# Singularities of Møller-Plesset series: example "o7"

## Molecule CH2. Basis cc-pVTZ-(f/d). 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 Plot of singularities  List of examples Mathematica programs Work in UMassD  Unpublished reports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`
`1.5895`
`0.403`  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`
`1.511`
`0.365` `2`
`2416.859`
`14.6 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`
`2.1945`
`1.97` `2`
`-4.8612`
`0.599` 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`
`-1.6556`
`0.246` `2`
`1.7987`
`0.698`
`3`
`-1.9982`
`0.232 i` 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`
`1.3217`
`0.126` `2`
`-2.5341 + 2.2023 i`
`0.162 + 0.0427 i`
`3`
`-2.5341 - 2.2023 i`
`0.162 - 0.0427 i`
`4`
`4.7752`
`1.19 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`
`1.205 + 0.2792 i`
`0.0368 - 0.0303 i` `2`
`1.205 - 0.2792 i`
`0.0368 + 0.0303 i`
`3`
`1.8615`
`0.103`
`4`
`-2.3641`
`0.0995` 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`
`1.156`
`0.0555` `2`
`-1.9784 + 1.169 i`
`0.0409 + 0.022 i`
`3`
`-1.9784 - 1.169 i`
`0.0409 - 0.022 i`
`4`
`2.5849`
`0.236 i`
`5`
`-10.9544`
`0.221` 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`
`1.0421 + 0.1771 i`
`0.00613 - 0.0138 i` `2`
`1.0421 - 0.1771 i`
`0.00613 + 0.0138 i`
`3`
`1.1399`
`0.0124`
`4`
`-2.5123`
`0.117`
`5`
`18.3276`
`1.5 i`
`6`
`-22.3891`
`0.675 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`
`1.1266`
`0.048` `2`
`1.4467 + 1.4772 i`
`0.00735 + 0.0515 i`
`3`
`1.4467 - 1.4772 i`
`0.00735 - 0.0515 i`
`4`
`1.8193 + 1.3656 i`
`0.0572 - 0.024 i`
`5`
`1.8193 - 1.3656 i`
`0.0572 + 0.024 i`
`6`
`-2.6517`
`0.216` 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`
`1.0848`
`0.0258` `2`
`-0.1243 + 1.1305 i`
`0.00292 + 0.000857 i`
`3`
`-0.1243 - 1.1305 i`
`0.00292 - 0.000857 i`
`4`
`-0.1429 + 1.141 i`
`0.000913 - 0.00295 i`
`5`
`-0.1429 - 1.141 i`
`0.000913 + 0.00295 i`
`6`
`-2.656`
`0.344`
`7`
`14.6598`
`0.643 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`
`-0.5019 + 0.0001 i`
`0.000117 + 0.000117 i` `2`
`-0.5019 - 0.0001 i`
`0.000117 - 0.000117 i`
`3`
`1.0984`
`0.0307`
`4`
`-2.3669`
`0.0433`
`5`
`3.7816`
`1.78 i`
`6`
`-4.794`
`2.32 i`
`7`
`3.1913 + 7.4017 i`
`0.105 - 0.351 i`
`8`
`3.1913 - 7.4017 i`
`0.105 + 0.351 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`
`-0.4766 + 0.3568 i`
`0.0000398 + 0.000138 i` `2`
`-0.4766 - 0.3568 i`
`0.0000398 - 0.000138 i`
`3`
`-0.4775 + 0.3566 i`
`0.000138 - 0.0000395 i`
`4`
`-0.4775 - 0.3566 i`
`0.000138 + 0.0000395 i`
`5`
`1.1065`
`0.0304`
`6`
`1.533`
`0.869 i`
`7`
`1.6936`
`0.161`
`8`
`-2.9744`
`4.56` 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`
`-0.8866 + 0.5063 i`
`0.0000416 + 0.00125 i` `2`
`-0.8866 - 0.5063 i`
`0.0000416 - 0.00125 i`
`3`
`-0.902 + 0.494 i`
`0.00124 - 9.11e-6 i`
`4`
`-0.902 - 0.494 i`
`0.00124 + 9.11e-6 i`
`5`
`1.1668 + 0.0734 i`
`0.0524 + 0.061 i`
`6`
`1.1668 - 0.0734 i`
`0.0524 - 0.061 i`
`7`
`1.3176`
`0.0492`
`8`
`-2.8203`
`7.66`
`9`
`13.8402`
`0.776 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`
`-0.8758 + 0.5019 i`
`0.0000355 + 0.00116 i` `2`
`-0.8758 - 0.5019 i`
`0.0000355 - 0.00116 i`
`3`
`-0.8904 + 0.4903 i`
`0.00115 - 6.61e-6 i`
`4`
`-0.8904 - 0.4903 i`
`0.00115 + 6.61e-6 i`
`5`
`1.159 + 0.0777 i`
`0.0452 + 0.0562 i`
`6`
`1.159 - 0.0777 i`
`0.0452 - 0.0562 i`
`7`
`1.2971`
`0.046`
`8`
`-2.857`
`18.`
`9`
`14.1672`
`0.729 i`
`10`
`-11742.8705`
`8.93 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`
`-0.8667 + 0.5797 i`
`0.000197 - 0.000399 i` `2`
`-0.8667 - 0.5797 i`
`0.000197 + 0.000399 i`
`3`
`1.0852 + 0.0563 i`
`0.0279 - 0.00101 i`
`4`
`1.0852 - 0.0563 i`
`0.0279 + 0.00101 i`
`5`
`-0.9101 + 0.5989 i`
`0.000397 + 0.000255 i`
`6`
`-0.9101 - 0.5989 i`
`0.000397 - 0.000255 i`
`7`
`-1.1149 + 0.0287 i`
`0.000214 + 0.000193 i`
`8`
`-1.1149 - 0.0287 i`
`0.000214 - 0.000193 i`
`9`
`1.1332`
`0.093`
`10`
`-2.2927`
`0.0246` 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`
`-0.8732 + 0.5813 i`
`0.000183 - 0.000408 i` `2`
`-0.8732 - 0.5813 i`
`0.000183 + 0.000408 i`
`3`
`1.0833 + 0.055 i`
`0.0272 - 0.00182 i`
`4`
`1.0833 - 0.055 i`
`0.0272 + 0.00182 i`
`5`
`-0.9207 + 0.6005 i`
`0.000403 + 0.000243 i`
`6`
`-0.9207 - 0.6005 i`
`0.000403 - 0.000243 i`
`7`
`1.1305`
`0.104`
`8`
`-1.171 + 0.0386 i`
`0.000256 + 0.00022 i`
`9`
`-1.171 - 0.0386 i`
`0.000256 - 0.00022 i`
`10`
`-2.249`
`0.0188`
`11`
`-291.0982`
`0.285 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`
`0.1417 + 0.e-5 i`
`1.46e-8 - 1.46e-8 i` `2`
`0.1417 - 0.e-5 i`
`1.46e-8 + 1.46e-8 i`
`3`
`0.9859 + 0.0188 i`
`0.00333 - 0.00266 i`
`4`
`0.9859 - 0.0188 i`
`0.00333 + 0.00266 i`
`5`
`-0.923 + 0.5399 i`
`0.0007 + 0.000253 i`
`6`
`-0.923 - 0.5399 i`
`0.0007 - 0.000253 i`
`7`
`-0.9501 + 0.5773 i`
`0.000282 - 0.000771 i`
`8`
`-0.9501 - 0.5773 i`
`0.000282 + 0.000771 i`
`9`
`1.1476`
`3.57`
`10`
`-2.0916`
`0.0167`
`11`
`-31.2422`
`0.429 i`
`12`
`59.2998`
`0.773 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`
`0.0664 + 0.4183 i`
`3.25e-7 + 3.72e-6 i` `2`
`0.0664 - 0.4183 i`
`3.25e-7 - 3.72e-6 i`
`3`
`0.0664 + 0.4183 i`
`3.72e-6 - 3.25e-7 i`
`4`
`0.0664 - 0.4183 i`
`3.72e-6 + 3.25e-7 i`
`5`
`-0.9008 + 0.5176 i`
`0.000391 + 0.0000883 i`
`6`
`-0.9008 - 0.5176 i`
`0.000391 - 0.0000883 i`
`7`
`-0.9424 + 0.5557 i`
`0.000106 - 0.000447 i`
`8`
`-0.9424 - 0.5557 i`
`0.000106 + 0.000447 i`
`9`
`1.1407 + 0.0956 i`
`0.00164 - 0.0647 i`
`10`
`1.1407 - 0.0956 i`
`0.00164 + 0.0647 i`
`11`
`1.2236`
`0.0361`
`12`
`-2.1716`
`0.0182` 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`
`0.1452 + 0.6517 i`
`0.0000641 + 0.0000687 i` `2`
`0.1452 - 0.6517 i`
`0.0000641 - 0.0000687 i`
`3`
`0.1454 + 0.6519 i`
`0.0000688 - 0.0000641 i`
`4`
`0.1454 - 0.6519 i`
`0.0000688 + 0.0000641 i`
`5`
`-0.8995 + 0.5073 i`
`0.000116 + 0.000516 i`
`6`
`-0.8995 - 0.5073 i`
`0.000116 - 0.000516 i`
`7`
`-0.923 + 0.4769 i`
`0.000515 - 0.0000778 i`
`8`
`-0.923 - 0.4769 i`
`0.000515 + 0.0000778 i`
`9`
`1.1778 + 0.0674 i`
`0.0612 + 0.0568 i`
`10`
`1.1778 - 0.0674 i`
`0.0612 - 0.0568 i`
`11`
`1.3748`
`0.0554`
`12`
`-3.2759`
`0.402`
`13`
`8.0068`
`4.22 i` 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`
`-0.5838`
`8.18e-7` `2`
`-0.5852`
`8.18e-7 i`
`3`
`0.0975 + 0.8434 i`
`0.0000249 - 2.71e-6 i`
`4`
`0.0975 - 0.8434 i`
`0.0000249 + 2.71e-6 i`
`5`
`0.0904 + 0.8486 i`
`2.e-6 + 0.0000248 i`
`6`
`0.0904 - 0.8486 i`
`2.e-6 - 0.0000248 i`
`7`
`-0.8465 + 0.3674 i`
`0.0000145 - 0.0000129 i`
`8`
`-0.8465 - 0.3674 i`
`0.0000145 + 0.0000129 i`
`9`
`-0.8651 + 0.5241 i`
`0.0000287 + 0.0000151 i`
`10`
`-0.8651 - 0.5241 i`
`0.0000287 - 0.0000151 i`
`11`
`1.2731`
`0.096`
`12`
`1.8211`
`0.0735 i`
`13`
`4.2877 + 3.7926 i`
`0.0674 + 0.114 i`
`14`
`4.2877 - 3.7926 i`
`0.0674 - 0.114 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.