Singularities of Møller-Plesset series: example "a44"

Molecule BO+. 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Unpublished reports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	-2.8519	1.55

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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	-0.3684	0.0571
2	-0.8976	0.0892 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	-0.6746	0.212
2	8.4475	1.57

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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	-0.4915	0.0542
2	0.7452 + 1.0275 i	0.103 + 0.00756 i
3	0.7452 - 1.0275 i	0.103 - 0.00756 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	-0.5089	0.0646
2	0.9609 + 1.0451 i	0.135 + 0.0076 i
3	0.9609 - 1.0451 i	0.135 - 0.0076 i
4	-64.8418	3.84 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	0.0719	0.0000567
2	0.0719	0.0000567 i
3	-0.4707	0.0329
4	1.2306	0.121

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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	0.2387 + 0.0002 i	0.000526 - 0.000525 i
2	0.2387 - 0.0002 i	0.000526 + 0.000525 i
3	-0.4833	0.0419
4	1.5107	0.236
5	4.3914	4.36 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	-0.4859	0.0448
2	0.5252 + 0.0064 i	0.00309 - 0.00305 i
3	0.5252 - 0.0064 i	0.00309 + 0.00305 i
4	1.3395	0.0918
5	3.9881	90.3 i
6	-140.185	13.2 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	-0.4867	0.0479
2	-0.1053 + 0.6566 i	0.00849 - 0.0148 i
3	-0.1053 - 0.6566 i	0.00849 + 0.0148 i
4	-0.128 + 0.6598 i	0.0144 + 0.00884 i
5	-0.128 - 0.6598 i	0.0144 - 0.00884 i
6	1.388	0.273

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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	-0.3605 + 0.0006 i	0.00188 + 0.00189 i
2	-0.3605 - 0.0006 i	0.00188 - 0.00189 i
3	-0.4733	0.0252
4	1.3289	0.286
5	0.0714 + 2.1233 i	0.0207 + 0.27 i
6	0.0714 - 2.1233 i	0.0207 - 0.27 i
7	-14.3235	1.2 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	-0.0406	2.41e-7
2	-0.0406	2.41e-7 i
3	-0.4755	0.0317
4	1.2025	0.0986
5	-1.5307	0.507 i
6	-1.9408 + 1.7912 i	0.409 + 0.863 i
7	-1.9408 - 1.7912 i	0.409 - 0.863 i
8	11.0478	499. 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	-0.502	0.131
2	-0.5608	0.331 i
3	-0.6432	0.263
4	-1.1901	0.264 i
5	1.2658 + 0.1781 i	0.109 - 0.102 i
6	1.2658 - 0.1781 i	0.109 + 0.102 i
7	1.7864	0.285
8	-2.2802	832.

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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	-0.5093	0.219
2	-0.5399	1.58 i
3	-0.6224	0.171
4	-1.2326	0.261 i
5	1.2629 + 0.1703 i	0.108 - 0.106 i
6	1.2629 - 0.1703 i	0.108 + 0.106 i
7	1.752	0.27
8	-2.488	97.7
9	-34.9609	0.883 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	-0.5158 + 0.0348 i	0.089 + 0.0382 i
2	-0.5158 - 0.0348 i	0.089 - 0.0382 i
3	-0.6168	0.238
4	-0.4255 + 0.6014 i	0.00521 - 0.00544 i
5	-0.4255 - 0.6014 i	0.00521 + 0.00544 i
6	-0.4137 + 0.614 i	0.00554 + 0.00507 i
7	-0.4137 - 0.614 i	0.00554 - 0.00507 i
8	1.1637	0.0781
9	4.9348	7.03 i
10	-5.2832	9.18 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	0.1005	3.23e-7
2	0.1005	3.23e-7 i
3	-0.5172	1.08
4	0.5312	0.00045
5	0.5327	0.000451 i
6	-0.5384	0.643 i
7	-0.6385	0.336
8	1.0829	0.0279
9	-1.5288	0.196 i
10	-4.6261	0.898

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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	-0.5163 + 0.0213 i	0.0427 + 0.212 i
2	-0.5163 - 0.0213 i	0.0427 - 0.212 i
3	-0.5987	0.114
4	0.9051 + 0.7127 i	0.0136 - 0.0259 i
5	0.9051 - 0.7127 i	0.0136 + 0.0259 i
6	1.1611	0.0685
7	0.8977 + 0.7623 i	0.0273 + 0.0151 i
8	0.8977 - 0.7623 i	0.0273 - 0.0151 i
9	-1.4093	0.223 i
10	-4.2368 + 2.6667 i	0.288 + 0.732 i
11	-4.2368 - 2.6667 i	0.288 - 0.732 i

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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	0.4349 + 0.e-4 i	0.000137 - 0.000137 i
2	0.4349 - 0.e-4 i	0.000137 + 0.000137 i
3	-0.5206 + 0.0187 i	0.165 + 0.25 i
4	-0.5206 - 0.0187 i	0.165 - 0.25 i
5	-0.6158	0.176
6	1.4456 + 0.1362 i	0.161 + 0.238 i
7	1.4456 - 0.1362 i	0.161 - 0.238 i
8	-1.8967	0.196 i
9	-0.3142 + 2.2012 i	0.0564 + 0.0849 i
10	-0.3142 - 2.2012 i	0.0564 - 0.0849 i
11	1.1813 + 2.0801 i	0.0122 - 0.113 i
12	1.1813 - 2.0801 i	0.0122 + 0.113 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	0.1106	1.07e-7
2	0.1106	1.07e-7 i
3	-0.5234 + 0.0173 i	0.293 + 0.195 i
4	-0.5234 - 0.0173 i	0.293 - 0.195 i
5	-0.6275	0.266
6	1.1932 + 0.2526 i	0.0546 - 0.023 i
7	1.1932 - 0.2526 i	0.0546 + 0.023 i
8	-1.3844 + 1.09 i	0.0607 + 0.105 i
9	-1.3844 - 1.09 i	0.0607 - 0.105 i
10	-0.9386 + 1.5147 i	0.0386 - 0.108 i
11	-0.9386 - 1.5147 i	0.0386 + 0.108 i
12	1.7889	0.22

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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	-0.523 + 0.0132 i	0.192 + 0.471 i
2	-0.523 - 0.0132 i	0.192 - 0.471 i
3	-0.6467	0.3
4	-0.7566	1.72 i
5	-0.8581	0.49
6	1.2494 + 0.2224 i	0.0866 - 0.089 i
7	1.2494 - 0.2224 i	0.0866 + 0.089 i
8	-1.2847	0.368 i
9	1.9754 + 0.5013 i	0.299 + 0.148 i
10	1.9754 - 0.5013 i	0.299 - 0.148 i
11	-1.8867 + 1.5953 i	0.881 + 0.0895 i
12	-1.8867 - 1.5953 i	0.881 - 0.0895 i
13	6.9332	0.957

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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	-0.5227 + 0.0131 i	0.142 + 0.486 i
2	-0.5227 - 0.0131 i	0.142 - 0.486 i
3	-0.6463	0.274
4	-0.7442	4.4 i
5	-0.8385	0.373
6	1.2482 + 0.2268 i	0.087 - 0.0774 i
7	1.2482 - 0.2268 i	0.087 + 0.0774 i
8	-1.301	0.381 i
9	2.0444 + 0.554 i	0.303 + 0.254 i
10	2.0444 - 0.554 i	0.303 - 0.254 i
11	-1.9223 + 1.4827 i	1.02 + 0.481 i
12	-1.9223 - 1.4827 i	1.02 - 0.481 i
13	11.4624	2.14
14	86.4314	73.7 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.