the mapping.

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Branch points locations of the ground‐state Møller–Plesset energy function23 for the C2 molecule (cc‐pVDZ basis), the chloride ion (aug‐cc‐pVDZ), the N2 molecule (cc‐pVDZ), and the boron hydride molecule (aug‐cc‐pVQZ). C2 and N2 are of class α|α, Cl is of class‐β|α, and BH has a complicated singularity structure, technically of class β|β but best described as x|α. The circle marks the physical point, z = 1.

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Avoided crossings in a 5 × 5 matrix eigenvalue perturbation theory as a function of the perturbation parameter z along the real axis.

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Electronic Structure Theory > Ab Initio Electronic Structure Methods

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