Singularities of Møller-Plesset series: example "a62"

Molecule N2. Basis cc-pVDZ. Structure "mpn_Rfci"

Content

• Definition of quadratic approximants
  • Approximant [0, 0, 0]
  • 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]

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Unpublished reports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 [0, 0, 0] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane

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Table 2. 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

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Table 3. 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	-0.2483	0.0241
2	-0.3448	0.0284 i

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Table 4. 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	-1.3063	0.686
2	2.7606	9.36

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Table 5. 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	-1.4118 + 0.9359 i	0.588 + 0.0961 i
2	-1.4118 - 0.9359 i	0.588 - 0.0961 i
3	1.7485	0.499

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Table 6. 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	-1.7693 + 0.8268 i	1.21 + 0.15 i
2	-1.7693 - 0.8268 i	1.21 - 0.15 i
3	1.9532	0.765
4	1393.8296	32.7 i

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Table 7. 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	-1.7516 + 0.8249 i	1.19 + 0.182 i
2	-1.7516 - 0.8249 i	1.19 - 0.182 i
3	1.9646	0.785
4	-112.3942	29.9

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Table 8. 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	0.0551	0.0000596
2	0.0551	0.0000596 i
3	-1.7581 + 0.8829 i	1.06 - 0.0838 i
4	-1.7581 - 0.8829 i	1.06 + 0.0838 i
5	2.0068	0.952

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Table 9. 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	-1.2342 + 0.8547 i	0.194 + 0.0276 i
2	-1.2342 - 0.8547 i	0.194 - 0.0276 i
3	1.7234	0.329
4	-1.9422 + 1.2041 i	0.0267 + 0.354 i
5	-1.9422 - 1.2041 i	0.0267 - 0.354 i
6	13.5081	1.59e5 i

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Table 10. 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	-1.5903 + 0.6767 i	0.956 + 0.938 i
2	-1.5903 - 0.6767 i	0.956 - 0.938 i
3	1.7733	0.634
4	2.7898	0.891 i
5	-3.7545	1.1
6	5.9917	5.12

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Table 11. 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	-1.5428 + 0.745 i	0.806 + 0.275 i
2	-1.5428 - 0.745 i	0.806 - 0.275 i
3	1.7166	0.416
4	-2.7704	1.47
5	3.042	0.85 i
6	5.6666	3.93
7	-5.8719	1.78 i

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Table 12. 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	-0.1833 + 0.e-5 i	0.0000174 + 0.0000174 i
2	-0.1833 - 0.e-5 i	0.0000174 - 0.0000174 i
3	-1.5189 + 0.6622 i	0.219 + 0.636 i
4	-1.5189 - 0.6622 i	0.219 - 0.636 i
5	1.6595	0.24
6	-4.866 + 3.7463 i	0.785 + 1.23 i
7	-4.866 - 3.7463 i	0.785 - 1.23 i
8	7.2088	13.9 i

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Table 13. 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	1.6303 + 0.2927 i	0.0811 - 0.145 i
2	1.6303 - 0.2927 i	0.0811 + 0.145 i
3	-1.5106 + 0.6911 i	0.412 + 0.674 i
4	-1.5106 - 0.6911 i	0.412 - 0.674 i
5	1.9116	0.154
6	-2.9377 + 1.233 i	0.321 - 0.764 i
7	-2.9377 - 1.233 i	0.321 + 0.764 i
8	-7.8066	0.633

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Table 14. 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	-1.5049 + 0.639 i	0.0451 - 0.731 i
2	-1.5049 - 0.639 i	0.0451 + 0.731 i
3	1.6439 + 0.3105 i	0.101 - 0.133 i
4	1.6439 - 0.3105 i	0.101 + 0.133 i
5	2.0082	0.183
6	-2.2102 + 1.489 i	0.413 - 0.233 i
7	-2.2102 - 1.489 i	0.413 + 0.233 i
8	-4.5764	0.391
9	-7.4485	0.391 i

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Table 15. 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	-1.5042 + 0.6357 i	0.0716 - 0.719 i
2	-1.5042 - 0.6357 i	0.0716 + 0.719 i
3	1.6473 + 0.3119 i	0.104 - 0.133 i
4	1.6473 - 0.3119 i	0.104 + 0.133 i
5	2.0234	0.189
6	-2.1744 + 1.492 i	0.41 - 0.214 i
7	-2.1744 - 1.492 i	0.41 + 0.214 i
8	-4.7158	0.334
9	-6.7241	0.329 i
10	74381.264	92.6 i

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Table 16. 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	-1.5011 + 0.6444 i	0.000293 - 0.702 i
2	-1.5011 - 0.6444 i	0.000293 + 0.702 i
3	1.6577 + 0.3203 i	0.114 - 0.129 i
4	1.6577 - 0.3203 i	0.114 + 0.129 i
5	2.0989	0.219
6	-2.3029 + 1.6204 i	0.334 - 0.28 i
7	-2.3029 - 1.6204 i	0.334 + 0.28 i
8	-5.5917	0.368
9	11.2627	3.89 i
10	-45.5406	0.338 i

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Table 17. 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	0.3845	0.00522
2	0.3845	0.00522 i
3	-1.5046 + 0.6375 i	0.0571 - 0.727 i
4	-1.5046 - 0.6375 i	0.0571 + 0.727 i
5	1.6441 + 0.3109 i	0.101 - 0.132 i
6	1.6441 - 0.3109 i	0.101 + 0.132 i
7	2.01	0.184
8	-2.1962 + 1.4869 i	0.414 - 0.224 i
9	-2.1962 - 1.4869 i	0.414 + 0.224 i
10	-4.5382	0.39
11	-7.298	0.387 i

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Table 18. 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	1.4822	0.0472
2	-1.5979 + 0.015 i	2.62 + 0.738 i
3	-1.5979 - 0.015 i	2.62 - 0.738 i
4	1.6185	0.0342 i
5	1.6228 + 0.3946 i	0.149 + 0.0123 i
6	1.6228 - 0.3946 i	0.149 - 0.0123 i
7	-1.5377 + 0.704 i	0.755 + 0.574 i
8	-1.5377 - 0.704 i	0.755 - 0.574 i
9	1.7836	0.0559
10	-3.1762	1.41
11	-7.1121	3.15 i
12	43.6751	7.91 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.