# Singularities of Møller-Plesset series: example "a88"

## Molecule Ne. Basis aug-cc-pVDZ. Structure "mpn_Rfci"

### Content

 Examples a1 a2 a8 a16 a22 a30 a38 a44 a45 a51 a62 a69 a75 a83 a84 a85 a86 a87 a88 a90 a91 Molecule Ar BH BH BH BH BH BH BO+ C2 CN+ N2 HF HF HCl HCl F- Cl- Cl- Ne OH- SH- Basis aug-cc-pVDZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ aug-cc-pVDZ cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ

 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports

### Quadratic approximants

[n1n2n3] 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 zc is defined so that
f ~ c(1 - z/zc)1/2 at z -> zc.
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 = zc.

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`33.4218`
`13.8`
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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`0.2496`
`0.0115`
`2`
`0.2991`
`0.0126 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. zcc Location in the complex plane
`1`
`-1.4264`
`0.239`
`2`
`-25.9117`
`0.369 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. zcc Location in the complex plane
`1`
`-0.7753 + 0.6567 i`
`0.0265 + 0.00763 i`
`2`
`-0.7753 - 0.6567 i`
`0.0265 - 0.00763 i`
`3`
`1.3546`
`0.0409`
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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-1.2088`
`0.172`
`2`
`-3.1197`
`0.254 i`
`3`
`3.8843`
`16.8`
`4`
`9.5498`
`1.13 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. zcc Location in the complex plane
`1`
`-1.1091`
`0.0871`
`2`
`2.4781 + 1.1055 i`
`0.196 - 0.0351 i`
`3`
`2.4781 - 1.1055 i`
`0.196 + 0.0351 i`
`4`
`1045.7056`
`0.135`
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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`0.2566`
`0.00023`
`2`
`0.2568`
`0.00023 i`
`3`
`-1.069`
`0.0554`
`4`
`2.1176`
`0.101`
`5`
`-4.1264`
`0.718 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. zcc Location in the complex plane
`1`
`-1.0278`
`0.0697`
`2`
`-1.5038`
`0.0821 i`
`3`
`-1.779 + 0.9899 i`
`0.105 - 0.174 i`
`4`
`-1.779 - 0.9899 i`
`0.105 + 0.174 i`
`5`
`3.1821`
`1.89`
`6`
`15.1104`
`1.2 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. zcc Location in the complex plane
`1`
`-0.9837`
`0.0463`
`2`
`-1.2729`
`0.0637 i`
`3`
`-1.992`
`0.449`
`4`
`-3.2806`
`0.16 i`
`5`
`3.1704 + 1.6133 i`
`0.432 + 0.146 i`
`6`
`3.1704 - 1.6133 i`
`0.432 - 0.146 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. zcc Location in the complex plane
`1`
`-0.9332`
`0.017`
`2`
`0.0081 + 1.4805 i`
`0.00727 + 0.000729 i`
`3`
`0.0081 - 1.4805 i`
`0.00727 - 0.000729 i`
`4`
`-1.57`
`0.0891 i`
`5`
`0.0832 + 1.5785 i`
`0.000446 - 0.00792 i`
`6`
`0.0832 - 1.5785 i`
`0.000446 + 0.00792 i`
`7`
`4.3422`
`1.06`
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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.9205`
`0.0237`
`2`
`-1.0904`
`0.0296 i`
`3`
`-1.5794`
`0.231`
`4`
`-1.5342 + 1.5981 i`
`0.0204 - 0.0914 i`
`5`
`-1.5342 - 1.5981 i`
`0.0204 + 0.0914 i`
`6`
`-2.5134`
`0.0958 i`
`7`
`3.6433`
`70.1`
`8`
`11.4282`
`0.855 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. zcc Location in the complex plane
`1`
`-0.8827 + 0.0375 i`
`0.00537 + 0.00723 i`
`2`
`-0.8827 - 0.0375 i`
`0.00537 - 0.00723 i`
`3`
`-0.9926`
`0.0118`
`4`
`3.1234`
`0.962`
`5`
`-4.0205`
`4.92 i`
`6`
`-3.123 + 3.6201 i`
`0.776 + 0.228 i`
`7`
`-3.123 - 3.6201 i`
`0.776 - 0.228 i`
`8`
`33.3157`
`0.987 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. zcc Location in the complex plane
`1`
`-0.8748 + 0.0193 i`
`0.0108 + 0.015 i`
`2`
`-0.8748 - 0.0193 i`
`0.0108 - 0.015 i`
`3`
`-1.0089`
`0.0223`
`4`
`-2.0545`
`0.545 i`
`5`
`-0.87 + 2.6186 i`
`0.0583 + 0.0216 i`
`6`
`-0.87 - 2.6186 i`
`0.0583 - 0.0216 i`
`7`
`3.8497`
`8.98`
`8`
`0.055 + 4.0544 i`
`0.0127 - 0.0928 i`
`9`
`0.055 - 4.0544 i`
`0.0127 + 0.0928 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. zcc Location in the complex plane
`1`
`-0.8676 + 0.0231 i`
`0.007 + 0.00922 i`
`2`
`-0.8676 - 0.0231 i`
`0.007 - 0.00922 i`
`3`
`-0.9898`
`0.0159`
`4`
`-2.1178`
`0.854 i`
`5`
`-0.9433 + 2.6073 i`
`0.0706 + 0.0274 i`
`6`
`-0.9433 - 2.6073 i`
`0.0706 - 0.0274 i`
`7`
`3.6681`
`380.`
`8`
`-0.482 + 4.3166 i`
`0.03 - 0.114 i`
`9`
`-0.482 - 4.3166 i`
`0.03 + 0.114 i`
`10`
`158.5786`
`7.11 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. zcc Location in the complex plane
`1`
`-0.8328 + 0.0286 i`
`0.00161 + 0.00213 i`
`2`
`-0.8328 - 0.0286 i`
`0.00161 - 0.00213 i`
`3`
`-0.9062`
`0.00352`
`4`
`1.838 + 0.0176 i`
`0.00687 - 0.00664 i`
`5`
`1.838 - 0.0176 i`
`0.00687 + 0.00664 i`
`6`
`-2.9288 + 1.9513 i`
`0.209 - 0.123 i`
`7`
`-2.9288 - 1.9513 i`
`0.209 + 0.123 i`
`8`
`0.1158 + 3.9463 i`
`0.15 + 0.0291 i`
`9`
`0.1158 - 3.9463 i`
`0.15 - 0.0291 i`
`10`
`4.0492`
`1.4`
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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.8285 + 0.0273 i`
`0.00144 + 0.00191 i`
`2`
`-0.8285 - 0.0273 i`
`0.00144 - 0.00191 i`
`3`
`-0.8991`
`0.00314`
`4`
`-2.3518 + 1.5695 i`
`0.172 - 0.0131 i`
`5`
`-2.3518 - 1.5695 i`
`0.172 + 0.0131 i`
`6`
`3.3986 + 0.6183 i`
`0.227 + 0.537 i`
`7`
`3.3986 - 0.6183 i`
`0.227 - 0.537 i`
`8`
`0.3905 + 3.511 i`
`0.0604 + 0.0403 i`
`9`
`0.3905 - 3.511 i`
`0.0604 - 0.0403 i`
`10`
`2.6629 + 2.792 i`
`0.0244 + 0.112 i`
`11`
`2.6629 - 2.792 i`
`0.0244 - 0.112 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. zcc Location in the complex plane
`1`
`0.3042 + 0.e-5 i`
`4.16e-7 - 4.16e-7 i`
`2`
`0.3042 - 0.e-5 i`
`4.16e-7 + 4.16e-7 i`
`3`
`-0.8214 + 0.0296 i`
`0.000967 + 0.00131 i`
`4`
`-0.8214 - 0.0296 i`
`0.000967 - 0.00131 i`
`5`
`-0.8811`
`0.00203`
`6`
`1.3072 + 2.8509 i`
`0.00688 + 0.0439 i`
`7`
`1.3072 - 2.8509 i`
`0.00688 - 0.0439 i`
`8`
`-3.4019`
`21. i`
`9`
`-2.0582 + 2.7664 i`
`0.058 + 0.104 i`
`10`
`-2.0582 - 2.7664 i`
`0.058 - 0.104 i`
`11`
`3.1277 + 2.7404 i`
`0.0749 - 0.0663 i`
`12`
`3.1277 - 2.7404 i`
`0.0749 + 0.0663 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. zcc Location in the complex plane
`1`
`-0.8317 + 0.0028 i`
`0.0567 - 0.000758 i`
`2`
`-0.8317 - 0.0028 i`
`0.0567 + 0.000758 i`
`3`
`-0.9203 + 0.0525 i`
`0.00213 + 0.00674 i`
`4`
`-0.9203 - 0.0525 i`
`0.00213 - 0.00674 i`
`5`
`-0.9534`
`0.00556`
`6`
`3.0625 + 0.6706 i`
`0.0664 - 0.729 i`
`7`
`3.0625 - 0.6706 i`
`0.0664 + 0.729 i`
`8`
`-3.7494`
`0.952 i`
`9`
`4.3954`
`0.725`
`10`
`-0.7078 + 4.549 i`
`0.0728 + 0.643 i`
`11`
`-0.7078 - 4.549 i`
`0.0728 - 0.643 i`
`12`
`-29.654`
`0.616`
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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.8283 + 0.0115 i`
`0.00555 + 0.00472 i`
`2`
`-0.8283 - 0.0115 i`
`0.00555 - 0.00472 i`
`3`
`-0.9232 + 0.0419 i`
`0.000576 + 0.00923 i`
`4`
`-0.9232 - 0.0419 i`
`0.000576 - 0.00923 i`
`5`
`-0.9666`
`0.00755`
`6`
`3.0404 + 0.5651 i`
`0.397 + 0.555 i`
`7`
`3.0404 - 0.5651 i`
`0.397 - 0.555 i`
`8`
`4.3981`
`0.65`
`9`
`-0.3677 + 4.4135 i`
`0.229 + 0.361 i`
`10`
`-0.3677 - 4.4135 i`
`0.229 - 0.361 i`
`11`
`-5.028 + 0.5766 i`
`0.229 - 0.224 i`
`12`
`-5.028 - 0.5766 i`
`0.229 + 0.224 i`
`13`
`15.6351`
`8.71 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. zcc Location in the complex plane
`1`
`-0.7805`
`0.000274`
`2`
`-0.791 + 0.0659 i`
`0.000284 - 0.0000789 i`
`3`
`-0.791 - 0.0659 i`
`0.000284 + 0.0000789 i`
`4`
`-0.7963 + 0.0527 i`
`0.0000316 + 0.00024 i`
`5`
`-0.7963 - 0.0527 i`
`0.0000316 - 0.00024 i`
`6`
`2.6224`
`0.34`
`7`
`2.7231`
`0.745 i`
`8`
`-2.7829`
`41.2 i`
`9`
`-1.5653 + 3.1093 i`
`0.0954 + 0.0497 i`
`10`
`-1.5653 - 3.1093 i`
`0.0954 - 0.0497 i`
`11`
`1.5654 + 3.5217 i`
`0.0426 + 0.0998 i`
`12`
`1.5654 - 3.5217 i`
`0.0426 - 0.0998 i`
`13`
`3.9151`
`2.07`
`14`
`11.2879`
`0.77 i`
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 Examples a1 a2 a8 a16 a22 a30 a38 a44 a45 a51 a62 a69 a75 a83 a84 a85 a86 a87 a88 a90 a91 Molecule Ar BH BH BH BH BH BH BO+ C2 CN+ N2 HF HF HCl HCl F- Cl- Cl- Ne OH- SH- Basis aug-cc-pVDZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ aug-cc-pVDZ cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVDZ

 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports

Designed by A. Sergeev.