Singularities of Møller-Plesset series: example "H--cc-pV5Z"

Molecule H- ion. Basis AUG-CC-PV5Z. Structure ""

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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports

[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`
`1.3368`
`0.0434`
`2`
`2.1034`
`0.0871 i`
`3`
`10.0647`
`0.0755`
`4`
`-10.7919`
`2.73`
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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`
`1.3605`
`0.0516`
`2`
`2.1407`
`0.0961 i`
`3`
`-6.5591`
`0.568`
`4`
`6.0711 + 5.0139 i`
`0.0648 - 0.0736 i`
`5`
`6.0711 - 5.0139 i`
`0.0648 + 0.0736 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`
`0.664`
`0.000618`
`2`
`0.6705`
`0.000618 i`
`3`
`1.1971`
`0.00917`
`4`
`3.0455`
`0.741 i`
`5`
`-5.4311`
`0.068`
`6`
`-15.1764`
`0.822 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`
`-0.0973 + 0.e-5 i`
`1.7e-7 + 1.7e-7 i`
`2`
`-0.0973 - 0.e-5 i`
`1.7e-7 - 1.7e-7 i`
`3`
`1.1673 + 0.3612 i`
`0.00506 - 0.00132 i`
`4`
`1.1673 - 0.3612 i`
`0.00506 + 0.00132 i`
`5`
`1.7743`
`0.0199`
`6`
`32.989`
`0.0303 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`
`1.2991`
`0.0106`
`2`
`1.3956 + 0.6519 i`
`0.00943 - 0.015 i`
`3`
`1.3956 - 0.6519 i`
`0.00943 + 0.015 i`
`4`
`1.774`
`0.0364 i`
`5`
`2.9478`
`0.467`
`6`
`-3.4791 + 6.0367 i`
`0.0645 - 0.0435 i`
`7`
`-3.4791 - 6.0367 i`
`0.0645 + 0.0435 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`
`1.4096 + 0.371 i`
`0.02 + 0.000945 i`
`2`
`1.4096 - 0.371 i`
`0.02 - 0.000945 i`
`3`
`1.5742 + 1.7988 i`
`0.0148 + 0.00581 i`
`4`
`1.5742 - 1.7988 i`
`0.0148 - 0.00581 i`
`5`
`1.3609 + 2.9873 i`
`0.0151 - 0.00846 i`
`6`
`1.3609 - 2.9873 i`
`0.0151 + 0.00846 i`
`7`
`-4.6373 + 0.5599 i`
`0.0119 + 0.00778 i`
`8`
`-4.6373 - 0.5599 i`
`0.0119 - 0.00778 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`
`1.4489 + 0.281 i`
`0.0272 - 0.0439 i`
`2`
`1.4489 - 0.281 i`
`0.0272 + 0.0439 i`
`3`
`1.9501 + 0.9484 i`
`0.0238 + 0.0476 i`
`4`
`1.9501 - 0.9484 i`
`0.0238 - 0.0476 i`
`5`
`1.7157 + 2.8947 i`
`0.0272 + 0.0161 i`
`6`
`1.7157 - 2.8947 i`
`0.0272 - 0.0161 i`
`7`
`-0.0112 + 6.6159 i`
`0.0281 - 0.0231 i`
`8`
`-0.0112 - 6.6159 i`
`0.0281 + 0.0231 i`
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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`
`0.1016`
`5.39e-9 - 5.39e-9 i`
`2`
`0.1016`
`5.39e-9 + 5.39e-9 i`
`3`
`1.4142 + 0.4346 i`
`0.0112 + 0.00838 i`
`4`
`1.4142 - 0.4346 i`
`0.0112 - 0.00838 i`
`5`
`1.5689 + 1.9677 i`
`0.00186 + 0.0144 i`
`6`
`1.5689 - 1.9677 i`
`0.00186 - 0.0144 i`
`7`
`1.6725 + 5.3755 i`
`0.0113 + 0.023 i`
`8`
`1.6725 - 5.3755 i`
`0.0113 - 0.023 i`
`9`
`10.329`
`0.0803`
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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`
`1.4438 + 0.2602 i`
`0.00993 - 0.0522 i`
`2`
`1.4438 - 0.2602 i`
`0.00993 + 0.0522 i`
`3`
`1.8665 + 0.9321 i`
`0.0235 + 0.03 i`
`4`
`1.8665 - 0.9321 i`
`0.0235 - 0.03 i`
`5`
`1.6335 + 2.7804 i`
`0.0193 + 0.00888 i`
`6`
`1.6335 - 2.7804 i`
`0.0193 - 0.00888 i`
`7`
`-0.6951 + 5.5948 i`
`0.018 - 0.0164 i`
`8`
`-0.6951 - 5.5948 i`
`0.018 + 0.0164 i`
`9`
`12.9401 + 14.9992 i`
`0.0578 - 0.00181 i`
`10`
`12.9401 - 14.9992 i`
`0.0578 + 0.00181 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`
`1.3793 + 0.3146 i`
`0.013 - 0.0126 i`
`2`
`1.3793 - 0.3146 i`
`0.013 + 0.0126 i`
`3`
`1.6896 + 0.2609 i`
`0.0154 + 0.0305 i`
`4`
`1.6896 - 0.2609 i`
`0.0154 - 0.0305 i`
`5`
`1.7107 + 0.9299 i`
`0.0142 + 0.0201 i`
`6`
`1.7107 - 0.9299 i`
`0.0142 - 0.0201 i`
`7`
`1.6481 + 2.9057 i`
`0.0233 + 0.0113 i`
`8`
`1.6481 - 2.9057 i`
`0.0233 - 0.0113 i`
`9`
`-0.0018 + 6.1414 i`
`0.0245 - 0.0212 i`
`10`
`-0.0018 - 6.1414 i`
`0.0245 + 0.0212 i`
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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`
`1.4399 + 0.2821 i`
`0.0224 - 0.039 i`
`2`
`1.4399 - 0.2821 i`
`0.0224 + 0.039 i`
`3`
`1.8801 + 1.0814 i`
`0.0035 + 0.0364 i`
`4`
`1.8801 - 1.0814 i`
`0.0035 - 0.0364 i`
`5`
`1.5257 + 3.2079 i`
`0.00498 + 0.0186 i`
`6`
`1.5257 - 3.2079 i`
`0.00498 - 0.0186 i`
`7`
`-2.5468 + 3.9706 i`
`0.00574 + 0.00595 i`
`8`
`-2.5468 - 3.9706 i`
`0.00574 - 0.00595 i`
`9`
`-3.5578 + 3.5523 i`
`0.0081 - 0.0043 i`
`10`
`-3.5578 - 3.5523 i`
`0.0081 + 0.0043 i`
`11`
`-12.8551`
`0.384`
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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`
`1.4396 + 0.3151 i`
`0.0258 - 0.00663 i`
`2`
`1.4396 - 0.3151 i`
`0.0258 + 0.00663 i`
`3`
`1.501`
`0.0391`
`4`
`1.6535`
`3.24 i`
`5`
`2.136 + 1.4428 i`
`0.0483 + 0.0314 i`
`6`
`2.136 - 1.4428 i`
`0.0483 - 0.0314 i`
`7`
`2.7204`
`0.311`
`8`
`-3.5237 + 0.035 i`
`0.00168 + 0.00165 i`
`9`
`-3.5237 - 0.035 i`
`0.00168 - 0.00165 i`
`10`
`0.6449 + 4.3066 i`
`0.0212 + 0.0126 i`
`11`
`0.6449 - 4.3066 i`
`0.0212 - 0.0126 i`
`12`
`8.0551`
`0.086 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`
`1.4559 + 0.3036 i`
`0.0435 - 0.00935 i`
`2`
`1.4559 - 0.3036 i`
`0.0435 + 0.00935 i`
`3`
`1.7119`
`0.0621`
`4`
`2.0313 + 0.42 i`
`0.0498 + 0.125 i`
`5`
`2.0313 - 0.42 i`
`0.0498 - 0.125 i`
`6`
`2.1325 + 1.5777 i`
`0.0267 + 0.0394 i`
`7`
`2.1325 - 1.5777 i`
`0.0267 - 0.0394 i`
`8`
`-3.8379 + 0.0543 i`
`0.00236 + 0.00227 i`
`9`
`-3.8379 - 0.0543 i`
`0.00236 - 0.00227 i`
`10`
`0.5927 + 4.2771 i`
`0.02 + 0.0123 i`
`11`
`0.5927 - 4.2771 i`
`0.02 - 0.0123 i`
`12`
`9.0224`
`0.0761 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`
`1.4554 + 0.2996 i`
`0.043 - 0.016 i`
`2`
`1.4554 - 0.2996 i`
`0.043 + 0.016 i`
`3`
`1.6764`
`0.0712`
`4`
`1.9388`
`0.199 i`
`5`
`2.239 + 1.3475 i`
`0.108 + 0.0637 i`
`6`
`2.239 - 1.3475 i`
`0.108 - 0.0637 i`
`7`
`3.0428`
`0.24`
`8`
`0.7335 + 4.5262 i`
`0.0323 + 0.0189 i`
`9`
`0.7335 - 4.5262 i`
`0.0323 - 0.0189 i`
`10`
`-5.1375 + 0.2493 i`
`0.00971 + 0.00837 i`
`11`
`-5.1375 - 0.2493 i`
`0.00971 - 0.00837 i`
`12`
`19.6945`
`0.142 i`
`13`
`77.4099`
`2.2`
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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`
`-0.5573`
`7.4e-7`
`2`
`-0.5573`
`7.4e-7 i`
`3`
`1.4462 + 0.3171 i`
`0.0297 - 0.00226 i`
`4`
`1.4462 - 0.3171 i`
`0.0297 + 0.00226 i`
`5`
`1.5741`
`0.0496`
`6`
`2.0159 + 0.1787 i`
`0.122 + 0.0914 i`
`7`
`2.0159 - 0.1787 i`
`0.122 - 0.0914 i`
`8`
`2.1287 + 1.563 i`
`0.0265 + 0.0357 i`
`9`
`2.1287 - 1.563 i`
`0.0265 - 0.0357 i`
`10`
`-3.8948 + 0.0535 i`
`0.0029 + 0.00282 i`
`11`
`-3.8948 - 0.0535 i`
`0.0029 - 0.00282 i`
`12`
`0.5632 + 4.3231 i`
`0.0185 + 0.0146 i`
`13`
`0.5632 - 4.3231 i`
`0.0185 - 0.0146 i`
`14`
`7.7334`
`0.0796 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.