Singularities of Møller-Plesset series: example "o1"

Molecule O2-. Basis aug-cc-pVDZ. Structure ""

Content

• Definition of quadratic approximants
  • Approximant [2, 2, 2]
  • Approximant [2, 2, 3]
  • Approximant [2, 3, 3]
  • Approximant [3, 3, 3]
  • Approximant [3, 3, 4]
  • Approximant [3, 4, 4]
  • Approximant [4, 4, 4]
  • Approximant [4, 4, 5]
  • Approximant [4, 5, 5]
  • Approximant [5, 5, 5]
  • Approximant [5, 5, 6]
  • Approximant [5, 6, 6]
  • Approximant [6, 6, 6]
  • Approximant [6, 6, 7]
  • Approximant [6, 7, 7]

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Unpublished reports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 [2, 2, 2] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5863 + 0.0429 i	0.605 + 1.34 i
2	-0.5863 - 0.0429 i	0.605 - 1.34 i
3	-1.2334	0.77
4	2.4084	2.34

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Table 2. Singularities with their weights for the quadratic approximant [2, 2, 3] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5652 + 0.0714 i	0.207 + 0.552 i
2	-0.5652 - 0.0714 i	0.207 - 0.552 i
3	-1.0352	0.396
4	2.9253	30.8
5	7.1608	1.32 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.	zc	c	Location in the complex plane
1	-0.5867	17.3
2	-0.6228	3.16 i
3	1.9657	0.691
4	-2.0289	326.
5	-3.017	1.68 i
6	403.786	11.2 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.	zc	c	Location in the complex plane
1	-0.5917 + 0.0384 i	1.16 + 1.44 i
2	-0.5917 - 0.0384 i	1.16 - 1.44 i
3	0.8645	0.108
4	0.8811	0.109 i
5	-1.3193	1.1
6	2.7423	8.57

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Table 5. Singularities with their weights for the quadratic approximant [3, 3, 4] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5808 + 0.0633 i	0.649 + 0.434 i
2	-0.5808 - 0.0633 i	0.649 - 0.434 i
3	-0.9319	0.624
4	-1.6087	0.354 i
5	1.7353	0.348
6	-0.9951 + 1.6883 i	0.367 - 0.182 i
7	-0.9951 - 1.6883 i	0.367 + 0.182 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.	zc	c	Location in the complex plane
1	0.362	0.000328
2	0.3623	0.000328 i
3	-0.4648 + 0.0134 i	0.0166 + 0.013 i
4	-0.4648 - 0.0134 i	0.0166 - 0.013 i
5	-0.8439	0.0915
6	1.3718	0.0536
7	-2.2046	0.799 i
8	5.8892	3.94 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.	zc	c	Location in the complex plane
1	-0.5504 + 0.1338 i	0.0687 - 0.0038 i
2	-0.5504 - 0.1338 i	0.0687 + 0.0038 i
3	-0.629 + 0.1654 i	0.0253 + 0.0703 i
4	-0.629 - 0.1654 i	0.0253 - 0.0703 i
5	-1.4604	0.433
6	1.5615	0.183
7	3.7696	0.661 i
8	-4.9544	1.59 i

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Table 8. Singularities with their weights for the quadratic approximant [4, 4, 5] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5715 + 0.0931 i	0.239 + 0.0152 i
2	-0.5715 - 0.0931 i	0.239 - 0.0152 i
3	-0.7444 + 0.1046 i	0.184 + 0.232 i
4	-0.7444 - 0.1046 i	0.184 - 0.232 i
5	1.5379	0.201
6	-2.1686 + 0.9353 i	1.66 + 0.224 i
7	-2.1686 - 0.9353 i	1.66 - 0.224 i
8	2.3906	0.355 i
9	4.3953	4.93

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Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5942	0.375
2	-0.592 + 0.0723 i	0.0451 + 0.327 i
3	-0.592 - 0.0723 i	0.0451 - 0.327 i
4	-0.4983 + 0.5501 i	0.00797 - 0.00501 i
5	-0.4983 - 0.5501 i	0.00797 + 0.00501 i
6	-0.4912 + 0.5781 i	0.0053 + 0.00794 i
7	-0.4912 - 0.5781 i	0.0053 - 0.00794 i
8	-1.4804	0.149 i
9	1.4925	0.106
10	7.7509	12.3 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.	zc	c	Location in the complex plane
1	0.3452	0.0000641
2	0.3453	0.0000641 i
3	-0.4841 + 0.0164 i	0.0093 + 0.00639 i
4	-0.4841 - 0.0164 i	0.0093 - 0.00639 i
5	-0.5782 + 0.1969 i	0.0225 + 0.0126 i
6	-0.5782 - 0.1969 i	0.0225 - 0.0126 i
7	-0.6505	0.668
8	-0.8653	0.0528 i
9	1.414	0.0546
10	-25.9719	0.457

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Table 11. Singularities with their weights for the quadratic approximant [5, 5, 6] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5687 + 0.0692 i	0.377 + 0.417 i
2	-0.5687 - 0.0692 i	0.377 - 0.417 i
3	-0.8341	2.45
4	-0.8763	19.5 i
5	-1.4684 + 1.0037 i	0.307 + 0.668 i
6	-1.4684 - 1.0037 i	0.307 - 0.668 i
7	1.9305 + 0.5516 i	0.277 + 0.0516 i
8	1.9305 - 0.5516 i	0.277 - 0.0516 i
9	1.5205 + 1.3753 i	0.0955 - 0.205 i
10	1.5205 - 1.3753 i	0.0955 + 0.205 i
11	3.5009	0.393

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Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5696 + 0.0698 i	0.435 + 0.386 i
2	-0.5696 - 0.0698 i	0.435 - 0.386 i
3	-0.7769	0.652
4	-0.9631	0.56 i
5	1.5441 + 0.4115 i	0.0904 - 0.105 i
6	1.5441 - 0.4115 i	0.0904 + 0.105 i
7	-1.3204 + 1.2207 i	0.445 + 0.107 i
8	-1.3204 - 1.2207 i	0.445 - 0.107 i
9	2.0508 + 1.018 i	0.171 + 0.144 i
10	2.0508 - 1.018 i	0.171 - 0.144 i
11	-1.3832 + 6.1748 i	0.638 + 0.473 i
12	-1.3832 - 6.1748 i	0.638 - 0.473 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.	zc	c	Location in the complex plane
1	-0.5734 + 0.0706 i	0.671 + 0.209 i
2	-0.5734 - 0.0706 i	0.671 - 0.209 i
3	-0.7177	0.335
4	1.2516	0.0116
5	-1.2964	0.116 i
6	1.2896 + 0.6064 i	0.0106 + 0.0255 i
7	1.2896 - 0.6064 i	0.0106 - 0.0255 i
8	-1.6548	0.176
9	1.6722	0.0089 i
10	1.767	0.0116
11	-1.2203 + 1.3043 i	0.156 - 0.223 i
12	-1.2203 - 1.3043 i	0.156 + 0.223 i

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Table 14. Singularities with their weights for the quadratic approximant [6, 6, 7] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5665 + 0.0695 i	0.25 + 0.407 i
2	-0.5665 - 0.0695 i	0.25 - 0.407 i
3	-0.7731 + 0.1038 i	0.027 - 0.25 i
4	-0.7731 - 0.1038 i	0.027 + 0.25 i
5	-0.9073	0.188
6	-1.2048	0.597 i
7	-1.3706 + 0.8677 i	0.253 - 0.577 i
8	-1.3706 - 0.8677 i	0.253 + 0.577 i
9	1.5913 + 0.4808 i	0.116 - 0.0438 i
10	1.5913 - 0.4808 i	0.116 + 0.0438 i
11	1.7511 + 1.3545 i	0.0429 - 0.185 i
12	1.7511 - 1.3545 i	0.0429 + 0.185 i
13	52.4936	6.22

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Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7] The most stable singularity is highlighted.
No.	zc	c	Location in the complex plane
1	-0.5716 + 0.0739 i	0.457 + 0.138 i
2	-0.5716 - 0.0739 i	0.457 - 0.138 i
3	-0.6343	0.624
4	-0.6832	0.514 i
5	-0.8671 + 0.1302 i	0.288 + 0.167 i
6	-0.8671 - 0.1302 i	0.288 - 0.167 i
7	1.6233 + 0.4488 i	0.139 - 0.122 i
8	1.6233 - 0.4488 i	0.139 + 0.122 i
9	-1.3749 + 1.0848 i	0.345 + 0.32 i
10	-1.3749 - 1.0848 i	0.345 - 0.32 i
11	1.8232 + 1.1543 i	0.115 + 0.207 i
12	1.8232 - 1.1543 i	0.115 - 0.207 i
13	4.4191 + 10.3127 i	0.969 + 0.115 i
14	4.4191 - 10.3127 i	0.969 - 0.115 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.