# Møller-Plesset perturbation theory: example "o9"

## Molecule N2. Basis cc-pVDZ. Structure ""

### Content

 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 seriesMathematica programsWork in UMassDUnpublished reports

Coefficients of Møller-Plesset perturbation series
nEnPartial sum
1 -109.598 501  -109.598 501
2  0.305 288  -109.293 213
3 -0.004 04  -109.297 253
4  0.022 176  -109.275 077
5 -0.004 694  -109.279 771
6  0.003 565  -109.276 206
7 -0.000 853  -109.277 059
8  0.000 511  -109.276 548
9 -0.000 042  -109.276 59
10  0.000 029  -109.276 561
11  0.000 033  -109.276 528
12 -0.000 011  -109.276 539
13  0.000 014  -109.276 525
14 -0.000 004  -109.276 529
15  0.000 003  -109.276 526
Exact energy -109.276 527
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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 =  0.5305, b = -1.4253 and c =  1.3383
are chosen to make scaled coefficients of order of one in magnitude for all n.
Coefficient E1 = -109.60 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),
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 seriesMathematica programsWork in UMassDUnpublished reports

Designed by A. Sergeev.