Singularities of Møller-Plesset series: example "a69"

Molecule HF. Basis 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Unpublished reports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	17.4809	7.05

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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	0.5708	0.0399
2	0.8503	0.0487 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	-3.3094	1.41
2	3.6526	3.

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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	1.4969	0.0758
2	-1.2956 + 1.3496 i	0.0804 + 0.0117 i
3	-1.2956 - 1.3496 i	0.0804 - 0.0117 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	-1.7137	0.143
2	2.3823 + 1.699 i	0.341 + 0.0377 i
3	2.3823 - 1.699 i	0.341 - 0.0377 i
4	-9.7433	0.502 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	-1.8538 + 0.2686 i	0.249 + 0.195 i
2	-1.8538 - 0.2686 i	0.249 - 0.195 i
3	2.4733	0.573
4	-5.7519	2.42

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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	-0.301	0.000195
2	-0.3012	0.000195 i
3	-1.3747	0.0301
4	2.2109	0.148
5	4.7852	11.5 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	-1.6621	0.181
2	2.3965	0.576
3	-3.5688	0.207 i
4	-0.0334 + 4.721 i	0.173 - 0.382 i
5	-0.0334 - 4.721 i	0.173 + 0.382 i
6	61.8124	299. 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	-1.537	0.0782
2	2.4644	0.719
3	-1.4533 + 2.1532 i	0.0489 + 0.0777 i
4	-1.4533 - 2.1532 i	0.0489 - 0.0777 i
5	-0.8128 + 2.6333 i	0.0777 - 0.0882 i
6	-0.8128 - 2.6333 i	0.0777 + 0.0882 i

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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	-1.5556	0.0908
2	-0.509 + 2.2766 i	0.0575 - 0.0525 i
3	-0.509 - 2.2766 i	0.0575 + 0.0525 i
4	2.3492	0.379
5	-1.0047 + 2.3365 i	0.0399 + 0.062 i
6	-1.0047 - 2.3365 i	0.0399 - 0.062 i
7	8.0894	11.5 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	-1.3972	0.0332
2	-1.7312	0.046 i
3	2.6403	2.79
4	-2.5171 + 0.9666 i	0.0359 + 0.1 i
5	-2.5171 - 0.9666 i	0.0359 - 0.1 i
6	0.0669 + 3.5097 i	0.00164 + 0.159 i
7	0.0669 - 3.5097 i	0.00164 - 0.159 i
8	9013.5654	203. 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	-1.4458	0.0426
2	-2.2794	0.0978 i
3	-2.0268 + 1.3288 i	0.0298 + 0.0895 i
4	-2.0268 - 1.3288 i	0.0298 - 0.0895 i
5	2.7706	17.
6	0.4008 + 4.2354 i	0.0313 + 0.219 i
7	0.4008 - 4.2354 i	0.0313 - 0.219 i
8	6.0652	0.583 i

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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	1.4055	0.0343
2	1.4099	0.0347 i
3	-1.4267	0.0379
4	-1.9532	0.0664 i
5	2.549	1.2
6	-2.2967 + 1.2556 i	0.0361 + 0.0907 i
7	-2.2967 - 1.2556 i	0.0361 - 0.0907 i
8	-0.0521 + 3.6578 i	0.00341 - 0.174 i
9	-0.0521 - 3.6578 i	0.00341 + 0.174 i

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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	-1.3722	0.0159
2	0.2797 + 1.4127 i	0.00125 + 0.000111 i
3	0.2797 - 1.4127 i	0.00125 - 0.000111 i
4	0.2882 + 1.4218 i	0.00011 - 0.00126 i
5	0.2882 - 1.4218 i	0.00011 + 0.00126 i
6	-2.1223	0.118 i
7	-1.4281 + 2.3975 i	0.0146 - 0.0339 i
8	-1.4281 - 2.3975 i	0.0146 + 0.0339 i
9	3.6719 + 1.0508 i	0.155 - 0.179 i
10	3.6719 - 1.0508 i	0.155 + 0.179 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	-1.1566 + 0.0031 i	0.000605 + 0.000612 i
2	-1.1566 - 0.0031 i	0.000605 - 0.000612 i
3	-1.3291	0.00537
4	2.099 + 0.075 i	0.0372 - 0.0294 i
5	2.099 - 0.075 i	0.0372 + 0.0294 i
6	-2.6988	6.9 i
7	-1.5541 + 2.7724 i	0.0107 + 0.128 i
8	-1.5541 - 2.7724 i	0.0107 - 0.128 i
9	3.1826	0.65
10	122.3365	0.913 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	-1.3672	0.0174
2	-1.8326	0.0459 i
3	-1.5817 + 1.6119 i	0.015 + 0.0194 i
4	-1.5817 - 1.6119 i	0.015 - 0.0194 i
5	2.5655 + 0.6948 i	0.233 + 0.0835 i
6	2.5655 - 0.6948 i	0.233 - 0.0835 i
7	-2.4609 + 1.4587 i	0.044 - 0.0162 i
8	-2.4609 - 1.4587 i	0.044 + 0.0162 i
9	-0.2451 + 3.3329 i	0.012 - 0.0673 i
10	-0.2451 - 3.3329 i	0.012 + 0.0673 i
11	3.6728	3.07

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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	-1.24 + 0.0106 i	0.00114 + 0.00127 i
2	-1.24 - 0.0106 i	0.00114 - 0.00127 i
3	-1.3245	0.00363
4	2.5005 + 0.3885 i	0.182 + 0.594 i
5	2.5005 - 0.3885 i	0.182 - 0.594 i
6	-2.6785	2.98 i
7	3.084	0.509
8	-1.8181 + 2.6489 i	0.0858 - 0.127 i
9	-1.8181 - 2.6489 i	0.0858 + 0.127 i
10	0.7762 + 5.7411 i	0.478 - 0.0863 i
11	0.7762 - 5.7411 i	0.478 + 0.0863 i
12	16.1916	1.3 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	-1.3013 + 0.0371 i	0.000427 + 0.0027 i
2	-1.3013 - 0.0371 i	0.000427 - 0.0027 i
3	-1.3115	0.00198
4	2.4785 + 0.311 i	0.522 + 0.323 i
5	2.4785 - 0.311 i	0.522 - 0.323 i
6	-2.7454	3.53 i
7	2.9607	0.543
8	-2.0807 + 2.6727 i	0.189 - 0.133 i
9	-2.0807 - 2.6727 i	0.189 + 0.133 i
10	-1.1453 + 5.2323 i	0.764 + 0.583 i
11	-1.1453 - 5.2323 i	0.764 - 0.583 i
12	-14.1423	3.98

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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	0.3706	6.57e-7
2	0.3706	6.57e-7 i
3	-1.2706	0.00103
4	-1.2843 + 0.0248 i	0.000308 - 0.00136 i
5	-1.2843 - 0.0248 i	0.000308 + 0.00136 i
6	2.5138 + 0.3717 i	0.0907 - 1.04 i
7	2.5138 - 0.3717 i	0.0907 + 1.04 i
8	2.786	0.497
9	-3.152	26.6 i
10	-2.027 + 3.2379 i	0.125 + 0.495 i
11	-2.027 - 3.2379 i	0.125 - 0.495 i
12	-3.4662 + 5.0077 i	5.24 + 3.17 i
13	-3.4662 - 5.0077 i	5.24 - 3.17 i

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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	-1.2986 + 0.044 i	0.000144 + 0.00269 i
2	-1.2986 - 0.044 i	0.000144 - 0.00269 i
3	-1.2995	0.00191
4	2.4413 + 0.3113 i	0.246 + 0.0603 i
5	2.4413 - 0.3113 i	0.246 - 0.0603 i
6	3.0985 + 0.8629 i	0.291 + 0.0293 i
7	3.0985 - 0.8629 i	0.291 - 0.0293 i
8	-1.614 + 3.0946 i	0.0879 + 0.182 i
9	-1.614 - 3.0946 i	0.0879 - 0.182 i
10	-3.7137 + 0.7735 i	2.03 - 1.73 i
11	-3.7137 - 0.7735 i	2.03 + 1.73 i
12	4.9266	0.495
13	-5.7437	58.8 i
14	12.0981	3.25 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.