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

## Molecule X 1^Sigma+ State of BH. Basis CC-PVTZ. 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 Plot of singularities  List of examples Mathematica programs Work in UMassD  Unpublished reports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`
`1.2929 + 0.6041 i`
`0.266 - 0.0418 i` `2`
`1.2929 - 0.6041 i`
`0.266 + 0.0418 i`
`3`
`2.1451`
`0.618`
`4`
`-2.2454`
`0.107` Top of the page

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`
`1.3103 + 0.5121 i`
`0.371 - 0.222 i` `2`
`1.3103 - 0.5121 i`
`0.371 + 0.222 i`
`3`
`1.8646`
`0.411`
`4`
`-2.3075`
`0.134`
`5`
`-16.7243`
`0.186 i` Top of the page

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`
`1.3099 + 0.3338 i`
`0.522 + 0.488 i` `2`
`1.3099 - 0.3338 i`
`0.522 - 0.488 i`
`3`
`1.9076`
`0.385`
`4`
`-2.6544`
`0.394`
`5`
`-5.6942`
`0.288 i`
`6`
`254.3534`
`1.68 i` Top of the page

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`
`1.3072 + 0.3911 i`
`0.124 + 0.632 i` `2`
`1.3072 - 0.3911 i`
`0.124 - 0.632 i`
`3`
`1.8911`
`0.39`
`4`
`-2.3157 + 0.8012 i`
`0.0544 + 0.114 i`
`5`
`-2.3157 - 0.8012 i`
`0.0544 - 0.114 i`
`6`
`-3.9396`
`0.102` Top of the page

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`
`1.1937 + 0.3221 i`
`0.127 - 0.0041 i` `2`
`1.1937 - 0.3221 i`
`0.127 + 0.0041 i`
`3`
`1.6546`
`3.49`
`4`
`1.5325 + 0.6922 i`
`0.141 - 0.11 i`
`5`
`1.5325 - 0.6922 i`
`0.141 + 0.11 i`
`6`
`-3.1007 + 0.8417 i`
`0.323 + 0.00822 i`
`7`
`-3.1007 - 0.8417 i`
`0.323 - 0.00822 i` Top of the page

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`
`0.7554`
`0.00333` `2`
`0.7625`
`0.00335 i`
`3`
`1.2365`
`0.076`
`4`
`1.3566 + 0.6143 i`
`0.0551 + 0.163 i`
`5`
`1.3566 - 0.6143 i`
`0.0551 - 0.163 i`
`6`
`-3.0696 + 0.9471 i`
`0.244 - 0.0412 i`
`7`
`-3.0696 - 0.9471 i`
`0.244 + 0.0412 i`
`8`
`941.565`
`3.97 i` Top of the page

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`
`1.2101 + 0.4277 i`
`0.0565 - 0.208 i` `2`
`1.2101 - 0.4277 i`
`0.0565 + 0.208 i`
`3`
`1.4088 + 0.1202 i`
`0.201 - 0.0658 i`
`4`
`1.4088 - 0.1202 i`
`0.201 + 0.0658 i`
`5`
`2.3508`
`0.435`
`6`
`-2.6865 + 0.7713 i`
`0.111 + 0.231 i`
`7`
`-2.6865 - 0.7713 i`
`0.111 - 0.231 i`
`8`
`-6.489`
`0.189` Top of the page

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`
`1.1872 + 0.4546 i`
`0.0685 - 0.118 i` `2`
`1.1872 - 0.4546 i`
`0.0685 + 0.118 i`
`3`
`1.3929`
`0.0658`
`4`
`1.4384`
`0.0802 i`
`5`
`-2.7493 + 0.815 i`
`0.15 + 0.202 i`
`6`
`-2.7493 - 0.815 i`
`0.15 - 0.202 i`
`7`
`4.2244 + 0.9464 i`
`0.371 + 1.82 i`
`8`
`4.2244 - 0.9464 i`
`0.371 - 1.82 i`
`9`
`-23.8422`
`0.594` Top of the page

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`
`1.1716 + 0.4577 i`
`0.0543 - 0.0858 i` `2`
`1.1716 - 0.4577 i`
`0.0543 + 0.0858 i`
`3`
`1.2712`
`0.056`
`4`
`1.3689`
`0.0832 i`
`5`
`-2.7287 + 0.7391 i`
`0.0621 + 0.268 i`
`6`
`-2.7287 - 0.7391 i`
`0.0621 - 0.268 i`
`7`
`3.9084`
`4.17`
`8`
`5.5978`
`2.45 i`
`9`
`-15.3424`
`0.215`
`10`
`-93.9513`
`0.612 i` Top of the page

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`
`1.18 + 0.3935 i`
`0.0606 + 0.0502 i` `2`
`1.18 - 0.3935 i`
`0.0606 - 0.0502 i`
`3`
`1.1159 + 0.6584 i`
`0.0138 - 0.0352 i`
`4`
`1.1159 - 0.6584 i`
`0.0138 + 0.0352 i`
`5`
`1.1756 + 0.642 i`
`0.0398 + 0.00838 i`
`6`
`1.1756 - 0.642 i`
`0.0398 - 0.00838 i`
`7`
`2.2068`
`0.546`
`8`
`-2.7409 + 0.8731 i`
`0.19 + 0.128 i`
`9`
`-2.7409 - 0.8731 i`
`0.19 - 0.128 i`
`10`
`-8.0514`
`0.312` Top of the page

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`
`1.1913 + 0.4535 i`
`0.0598 - 0.147 i` `2`
`1.1913 - 0.4535 i`
`0.0598 + 0.147 i`
`3`
`1.4672`
`0.183`
`4`
`2.0639 + 0.2332 i`
`0.471 + 1.61 i`
`5`
`2.0639 - 0.2332 i`
`0.471 - 1.61 i`
`6`
`2.0509 + 1.2428 i`
`0.208 + 0.0969 i`
`7`
`2.0509 - 1.2428 i`
`0.208 - 0.0969 i`
`8`
`-2.7317 + 0.8371 i`
`0.171 + 0.167 i`
`9`
`-2.7317 - 0.8371 i`
`0.171 - 0.167 i`
`10`
`-7.1372`
`0.282`
`11`
`-24.0387`
`0.329 i` Top of the page

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`
`1.2011 + 0.4557 i`
`0.0811 - 0.186 i` `2`
`1.2011 - 0.4557 i`
`0.0811 + 0.186 i`
`3`
`1.6348`
`0.306`
`4`
`1.5739 + 0.5096 i`
`0.299 + 0.0922 i`
`5`
`1.5739 - 0.5096 i`
`0.299 - 0.0922 i`
`6`
`1.8761 + 0.7359 i`
`0.508 - 0.0492 i`
`7`
`1.8761 - 0.7359 i`
`0.508 + 0.0492 i`
`8`
`-2.7369 + 0.7936 i`
`0.146 + 0.225 i`
`9`
`-2.7369 - 0.7936 i`
`0.146 - 0.225 i`
`10`
`-8.836`
`0.22`
`11`
`31.6111`
`0.349 i`
`12`
`-145.59`
`0.585 i` Top of the page

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`
`1.1917 + 0.4688 i`
`0.102 - 0.117 i` `2`
`1.1917 - 0.4688 i`
`0.102 + 0.117 i`
`3`
`1.4166 + 0.2348 i`
`0.225 + 0.16 i`
`4`
`1.4166 - 0.2348 i`
`0.225 - 0.16 i`
`5`
`1.5351 + 0.3158 i`
`0.421 + 0.0896 i`
`6`
`1.5351 - 0.3158 i`
`0.421 - 0.0896 i`
`7`
`2.4354 + 1.2131 i`
`0.221 + 0.28 i`
`8`
`2.4354 - 1.2131 i`
`0.221 - 0.28 i`
`9`
`-2.7329 + 0.8036 i`
`0.147 + 0.21 i`
`10`
`-2.7329 - 0.8036 i`
`0.147 - 0.21 i`
`11`
`3.4541`
`0.431`
`12`
`-8.6519`
`0.265` Top of the page

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`
`1.218 + 0.4599 i`
`0.151 - 0.336 i` `2`
`1.218 - 0.4599 i`
`0.151 + 0.336 i`
`3`
`1.3601 + 0.4502 i`
`0.374 + 0.292 i`
`4`
`1.3601 - 0.4502 i`
`0.374 - 0.292 i`
`5`
`1.4859 + 0.405 i`
`1.63 + 1.01 i`
`6`
`1.4859 - 0.405 i`
`1.63 - 1.01 i`
`7`
`2.6581`
`0.485`
`8`
`-2.7306 + 0.8249 i`
`0.168 + 0.179 i`
`9`
`-2.7306 - 0.8249 i`
`0.168 - 0.179 i`
`10`
`5.1371`
`42.5 i`
`11`
`-5.8872`
`0.278`
`12`
`-9.1167`
`0.412 i`
`13`
`17.3669`
`0.844` Top of the page

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`
`1.2004 + 0.4655 i`
`0.12 - 0.149 i` `2`
`1.2004 - 0.4655 i`
`0.12 + 0.149 i`
`3`
`1.5523 + 0.3991 i`
`0.335 + 0.0231 i`
`4`
`1.5523 - 0.3991 i`
`0.335 - 0.0231 i`
`5`
`1.7892 + 0.4035 i`
`0.582 - 1.31 i`
`6`
`1.7892 - 0.4035 i`
`0.582 + 1.31 i`
`7`
`2.0058`
`0.325`
`8`
`-2.6892 + 0.8038 i`
`0.0937 + 0.191 i`
`9`
`-2.6892 - 0.8038 i`
`0.0937 - 0.191 i`
`10`
`-3.8094 + 0.2208 i`
`0.0442 - 0.344 i`
`11`
`-3.8094 - 0.2208 i`
`0.0442 + 0.344 i`
`12`
`-8.5368`
`0.126`
`13`
`16.2998`
`0.331 i`
`14`
`-40.4897`
`0.274 i` Top of the page

 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.