# Singularities of Møller-Plesset series: example "o9"

## Molecule N2. 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 Plot of singularities  List of examples Mathematica programs Work in UMassD  Unpublished reports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`
`-0.1659`
`0.0118` `2`
`-0.202`
`0.013 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`
`-1.3811`
`0.691` `2`
`2.8353`
`10.1` 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`
`-1.2094 + 0.9824 i`
`0.351 + 0.0607 i` `2`
`-1.2094 - 0.9824 i`
`0.351 - 0.0607 i`
`3`
`1.6115`
`0.345` 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`
`1.9942`
`0.78` `2`
`-1.86 + 0.8736 i`
`1.21 + 0.171 i`
`3`
`-1.86 - 0.8736 i`
`1.21 - 0.171 i`
`4`
`729.2936`
`24.6 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`
`1.9975`
`0.786` `2`
`-1.8542 + 0.8731 i`
`1.2 + 0.18 i`
`3`
`-1.8542 - 0.8731 i`
`1.2 - 0.18 i`
`4`
`-748.1765`
`16.8` 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`
`0.2374`
`0.00232` `2`
`0.2375`
`0.00232 i`
`3`
`-1.8356 + 0.9477 i`
`1.01 - 0.0998 i`
`4`
`-1.8356 - 0.9477 i`
`1.01 + 0.0998 i`
`5`
`2.0796`
`1.11` Top of the page

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`
`-1.3369 + 0.9254 i`
`0.227 + 0.0195 i` `2`
`-1.3369 - 0.9254 i`
`0.227 - 0.0195 i`
`3`
`1.762`
`0.339`
`4`
`-2.2078 + 1.4721 i`
`0.101 + 0.411 i`
`5`
`-2.2078 - 1.4721 i`
`0.101 - 0.411 i`
`6`
`12.0364`
`483. i` Top of the page

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`
`1.8106`
`0.626` `2`
`-1.6723 + 0.7292 i`
`0.959 + 0.839 i`
`3`
`-1.6723 - 0.7292 i`
`0.959 - 0.839 i`
`4`
`2.877`
`0.916 i`
`5`
`-4.077`
`1.15`
`6`
`5.9448`
`5.32` Top of the page

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`
`1.7494`
`0.398` `2`
`-1.6036 + 0.8087 i`
`0.706 + 0.194 i`
`3`
`-1.6036 - 0.8087 i`
`0.706 - 0.194 i`
`4`
`-2.8965`
`1.66`
`5`
`3.2263`
`0.92 i`
`6`
`-5.2801`
`2.48 i`
`7`
`6.1168`
`3.12` Top of the page

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`
`-1.5417 + 0.0288 i`
`0.0889 + 0.074 i` `2`
`-1.5417 - 0.0288 i`
`0.0889 - 0.074 i`
`3`
`-1.4942 + 0.739 i`
`0.213 + 0.192 i`
`4`
`-1.4942 - 0.739 i`
`0.213 - 0.192 i`
`5`
`1.7189`
`0.295`
`6`
`5.3194`
`3.14 i`
`7`
`-4.2112 + 7.6483 i`
`1.1 + 0.502 i`
`8`
`-4.2112 - 7.6483 i`
`1.1 - 0.502 i` Top of the page

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`
`0.9546`
`0.008` `2`
`0.9588`
`0.00799 i`
`3`
`1.6267`
`0.131`
`4`
`-1.6024 + 0.7419 i`
`0.433 + 0.641 i`
`5`
`-1.6024 - 0.7419 i`
`0.433 - 0.641 i`
`6`
`-3.7224 + 2.3496 i`
`0.0043 - 0.974 i`
`7`
`-3.7224 - 2.3496 i`
`0.0043 + 0.974 i`
`8`
`63.4829`
`0.991 i` Top of the page

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`
`-1.5328 + 0.5331 i`
`0.243 - 0.103 i` `2`
`-1.5328 - 0.5331 i`
`0.243 + 0.103 i`
`3`
`1.6579 + 0.3248 i`
`0.0931 - 0.1 i`
`4`
`1.6579 - 0.3248 i`
`0.0931 + 0.1 i`
`5`
`2.0224`
`0.164`
`6`
`-1.4867 + 1.3929 i`
`0.176 + 0.0986 i`
`7`
`-1.4867 - 1.3929 i`
`0.176 - 0.0986 i`
`8`
`-2.5504 + 1.2257 i`
`0.272 - 0.409 i`
`9`
`-2.5504 - 1.2257 i`
`0.272 + 0.409 i` 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.