Møller-Plesset perturbation theory: example "o5"

Molecule H2O. Basis cc-pVDZ(+). Structure ""

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

Examples	o1	o2	o3	o4	o5	o6	o7	o8	o9
Molecule	Ne	Ne	F-	HF	H2O	CH₂	CH₂	C₂	N₂
Basis	cc-pVDZ	cc-pVTZ-(f)	cc-pVTZ-(f)	cc-pVTZ-(f/d)	cc-pVDZ(+)	aug-cc-pVDZ	cc-pVTZ-(f/d)	cc-pVDZ(+)	cc-pVDZ

Examples of MP series

Mathematica programs

Work in UMassD

Unpublished reports

Coefficients of Møller-Plesset perturbation series
n	E_n	Partial sum
1	`-76.478 859`	`-76.478 859`
2	`0.208 023`	`-76.270 836`
3	`0.004 839`	`-76.265 997`
4	`0.007 302`	`-76.258 695`
5	`-0.000 523`	`-76.259 218`
6	`0.001 214`	`-76.258 004`
7	`-0.000 531`	`-76.258 535`
8	`0.000 517`	`-76.258 018`
9	`-0.000 366`	`-76.258 384`
10	`0.000 313`	`-76.258 071`
11	`-0.000 257`	`-76.258 328`
12	`0.000 223`	`-76.258 105`
13	`-0.000 195`	`-76.258 3`
14	`0.000 175`	`-76.258 125`
15	`-0.000 159`	`-76.258 284`
16	`0.000 147`	`-76.258 137`
17	`-0.000 138`	`-76.258 275`
18	`0.000 131`	`-76.258 144`
19	`-0.000 126`	`-76.258 27`
20	`0.000 123`	`-76.258 147`
21	`-0.000 121`	`-76.258 268`
22	`0.000 12`	`-76.258 148`
23	`-0.000 12`	`-76.258 268`
24	`0.000 12`	`-76.258 148`
25	`-0.000 122`	`-76.258 27`
26	`0.000 125`	`-76.258 145`
27	`-0.000 128`	`-76.258 273`
28	`0.000 133`	`-76.258 14`
29	`-0.000 14`	`-76.258 28`
30	`0.000 148`	`-76.258 132`
31	`-0.000 157`	`-76.258 289`
Exact energy		`-76.258 208`

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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 = 1.2360, b = -4.6439 and c = 1.7611 are chosen to make scaled coefficients of order of one in magnitude for all n. Coefficient E₁ = -76.48 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. 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. To view results of summation of a specific number of terms of the series, click on the right bar.

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Examples	o1	o2	o3	o4	o5	o6	o7	o8	o9
Molecule	Ne	Ne	F-	HF	H2O	CH₂	CH₂	C₂	N₂
Basis	cc-pVDZ	cc-pVTZ-(f)	cc-pVTZ-(f)	cc-pVTZ-(f/d)	cc-pVDZ(+)	aug-cc-pVDZ	cc-pVTZ-(f/d)	cc-pVDZ(+)	cc-pVDZ

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.

Examples of MP series

Mathematica programs

Work in UMassD

Unpublished reports