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

Molecule X 1^Sigma+ State of BH. Basis CC-PVDZ. 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
1.3157 + 0.3983 i
0.186 - 0.987 i
Singularities of quadratic [2, 2, 2] approximant
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
Singularities of quadratic [2, 2, 3] approximant
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
Singularities of quadratic [2, 3, 3] approximant
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
Singularities of quadratic [3, 3, 3] approximant
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
Singularities of quadratic [3, 3, 4] approximant
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
Singularities of quadratic [3, 4, 4] approximant
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
Singularities of quadratic [4, 4, 4] approximant
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
Singularities of quadratic [4, 4, 5] approximant
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
Singularities of quadratic [4, 5, 5] approximant
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
Singularities of quadratic [5, 5, 5] approximant
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
Singularities of quadratic [5, 5, 6] approximant
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
Singularities of quadratic [5, 6, 6] approximant
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
Singularities of quadratic [6, 6, 6] approximant
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
Singularities of quadratic [6, 6, 7] approximant
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
Singularities of quadratic [6, 7, 7] approximant
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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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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Designed by A. Sergeev.