Møller-Plesset perturbation theory: example "HF-cc-pVDZ-Re"

Coefficients of Møller-Plesset perturbation series
n	E_n	Partial sum
1	`-100.019 418 409 917 748`	`-100.019 418 409 917 748`
2	`-0.201 619 181 365 082`	`-100.221 037 591 282 83`
3	`-0.002 909 956 975 082`	`-100.223 947 548 257 912`
4	`-0.004 199 520 355 299`	`-100.228 147 068 613 211`
5	`-0.000 061 232 756 48`	`-100.228 208 301 369 691`
6	`-0.000 438 246 849 547`	`-100.228 646 548 219 238`
7	`0.000 078 691 820 312`	`-100.228 567 856 398 926`
8	`-0.000 084 114 801 069`	`-100.228 651 971 199 995`
9	`0.000 026 939 135 579`	`-100.228 625 032 064 416`
10	`-0.000 019 143 976 835`	`-100.228 644 176 041 251`
11	`0.000 008 797 330 617`	`-100.228 635 378 710 634`
12	`-0.000 005 656 443 509`	`-100.228 641 035 154 143`
13	`0.000 003 168 258 801`	`-100.228 637 866 895 342`
14	`-0.000 002 003 325 477`	`-100.228 639 870 220 819`
15	`0.000 001 233 070 289`	`-100.228 638 637 150 53`
16	`-0.000 000 792 839 777`	`-100.228 639 429 990 307`
17	`0.000 000 512 127 82`	`-100.228 638 917 862 487`
18	`-0.000 000 336 959 079`	`-100.228 639 254 821 566`
19	`0.000 000 223 922 701`	`-100.228 639 030 898 865`
20	`-0.000 000 150 376 468`	`-100.228 639 181 275 333`
21	`0.000 000 101 891 428`	`-100.228 639 079 383 905`
22	`-0.000 000 069 559 438`	`-100.228 639 148 943 343`
23	`0.000 000 047 821 572`	`-100.228 639 101 121 771`
24	`-0.000 000 033 069 549`	`-100.228 639 134 191 32`
25	`0.000 000 022 992 406`	`-100.228 639 111 198 914`
26	`-0.000 000 016 060 13`	`-100.228 639 127 259 044`
27	`0.000 000 011 265 205`	`-100.228 639 115 993 839`
28	`-0.000 000 007 930 91`	`-100.228 639 123 924 749`
29	`0.000 000 005 601 962`	`-100.228 639 118 322 787`
30	`-0.000 000 003 968 504`	`-100.228 639 122 291 291`
31	`0.000 000 002 818 729`	`-100.228 639 119 472 562`
32	`-0.000 000 002 006 782`	`-100.228 639 121 479 344`
33	`0.000 000 001 431 75`	`-100.228 639 120 047 594`
34	`-0.000 000 001 023 448`	`-100.228 639 121 071 042`
35	`0.000 000 000 732 859`	`-100.228 639 120 338 183`
36	`-0.000 000 000 525 613`	`-100.228 639 120 863 796`
37	`0.000 000 000 377 525`	`-100.228 639 120 486 271`
38	`-0.000 000 000 271 525`	`-100.228 639 120 757 796`
39	`0.000 000 000 195 533`	`-100.228 639 120 562 263`
40	`-0.000 000 000 140 974`	`-100.228 639 120 703 237`
41	`0.000 000 000 101 75`	`-100.228 639 120 601 487`
42	`-0.000 000 000 073 516`	`-100.228 639 120 675 003`
43	`0.000 000 000 053 169`	`-100.228 639 120 621 834`
44	`-0.000 000 000 038 49`	`-100.228 639 120 660 324`
45	`0.000 000 000 027 888`	`-100.228 639 120 632 436`
46	`-0.000 000 000 020 225`	`-100.228 639 120 652 661`
47	`0.000 000 000 014 679`	`-100.228 639 120 637 982`
48	`-0.000 000 000 010 663`	`-100.228 639 120 648 645`
49	`0.000 000 000 007 752`	`-100.228 639 120 640 893`
Exact energy		`-100.228 639 120 640 893`

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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 = 0.8342, b = -4.5927 and c = 4.6143 are chosen to make scaled coefficients of order of one in magnitude for all n. Coefficient E₁ = -100.02 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

HF-cc-pVDZ-Re

Molecule

X 1^Sigma+ State of BH

H- ion

X 1^Sigma+ State of HF

X 1^Sigma+ State of HF

X 1^Sigma+ State of O2-

Basis

CC-PVDZ

CC-PVQZ

CC-PVTZ

AUG-CC-PV5Z

AUG-CC-PVQZ

CC-PVDZ

CC-PVDZ

AUG-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