# Singularities of Møller-Plesset series: example "o2"

## Molecule Ne. Basis cc-pVTZ-(f). 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc 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. zcc 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. zcc Location in the complex plane
`1`
`-0.1605`
`0.00647`
`2`
`-0.1785`
`0.00683 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. zcc Location in the complex plane
`1`
`-1.9937`
`0.584`
`2`
`7.8597`
`51.2`
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Table 5. 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`
`-1.0786`
`0.0434`
`2`
`0.7163 + 1.4331 i`
`0.0594 + 0.00679 i`
`3`
`0.7163 - 1.4331 i`
`0.0594 - 0.00679 i`
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Table 6. 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`
`-1.6704`
`0.31`
`2`
`5.3355 + 2.8566 i`
`1.13 + 1.17 i`
`3`
`5.3355 - 2.8566 i`
`1.13 - 1.17 i`
`4`
`-17.7119`
`1.07 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. zcc Location in the complex plane
`1`
`0.5777 + 0.0009 i`
`0.00217 - 0.00217 i`
`2`
`0.5777 - 0.0009 i`
`0.00217 + 0.00217 i`
`3`
`-1.5528`
`0.168`
`4`
`3.3052`
`0.546`
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Table 8. 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`
`-1.6335`
`0.334`
`2`
`3.2846`
`1.1`
`3`
`-2.0852 + 3.9416 i`
`0.237 - 0.494 i`
`4`
`-2.0852 - 3.9416 i`
`0.237 + 0.494 i`
`5`
`-5.2395`
`0.464 i`
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Table 9. 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`
`-1.6705`
`0.46`
`2`
`3.2425`
`0.926`
`3`
`-3.4775`
`0.385 i`
`4`
`-2.692 + 3.3672 i`
`0.488 - 0.409 i`
`5`
`-2.692 - 3.3672 i`
`0.488 + 0.409 i`
`6`
`5259.0761`
`87.8 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. zcc Location in the complex plane
`1`
`-1.8616 + 0.1892 i`
`0.78 + 0.707 i`
`2`
`-1.8616 - 0.1892 i`
`0.78 - 0.707 i`
`3`
`3.2115`
`0.92`
`4`
`-3.2177`
`1.9`
`5`
`11.2609`
`2.68 i`
`6`
`-14.3388`
`1.14 i`
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Table 11. 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`
`0.3943`
`0.00131`
`2`
`0.3943`
`0.00131 i`
`3`
`-1.6763`
`0.494`
`4`
`3.3048`
`1.13`
`5`
`-3.5777`
`0.368 i`
`6`
`-2.4909 + 3.3804 i`
`0.397 - 0.386 i`
`7`
`-2.4909 - 3.3804 i`
`0.397 + 0.386 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. zcc Location in the complex plane
`1`
`-1.6316`
`0.365`
`2`
`-0.1141 + 1.8709 i`
`0.00797 - 0.0317 i`
`3`
`-0.1141 - 1.8709 i`
`0.00797 + 0.0317 i`
`4`
`0.0093 + 1.9606 i`
`0.0324 + 0.006 i`
`5`
`0.0093 - 1.9606 i`
`0.0324 - 0.006 i`
`6`
`2.4883`
`0.119`
`7`
`6.4486`
`1.47 i`
`8`
`-34.2029`
`49. 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. zcc Location in the complex plane
`1`
`-0.0212 + 1.1594 i`
`0.00345 - 0.00382 i`
`2`
`-0.0212 - 1.1594 i`
`0.00345 + 0.00382 i`
`3`
`-0.0092 + 1.1655 i`
`0.00387 + 0.00342 i`
`4`
`-0.0092 - 1.1655 i`
`0.00387 - 0.00342 i`
`5`
`-1.7271`
`1.81`
`6`
`2.6743`
`0.155`
`7`
`-2.8558`
`0.351 i`
`8`
`-5.018`
`1.87`
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Table 14. 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`
`0.0821`
`9.61e-8`
`2`
`0.0821`
`9.61e-8 i`
`3`
`-1.5803`
`0.169`
`4`
`2.0383`
`0.0271`
`5`
`-0.6984 + 2.2492 i`
`0.0591 + 0.0000469 i`
`6`
`-0.6984 - 2.2492 i`
`0.0591 - 0.0000469 i`
`7`
`-0.2307 + 2.8661 i`
`0.0286 + 0.0521 i`
`8`
`-0.2307 - 2.8661 i`
`0.0286 - 0.0521 i`
`9`
`2.946`
`0.066 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.