# Singularities of Møller-Plesset series: example "o1"

## Molecule Ne. 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Top of the page

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
Top of the page

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`
`0.8061`
`0.0558`
`2`
`1.2672`
`0.07 i`
Top of the page

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`
`-2.7839`
`0.557`
`2`
`10.8432`
`17.`
Top of the page

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`
`-1.4628 + 1.1392 i`
`0.0579 + 0.017 i`
`2`
`-1.4628 - 1.1392 i`
`0.0579 - 0.017 i`
`3`
`1.9226`
`0.0753`
Top of the page

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`
`-2.1582 + 0.9046 i`
`0.186 + 0.0777 i`
`2`
`-2.1582 - 0.9046 i`
`0.186 - 0.0777 i`
`3`
`2.5034`
`0.223`
`4`
`47.3754`
`2.55 i`
Top of the page

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`
`-1.5464 + 0.8795 i`
`0.0501 + 0.0301 i`
`2`
`-1.5464 - 0.8795 i`
`0.0501 - 0.0301 i`
`3`
`2.4197`
`0.147`
`4`
`-5.1819`
`0.191`
Top of the page

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`
`-1.9502 + 0.9635 i`
`0.125 + 0.0763 i`
`2`
`-1.9502 - 0.9635 i`
`0.125 - 0.0763 i`
`3`
`2.5557 + 1.0539 i`
`0.0533 - 0.286 i`
`4`
`2.5557 - 1.0539 i`
`0.0533 + 0.286 i`
`5`
`5.8211`
`0.291`
Top of the page

 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.