# Singularities of Møller-Plesset series: example "o3"

## Molecule F-. 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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`
`-0.4351`
`0.0373`
`2`
`-0.6483`
`0.0456 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`
`-1.8087`
`0.671`
`2`
`4.9355`
`45.9`
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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`
`-1.5549`
`0.322`
`2`
`2.5855 + 3.2621 i`
`0.743 + 0.225 i`
`3`
`2.5855 - 3.2621 i`
`0.743 - 0.225 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`
`-2.2237`
`5.05`
`2`
`2.8386`
`0.96`
`3`
`-4.0505`
`1.48 i`
`4`
`90.2541`
`4.32 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`
`-2.012`
`1.64`
`2`
`2.6096`
`0.765`
`3`
`7.26`
`1.28 i`
`4`
`-9.5994`
`0.781 i`
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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`
`-1.9851`
`1.17`
`2`
`2.0338 + 0.1298 i`
`0.0498 - 0.0726 i`
`3`
`2.0338 - 0.1298 i`
`0.0498 + 0.0726 i`
`4`
`2.3362`
`0.103`
`5`
`-5.1074`
`1.84 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`
`-2.0947`
`3.28`
`2`
`2.4434`
`0.46`
`3`
`-4.4385 + 8.8953 i`
`0.503 - 0.368 i`
`4`
`-4.4385 - 8.8953 i`
`0.503 + 0.368 i`
`5`
`9.9435`
`2.13 i`
`6`
`-12.3749`
`0.547 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`
`-2.1747`
`13.9`
`2`
`2.2677`
`0.219`
`3`
`0.9268 + 3.5737 i`
`0.177 - 0.017 i`
`4`
`0.9268 - 3.5737 i`
`0.177 + 0.017 i`
`5`
`-0.0148 + 3.9484 i`
`0.0311 - 0.184 i`
`6`
`-0.0148 - 3.9484 i`
`0.0311 + 0.184 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`
`-0.4332 + 0.e-4 i`
`0.000615 + 0.000615 i`
`2`
`-0.4332 - 0.e-4 i`
`0.000615 - 0.000615 i`
`3`
`-2.0783`
`3.28`
`4`
`2.2047 + 0.5186 i`
`0.0805 - 0.118 i`
`5`
`2.2047 - 0.5186 i`
`0.0805 + 0.118 i`
`6`
`2.5886`
`0.131`
`7`
`-6.9014`
`1.16 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`
`0.4224`
`0.000224`
`2`
`0.4224`
`0.000224 i`
`3`
`-0.8618 + 0.0015 i`
`0.00374 + 0.00372 i`
`4`
`-0.8618 - 0.0015 i`
`0.00374 - 0.00372 i`
`5`
`-2.0847`
`5.48`
`6`
`2.2643`
`0.186`
`7`
`-5.8915`
`2.05 i`
`8`
`327.0342`
`20.8 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.