Singularities of Møller-Plesset series: example "a1"

Molecule Ar. 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Unpublished reports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	3.0375	0.938

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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	3.9562	1.64
2	198.2823	11.6 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	3.0922	0.68
2	13.2156	3.46 i

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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	0.5998	0.0176
2	0.6174	0.0179 i
3	5.9286	1.19e6

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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	-1.195 + 0.0923 i	0.0108 + 0.00973 i
2	-1.195 - 0.0923 i	0.0108 - 0.00973 i
3	1.8229	0.0712
4	6.8976	0.619 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	-1.2759	0.00512
2	1.157 + 0.9604 i	0.0107 + 0.000432 i
3	1.157 - 0.9604 i	0.0107 - 0.000432 i
4	-3.4477	0.011 i

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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	-1.9294	0.0294
2	-1.866 + 2.0303 i	0.0219 + 0.0456 i
3	-1.866 - 2.0303 i	0.0219 - 0.0456 i
4	2.855 + 1.6465 i	0.195 + 0.148 i
5	2.855 - 1.6465 i	0.195 - 0.148 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	-1.6512 + 0.6966 i	0.0106 + 0.00482 i
2	-1.6512 - 0.6966 i	0.0106 - 0.00482 i
3	1.9108	0.0281
4	0.4957 + 2.3486 i	0.0241 + 0.0145 i
5	0.4957 - 2.3486 i	0.0241 - 0.0145 i
6	4.2424	0.368 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	-1.6298	0.0117
2	-1.3782 + 1.929 i	0.00825 + 0.0174 i
3	-1.3782 - 1.929 i	0.00825 - 0.0174 i
4	2.686	1.27
5	1.3465 + 2.6048 i	0.0118 - 0.0619 i
6	1.3465 - 2.6048 i	0.0118 + 0.0619 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	-1.6889	0.0166
2	-1.5927 + 2.0658 i	0.00337 + 0.0279 i
3	-1.5927 - 2.0658 i	0.00337 - 0.0279 i
4	2.6961	0.688
5	1.0702 + 3.1952 i	0.0712 - 0.0787 i
6	1.0702 - 3.1952 i	0.0712 + 0.0787 i
7	-6.6937	0.0655 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	-1.0018 + 0.0048 i	0.00045 + 0.00045 i
2	-1.0018 - 0.0048 i	0.00045 - 0.00045 i
3	-1.6577	0.00736
4	2.4299	0.122
5	-0.5261 + 3.0016 i	0.0296 - 0.053 i
6	-0.5261 - 3.0016 i	0.0296 + 0.053 i
7	-5.2065	0.892 i
8	7.7061	4.59 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	-1.7104 + 0.1328 i	0.0198 + 0.00729 i
2	-1.7104 - 0.1328 i	0.0198 - 0.00729 i
3	-2.031 + 0.9011 i	0.0152 + 0.0162 i
4	-2.031 - 0.9011 i	0.0152 - 0.0162 i
5	2.4462	0.126
6	-0.1145 + 2.8777 i	0.0137 + 0.0494 i
7	-0.1145 - 2.8777 i	0.0137 - 0.0494 i
8	5.4292	124. 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	-1.4637	0.0152
2	-1.5446	0.0154 i
3	-2.3643	1.94e3
4	2.6486	0.412
5	-2.9789	0.0766 i
6	0.4367 + 3.7301 i	0.225 - 0.0243 i
7	0.4367 - 3.7301 i	0.225 + 0.0243 i
8	-2.4335 + 2.9224 i	0.0791 - 0.0621 i
9	-2.4335 - 2.9224 i	0.0791 + 0.0621 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	-1.397	0.00234
2	-1.5744 + 0.6958 i	0.0023 - 0.00201 i
3	-1.5744 - 0.6958 i	0.0023 + 0.00201 i
4	-1.894 + 0.6655 i	0.000636 + 0.00347 i
5	-1.894 - 0.6655 i	0.000636 - 0.00347 i
6	-2.089	0.00406 i
7	2.6244	0.364
8	0.1774 + 3.3018 i	0.0901 + 0.0489 i
9	0.1774 - 3.3018 i	0.0901 - 0.0489 i
10	34.1213	1.98 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	0.0661	3.7e-10
2	0.0661	3.7e-10 i
3	-1.2827 + 0.0346 i	0.000523 + 0.000557 i
4	-1.2827 - 0.0346 i	0.000523 - 0.000557 i
5	-1.5293	0.00186
6	2.6562	0.495
7	0.0645 + 3.2991 i	0.0583 + 0.0659 i
8	0.0645 - 3.2991 i	0.0583 - 0.0659 i
9	-4.9233	7.72 i
10	-24.2892	1.75

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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	-1.3135 + 0.0367 i	0.000816 + 0.000889 i
2	-1.3135 - 0.0367 i	0.000816 - 0.000889 i
3	1.5158	0.00215
4	1.5199	0.00217 i
5	-1.5784	0.00276
6	2.7664	2.55
7	-0.0123 + 3.2403 i	0.0296 + 0.0665 i
8	-0.0123 - 3.2403 i	0.0296 - 0.0665 i
9	-7.7849	2.87 i
10	-9.787	3.73
11	13.7243	5.99 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	-1.2869 + 0.0302 i	0.000666 + 0.00073 i
2	-1.2869 - 0.0302 i	0.000666 - 0.00073 i
3	-1.5415	0.00229
4	-0.4454 + 2.1561 i	0.00528 - 0.00375 i
5	-0.4454 - 2.1561 i	0.00528 + 0.00375 i
6	-0.4189 + 2.2157 i	0.00382 + 0.00559 i
7	-0.4189 - 2.2157 i	0.00382 - 0.00559 i
8	2.6073	0.305
9	0.3891 + 3.2607 i	0.0839 - 0.0242 i
10	0.3891 - 3.2607 i	0.0839 + 0.0242 i
11	-3.9611	1.74 i
12	39.2207	2.12 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	-1.2441 + 0.0143 i	0.000842 + 0.000888 i
2	-1.2441 - 0.0143 i	0.000842 - 0.000888 i
3	-1.5143	0.00353
4	-1.8233	0.0114 i
5	-2.1607	0.331
6	2.5747	0.232
7	0.0033 + 3.6511 i	0.0582 + 0.268 i
8	0.0033 - 3.6511 i	0.0582 - 0.268 i
9	-3.0654 + 3.5536 i	0.0123 + 0.172 i
10	-3.0654 - 3.5536 i	0.0123 - 0.172 i
11	5.7093	3.79 i
12	18.9284	2.46

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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	-1.2357 + 0.0212 i	0.000382 + 0.000416 i
2	-1.2357 - 0.0212 i	0.000382 - 0.000416 i
3	-1.4586	0.00137
4	-2.201	0.0157 i
5	2.5765	0.227
6	-3.0595	0.0462
7	-0.5928 + 3.0397 i	0.0299 - 0.00379 i
8	-0.5928 - 3.0397 i	0.0299 + 0.00379 i
9	-1.219 + 3.241 i	0.0145 - 0.0311 i
10	-1.219 - 3.241 i	0.0145 + 0.0311 i
11	1.154 + 3.9779 i	0.173 + 0.154 i
12	1.154 - 3.9779 i	0.173 - 0.154 i
13	114.4447	11.1 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	-0.9711	0.0000275
2	-0.9722	0.0000274 i
3	-1.2431 + 0.0486 i	0.000143 + 0.000127 i
4	-1.2431 - 0.0486 i	0.000143 - 0.000127 i
5	-1.4058	0.000435
6	2.5997 + 0.1291 i	0.114 - 0.2 i
7	2.5997 - 0.1291 i	0.114 + 0.2 i
8	2.712	0.196
9	0.3373 + 3.5701 i	0.147 + 0.00636 i
10	0.3373 - 3.5701 i	0.147 - 0.00636 i
11	-3.6341	7.98 i
12	-3.4095 + 5.8373 i	0.32 + 0.586 i
13	-3.4095 - 5.8373 i	0.32 - 0.586 i
14	66.4601	19.5 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.