Møller-Plesset perturbation theory: example "a86"

Content

• Table of coefficients and partial sums
• Semilogarithmic plot of coefficients
• Plot of scaled coefficients
• Convergence of summation approximants
• Singularities of approximants in complex plane
• Table of singularities with their weights
• Plot of the energy function found by summation of the series
• Known inaccuracies

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Coefficients of Moller-Plesset perturbation theory, semilogarithmic plot. Red/blue dots correspond to positive/negative coefficients

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Scaled coefficients of Møller-Plesset perturbation theory. Parameters a =  0.3529, b = -1.9273 and c =  2.0114 are chosen to make scaled coefficients of order of one in magnitude for all n. Coefficient E1 = -459.54 is not shown because it is too small and out of scale

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Convergence of summation approximants for the Møller - Plesset series measured in growth of number of accurate decimal digits of summation results with increase of n, number of used coefficients. The summation methods are partial sums (red connected disks), Pade approximants (blue circles), quadratic approximants (green boxes), cubic, quartic, fifth and sixth degree approximants (triangles, diamonds, pentagonal and hexagonal stars respectively).

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Location of singularities in the complex plane of the parameter z. Left panel refers to quadratic approximants, right panel to differential approximants. Encircled areas are subjectively estimated locations of the dominant zc = 2.6 + 0.2 i and a subdominant z'c = -2.7 + 3.6 i singularities. To view an individual approximant, click on the right bar. To view all singularities with their weights, see this table.

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The function E(z) found by summation of its power series. Dashed line indicates that the approximant is complex valued. Red dot marks exact physical energy at z = 1. Red circle marks the lowest excited energy level at z = 1. To view results of summation of a specific number of terms of the series, click on the right bar.

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Known inaccuracies

• We always sum the series
  F(z) = MP1 + (MP2-MP1) z + (MP3-MP2) z² + ... rather than a complete Møller-Plesset series
  E(z) = MP0 + (MP1-MP0) z + (MP2-MP1) z² + ... where F(z) is expressed through E(z) by a formula
  F(z) = MP0 + [E(z) - MP0] z^-1.
• On the page, table of singularities with their weights for differential approximants is absent.
• The lowest excited energy level shown on the plot of the function E(z) was calculated using data for "HOMO-LUMO gap in canonical RHF orbitals" from the paper J. Chem. Phys., vol. 112, p. 9213. It likely corresponds to z = 0 rather than z = 1 as marked on the plot.

Examples of MP seriesMathematica programsWork in UMassDUnpublished reports

Designed by A. Sergeev.