Singularities of Møller-Plesset series: example "HF-cc-pVDZ-Re"

Molecule X 1^Sigma+ State of HF. Basis CC-PVDZ. 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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`
`-1.8622 + 0.2673 i`
`0.256 + 0.2 i`
`2`
`-1.8622 - 0.2673 i`
`0.256 - 0.2 i`
`3`
`2.4813`
`0.577`
`4`
`-5.8362`
`2.55`
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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`
`0.0497 + 0.e-4 i`
`9.32e-7 - 9.32e-7 i`
`2`
`0.0497 - 0.e-4 i`
`9.32e-7 + 9.32e-7 i`
`3`
`-1.4168`
`0.038`
`4`
`2.1869`
`0.115`
`5`
`4.1308`
`2.42 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`
`-1.6674`
`0.183`
`2`
`2.4018`
`0.574`
`3`
`-3.5917`
`0.207 i`
`4`
`-0.0381 + 4.7216 i`
`0.176 - 0.381 i`
`5`
`-0.0381 - 4.7216 i`
`0.176 + 0.381 i`
`6`
`71.6813`
`92.3 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`
`-1.5415`
`0.0786`
`2`
`2.4699`
`0.717`
`3`
`-1.4596 + 2.1472 i`
`0.0481 + 0.0783 i`
`4`
`-1.4596 - 2.1472 i`
`0.0481 - 0.0783 i`
`5`
`-0.8267 + 2.639 i`
`0.0796 - 0.0876 i`
`6`
`-0.8267 - 2.639 i`
`0.0796 + 0.0876 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`
`-1.5082`
`0.0623`
`2`
`2.545`
`1.19`
`3`
`-1.8594 + 1.7986 i`
`0.0448 + 0.0833 i`
`4`
`-1.8594 - 1.7986 i`
`0.0448 - 0.0833 i`
`5`
`-0.7929 + 3.4832 i`
`0.0818 - 0.146 i`
`6`
`-0.7929 - 3.4832 i`
`0.0818 + 0.146 i`
`7`
`-4.1492`
`0.952 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`
`-1.4023`
`0.0334`
`2`
`-1.7432`
`0.0467 i`
`3`
`2.647`
`2.8`
`4`
`-2.5132 + 0.9888 i`
`0.0358 + 0.0997 i`
`5`
`-2.5132 - 0.9888 i`
`0.0358 - 0.0997 i`
`6`
`0.0673 + 3.5199 i`
`0.0011 + 0.159 i`
`7`
`0.0673 - 3.5199 i`
`0.0011 - 0.159 i`
`8`
`48727.1801`
`50.6 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`
`-1.45`
`0.0427`
`2`
`-2.2837`
`0.098 i`
`3`
`-2.0327 + 1.3351 i`
`0.0301 + 0.0896 i`
`4`
`-2.0327 - 1.3351 i`
`0.0301 - 0.0896 i`
`5`
`2.7746`
`16.`
`6`
`0.3996 + 4.2325 i`
`0.0305 + 0.218 i`
`7`
`0.3996 - 4.2325 i`
`0.0305 - 0.218 i`
`8`
`6.1602`
`0.59 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`
`1.371`
`0.0561`
`2`
`1.3734`
`0.0567 i`
`3`
`-1.4328`
`0.0393`
`4`
`-1.951`
`0.0662 i`
`5`
`2.5753`
`1.49`
`6`
`-2.2977 + 1.2315 i`
`0.032 + 0.0953 i`
`7`
`-2.2977 - 1.2315 i`
`0.032 - 0.0953 i`
`8`
`-0.0899 + 3.6685 i`
`0.0168 - 0.176 i`
`9`
`-0.0899 - 3.6685 i`
`0.0168 + 0.176 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`
`0.303 + 1.2977 i`
`0.000884 - 0.0000599 i`
`2`
`0.303 - 1.2977 i`
`0.000884 + 0.0000599 i`
`3`
`0.3089 + 1.3019 i`
`0.0000624 + 0.000888 i`
`4`
`0.3089 - 1.3019 i`
`0.0000624 - 0.000888 i`
`5`
`-1.3755`
`0.0157`
`6`
`-2.1414`
`0.125 i`
`7`
`-1.4287 + 2.4223 i`
`0.014 - 0.0365 i`
`8`
`-1.4287 - 2.4223 i`
`0.014 + 0.0365 i`
`9`
`3.7964 + 1.0061 i`
`0.168 - 0.173 i`
`10`
`3.7964 - 1.0061 i`
`0.168 + 0.173 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`
`-1.1546 + 0.0029 i`
`0.000583 + 0.000589 i`
`2`
`-1.1546 - 0.0029 i`
`0.000583 - 0.000589 i`
`3`
`-1.3336`
`0.00551`
`4`
`2.0992 + 0.0735 i`
`0.0362 - 0.0288 i`
`5`
`2.0992 - 0.0735 i`
`0.0362 + 0.0288 i`
`6`
`-2.6957`
`5.88 i`
`7`
`-1.5533 + 2.7765 i`
`0.0101 + 0.127 i`
`8`
`-1.5533 - 2.7765 i`
`0.0101 - 0.127 i`
`9`
`3.1926`
`0.653`
`10`
`111.4405`
`0.976 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`
`-1.3706`
`0.017`
`2`
`-1.8698`
`0.0503 i`
`3`
`-1.5221 + 1.6219 i`
`0.0142 + 0.0154 i`
`4`
`-1.5221 - 1.6219 i`
`0.0142 - 0.0154 i`
`5`
`2.5777 + 0.714 i`
`0.215 + 0.0955 i`
`6`
`2.5777 - 0.714 i`
`0.215 - 0.0955 i`
`7`
`-2.303 + 1.6594 i`
`0.0317 - 0.0208 i`
`8`
`-2.303 - 1.6594 i`
`0.0317 + 0.0208 i`
`9`
`-0.2328 + 3.2604 i`
`0.0121 - 0.0582 i`
`10`
`-0.2328 - 3.2604 i`
`0.0121 + 0.0582 i`
`11`
`3.7736`
`4.8`
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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`
`-1.2369 + 0.0097 i`
`0.00113 + 0.00123 i`
`2`
`-1.2369 - 0.0097 i`
`0.00113 - 0.00123 i`
`3`
`-1.3298`
`0.00391`
`4`
`2.5099 + 0.3973 i`
`0.138 + 0.6 i`
`5`
`2.5099 - 0.3973 i`
`0.138 - 0.6 i`
`6`
`-2.6719`
`2.66 i`
`7`
`3.1153`
`0.516`
`8`
`-1.8069 + 2.6526 i`
`0.0821 - 0.125 i`
`9`
`-1.8069 - 2.6526 i`
`0.0821 + 0.125 i`
`10`
`0.9158 + 5.7184 i`
`0.456 - 0.106 i`
`11`
`0.9158 - 5.7184 i`
`0.456 + 0.106 i`
`12`
`14.0242`
`1.26 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`
`-1.3049 + 0.0369 i`
`0.000366 + 0.00267 i`
`2`
`-1.3049 - 0.0369 i`
`0.000366 - 0.00267 i`
`3`
`-1.3135`
`0.00194`
`4`
`2.4861 + 0.3113 i`
`0.526 + 0.325 i`
`5`
`2.4861 - 0.3113 i`
`0.526 - 0.325 i`
`6`
`-2.742`
`3.2 i`
`7`
`2.9698`
`0.546`
`8`
`-2.0836 + 2.6771 i`
`0.19 - 0.131 i`
`9`
`-2.0836 - 2.6771 i`
`0.19 + 0.131 i`
`10`
`-1.1489 + 5.2214 i`
`0.754 + 0.594 i`
`11`
`-1.1489 - 5.2214 i`
`0.754 - 0.594 i`
`12`
`-13.7663`
`3.62`
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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`
`0.4511`
`1.93e-6`
`2`
`0.4511`
`1.93e-6 i`
`3`
`-1.2724`
`0.00106`
`4`
`-1.288 + 0.0246 i`
`0.000352 - 0.00136 i`
`5`
`-1.288 - 0.0246 i`
`0.000352 + 0.00136 i`
`6`
`2.5235 + 0.3758 i`
`0.19 - 1.05 i`
`7`
`2.5235 - 0.3758 i`
`0.19 + 1.05 i`
`8`
`2.7833`
`0.494`
`9`
`-3.1586`
`28.6 i`
`10`
`-1.9982 + 3.2615 i`
`0.149 + 0.466 i`
`11`
`-1.9982 - 3.2615 i`
`0.149 - 0.466 i`
`12`
`-3.6285 + 4.9255 i`
`6.31 + 2.52 i`
`13`
`-3.6285 - 4.9255 i`
`6.31 - 2.52 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`
`-1.3019`
`0.0019`
`2`
`-1.3022 + 0.0439 i`
`0.000103 + 0.00268 i`
`3`
`-1.3022 - 0.0439 i`
`0.000103 - 0.00268 i`
`4`
`2.4505 + 0.314 i`
`0.252 + 0.0665 i`
`5`
`2.4505 - 0.314 i`
`0.252 - 0.0665 i`
`6`
`3.1119 + 0.856 i`
`0.297 + 0.0313 i`
`7`
`3.1119 - 0.856 i`
`0.297 - 0.0313 i`
`8`
`-1.6236 + 3.1011 i`
`0.0853 + 0.187 i`
`9`
`-1.6236 - 3.1011 i`
`0.0853 - 0.187 i`
`10`
`-3.8269 + 0.6813 i`
`2.14 - 2.09 i`
`11`
`-3.8269 - 0.6813 i`
`2.14 + 2.09 i`
`12`
`4.8758`
`0.513`
`13`
`-5.5228`
`238. i`
`14`
`13.1502`
`4.67 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.