# Singularities of Møller-Plesset series: example "o4"

## Molecule HF. Basis cc-pVTZ-(f/d). 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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`
`-0.1506`
`0.0068`
`2`
`-0.1704`
`0.00723 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`
`-1.7346`
`0.584`
`2`
`5.0764`
`527.`
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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`
`-1.0433`
`0.0643`
`2`
`0.7594 + 1.4229 i`
`0.0906 + 0.00984 i`
`3`
`0.7594 - 1.4229 i`
`0.0906 - 0.00984 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`
`-1.5561`
`0.417`
`2`
`3.8669`
`53.1`
`3`
`12.3549`
`1.31 i`
`4`
`-14.9908`
`0.965 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`
`0.5717 + 0.0021 i`
`0.0054 - 0.00538 i`
`2`
`0.5717 - 0.0021 i`
`0.0054 + 0.00538 i`
`3`
`-1.4466`
`0.23`
`4`
`2.9308`
`0.801`
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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`
`-1.416 + 0.1652 i`
`0.188 + 0.162 i`
`2`
`-1.416 - 0.1652 i`
`0.188 - 0.162 i`
`3`
`-2.3349`
`0.973`
`4`
`2.5789`
`0.681`
`5`
`-3.7513`
`8.4 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`
`-1.2161 + 0.1558 i`
`0.0236 + 0.0399 i`
`2`
`-1.2161 - 0.1558 i`
`0.0236 - 0.0399 i`
`3`
`-1.3931`
`0.0453`
`4`
`2.4802`
`0.5`
`5`
`-5.1513`
`4.96 i`
`6`
`297.2105`
`19.6 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`
`-1.2458`
`0.121`
`2`
`-1.6405`
`0.182 i`
`3`
`-2.401`
`11.3`
`4`
`2.4221 + 0.8107 i`
`0.256 - 0.126 i`
`5`
`2.4221 - 0.8107 i`
`0.256 + 0.126 i`
`6`
`4.2288`
`0.88`
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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`
`-0.9824 + 0.0175 i`
`0.00654 + 0.00695 i`
`2`
`-0.9824 - 0.0175 i`
`0.00654 - 0.00695 i`
`3`
`-1.1939`
`0.0265`
`4`
`2.493`
`0.504`
`5`
`-3.423`
`1.9 i`
`6`
`-5.4278 + 1.6789 i`
`1. + 0.972 i`
`7`
`-5.4278 - 1.6789 i`
`1. - 0.972 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`
`-0.0519`
`4.75e-8 + 4.75e-8 i`
`2`
`-0.0519`
`4.75e-8 - 4.75e-8 i`
`3`
`-0.6221 + 0.001 i`
`0.000435 + 0.000434 i`
`4`
`-0.6221 - 0.001 i`
`0.000435 - 0.000434 i`
`5`
`-1.1611`
`0.0262`
`6`
`2.5476`
`0.712`
`7`
`-4.2108`
`144. i`
`8`
`175.0269`
`21.6 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`
`-1.2735`
`0.325`
`2`
`-1.447`
`0.279 i`
`3`
`0.3243 + 1.7495 i`
`0.0394 + 0.00286 i`
`4`
`0.3243 - 1.7495 i`
`0.0394 - 0.00286 i`
`5`
`0.2342 + 1.7702 i`
`0.00468 - 0.0397 i`
`6`
`0.2342 - 1.7702 i`
`0.00468 + 0.0397 i`
`7`
`-2.1318`
`30.5`
`8`
`2.2735`
`0.213`
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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`
`-0.0472`
`1.81e-9 + 1.81e-9 i`
`2`
`-0.0472`
`1.81e-9 - 1.81e-9 i`
`3`
`0.0821 + 0.9505 i`
`0.000759 - 0.00132 i`
`4`
`0.0821 - 0.9505 i`
`0.000759 + 0.00132 i`
`5`
`0.0756 + 0.9548 i`
`0.00132 + 0.000786 i`
`6`
`0.0756 - 0.9548 i`
`0.00132 - 0.000786 i`
`7`
`-1.1498`
`0.0259`
`8`
`2.3405`
`0.216`
`9`
`-4.4604`
`7.42 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.