# Singularities of Møller-Plesset series: example "o5"

## Molecule H2O. Basis cc-pVDZ(+). Structure ""

### Content

 Examples o1 o2 o3 o4 o5 o6 o7 o8 o9 Molecule Ne Ne F- HF H2O CH2 CH2 C2 N2 Basis cc-pVDZ cc-pVTZ-(f) cc-pVTZ-(f) cc-pVTZ-(f/d) cc-pVDZ(+) aug-cc-pVDZ cc-pVTZ-(f/d) cc-pVDZ(+) cc-pVDZ

 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports

[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`
`10.7472`
`4.47`
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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`
`0.5317`
`0.044`
`2`
`0.8794`
`0.0566 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`
`-2.0348`
`0.582`
`2`
`3.9655`
`24.3`
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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`
`1.4314`
`0.109`
`2`
`-1.2214 + 0.9806 i`
`0.0974 + 0.0246 i`
`3`
`-1.2214 - 0.9806 i`
`0.0974 - 0.0246 i`
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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`
`-1.2447`
`0.103`
`2`
`1.8738 + 1.8073 i`
`0.266 + 0.0843 i`
`3`
`1.8738 - 1.8073 i`
`0.266 - 0.0843 i`
`4`
`-7.9656`
`0.391 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`
`-1.0951 + 0.1958 i`
`0.0287 + 0.0337 i`
`2`
`-1.0951 - 0.1958 i`
`0.0287 - 0.0337 i`
`3`
`-1.6277`
`0.0649`
`4`
`2.2167`
`0.487`
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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`
`-1.1388 + 0.1907 i`
`0.0428 + 0.0456 i`
`2`
`-1.1388 - 0.1907 i`
`0.0428 - 0.0456 i`
`3`
`-1.871`
`0.114`
`4`
`2.2211`
`0.517`
`5`
`-38.1604`
`1.51 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`
`-0.8881 + 0.1497 i`
`0.00114 + 0.00804 i`
`2`
`-0.8881 - 0.1497 i`
`0.00114 - 0.00804 i`
`3`
`-0.9012`
`0.00576`
`4`
`1.9972`
`0.21`
`5`
`9.3489`
`27.7 i`
`6`
`-57.9182`
`12.2 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`
`-0.2506 + 0.e-5 i`
`0.0000769 + 0.0000769 i`
`2`
`-0.2506 - 0.e-5 i`
`0.0000769 - 0.0000769 i`
`3`
`-1.0457`
`0.0397`
`4`
`-1.7698`
`0.128 i`
`5`
`2.1179`
`0.322`
`6`
`-2.892`
`3.92`
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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`
`-1.0167`
`0.0467`
`2`
`-1.3454`
`0.0502 i`
`3`
`-1.7046 + 0.4529 i`
`0.0803 - 0.102 i`
`4`
`-1.7046 - 0.4529 i`
`0.0803 + 0.102 i`
`5`
`2.3388`
`1.08`
`6`
`-2.0343 + 3.0299 i`
`0.266 - 0.203 i`
`7`
`-2.0343 - 3.0299 i`
`0.266 + 0.203 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. zcc Location in the complex plane
`1`
`-0.8822 + 0.0127 i`
`0.0028 + 0.00328 i`
`2`
`-0.8822 - 0.0127 i`
`0.0028 - 0.00328 i`
`3`
`-0.966`
`0.00769`
`4`
`2.3418`
`1.`
`5`
`-3.7198`
`0.207 i`
`6`
`-1.0867 + 3.8914 i`
`0.0833 + 0.379 i`
`7`
`-1.0867 - 3.8914 i`
`0.0833 - 0.379 i`
`8`
`121.8585`
`4.93 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`
`-0.721 + 0.0022 i`
`0.00136 + 0.00137 i`
`2`
`-0.721 - 0.0022 i`
`0.00136 - 0.00137 i`
`3`
`-0.9813`
`0.0152`
`4`
`2.3066`
`0.791`
`5`
`-0.7361 + 3.412 i`
`0.143 + 0.284 i`
`6`
`-0.7361 - 3.412 i`
`0.143 - 0.284 i`
`7`
`-4.0823 + 4.0349 i`
`0.319 + 0.296 i`
`8`
`-4.0823 - 4.0349 i`
`0.319 - 0.296 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`
`0.37`
`0.0000691`
`2`
`0.37`
`0.0000691 i`
`3`
`-0.7284 + 0.0024 i`
`0.000882 + 0.000886 i`
`4`
`-0.7284 - 0.0024 i`
`0.000882 - 0.000886 i`
`5`
`-0.9667`
`0.0113`
`6`
`2.4023`
`1.93`
`7`
`-3.4514`
`0.209 i`
`8`
`-1.6235 + 3.6419 i`
`0.234 - 0.432 i`
`9`
`-1.6235 - 3.6419 i`
`0.234 + 0.432 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`
`0.4271 + 0.e-5 i`
`0.000014 - 0.000014 i`
`2`
`0.4271 - 0.e-5 i`
`0.000014 + 0.000014 i`
`3`
`-0.9548`
`0.0136`
`4`
`-1.2509`
`0.0415 i`
`5`
`-1.4408`
`0.13`
`6`
`1.6461`
`0.0149`
`7`
`1.9576`
`0.0284 i`
`8`
`2.4586 + 2.8112 i`
`0.0958 + 0.143 i`
`9`
`2.4586 - 2.8112 i`
`0.0958 - 0.143 i`
`10`
`-4.4729`
`0.366 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`
`-0.2512 + 0.e-5 i`
`1.65e-6 + 1.65e-6 i`
`2`
`-0.2512 - 0.e-5 i`
`1.65e-6 - 1.65e-6 i`
`3`
`-0.9506`
`0.0108`
`4`
`1.4332`
`0.00834`
`5`
`-1.4993`
`0.23 i`
`6`
`1.5314`
`0.00974 i`
`7`
`-1.9808`
`0.0693`
`8`
`2.8358 + 1.9931 i`
`0.197 - 0.029 i`
`9`
`2.8358 - 1.9931 i`
`0.197 + 0.029 i`
`10`
`-6.5942`
`3.05 i`
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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`
`-0.8217 + 0.0061 i`
`0.00149 + 0.00155 i`
`2`
`-0.8217 - 0.0061 i`
`0.00149 - 0.00155 i`
`3`
`0.5786 + 0.7317 i`
`0.00484 + 0.0018 i`
`4`
`0.5786 - 0.7317 i`
`0.00484 - 0.0018 i`
`5`
`0.5796 + 0.7317 i`
`0.0018 - 0.00485 i`
`6`
`0.5796 - 0.7317 i`
`0.0018 + 0.00485 i`
`7`
`-0.9568`
`0.00793`
`8`
`2.3431`
`1.13`
`9`
`-3.0128`
`0.191 i`
`10`
`-1.9095 + 3.581 i`
`0.293 - 0.41 i`
`11`
`-1.9095 - 3.581 i`
`0.293 + 0.41 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`
`-0.8767 + 0.014 i`
`0.00219 + 0.0025 i`
`2`
`-0.8767 - 0.014 i`
`0.00219 - 0.0025 i`
`3`
`0.4199 + 0.846 i`
`0.00146 + 0.00035 i`
`4`
`0.4199 - 0.846 i`
`0.00146 - 0.00035 i`
`5`
`0.4184 + 0.848 i`
`0.000354 - 0.00146 i`
`6`
`0.4184 - 0.848 i`
`0.000354 + 0.00146 i`
`7`
`-0.9593`
`0.00625`
`8`
`2.1319`
`0.202`
`9`
`-0.2568 + 3.5707 i`
`0.125 + 0.0989 i`
`10`
`-0.2568 - 3.5707 i`
`0.125 - 0.0989 i`
`11`
`-3.7238`
`0.196 i`
`12`
`13.4387`
`0.899 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`
`-0.8715 + 0.0132 i`
`0.00199 + 0.00224 i`
`2`
`-0.8715 - 0.0132 i`
`0.00199 - 0.00224 i`
`3`
`0.4134 + 0.842 i`
`0.00127 + 0.000344 i`
`4`
`0.4134 - 0.842 i`
`0.00127 - 0.000344 i`
`5`
`0.4119 + 0.844 i`
`0.000348 - 0.00127 i`
`6`
`0.4119 - 0.844 i`
`0.000348 + 0.00127 i`
`7`
`-0.9566`
`0.00601`
`8`
`2.1162`
`0.179`
`9`
`-3.5836`
`0.188 i`
`10`
`-0.1901 + 3.6035 i`
`0.125 + 0.0903 i`
`11`
`-0.1901 - 3.6035 i`
`0.125 - 0.0903 i`
`12`
`10.28`
`0.859 i`
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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`
`-0.8961 + 0.0137 i`
`0.00454 + 0.00579 i`
`2`
`-0.8961 - 0.0137 i`
`0.00454 - 0.00579 i`
`3`
`0.4209 + 0.8125 i`
`0.0015 - 0.000989 i`
`4`
`0.4209 - 0.8125 i`
`0.0015 + 0.000989 i`
`5`
`0.4223 + 0.8132 i`
`0.000992 + 0.0015 i`
`6`
`0.4223 - 0.8132 i`
`0.000992 - 0.0015 i`
`7`
`-0.9817`
`0.0107`
`8`
`1.9073 + 0.0317 i`
`0.216 - 0.145 i`
`9`
`1.9073 - 0.0317 i`
`0.216 + 0.145 i`
`10`
`2.5574`
`5.92`
`11`
`-3.8214`
`0.223 i`
`12`
`-1.624 + 3.7286 i`
`0.291 - 0.602 i`
`13`
`-1.624 - 3.7286 i`
`0.291 + 0.602 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`
`-0.3012 + 0.e-4 i`
`1.07e-6 + 1.07e-6 i`
`2`
`-0.3012 - 0.e-4 i`
`1.07e-6 - 1.07e-6 i`
`3`
`-0.8472 + 0.0109 i`
`0.00112 + 0.00116 i`
`4`
`-0.8472 - 0.0109 i`
`0.00112 - 0.00116 i`
`5`
`0.4067 + 0.8395 i`
`0.00125 - 0.000415 i`
`6`
`0.4067 - 0.8395 i`
`0.00125 + 0.000415 i`
`7`
`0.4045 + 0.8405 i`
`0.000411 + 0.00125 i`
`8`
`0.4045 - 0.8405 i`
`0.000411 - 0.00125 i`
`9`
`-0.9433`
`0.00456`
`10`
`2.1225`
`0.199`
`11`
`0.0425 + 3.578 i`
`0.136 + 0.0405 i`
`12`
`0.0425 - 3.578 i`
`0.136 - 0.0405 i`
`13`
`-3.6361`
`0.186 i`
`14`
`9.7538`
`0.846 i`
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 Examples o1 o2 o3 o4 o5 o6 o7 o8 o9 Molecule Ne Ne F- HF H2O CH2 CH2 C2 N2 Basis cc-pVDZ cc-pVTZ-(f) cc-pVTZ-(f) cc-pVTZ-(f/d) cc-pVDZ(+) aug-cc-pVDZ cc-pVTZ-(f/d) cc-pVDZ(+) cc-pVDZ

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

Designed by A. Sergeev.