Singularities of Møller-Plesset series: example "BH-cc-pVQZ-Re"

Molecule X 1^Sigma+ State of BH. Basis CC-PVQZ. Structure ""

Content

 Examples BH-cc-pVDZ-1.5Re BH-cc-pVDZ-2Re BH-cc-pVDZ-Re BH-cc-pVQZ-1.5Re BH-cc-pVQZ-2Re BH-cc-pVQZ-Re BH-cc-pVTZ-1.5Re BH-cc-pVTZ-2Re BH-cc-pVTZ-Re H--cc-pV5Z H--cc-pVQZ HF-cc-pVDZ-1.5Re HF-cc-pVDZ-2Re HF-cc-pVDZ-Re O2--aug-cc-pVDZ Molecule X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH H- ion H- ion X 1^Sigma+ State of HF X 1^Sigma+ State of HF X 1^Sigma+ State of HF X 1^Sigma+ State of O2- Basis CC-PVDZ CC-PVDZ CC-PVDZ CC-PVQZ CC-PVQZ CC-PVQZ CC-PVTZ CC-PVTZ CC-PVTZ AUG-CC-PV5Z AUG-CC-PVQZ CC-PVDZ CC-PVDZ CC-PVDZ AUG-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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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 [2, 2, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.5137`
`0.499`
`2`
`3.1207`
`0.616 i`
`3`
`4.4818`
`10.7`
`4`
`-5.443`
`0.36`
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Table 2. Singularities with their weights for the quadratic approximant [2, 2, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.3496`
`0.0979`
`2`
`1.3339 + 0.2174 i`
`0.00204 - 0.141 i`
`3`
`1.3339 - 0.2174 i`
`0.00204 + 0.141 i`
`4`
`-5.0169`
`0.335`
`5`
`-20.1982`
`0.327 i`
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Table 3. Singularities with their weights for the quadratic approximant [2, 3, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.5719 + 0.314 i`
`0.28 - 0.222 i`
`2`
`1.5719 - 0.314 i`
`0.28 + 0.222 i`
`3`
`2.1468`
`0.506`
`4`
`-4.2318`
`0.157`
`5`
`-17.9777`
`0.22 i`
`6`
`42.9622`
`0.475 i`
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Table 4. 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`
`1.599 + 0.3238 i`
`0.322 - 0.174 i`
`2`
`1.599 - 0.3238 i`
`0.322 + 0.174 i`
`3`
`2.3355`
`0.834`
`4`
`-4.2686`
`0.152`
`5`
`11.4329`
`0.49 i`
`6`
`-224.3151`
`0.111 i`
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Table 5. Singularities with their weights for the quadratic approximant [3, 3, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.6051 + 0.3334 i`
`0.32 - 0.13 i`
`2`
`1.6051 - 0.3334 i`
`0.32 + 0.13 i`
`3`
`2.4038`
`1.11`
`4`
`-4.1648`
`0.128`
`5`
`13.4634`
`0.45 i`
`6`
`-0.2111 + 17.9753 i`
`0.07 + 0.279 i`
`7`
`-0.2111 - 17.9753 i`
`0.07 - 0.279 i`
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Table 6. Singularities with their weights for the quadratic approximant [3, 4, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.6046 + 0.3392 i`
`0.315 - 0.0996 i`
`2`
`1.6046 - 0.3392 i`
`0.315 + 0.0996 i`
`3`
`2.365`
`1.07`
`4`
`-3.0738`
`0.0273`
`5`
`-3.3638`
`0.0286 i`
`6`
`-6.6791 + 2.0373 i`
`0.0594 - 0.224 i`
`7`
`-6.6791 - 2.0373 i`
`0.0594 + 0.224 i`
`8`
`58.4353`
`0.6 i`
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Table 7. 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`
`1.5999 + 0.3101 i`
`0.314 - 0.256 i`
`2`
`1.5999 - 0.3101 i`
`0.314 + 0.256 i`
`3`
`-1.8672 + 0.0011 i`
`0.00341 + 0.00342 i`
`4`
`-1.8672 - 0.0011 i`
`0.00341 - 0.00342 i`
`5`
`2.3937`
`0.811`
`6`
`-4.0805`
`0.0956`
`7`
`5.9291`
`0.818 i`
`8`
`13.8429`
`8.52`
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Table 8. Singularities with their weights for the quadratic approximant [4, 4, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`-0.2473 + 0.e-5 i`
`2.7e-6 + 2.7e-6 i`
`2`
`-0.2473 - 0.e-5 i`
`2.7e-6 - 2.7e-6 i`
`3`
`1.5973 + 0.3062 i`
`0.285 - 0.279 i`
`4`
`1.5973 - 0.3062 i`
`0.285 + 0.279 i`
`5`
`2.4282`
`0.883`
`6`
`-4.0874`
`0.105`
`7`
`6.5316`
`0.578 i`
`8`
`12.2942 + 16.0297 i`
`0.137 + 0.446 i`
`9`
`12.2942 - 16.0297 i`
`0.137 - 0.446 i`
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Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.2258 + 0.0048 i`
`0.0213 - 0.0215 i`
`2`
`1.2258 - 0.0048 i`
`0.0213 + 0.0215 i`
`3`
`1.6422 + 0.3741 i`
`0.322 + 0.12 i`
`4`
`1.6422 - 0.3741 i`
`0.322 - 0.12 i`
`5`
`2.3724`
`1.51`
`6`
`-3.1434`
`0.043`
`7`
`-3.3221`
`0.0445 i`
`8`
`-5.4841`
`27.9`
`9`
`-10.1229`
`0.219 i`
`10`
`75.7234`
`0.695 i`
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Table 10. 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`
`1.2942 + 0.0094 i`
`0.0303 - 0.0307 i`
`2`
`1.2942 - 0.0094 i`
`0.0303 + 0.0307 i`
`3`
`1.6568 + 0.3847 i`
`0.293 + 0.182 i`
`4`
`1.6568 - 0.3847 i`
`0.293 - 0.182 i`
`5`
`2.3818`
`1.85`
`6`
`-3.5478`
`0.0492`
`7`
`-4.7808`
`0.0541 i`
`8`
`-4.9641 + 3.2827 i`
`0.0704 - 0.111 i`
`9`
`-4.9641 - 3.2827 i`
`0.0704 + 0.111 i`
`10`
`-22.0044`
`4.01`
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Table 11. Singularities with their weights for the quadratic approximant [5, 5, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.3932 + 0.0365 i`
`0.0338 - 0.0298 i`
`2`
`1.3932 - 0.0365 i`
`0.0338 + 0.0298 i`
`3`
`1.7338 + 0.4579 i`
`0.0817 - 0.28 i`
`4`
`1.7338 - 0.4579 i`
`0.0817 + 0.28 i`
`5`
`2.0833`
`1.31`
`6`
`-3.0502 + 0.0722 i`
`0.0161 + 0.0163 i`
`7`
`-3.0502 - 0.0722 i`
`0.0161 - 0.0163 i`
`8`
`-4.371`
`0.0908`
`9`
`4.9312`
`86.2 i`
`10`
`7.6856`
`0.644`
`11`
`-11.1518`
`0.499 i`
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Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.4418 + 0.0661 i`
`0.0502 - 0.038 i`
`2`
`1.4418 - 0.0661 i`
`0.0502 + 0.038 i`
`3`
`1.6875 + 0.5343 i`
`0.0302 - 0.147 i`
`4`
`1.6875 - 0.5343 i`
`0.0302 + 0.147 i`
`5`
`2.0476 + 0.2382 i`
`0.64 - 0.00592 i`
`6`
`2.0476 - 0.2382 i`
`0.64 + 0.00592 i`
`7`
`3.0082`
`582.`
`8`
`-3.2416`
`0.174`
`9`
`-3.3216`
`0.172 i`
`10`
`-5.1783`
`0.986`
`11`
`-10.0571`
`0.284 i`
`12`
`133.3694`
`1.05 i`
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Table 13. 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`
`1.5739 + 0.1599 i`
`1.42 - 3.9 i`
`2`
`1.5739 - 0.1599 i`
`1.42 + 3.9 i`
`3`
`1.577 + 0.436 i`
`0.159 - 0.0536 i`
`4`
`1.577 - 0.436 i`
`0.159 + 0.0536 i`
`5`
`1.6681 + 0.1793 i`
`0.0138 - 0.205 i`
`6`
`1.6681 - 0.1793 i`
`0.0138 + 0.205 i`
`7`
`1.9698`
`0.184`
`8`
`-3.1442`
`0.0333`
`9`
`-3.3514`
`0.0359 i`
`10`
`-5.2287`
`5.81`
`11`
`17.7283`
`0.469 i`
`12`
`-17.7717`
`0.176 i`
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Table 14. Singularities with their weights for the quadratic approximant [6, 6, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.466`
`0.0527`
`2`
`1.4752 + 0.3131 i`
`0.0292 + 0.00507 i`
`3`
`1.4752 - 0.3131 i`
`0.0292 - 0.00507 i`
`4`
`1.4705 + 0.4241 i`
`0.014 - 0.0237 i`
`5`
`1.4705 - 0.4241 i`
`0.014 + 0.0237 i`
`6`
`1.9415 + 0.5142 i`
`0.0807 + 0.124 i`
`7`
`1.9415 - 0.5142 i`
`0.0807 - 0.124 i`
`8`
`-3.1118`
`0.0144`
`9`
`-3.5455`
`0.0171 i`
`10`
`-7.5919`
`0.108`
`11`
`-4.4657 + 9.1025 i`
`0.00285 + 0.078 i`
`12`
`-4.4657 - 9.1025 i`
`0.00285 - 0.078 i`
`13`
`-32.3506`
`0.2 i`
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Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
`1`
`1.3979 + 0.195 i`
`0.0564 + 0.0726 i`
`2`
`1.3979 - 0.195 i`
`0.0564 - 0.0726 i`
`3`
`1.4384 + 0.2008 i`
`0.0676 - 0.0508 i`
`4`
`1.4384 - 0.2008 i`
`0.0676 + 0.0508 i`
`5`
`1.6765 + 0.4204 i`
`0.108 + 0.38 i`
`6`
`1.6765 - 0.4204 i`
`0.108 - 0.38 i`
`7`
`2.0722`
`0.648`
`8`
`-3.215`
`0.25`
`9`
`-3.2693`
`0.263 i`
`10`
`4.3451`
`11.5 i`
`11`
`-4.9945`
`0.545`
`12`
`5.3458`
`0.827`
`13`
`-10.7331`
`0.296 i`
`14`
`224.8864`
`1.54 i`
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 Examples BH-cc-pVDZ-1.5Re BH-cc-pVDZ-2Re BH-cc-pVDZ-Re BH-cc-pVQZ-1.5Re BH-cc-pVQZ-2Re BH-cc-pVQZ-Re BH-cc-pVTZ-1.5Re BH-cc-pVTZ-2Re BH-cc-pVTZ-Re H--cc-pV5Z H--cc-pVQZ HF-cc-pVDZ-1.5Re HF-cc-pVDZ-2Re HF-cc-pVDZ-Re O2--aug-cc-pVDZ Molecule X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH X 1^Sigma+ State of BH H- ion H- ion X 1^Sigma+ State of HF X 1^Sigma+ State of HF X 1^Sigma+ State of HF X 1^Sigma+ State of O2- Basis CC-PVDZ CC-PVDZ CC-PVDZ CC-PVQZ CC-PVQZ CC-PVQZ CC-PVTZ CC-PVTZ CC-PVTZ AUG-CC-PV5Z AUG-CC-PVQZ CC-PVDZ CC-PVDZ CC-PVDZ AUG-CC-PVDZ

 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports

Designed by A. Sergeev.