# Singularities of Møller-Plesset series: example "BH-cc-pVDZ-2Re"

## Molecule X 1^Sigma+ State of BH. Basis CC-PVDZ. 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 Plot of singularities List of examples Mathematica programs Work in UMassD Unpublished reports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`
`1.3157 + 0.3983 i`
`0.186 - 0.987 i`
`2`
`1.3157 - 0.3983 i`
`0.186 + 0.987 i`
`3`
`1.6397`
`0.439`
`4`
`-3.8802`
`0.168`
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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`
`1.2765 + 0.5532 i`
`0.319 - 0.128 i`
`2`
`1.2765 - 0.5532 i`
`0.319 + 0.128 i`
`3`
`1.7375`
`0.394`
`4`
`-3.9366`
`0.14`
`5`
`13.0582`
`0.221 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`
`1.2937 + 0.3671 i`
`0.4 + 0.343 i`
`2`
`1.2937 - 0.3671 i`
`0.4 - 0.343 i`
`3`
`2.4176`
`0.36`
`4`
`-5.4638 + 3.5195 i`
`0.346 + 0.143 i`
`5`
`-5.4638 - 3.5195 i`
`0.346 - 0.143 i`
`6`
`8.8182`
`3.39 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`
`1.3175 + 0.454 i`
`0.365 - 0.645 i`
`2`
`1.3175 - 0.454 i`
`0.365 + 0.645 i`
`3`
`1.758`
`0.419`
`4`
`-2.8057`
`0.0903`
`5`
`-2.9621 + 0.7466 i`
`0.0533 - 0.0852 i`
`6`
`-2.9621 - 0.7466 i`
`0.0533 + 0.0852 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`
`-0.3162`
`0.0000124`
`2`
`-0.3162`
`0.0000124 i`
`3`
`1.0558`
`0.0185`
`4`
`1.0266 + 0.7744 i`
`0.00969 + 0.0138 i`
`5`
`1.0266 - 0.7744 i`
`0.00969 - 0.0138 i`
`6`
`-1.4942 + 2.0103 i`
`0.00808 - 0.00618 i`
`7`
`-1.4942 - 2.0103 i`
`0.00808 + 0.00618 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`
`1.1646 + 0.3738 i`
`0.106 + 0.0418 i`
`2`
`1.1646 - 0.3738 i`
`0.106 - 0.0418 i`
`3`
`1.7909 + 0.8274 i`
`0.127 + 0.0315 i`
`4`
`1.7909 - 0.8274 i`
`0.127 - 0.0315 i`
`5`
`2.8199`
`0.189`
`6`
`-2.789 + 2.9366 i`
`0.0755 + 0.0155 i`
`7`
`-2.789 - 2.9366 i`
`0.0755 - 0.0155 i`
`8`
`10.7985`
`8.44 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`
`1.2502`
`0.0432`
`2`
`1.1618 + 0.4721 i`
`0.0508 - 0.0941 i`
`3`
`1.1618 - 0.4721 i`
`0.0508 + 0.0941 i`
`4`
`1.3085`
`0.0532 i`
`5`
`3.565`
`1.1`
`6`
`-3.5389 + 2.2055 i`
`0.00178 + 0.18 i`
`7`
`-3.5389 - 2.2055 i`
`0.00178 - 0.18 i`
`8`
`-27.6148`
`0.0746`
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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`
`0.4266`
`0.0000425`
`2`
`0.4266`
`0.0000425 i`
`3`
`1.1393 + 0.4274 i`
`0.0183 + 0.0568 i`
`4`
`1.1393 - 0.4274 i`
`0.0183 - 0.0568 i`
`5`
`-1.8702 + 0.0398 i`
`0.00155 + 0.00149 i`
`6`
`-1.8702 - 0.0398 i`
`0.00155 - 0.00149 i`
`7`
`2.3973`
`0.744`
`8`
`-1.5261 + 3.5883 i`
`0.00587 + 0.0379 i`
`9`
`-1.5261 - 3.5883 i`
`0.00587 - 0.0379 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`
`1.1612 + 0.471 i`
`0.0433 - 0.105 i`
`2`
`1.1612 - 0.471 i`
`0.0433 + 0.105 i`
`3`
`1.2896`
`0.078`
`4`
`1.4267`
`0.132 i`
`5`
`3.0969 + 2.3187 i`
`0.0864 - 0.236 i`
`6`
`3.0969 - 2.3187 i`
`0.0864 + 0.236 i`
`7`
`-2.9872 + 2.4898 i`
`0.0641 + 0.0573 i`
`8`
`-2.9872 - 2.4898 i`
`0.0641 - 0.0573 i`
`9`
`5.7258 + 18.1768 i`
`0.205 - 0.231 i`
`10`
`5.7258 - 18.1768 i`
`0.205 + 0.231 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`
`1.1899 + 0.45 i`
`0.0843 + 0.175 i`
`2`
`1.1899 - 0.45 i`
`0.0843 - 0.175 i`
`3`
`1.3683 + 0.6412 i`
`0.179 + 0.00106 i`
`4`
`1.3683 - 0.6412 i`
`0.179 - 0.00106 i`
`5`
`1.6392 + 0.6696 i`
`0.303 - 0.379 i`
`6`
`1.6392 - 0.6696 i`
`0.303 + 0.379 i`
`7`
`1.9415`
`0.435`
`8`
`-3.2089 + 2.5267 i`
`0.092 + 0.083 i`
`9`
`-3.2089 - 2.5267 i`
`0.092 - 0.083 i`
`10`
`-21.7813`
`0.11`
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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`
`1.1912 + 0.4349 i`
`0.123 + 0.104 i`
`2`
`1.1912 - 0.4349 i`
`0.123 - 0.104 i`
`3`
`1.2773 + 0.6293 i`
`0.137 - 0.0855 i`
`4`
`1.2773 - 0.6293 i`
`0.137 + 0.0855 i`
`5`
`1.4485 + 0.5817 i`
`0.399 + 0.219 i`
`6`
`1.4485 - 0.5817 i`
`0.399 - 0.219 i`
`7`
`2.3129`
`0.414`
`8`
`-3.1675 + 2.5205 i`
`0.0876 + 0.0749 i`
`9`
`-3.1675 - 2.5205 i`
`0.0876 - 0.0749 i`
`10`
`-18.326`
`0.134`
`11`
`33.6163`
`0.257 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`
`1.2051 + 0.4501 i`
`0.157 + 0.219 i`
`2`
`1.2051 - 0.4501 i`
`0.157 - 0.219 i`
`3`
`1.3119 + 0.5831 i`
`0.289 - 0.11 i`
`4`
`1.3119 - 0.5831 i`
`0.289 + 0.11 i`
`5`
`1.4927 + 0.5375 i`
`1.44 + 0.248 i`
`6`
`1.4927 - 0.5375 i`
`1.44 - 0.248 i`
`7`
`2.3859`
`0.431`
`8`
`-3.1993 + 2.5018 i`
`0.0841 + 0.0883 i`
`9`
`-3.1993 - 2.5018 i`
`0.0841 - 0.0883 i`
`10`
`-10.8385 + 25.0216 i`
`0.162 + 0.0444 i`
`11`
`-10.8385 - 25.0216 i`
`0.162 - 0.0444 i`
`12`
`30.2653`
`0.227 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`
`-0.779`
`0.0000749`
`2`
`-0.779`
`0.0000749 i`
`3`
`1.2021 + 0.4834 i`
`0.172 - 0.261 i`
`4`
`1.2021 - 0.4834 i`
`0.172 + 0.261 i`
`5`
`1.4136 + 0.4964 i`
`0.359 + 0.0796 i`
`6`
`1.4136 - 0.4964 i`
`0.359 - 0.0796 i`
`7`
`1.6744 + 0.5068 i`
`5.21 + 0.507 i`
`8`
`1.6744 - 0.5068 i`
`5.21 - 0.507 i`
`9`
`2.0649`
`0.39`
`10`
`-3.1992 + 2.5779 i`
`0.11 + 0.0687 i`
`11`
`-3.1992 - 2.5779 i`
`0.11 - 0.0687 i`
`12`
`-16.6362`
`0.111`
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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`
`1.1889 + 0.49 i`
`0.138 - 0.142 i`
`2`
`1.1889 - 0.49 i`
`0.138 + 0.142 i`
`3`
`1.5434 + 0.4145 i`
`0.326 + 0.00509 i`
`4`
`1.5434 - 0.4145 i`
`0.326 - 0.00509 i`
`5`
`1.8077`
`0.311`
`6`
`1.8672 + 0.5401 i`
`1.29 + 0.743 i`
`7`
`1.8672 - 0.5401 i`
`1.29 - 0.743 i`
`8`
`-3.9213`
`0.0751`
`9`
`-3.8712 + 2.521 i`
`0.892 + 3.14 i`
`10`
`-3.8712 - 2.521 i`
`0.892 - 3.14 i`
`11`
`-5.3726`
`0.0497 i`
`12`
`-8.7483 + 1.6189 i`
`0.039 + 0.0288 i`
`13`
`-8.7483 - 1.6189 i`
`0.039 - 0.0288 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`
`1.192 + 0.4738 i`
`0.000852 - 0.211 i`
`2`
`1.192 - 0.4738 i`
`0.000852 + 0.211 i`
`3`
`1.4221 + 0.4209 i`
`1.23 - 1.23 i`
`4`
`1.4221 - 0.4209 i`
`1.23 + 1.23 i`
`5`
`1.4011 + 0.5357 i`
`0.172 + 0.221 i`
`6`
`1.4011 - 0.5357 i`
`0.172 - 0.221 i`
`7`
`-1.7632 + 0.0005 i`
`0.000613 + 0.000613 i`
`8`
`-1.7632 - 0.0005 i`
`0.000613 - 0.000613 i`
`9`
`3.1078 + 1.6791 i`
`0.113 - 0.232 i`
`10`
`3.1078 - 1.6791 i`
`0.113 + 0.232 i`
`11`
`-3.1646 + 2.2506 i`
`0.00835 + 0.069 i`
`12`
`-3.1646 - 2.2506 i`
`0.00835 - 0.069 i`
`13`
`3.4939 + 6.438 i`
`0.0472 - 0.144 i`
`14`
`3.4939 - 6.438 i`
`0.0472 + 0.144 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.