# Singularities of Møller-Plesset series: example "o6"

## Molecule CH2. Basis aug-cc-pVDZ. Structure "theta=104.5`256"

### 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 Plot of singularities  List of examples Mathematica programs Work in UMassD  Unpublished reports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 [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.3682`
`0.317`  Top of the page

Table 2. 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`
`1.6956`
`0.496` `2`
`132.1892`
`4.38 i` Top of the page

Table 3. 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`
`1.9495`
`0.927` `2`
`-14.4441`
`1.03` Top of the page

Table 4. 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`
`-0.1168`
`0.000683` `2`
`-0.117`
`0.000683 i`
`3`
`2.143`
`3.83` Top of the page

Table 5. 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`
`1.3037`
`0.123` `2`
`3.5333`
`0.555 i`
`3`
`-3.774 + 3.7843 i`
`0.215 + 0.0478 i`
`4`
`-3.774 - 3.7843 i`
`0.215 - 0.0478 i` Top of the page

Table 6. 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`
`0.606 + 0.0029 i`
`0.000649 - 0.000648 i` `2`
`0.606 - 0.0029 i`
`0.000649 + 0.000648 i`
`3`
`1.0435`
`0.0133`
`4`
`-2.6308`
`0.0452` Top of the page

Table 7. 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`
`1.1092 + 0.2492 i`
`0.0167 - 0.0183 i` `2`
`1.1092 - 0.2492 i`
`0.0167 + 0.0183 i`
`3`
`1.4353`
`0.0316`
`4`
`-2.6768`
`0.0643`
`5`
`-11.2388`
`0.148 i` Top of the page

Table 8. 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`
`-0.4484 + 0.0002 i`
`0.0000812 + 0.0000811 i` `2`
`-0.4484 - 0.0002 i`
`0.0000812 - 0.0000811 i`
`3`
`1.0953`
`0.0269`
`4`
`-2.1391`
`0.015`
`5`
`9.2922`
`0.9 i`
`6`
`-41.7268`
`0.302 i` Top of the page

Table 9. 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`
`-0.3458 + 0.e-4 i`
`0.0000215 + 0.0000215 i` `2`
`-0.3458 - 0.e-4 i`
`0.0000215 - 0.0000215 i`
`3`
`1.0872`
`0.0244`
`4`
`-1.954`
`0.0106`
`5`
`-4.5661`
`0.042 i`
`6`
`-20.3596`
`0.169` Top of the page

Table 10. 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`
`1.1409 + 0.1088 i`
`0.00645 - 0.0467 i` `2`
`1.1409 - 0.1088 i`
`0.00645 + 0.0467 i`
`3`
`1.2694`
`0.0344`
`4`
`-1.8455 + 0.2343 i`
`0.00371 + 0.00545 i`
`5`
`-1.8455 - 0.2343 i`
`0.00371 - 0.00545 i`
`6`
`-2.3056`
`0.00746`
`7`
`8.051`
`3.19 i` Top of the page

Table 11. 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`
`1.1212`
`0.0421` `2`
`1.7666`
`0.134 i`
`3`
`-1.8969 + 0.3188 i`
`0.00437 + 0.00605 i`
`4`
`-1.8969 - 0.3188 i`
`0.00437 - 0.00605 i`
`5`
`2.1383`
`46.4`
`6`
`-2.4751`
`0.00867`
`7`
`6.0147`
`114. i`
`8`
`-326.6589`
`2.78 i` Top of the page

Table 12. 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`
`-0.1546`
`6.89e-7` `2`
`-0.1546`
`6.89e-7 i`
`3`
`0.9126`
`0.0529`
`4`
`0.9174`
`0.0333 i`
`5`
`1.071`
`0.0206`
`6`
`-1.8675`
`0.00987`
`7`
`-3.08`
`0.022 i`
`8`
`-7.3903`
`0.522` Top of the page

Table 13. 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`
`-0.1337`
`1.06e-7` `2`
`-0.1337`
`1.06e-7 i`
`3`
`0.2644 + 0.e-4 i`
`2.26e-6 - 2.26e-6 i`
`4`
`0.2644 - 0.e-4 i`
`2.26e-6 + 2.26e-6 i`
`5`
`1.1131`
`0.0416`
`6`
`-1.8221`
`0.00579`
`7`
`1.2383 + 3.8446 i`
`0.0446 - 0.044 i`
`8`
`1.2383 - 3.8446 i`
`0.0446 + 0.044 i`
`9`
`253.2882`
`4.56 i` Top of the page

Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.601 + 0.3795 i`
`0.0000188 + 0.000105 i` `2`
`-0.601 - 0.3795 i`
`0.0000188 - 0.000105 i`
`3`
`-0.6029 + 0.3803 i`
`0.000106 - 0.0000185 i`
`4`
`-0.6029 - 0.3803 i`
`0.000106 + 0.0000185 i`
`5`
`1.1606 + 0.1357 i`
`0.0318 - 0.0288 i`
`6`
`1.1606 - 0.1357 i`
`0.0318 + 0.0288 i`
`7`
`1.3799`
`0.0541`
`8`
`-1.8497`
`0.00484`
`9`
`5.2297`
`16.5 i`
`10`
`-795.5416`
`2.88 i` Top of the page

Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.8653 + 0.0776 i`
`0.0000338 + 0.0000702 i` `2`
`-0.8653 - 0.0776 i`
`0.0000338 - 0.0000702 i`
`3`
`-0.908 + 0.1095 i`
`0.0000912 - 0.0000209 i`
`4`
`-0.908 - 0.1095 i`
`0.0000912 + 0.0000209 i`
`5`
`1.1551`
`0.115`
`6`
`1.112 + 0.9766 i`
`0.00362 + 0.00905 i`
`7`
`1.112 - 0.9766 i`
`0.00362 - 0.00905 i`
`8`
`1.2599 + 0.9837 i`
`0.011 - 0.00535 i`
`9`
`1.2599 - 0.9837 i`
`0.011 + 0.00535 i`
`10`
`-1.8613`
`0.00291` Top of the page

Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.6884 + 0.298 i`
`0.0000154 - 0.0000694 i` `2`
`-0.6884 - 0.298 i`
`0.0000154 + 0.0000694 i`
`3`
`-0.6922 + 0.2948 i`
`0.0000688 + 0.000016 i`
`4`
`-0.6922 - 0.2948 i`
`0.0000688 - 0.000016 i`
`5`
`0.7977 + 0.4317 i`
`0.00106 + 0.000812 i`
`6`
`0.7977 - 0.4317 i`
`0.00106 - 0.000812 i`
`7`
`0.8029 + 0.441 i`
`0.000853 - 0.00107 i`
`8`
`0.8029 - 0.441 i`
`0.000853 + 0.00107 i`
`9`
`1.1155`
`0.0549`
`10`
`-1.7904`
`0.00318`
`11`
`7.9057`
`4. i` Top of the page

Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.5471 + 0.2361 i`
`3.31e-6 - 0.0000102 i` `2`
`-0.5471 - 0.2361 i`
`3.31e-6 + 0.0000102 i`
`3`
`-0.548 + 0.2348 i`
`0.0000101 + 3.33e-6 i`
`4`
`-0.548 - 0.2348 i`
`0.0000101 - 3.33e-6 i`
`5`
`0.718 + 0.8703 i`
`0.0000639 - 0.00105 i`
`6`
`0.718 - 0.8703 i`
`0.0000639 + 0.00105 i`
`7`
`1.161`
`0.234`
`8`
`0.7523 + 0.8912 i`
`0.00116 + 0.0000849 i`
`9`
`0.7523 - 0.8912 i`
`0.00116 - 0.0000849 i`
`10`
`-1.5983`
`0.0013`
`11`
`10.6894`
`10.6 i`
`12`
`-27.3287`
`2.29 i` Top of the page

Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`0.5429`
`3.11e-6` `2`
`0.5499`
`2.97e-6 i`
`3`
`0.6376`
`8.04e-6`
`4`
`-0.5898 + 0.266 i`
`8.5e-8 - 7.92e-6 i`
`5`
`-0.5898 - 0.266 i`
`8.5e-8 + 7.92e-6 i`
`6`
`-0.5926 + 0.2625 i`
`7.88e-6 + 1.87e-7 i`
`7`
`-0.5926 - 0.2625 i`
`7.88e-6 - 1.87e-7 i`
`8`
`0.6867`
`0.0000137 i`
`9`
`0.9492`
`0.000344`
`10`
`0.9439 + 0.2988 i`
`0.0000821 - 0.000669 i`
`11`
`0.9439 - 0.2988 i`
`0.0000821 + 0.000669 i`
`12`
`-1.5567`
`0.0008` Top of the page

Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`0.0316 + 0.592 i`
`3.24e-6 + 4.69e-6 i` `2`
`0.0316 - 0.592 i`
`3.24e-6 - 4.69e-6 i`
`3`
`0.0321 + 0.5926 i`
`4.7e-6 - 3.24e-6 i`
`4`
`0.0321 - 0.5926 i`
`4.7e-6 + 3.24e-6 i`
`5`
`-0.7369 + 0.349 i`
`0.0000192 + 7.06e-6 i`
`6`
`-0.7369 - 0.349 i`
`0.0000192 - 7.06e-6 i`
`7`
`-0.7556 + 0.3739 i`
`7.32e-6 - 0.000021 i`
`8`
`-0.7556 - 0.3739 i`
`7.32e-6 + 0.000021 i`
`9`
`1.0615`
`0.00482`
`10`
`1.2421`
`0.0404 i`
`11`
`1.7137 + 0.6854 i`
`0.0331 - 0.0174 i`
`12`
`1.7137 - 0.6854 i`
`0.0331 + 0.0174 i`
`13`
`-3.0474`
`0.028` Top of the page

Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.6268 + 0.0042 i`
`7.38e-7 + 7.26e-7 i` `2`
`-0.6268 - 0.0042 i`
`7.38e-7 - 7.26e-7 i`
`3`
`0.1154 + 0.7213 i`
`6.47e-6 - 7.64e-6 i`
`4`
`0.1154 - 0.7213 i`
`6.47e-6 + 7.64e-6 i`
`5`
`0.12 + 0.7257 i`
`7.84e-6 + 6.65e-6 i`
`6`
`0.12 - 0.7257 i`
`7.84e-6 - 6.65e-6 i`
`7`
`-0.604 + 0.4273 i`
`3.32e-7 - 3.69e-6 i`
`8`
`-0.604 - 0.4273 i`
`3.32e-7 + 3.69e-6 i`
`9`
`-0.6091 + 0.4549 i`
`4.13e-6 + 2.73e-7 i`
`10`
`-0.6091 - 0.4549 i`
`4.13e-6 - 2.73e-7 i`
`11`
`1.246`
`0.18`
`12`
`2.5693`
`28.1 i`
`13`
`-4.5436 + 2.5005 i`
`0.015 - 0.00917 i`
`14`
`-4.5436 - 2.5005 i`
`0.015 + 0.00917 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.