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

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

Content


ExamplesBH-cc-pVDZ-1.5ReBH-cc-pVDZ-2ReBH-cc-pVDZ-ReBH-cc-pVQZ-1.5ReBH-cc-pVQZ-2ReBH-cc-pVQZ-ReBH-cc-pVTZ-1.5ReBH-cc-pVTZ-2ReBH-cc-pVTZ-ReH--cc-pV5ZH--cc-pVQZHF-cc-pVDZ-1.5ReHF-cc-pVDZ-2ReHF-cc-pVDZ-ReO2--aug-cc-pVDZ
MoleculeX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHH- ionH- ionX 1^Sigma+ State of HFX 1^Sigma+ State of HFX 1^Sigma+ State of HFX 1^Sigma+ State of O2-
BasisCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZCC-PVDZCC-PVDZCC-PVDZAUG-CC-PVDZ

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Quadratic approximants

[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
Singularities of quadratic [2, 2, 2] approximant
2
1.2929 - 0.6041 i
0.266 + 0.0418 i
3
2.1451
0.618
4
-2.2454
0.107
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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
1.3103 + 0.5121 i
0.371 - 0.222 i
Singularities of quadratic [2, 2, 3] approximant
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
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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
1.3099 + 0.3338 i
0.522 + 0.488 i
Singularities of quadratic [2, 3, 3] approximant
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
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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
1.3072 + 0.3911 i
0.124 + 0.632 i
Singularities of quadratic [3, 3, 3] approximant
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
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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
1.1937 + 0.3221 i
0.127 - 0.0041 i
Singularities of quadratic [3, 3, 4] approximant
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
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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
0.7554
0.00333
Singularities of quadratic [3, 4, 4] approximant
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
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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
1.2101 + 0.4277 i
0.0565 - 0.208 i
Singularities of quadratic [4, 4, 4] approximant
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
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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
1.1872 + 0.4546 i
0.0685 - 0.118 i
Singularities of quadratic [4, 4, 5] approximant
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
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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
1.1716 + 0.4577 i
0.0543 - 0.0858 i
Singularities of quadratic [4, 5, 5] approximant
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
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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
1.18 + 0.3935 i
0.0606 + 0.0502 i
Singularities of quadratic [5, 5, 5] approximant
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
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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
1.1913 + 0.4535 i
0.0598 - 0.147 i
Singularities of quadratic [5, 5, 6] approximant
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
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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
1.2011 + 0.4557 i
0.0811 - 0.186 i
Singularities of quadratic [5, 6, 6] approximant
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
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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
1.1917 + 0.4688 i
0.102 - 0.117 i
Singularities of quadratic [6, 6, 6] approximant
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
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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
1.218 + 0.4599 i
0.151 - 0.336 i
Singularities of quadratic [6, 6, 7] approximant
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
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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
1.2004 + 0.4655 i
0.12 - 0.149 i
Singularities of quadratic [6, 7, 7] approximant
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
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ExamplesBH-cc-pVDZ-1.5ReBH-cc-pVDZ-2ReBH-cc-pVDZ-ReBH-cc-pVQZ-1.5ReBH-cc-pVQZ-2ReBH-cc-pVQZ-ReBH-cc-pVTZ-1.5ReBH-cc-pVTZ-2ReBH-cc-pVTZ-ReH--cc-pV5ZH--cc-pVQZHF-cc-pVDZ-1.5ReHF-cc-pVDZ-2ReHF-cc-pVDZ-ReO2--aug-cc-pVDZ
MoleculeX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHX 1^Sigma+ State of BHH- ionH- ionX 1^Sigma+ State of HFX 1^Sigma+ State of HFX 1^Sigma+ State of HFX 1^Sigma+ State of O2-
BasisCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZCC-PVDZCC-PVDZCC-PVDZAUG-CC-PVDZ

Plot of singularities Blank Molecule - icon for Allen-dataList of examples Blank Mathematica programs Blank Work in UMassD Blank Waste iconUnpublished reports

Designed by A. Sergeev.