# Singularities of Møller-Plesset series: example "o8"

## Molecule C2. 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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 [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-2.3972`
`1.49`
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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`
`-0.326`
`0.0592`
`2`
`-0.818`
`0.0938 i`
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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`
`-0.7969`
`0.499`
`2`
`1.6932`
`2.98`
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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`
`-0.9611`
`1.7`
`2`
`1.4562`
`0.85`
`3`
`-2.9074`
`0.832 i`
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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`
`1.2435`
`0.446`
`2`
`-1.2758 + 0.3149 i`
`1.54 - 0.765 i`
`3`
`-1.2758 - 0.3149 i`
`1.54 + 0.765 i`
`4`
`129.4701`
`4.83 i`
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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`
`-1.2018 + 0.3605 i`
`1.45 - 0.2 i`
`2`
`-1.2018 - 0.3605 i`
`1.45 + 0.2 i`
`3`
`1.2731`
`0.493`
`4`
`-16.7027`
`312.`
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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`
`-0.8005`
`0.141`
`2`
`-0.8487 + 0.2074 i`
`0.0791 + 0.201 i`
`3`
`-0.8487 - 0.2074 i`
`0.0791 - 0.201 i`
`4`
`1.2043`
`0.344`
`5`
`-1.432`
`3.14e3 i`
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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`
`-0.8607 + 0.3816 i`
`0.205 + 0.0876 i`
`2`
`-0.8607 - 0.3816 i`
`0.205 - 0.0876 i`
`3`
`1.1833`
`0.304`
`4`
`-1.2658 + 0.3012 i`
`0.31 - 0.477 i`
`5`
`-1.2658 - 0.3012 i`
`0.31 + 0.477 i`
`6`
`309.2916`
`14.6 i`
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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`
`0.0817`
`0.0000147`
`2`
`0.0817`
`0.0000147 i`
`3`
`-0.8143 + 0.1002 i`
`0.106 + 0.0428 i`
`4`
`-0.8143 - 0.1002 i`
`0.106 - 0.0428 i`
`5`
`1.1099`
`0.154`
`6`
`-3.1921`
`0.473`
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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`
`-0.8959 + 0.1394 i`
`0.237 + 0.0481 i`
`2`
`-0.8959 - 0.1394 i`
`0.237 - 0.0481 i`
`3`
`-0.1202 + 1.1617 i`
`0.0283 + 0.0689 i`
`4`
`-0.1202 - 1.1617 i`
`0.0283 - 0.0689 i`
`5`
`-0.282 + 1.2106 i`
`0.074 - 0.0217 i`
`6`
`-0.282 - 1.2106 i`
`0.074 + 0.0217 i`
`7`
`1.2765`
`0.746`
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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`
`-0.9834 + 0.3334 i`
`0.528 + 0.765 i`
`2`
`-0.9834 - 0.3334 i`
`0.528 - 0.765 i`
`3`
`1.2029 + 0.325 i`
`0.167 - 0.243 i`
`4`
`1.2029 - 0.325 i`
`0.167 + 0.243 i`
`5`
`1.8051`
`0.332`
`6`
`-1.9994`
`1.47`
`7`
`-3.2313`
`10.5 i`
`8`
`13.781`
`11.5 i`
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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`
`-0.9378 + 0.2816 i`
`0.285 - 0.302 i`
`2`
`-0.9378 - 0.2816 i`
`0.285 + 0.302 i`
`3`
`1.2257 + 0.3889 i`
`0.214 - 0.0909 i`
`4`
`1.2257 - 0.3889 i`
`0.214 + 0.0909 i`
`5`
`-1.2299 + 0.6983 i`
`0.266 - 0.0437 i`
`6`
`-1.2299 - 0.6983 i`
`0.266 + 0.0437 i`
`7`
`-1.661`
`0.816`
`8`
`2.638`
`1.81`
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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`
`-0.9286 + 0.2674 i`
`0.287 - 0.182 i`
`2`
`-0.9286 - 0.2674 i`
`0.287 + 0.182 i`
`3`
`1.2251 + 0.3912 i`
`0.213 - 0.0843 i`
`4`
`1.2251 - 0.3912 i`
`0.213 + 0.0843 i`
`5`
`-1.1815 + 0.6698 i`
`0.254 + 0.0104 i`
`6`
`-1.1815 - 0.6698 i`
`0.254 - 0.0104 i`
`7`
`-1.4412`
`2.58`
`8`
`2.6362`
`1.88`
`9`
`-73.4278`
`0.596 i`
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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`
`-0.9744 + 0.3447 i`
`0.538 + 0.588 i`
`2`
`-0.9744 - 0.3447 i`
`0.538 - 0.588 i`
`3`
`1.233 + 0.3392 i`
`0.208 - 0.258 i`
`4`
`1.233 - 0.3392 i`
`0.208 + 0.258 i`
`5`
`-1.8034`
`1.93`
`6`
`2.2204`
`0.497`
`7`
`-2.2484`
`65.7 i`
`8`
`5.5577`
`125. i`
`9`
`-6.858 + 2.9665 i`
`1.91 + 2.66 i`
`10`
`-6.858 - 2.9665 i`
`1.91 - 2.66 i`
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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`
`-0.9743 + 0.3495 i`
`0.571 + 0.505 i`
`2`
`-0.9743 - 0.3495 i`
`0.571 - 0.505 i`
`3`
`1.2301 + 0.346 i`
`0.208 - 0.228 i`
`4`
`1.2301 - 0.346 i`
`0.208 + 0.228 i`
`5`
`-1.5804`
`1.3`
`6`
`2.2875`
`0.6`
`7`
`-3.5535`
`0.48 i`
`8`
`-2.3429 + 3.0773 i`
`0.244 + 0.371 i`
`9`
`-2.3429 - 3.0773 i`
`0.244 - 0.371 i`
`10`
`91.1886`
`0.183 i`
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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`
`-0.9743 + 0.3469 i`
`0.546 + 0.548 i`
`2`
`-0.9743 - 0.3469 i`
`0.546 - 0.548 i`
`3`
`1.2268 + 0.3521 i`
`0.209 - 0.2 i`
`4`
`1.2268 - 0.3521 i`
`0.209 + 0.2 i`
`5`
`-1.6616`
`1.4`
`6`
`2.2617`
`0.662`
`7`
`-3.307`
`0.976 i`
`8`
`-0.8513 + 3.4551 i`
`0.226 + 0.345 i`
`9`
`-0.8513 - 3.4551 i`
`0.226 - 0.345 i`
`10`
`0.7268 + 4.2062 i`
`0.414 - 0.297 i`
`11`
`0.7268 - 4.2062 i`
`0.414 + 0.297 i`
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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`
`0.8928`
`0.119`
`2`
`0.8935`
`0.117 i`
`3`
`-0.9751 + 0.348 i`
`0.58 + 0.538 i`
`4`
`-0.9751 - 0.348 i`
`0.58 - 0.538 i`
`5`
`1.2319 + 0.3374 i`
`0.214 - 0.266 i`
`6`
`1.2319 - 0.3374 i`
`0.214 + 0.266 i`
`7`
`-1.6619`
`1.47`
`8`
`2.1365`
`0.452`
`9`
`-2.4431`
`4.21 i`
`10`
`5.6796`
`78.4 i`
`11`
`-5.4737 + 3.1874 i`
`1.44 + 1.23 i`
`12`
`-5.4737 - 3.1874 i`
`1.44 - 1.23 i`
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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`
`0.5168`
`0.000729`
`2`
`0.5169`
`0.000729 i`
`3`
`-0.9792 + 0.3481 i`
`0.7 + 0.563 i`
`4`
`-0.9792 - 0.3481 i`
`0.7 - 0.563 i`
`5`
`1.2373 + 0.3574 i`
`0.295 - 0.153 i`
`6`
`1.2373 - 0.3574 i`
`0.295 + 0.153 i`
`7`
`-1.6058 + 0.2918 i`
`0.953 - 0.652 i`
`8`
`-1.6058 - 0.2918 i`
`0.953 + 0.652 i`
`9`
`1.8512`
`0.391`
`10`
`-2.2159`
`0.743`
`11`
`3.0572`
`1.08 i`
`12`
`11.9772`
`0.687`
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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`
`1.0203 + 0.1757 i`
`0.0109 - 0.0131 i`
`2`
`1.0203 - 0.1757 i`
`0.0109 + 0.0131 i`
`3`
`-0.9845 + 0.3622 i`
`0.812 + 0.125 i`
`4`
`-0.9845 - 0.3622 i`
`0.812 - 0.125 i`
`5`
`1.0771 + 0.1805 i`
`0.0155 + 0.0126 i`
`6`
`1.0771 - 0.1805 i`
`0.0155 - 0.0126 i`
`7`
`-1.1789 + 0.0495 i`
`1.38 - 0.64 i`
`8`
`-1.1789 - 0.0495 i`
`1.38 + 0.64 i`
`9`
`1.191 + 0.4241 i`
`0.0707 + 0.0196 i`
`10`
`1.191 - 0.4241 i`
`0.0707 - 0.0196 i`
`11`
`-1.4141`
`0.797`
`12`
`4.3113`
`8.65`
`13`
`-18.5382`
`0.5 i`
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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`
`0.5365 + 0.3368 i`
`0.000309 - 0.000699 i`
`2`
`0.5365 - 0.3368 i`
`0.000309 + 0.000699 i`
`3`
`0.5366 + 0.3368 i`
`0.000699 + 0.000309 i`
`4`
`0.5366 - 0.3368 i`
`0.000699 - 0.000309 i`
`5`
`-0.9854 + 0.35 i`
`0.949 + 0.521 i`
`6`
`-0.9854 - 0.35 i`
`0.949 - 0.521 i`
`7`
`1.2134 + 0.3419 i`
`0.105 - 0.183 i`
`8`
`1.2134 - 0.3419 i`
`0.105 + 0.183 i`
`9`
`-1.4532 + 0.2821 i`
`0.789 - 0.411 i`
`10`
`-1.4532 - 0.2821 i`
`0.789 + 0.411 i`
`11`
`-1.825`
`0.74`
`12`
`2.6175`
`1.23`
`13`
`-15.2158`
`1.19 i`
`14`
`16.2744`
`2.12 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.