Singularities of Møller-Plesset series: example "BH-cc-pVTZ-2Re"

[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 [2, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2929 + 0.6041 i

0.266 - 0.0418 i
Singularities of quadratic [2, 2, 2] approximant

2

1.2929 - 0.6041 i

0.266 + 0.0418 i

3

2.1451

0.618

4

-2.2454

0.107

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Table 2. Singularities with their weights for the quadratic approximant [2, 2, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3103 + 0.5121 i

0.371 - 0.222 i
Singularities of quadratic [2, 2, 3] approximant

2

1.3103 - 0.5121 i

0.371 + 0.222 i

3

1.8646

0.411

4

-2.3075

0.134

5

-16.7243

0.186 i

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Table 3. Singularities with their weights for the quadratic approximant [2, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3099 + 0.3338 i

0.522 + 0.488 i
Singularities of quadratic [2, 3, 3] approximant

2

1.3099 - 0.3338 i

0.522 - 0.488 i

3

1.9076

0.385

4

-2.6544

0.394

5

-5.6942

0.288 i

6

254.3534

1.68 i

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Table 4. Singularities with their weights for the quadratic approximant [3, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3072 + 0.3911 i

0.124 + 0.632 i
Singularities of quadratic [3, 3, 3] approximant

2

1.3072 - 0.3911 i

0.124 - 0.632 i

3

1.8911

0.39

4

-2.3157 + 0.8012 i

0.0544 + 0.114 i

5

-2.3157 - 0.8012 i

0.0544 - 0.114 i

6

-3.9396

0.102

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Table 5. Singularities with their weights for the quadratic approximant [3, 3, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1937 + 0.3221 i

0.127 - 0.0041 i
Singularities of quadratic [3, 3, 4] approximant

2

1.1937 - 0.3221 i

0.127 + 0.0041 i

3

1.6546

3.49

4

1.5325 + 0.6922 i

0.141 - 0.11 i

5

1.5325 - 0.6922 i

0.141 + 0.11 i

6

-3.1007 + 0.8417 i

0.323 + 0.00822 i

7

-3.1007 - 0.8417 i

0.323 - 0.00822 i

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Table 6. Singularities with their weights for the quadratic approximant [3, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.7554

0.00333
Singularities of quadratic [3, 4, 4] approximant

2

0.7625

0.00335 i

3

1.2365

0.076

4

1.3566 + 0.6143 i

0.0551 + 0.163 i

5

1.3566 - 0.6143 i

0.0551 - 0.163 i

6

-3.0696 + 0.9471 i

0.244 - 0.0412 i

7

-3.0696 - 0.9471 i

0.244 + 0.0412 i

8

941.565

3.97 i

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Table 7. Singularities with their weights for the quadratic approximant [4, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2101 + 0.4277 i

0.0565 - 0.208 i
Singularities of quadratic [4, 4, 4] approximant

2

1.2101 - 0.4277 i

0.0565 + 0.208 i

3

1.4088 + 0.1202 i

0.201 - 0.0658 i

4

1.4088 - 0.1202 i

0.201 + 0.0658 i

5

2.3508

0.435

6

-2.6865 + 0.7713 i

0.111 + 0.231 i

7

-2.6865 - 0.7713 i

0.111 - 0.231 i

8

-6.489

0.189

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Table 8. Singularities with their weights for the quadratic approximant [4, 4, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1872 + 0.4546 i

0.0685 - 0.118 i
Singularities of quadratic [4, 4, 5] approximant

2

1.1872 - 0.4546 i

0.0685 + 0.118 i

3

1.3929

0.0658

4

1.4384

0.0802 i

5

-2.7493 + 0.815 i

0.15 + 0.202 i

6

-2.7493 - 0.815 i

0.15 - 0.202 i

7

4.2244 + 0.9464 i

0.371 + 1.82 i

8

4.2244 - 0.9464 i

0.371 - 1.82 i

9

-23.8422

0.594

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Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1716 + 0.4577 i

0.0543 - 0.0858 i
Singularities of quadratic [4, 5, 5] approximant

2

1.1716 - 0.4577 i

0.0543 + 0.0858 i

3

1.2712

0.056

4

1.3689

0.0832 i

5

-2.7287 + 0.7391 i

0.0621 + 0.268 i

6

-2.7287 - 0.7391 i

0.0621 - 0.268 i

7

3.9084

4.17

8

5.5978

2.45 i

9

-15.3424

0.215

10

-93.9513

0.612 i

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Table 10. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.18 + 0.3935 i

0.0606 + 0.0502 i
Singularities of quadratic [5, 5, 5] approximant

2

1.18 - 0.3935 i

0.0606 - 0.0502 i

3

1.1159 + 0.6584 i

0.0138 - 0.0352 i

4

1.1159 - 0.6584 i

0.0138 + 0.0352 i

5

1.1756 + 0.642 i

0.0398 + 0.00838 i

6

1.1756 - 0.642 i

0.0398 - 0.00838 i

7

2.2068

0.546

8

-2.7409 + 0.8731 i

0.19 + 0.128 i

9

-2.7409 - 0.8731 i

0.19 - 0.128 i

10

-8.0514

0.312

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Table 11. Singularities with their weights for the quadratic approximant [5, 5, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1913 + 0.4535 i

0.0598 - 0.147 i
Singularities of quadratic [5, 5, 6] approximant

2

1.1913 - 0.4535 i

0.0598 + 0.147 i

3

1.4672

0.183

4

2.0639 + 0.2332 i

0.471 + 1.61 i

5

2.0639 - 0.2332 i

0.471 - 1.61 i

6

2.0509 + 1.2428 i

0.208 + 0.0969 i

7

2.0509 - 1.2428 i

0.208 - 0.0969 i

8

-2.7317 + 0.8371 i

0.171 + 0.167 i

9

-2.7317 - 0.8371 i

0.171 - 0.167 i

10

-7.1372

0.282

11

-24.0387

0.329 i

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Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2011 + 0.4557 i

0.0811 - 0.186 i
Singularities of quadratic [5, 6, 6] approximant

2

1.2011 - 0.4557 i

0.0811 + 0.186 i

3

1.6348

0.306

4

1.5739 + 0.5096 i

0.299 + 0.0922 i

5

1.5739 - 0.5096 i

0.299 - 0.0922 i

6

1.8761 + 0.7359 i

0.508 - 0.0492 i

7

1.8761 - 0.7359 i

0.508 + 0.0492 i

8

-2.7369 + 0.7936 i

0.146 + 0.225 i

9

-2.7369 - 0.7936 i

0.146 - 0.225 i

10

-8.836

0.22

11

31.6111

0.349 i

12

-145.59

0.585 i

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Table 13. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1917 + 0.4688 i

0.102 - 0.117 i
Singularities of quadratic [6, 6, 6] approximant

2

1.1917 - 0.4688 i

0.102 + 0.117 i

3

1.4166 + 0.2348 i

0.225 + 0.16 i

4

1.4166 - 0.2348 i

0.225 - 0.16 i

5

1.5351 + 0.3158 i

0.421 + 0.0896 i

6

1.5351 - 0.3158 i

0.421 - 0.0896 i

7

2.4354 + 1.2131 i

0.221 + 0.28 i

8

2.4354 - 1.2131 i

0.221 - 0.28 i

9

-2.7329 + 0.8036 i

0.147 + 0.21 i

10

-2.7329 - 0.8036 i

0.147 - 0.21 i

11

3.4541

0.431

12

-8.6519

0.265

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Table 14. Singularities with their weights for the quadratic approximant [6, 6, 7]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.218 + 0.4599 i

0.151 - 0.336 i
Singularities of quadratic [6, 6, 7] approximant

2

1.218 - 0.4599 i

0.151 + 0.336 i

3

1.3601 + 0.4502 i

0.374 + 0.292 i

4

1.3601 - 0.4502 i

0.374 - 0.292 i

5

1.4859 + 0.405 i

1.63 + 1.01 i

6

1.4859 - 0.405 i

1.63 - 1.01 i

7

2.6581

0.485

8

-2.7306 + 0.8249 i

0.168 + 0.179 i

9

-2.7306 - 0.8249 i

0.168 - 0.179 i

10

5.1371

42.5 i

11

-5.8872

0.278

12

-9.1167

0.412 i

13

17.3669

0.844

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Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2004 + 0.4655 i

0.12 - 0.149 i
Singularities of quadratic [6, 7, 7] approximant

2

1.2004 - 0.4655 i

0.12 + 0.149 i

3

1.5523 + 0.3991 i

0.335 + 0.0231 i

4

1.5523 - 0.3991 i

0.335 - 0.0231 i

5

1.7892 + 0.4035 i

0.582 - 1.31 i

6

1.7892 - 0.4035 i

0.582 + 1.31 i

7

2.0058

0.325

8

-2.6892 + 0.8038 i

0.0937 + 0.191 i

9

-2.6892 - 0.8038 i

0.0937 - 0.191 i

10

-3.8094 + 0.2208 i

0.0442 - 0.344 i

11

-3.8094 - 0.2208 i

0.0442 + 0.344 i

12

-8.5368

0.126

13

16.2998

0.331 i

14

-40.4897

0.274 i

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Table 1. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2929 + 0.6041 i	0.266 - 0.0418 i
2	1.2929 - 0.6041 i	0.266 + 0.0418 i
3	2.1451	0.618
4	-2.2454	0.107

Table 2. Singularities with their weights for the quadratic approximant [2, 2, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3103 + 0.5121 i	0.371 - 0.222 i
2	1.3103 - 0.5121 i	0.371 + 0.222 i
3	1.8646	0.411
4	-2.3075	0.134
5	-16.7243	0.186 i

Table 3. Singularities with their weights for the quadratic approximant [2, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3099 + 0.3338 i	0.522 + 0.488 i
2	1.3099 - 0.3338 i	0.522 - 0.488 i
3	1.9076	0.385
4	-2.6544	0.394
5	-5.6942	0.288 i
6	254.3534	1.68 i

Table 4. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3072 + 0.3911 i	0.124 + 0.632 i
2	1.3072 - 0.3911 i	0.124 - 0.632 i
3	1.8911	0.39
4	-2.3157 + 0.8012 i	0.0544 + 0.114 i
5	-2.3157 - 0.8012 i	0.0544 - 0.114 i
6	-3.9396	0.102

Table 5. Singularities with their weights for the quadratic approximant [3, 3, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1937 + 0.3221 i	0.127 - 0.0041 i
2	1.1937 - 0.3221 i	0.127 + 0.0041 i
3	1.6546	3.49
4	1.5325 + 0.6922 i	0.141 - 0.11 i
5	1.5325 - 0.6922 i	0.141 + 0.11 i
6	-3.1007 + 0.8417 i	0.323 + 0.00822 i
7	-3.1007 - 0.8417 i	0.323 - 0.00822 i

Table 6. Singularities with their weights for the quadratic approximant [3, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.7554	0.00333
2	0.7625	0.00335 i
3	1.2365	0.076
4	1.3566 + 0.6143 i	0.0551 + 0.163 i
5	1.3566 - 0.6143 i	0.0551 - 0.163 i
6	-3.0696 + 0.9471 i	0.244 - 0.0412 i
7	-3.0696 - 0.9471 i	0.244 + 0.0412 i
8	941.565	3.97 i

Table 7. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2101 + 0.4277 i	0.0565 - 0.208 i
2	1.2101 - 0.4277 i	0.0565 + 0.208 i
3	1.4088 + 0.1202 i	0.201 - 0.0658 i
4	1.4088 - 0.1202 i	0.201 + 0.0658 i
5	2.3508	0.435
6	-2.6865 + 0.7713 i	0.111 + 0.231 i
7	-2.6865 - 0.7713 i	0.111 - 0.231 i
8	-6.489	0.189

Table 8. Singularities with their weights for the quadratic approximant [4, 4, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1872 + 0.4546 i	0.0685 - 0.118 i
2	1.1872 - 0.4546 i	0.0685 + 0.118 i
3	1.3929	0.0658
4	1.4384	0.0802 i
5	-2.7493 + 0.815 i	0.15 + 0.202 i
6	-2.7493 - 0.815 i	0.15 - 0.202 i
7	4.2244 + 0.9464 i	0.371 + 1.82 i
8	4.2244 - 0.9464 i	0.371 - 1.82 i
9	-23.8422	0.594

Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1716 + 0.4577 i	0.0543 - 0.0858 i
2	1.1716 - 0.4577 i	0.0543 + 0.0858 i
3	1.2712	0.056
4	1.3689	0.0832 i
5	-2.7287 + 0.7391 i	0.0621 + 0.268 i
6	-2.7287 - 0.7391 i	0.0621 - 0.268 i
7	3.9084	4.17
8	5.5978	2.45 i
9	-15.3424	0.215
10	-93.9513	0.612 i

Table 10. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.18 + 0.3935 i	0.0606 + 0.0502 i
2	1.18 - 0.3935 i	0.0606 - 0.0502 i
3	1.1159 + 0.6584 i	0.0138 - 0.0352 i
4	1.1159 - 0.6584 i	0.0138 + 0.0352 i
5	1.1756 + 0.642 i	0.0398 + 0.00838 i
6	1.1756 - 0.642 i	0.0398 - 0.00838 i
7	2.2068	0.546
8	-2.7409 + 0.8731 i	0.19 + 0.128 i
9	-2.7409 - 0.8731 i	0.19 - 0.128 i
10	-8.0514	0.312

Table 11. Singularities with their weights for the quadratic approximant [5, 5, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1913 + 0.4535 i	0.0598 - 0.147 i
2	1.1913 - 0.4535 i	0.0598 + 0.147 i
3	1.4672	0.183
4	2.0639 + 0.2332 i	0.471 + 1.61 i
5	2.0639 - 0.2332 i	0.471 - 1.61 i
6	2.0509 + 1.2428 i	0.208 + 0.0969 i
7	2.0509 - 1.2428 i	0.208 - 0.0969 i
8	-2.7317 + 0.8371 i	0.171 + 0.167 i
9	-2.7317 - 0.8371 i	0.171 - 0.167 i
10	-7.1372	0.282
11	-24.0387	0.329 i

Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2011 + 0.4557 i	0.0811 - 0.186 i
2	1.2011 - 0.4557 i	0.0811 + 0.186 i
3	1.6348	0.306
4	1.5739 + 0.5096 i	0.299 + 0.0922 i
5	1.5739 - 0.5096 i	0.299 - 0.0922 i
6	1.8761 + 0.7359 i	0.508 - 0.0492 i
7	1.8761 - 0.7359 i	0.508 + 0.0492 i
8	-2.7369 + 0.7936 i	0.146 + 0.225 i
9	-2.7369 - 0.7936 i	0.146 - 0.225 i
10	-8.836	0.22
11	31.6111	0.349 i
12	-145.59	0.585 i

Table 13. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1917 + 0.4688 i	0.102 - 0.117 i
2	1.1917 - 0.4688 i	0.102 + 0.117 i
3	1.4166 + 0.2348 i	0.225 + 0.16 i
4	1.4166 - 0.2348 i	0.225 - 0.16 i
5	1.5351 + 0.3158 i	0.421 + 0.0896 i
6	1.5351 - 0.3158 i	0.421 - 0.0896 i
7	2.4354 + 1.2131 i	0.221 + 0.28 i
8	2.4354 - 1.2131 i	0.221 - 0.28 i
9	-2.7329 + 0.8036 i	0.147 + 0.21 i
10	-2.7329 - 0.8036 i	0.147 - 0.21 i
11	3.4541	0.431
12	-8.6519	0.265

Table 14. Singularities with their weights for the quadratic approximant [6, 6, 7] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.218 + 0.4599 i	0.151 - 0.336 i
2	1.218 - 0.4599 i	0.151 + 0.336 i
3	1.3601 + 0.4502 i	0.374 + 0.292 i
4	1.3601 - 0.4502 i	0.374 - 0.292 i
5	1.4859 + 0.405 i	1.63 + 1.01 i
6	1.4859 - 0.405 i	1.63 - 1.01 i
7	2.6581	0.485
8	-2.7306 + 0.8249 i	0.168 + 0.179 i
9	-2.7306 - 0.8249 i	0.168 - 0.179 i
10	5.1371	42.5 i
11	-5.8872	0.278
12	-9.1167	0.412 i
13	17.3669	0.844

Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2004 + 0.4655 i	0.12 - 0.149 i
2	1.2004 - 0.4655 i	0.12 + 0.149 i
3	1.5523 + 0.3991 i	0.335 + 0.0231 i
4	1.5523 - 0.3991 i	0.335 - 0.0231 i
5	1.7892 + 0.4035 i	0.582 - 1.31 i
6	1.7892 - 0.4035 i	0.582 + 1.31 i
7	2.0058	0.325
8	-2.6892 + 0.8038 i	0.0937 + 0.191 i
9	-2.6892 - 0.8038 i	0.0937 - 0.191 i
10	-3.8094 + 0.2208 i	0.0442 - 0.344 i
11	-3.8094 - 0.2208 i	0.0442 + 0.344 i
12	-8.5368	0.126
13	16.2998	0.331 i
14	-40.4897	0.274 i