Singularities of Møller-Plesset series: example "a90"

Molecule OH-. Basis aug-cc-pVDZ. Structure "mpn_Rfci"

Content

• Definition of quadratic approximants
  • Approximant [1, 0, 0]
  • Approximant [1, 1, 0]
  • Approximant [1, 1, 1]
  • Approximant [2, 1, 1]
  • Approximant [2, 2, 1]
  • Approximant [2, 2, 2]
  • Approximant [3, 2, 2]
  • Approximant [3, 3, 2]
  • Approximant [3, 3, 3]
  • Approximant [4, 3, 3]
  • Approximant [4, 4, 3]
  • Approximant [4, 4, 4]
  • Approximant [5, 4, 4]
  • Approximant [5, 5, 4]
  • Approximant [5, 5, 5]
  • Approximant [6, 5, 5]
  • Approximant [6, 6, 5]
  • Approximant [6, 6, 6]
  • Approximant [7, 6, 6]
  • Approximant [7, 7, 6]

Plot of singularities

List of examples

Mathematica programs

Work in UMassD

Unpublished reports

Quadratic approximants

[n₁, n₂, n₃] approximant is defined as a solution of the quadratic equation
A(z)f² +  B(z)f +  C(z) = 0
with polynomial coefficients A(z), B(z) and C(z) of degree n₃, n₂ and n₁ respectively.

Square-root singularities are determined as zeroes of the discriminant
D(z) = B²(z) - 4A(z)C(z).
The weight c of the singularity z_c is defined so that
f ~ c(1 - z/z_c)^1/2 at z -> z_c.
The weight is calculated by formula
c = 1/2[-z(D/A²)']^1/2
where r. h. s. of the above equation is evaluated at z = z_c.

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-7.8958	3.81

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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.3163	0.0283
2	-0.4944	0.0353 i

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Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.9395	0.237
2	13.4259	1.1

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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6024	0.0356
2	0.5924 + 1.0918 i	0.0599 + 0.00836 i
3	0.5924 - 1.0918 i	0.0599 - 0.00836 i

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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6772	0.0624
2	1.1411 + 1.4055 i	0.137 + 0.0235 i
3	1.1411 - 1.4055 i	0.137 - 0.0235 i
4	-20.7838	1.18 i

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Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.4456	0.0047
2	-0.4992 + 0.064 i	0.00172 - 0.00597 i
3	-0.4992 - 0.064 i	0.00172 + 0.00597 i
4	1.9049	0.316

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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6642	0.0565
2	0.7553 + 1.8063 i	0.111 + 0.0504 i
3	0.7553 - 1.8063 i	0.111 - 0.0504 i
4	2.1002	22.3
5	3.1396	0.274 i

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Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6775	0.0711
2	-2.4382	0.18 i
3	2.7694	3.57e4
4	-0.7364 + 2.9562 i	0.0674 + 0.231 i
5	-0.7364 - 2.9562 i	0.0674 - 0.231 i
6	14.6157	0.907 i

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Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6828	0.0924
2	-0.9616 + 0.5508 i	0.0499 - 0.0517 i
3	-0.9616 - 0.5508 i	0.0499 + 0.0517 i
4	-1.1169 + 0.6652 i	0.0481 + 0.0722 i
5	-1.1169 - 0.6652 i	0.0481 - 0.0722 i
6	2.2951	1.06

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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6821	0.0894
2	-0.9695 + 0.5853 i	0.0441 - 0.0558 i
3	-0.9695 - 0.5853 i	0.0441 + 0.0558 i
4	-1.1053 + 0.7221 i	0.0584 + 0.066 i
5	-1.1053 - 0.7221 i	0.0584 - 0.066 i
6	2.2991	1.08
7	535.1893	1.35 i

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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6675	0.0849
2	-0.7001	0.349 i
3	-0.7206	0.18
4	-1.9567	0.153 i
5	2.686	32.9
6	-1.3377 + 3.0196 i	0.0303 + 0.302 i
7	-1.3377 - 3.0196 i	0.0303 - 0.302 i
8	17.1222	1.04 i

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Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.6604	0.0843
2	-0.683	0.259 i
3	-0.7081	0.149
4	-2.1904	0.152 i
5	2.602	9.97
6	-1.1915 + 2.607 i	0.0106 - 0.255 i
7	-1.1915 - 2.607 i	0.0106 + 0.255 i
8	-20.2966	76.

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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5584 + 0.0012 i	0.0081 + 0.00831 i
2	-0.5584 - 0.0012 i	0.0081 - 0.00831 i
3	-0.6687	0.0474
4	-1.8007	0.163 i
5	2.6051	6.94
6	-1.8543 + 2.8684 i	0.0997 - 0.474 i
7	-1.8543 - 2.8684 i	0.0997 + 0.474 i
8	6.2307	1.46 i
9	18.3346	53.4

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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	0.4221	0.000111
2	0.4221	0.000111 i
3	-0.5485 + 0.0011 i	0.00575 + 0.00583 i
4	-0.5485 - 0.0011 i	0.00575 - 0.00583 i
5	-0.6674	0.0449
6	-1.824	0.155 i
7	2.691	52.7
8	-1.5444 + 2.8719 i	0.0455 - 0.325 i
9	-1.5444 - 2.8719 i	0.0455 + 0.325 i
10	22.4239	1.26 i

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Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.0577	2.33e-8 + 2.33e-8 i
2	-0.0577	2.33e-8 - 2.33e-8 i
3	-0.5412 + 0.0011 i	0.00439 + 0.00443 i
4	-0.5412 - 0.0011 i	0.00439 - 0.00443 i
5	-0.6666	0.0432
6	-1.8349	0.153 i
7	2.6768	32.9
8	-1.5031 + 2.899 i	0.026 - 0.317 i
9	-1.5031 - 2.899 i	0.026 + 0.317 i
10	21.4157	1.2 i

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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5851	0.198
2	-0.5859	0.3 i
3	-0.6746	0.0628
4	1.0719 + 0.1397 i	0.0000535 + 0.00112 i
5	1.0719 - 0.1397 i	0.0000535 - 0.00112 i
6	1.0786 + 0.1372 i	0.00111 - 0.0000834 i
7	1.0786 - 0.1372 i	0.00111 + 0.0000834 i
8	-1.7209	0.193 i
9	-2.2578 + 2.2834 i	0.877 - 0.316 i
10	-2.2578 - 2.2834 i	0.877 + 0.316 i
11	3.404	1.19

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Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5834 + 0.0005 i	0.0969 + 0.133 i
2	-0.5834 - 0.0005 i	0.0969 - 0.133 i
3	-0.6739	0.0607
4	0.7771 + 0.5115 i	0.000638 - 0.00143 i
5	0.7771 - 0.5115 i	0.000638 + 0.00143 i
6	0.7781 + 0.5115 i	0.00143 + 0.000636 i
7	0.7781 - 0.5115 i	0.00143 - 0.000636 i
8	-1.74	0.183 i
9	2.9912	4.28
10	-2.063 + 2.4273 i	0.519 - 0.367 i
11	-2.063 - 2.4273 i	0.519 + 0.367 i
12	195.6441	6.63 i

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Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5778 + 0.0016 i	0.0129 + 0.0136 i
2	-0.5778 - 0.0016 i	0.0129 - 0.0136 i
3	-0.6707	0.0501
4	2.0489	0.884
5	2.1693	0.594 i
6	-2.2979 + 0.1721 i	0.0707 - 0.0957 i
7	-2.2979 - 0.1721 i	0.0707 + 0.0957 i
8	2.8525 + 1.032 i	0.712 + 0.621 i
9	2.8525 - 1.032 i	0.712 - 0.621 i
10	-2.2848 + 2.0031 i	0.22 - 0.765 i
11	-2.2848 - 2.0031 i	0.22 + 0.765 i
12	9.8341	7.95

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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5764 + 0.0016 i	0.0117 + 0.0123 i
2	-0.5764 - 0.0016 i	0.0117 - 0.0123 i
3	-0.6699	0.0482
4	-1.3626	0.269 i
5	-1.4727 + 0.1802 i	0.0319 - 0.272 i
6	-1.4727 - 0.1802 i	0.0319 + 0.272 i
7	1.4004 + 0.7144 i	0.00358 - 0.0201 i
8	1.4004 - 0.7144 i	0.00358 + 0.0201 i
9	1.4223 + 0.6986 i	0.0202 + 0.00306 i
10	1.4223 - 0.6986 i	0.0202 - 0.00306 i
11	-2.3208 + 2.2486 i	0.772 - 0.311 i
12	-2.3208 - 2.2486 i	0.772 + 0.311 i
13	3.2638	1.41

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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5748 + 0.0023 i	0.00647 + 0.00659 i
2	-0.5748 - 0.0023 i	0.00647 - 0.00659 i
3	-0.6678	0.0408
4	-0.3044 + 1.4816 i	0.00494 - 0.00759 i
5	-0.3044 - 1.4816 i	0.00494 + 0.00759 i
6	-0.2922 + 1.5145 i	0.00773 + 0.00489 i
7	-0.2922 - 1.5145 i	0.00773 - 0.00489 i
8	-1.6524	4.3 i
9	-1.7772 + 0.5243 i	0.0952 + 0.11 i
10	-1.7772 - 0.5243 i	0.0952 - 0.11 i
11	1.8881	0.0593
12	2.1767	0.0958 i
13	2.5539 + 4.209 i	0.0813 + 0.301 i
14	2.5539 - 4.209 i	0.0813 - 0.301 i

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Plot of singularities

List of examples

Mathematica programs

Work in UMassD

Unpublished reports

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Designed by A. Sergeev.