Singularities of Møller-Plesset series: example "H--cc-pVQZ"

Molecule H- ion. Basis AUG-CC-PVQZ. 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

[n1, n2, n3] 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
 
1.3885
 
0.0483
 Singularities of quadratic [2, 2, 2] approximant 
2
 
2.4272
 
0.131 i
 
3
 
-2.732 + 12.0437 i
 
0.0946 - 0.053 i
 
4
 
-2.732 - 12.0437 i
 
0.0946 + 0.053 i
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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
 
1.4464
 
0.0772
 Singularities of quadratic [2, 2, 3] approximant 
2
 
2.3971
 
0.135 i
 
3
 
3.7586 + 4.2208 i
 
0.0537 - 0.119 i
 
4
 
3.7586 - 4.2208 i
 
0.0537 + 0.119 i
 
5
 
-6.5362
 
0.175
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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
 
1.2729 + 0.2991 i
 
0.00862 - 0.0123 i
 Singularities of quadratic [2, 3, 3] approximant 
2
 
1.2729 - 0.2991 i
 
0.00862 + 0.0123 i
 
3
 
1.547
 
0.0153
 
4
 
3.25
 
1.78 i
 
5
 
-9.3748 + 4.008 i
 
0.269 - 0.105 i
 
6
 
-9.3748 - 4.008 i
 
0.269 + 0.105 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
 
1.3534 + 0.3682 i
 
0.0158 - 0.00653 i
 Singularities of quadratic [3, 3, 3] approximant 
2
 
1.3534 - 0.3682 i
 
0.0158 + 0.00653 i
 
3
 
2.4158
 
0.171
 
4
 
-3.2775 + 0.1918 i
 
0.00483 + 0.00434 i
 
5
 
-3.2775 - 0.1918 i
 
0.00483 - 0.00434 i
 
6
 
5.796
 
0.0971 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
 
0.381
 
0.0000158
 Singularities of quadratic [3, 3, 4] approximant 
2
 
0.381
 
0.0000158 i
 
3
 
1.3645 + 0.4434 i
 
0.0114 + 0.00423 i
 
4
 
1.3645 - 0.4434 i
 
0.0114 - 0.00423 i
 
5
 
3.7939
 
0.209
 
6
 
-0.657 + 7.9333 i
 
0.0162 - 0.0456 i
 
7
 
-0.657 - 7.9333 i
 
0.0162 + 0.0456 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
 
1.4201 + 0.4445 i
 
0.0139 + 0.0047 i
 Singularities of quadratic [3, 4, 4] approximant 
2
 
1.4201 - 0.4445 i
 
0.0139 - 0.0047 i
 
3
 
1.0054 + 1.8291 i
 
0.00341 + 0.00265 i
 
4
 
1.0054 - 1.8291 i
 
0.00341 - 0.00265 i
 
5
 
0.8415 + 2.0518 i
 
0.00326 - 0.00283 i
 
6
 
0.8415 - 2.0518 i
 
0.00326 + 0.00283 i
 
7
 
-5.5243 + 0.9471 i
 
0.0128 + 0.00657 i
 
8
 
-5.5243 - 0.9471 i
 
0.0128 - 0.00657 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
 
1.4627 + 0.3492 i
 
0.0284 - 0.0196 i
 Singularities of quadratic [4, 4, 4] approximant 
2
 
1.4627 - 0.3492 i
 
0.0284 + 0.0196 i
 
3
 
1.6751 + 1.3144 i
 
0.002 - 0.0215 i
 
4
 
1.6751 - 1.3144 i
 
0.002 + 0.0215 i
 
5
 
1.6947 + 1.9192 i
 
0.0191 + 0.00744 i
 
6
 
1.6947 - 1.9192 i
 
0.0191 - 0.00744 i
 
7
 
2.5361 + 7.8391 i
 
0.0284 - 0.0251 i
 
8
 
2.5361 - 7.8391 i
 
0.0284 + 0.0251 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
 
1.4415 + 0.4181 i
 
0.0193 + 0.00157 i
 Singularities of quadratic [4, 4, 5] approximant 
2
 
1.4415 - 0.4181 i
 
0.0193 - 0.00157 i
 
3
 
1.469 + 1.5851 i
 
0.0118 - 0.00139 i
 
4
 
1.469 - 1.5851 i
 
0.0118 + 0.00139 i
 
5
 
1.5504 + 2.0885 i
 
0.00269 - 0.0141 i
 
6
 
1.5504 - 2.0885 i
 
0.00269 + 0.0141 i
 
7
 
-9.2828
 
0.0831
 
8
 
-21.3051 + 7.417 i
 
0.597 + 0.815 i
 
9
 
-21.3051 - 7.417 i
 
0.597 - 0.815 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
 
1.4358 + 0.1077 i
 
0.018 - 0.00905 i
 Singularities of quadratic [4, 5, 5] approximant 
2
 
1.4358 - 0.1077 i
 
0.018 + 0.00905 i
 
3
 
1.6068 + 0.6107 i
 
0.0191 - 0.0226 i
 
4
 
1.6068 - 0.6107 i
 
0.0191 + 0.0226 i
 
5
 
1.5522 + 2.1789 i
 
0.00768 - 0.0221 i
 
6
 
1.5522 - 2.1789 i
 
0.00768 + 0.0221 i
 
7
 
1.4766 + 2.7414 i
 
0.0247 + 0.0137 i
 
8
 
1.4766 - 2.7414 i
 
0.0247 - 0.0137 i
 
9
 
-10.3097 + 3.5996 i
 
0.343 - 0.0356 i
 
10
 
-10.3097 - 3.5996 i
 
0.343 + 0.0356 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
 
1.4964 + 0.2976 i
 
0.0207 - 0.0629 i
 Singularities of quadratic [5, 5, 5] approximant 
2
 
1.4964 - 0.2976 i
 
0.0207 + 0.0629 i
 
3
 
1.9055 + 0.8803 i
 
0.0418 + 0.0268 i
 
4
 
1.9055 - 0.8803 i
 
0.0418 - 0.0268 i
 
5
 
1.8017 + 2.1756 i
 
0.0275 - 0.013 i
 
6
 
1.8017 - 2.1756 i
 
0.0275 + 0.013 i
 
7
 
1.9859 + 3.6028 i
 
0.0044 + 0.0325 i
 
8
 
1.9859 - 3.6028 i
 
0.0044 - 0.0325 i
 
9
 
-4.6853 + 0.163 i
 
0.00949 + 0.00919 i
 
10
 
-4.6853 - 0.163 i
 
0.00949 - 0.00919 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
 
1.4876 + 0.3085 i
 
0.0233 - 0.0496 i
 Singularities of quadratic [5, 5, 6] approximant 
2
 
1.4876 - 0.3085 i
 
0.0233 + 0.0496 i
 
3
 
1.8796 + 1.0024 i
 
0.0231 + 0.0345 i
 
4
 
1.8796 - 1.0024 i
 
0.0231 - 0.0345 i
 
5
 
1.8801 + 2.3154 i
 
0.0293 + 0.00538 i
 
6
 
1.8801 - 2.3154 i
 
0.0293 - 0.00538 i
 
7
 
1.8634 + 4.4229 i
 
0.0214 - 0.0309 i
 
8
 
1.8634 - 4.4229 i
 
0.0214 + 0.0309 i
 
9
 
-9.2236 + 2.3426 i
 
0.516 + 1.18 i
 
10
 
-9.2236 - 2.3426 i
 
0.516 - 1.18 i
 
11
 
-15.9494
 
0.236
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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
 
1.4851 + 0.3043 i
 
0.0157 - 0.0484 i
 Singularities of quadratic [5, 6, 6] approximant 
2
 
1.4851 - 0.3043 i
 
0.0157 + 0.0484 i
 
3
 
1.9151 + 1.0461 i
 
0.00741 + 0.0417 i
 
4
 
1.9151 - 1.0461 i
 
0.00741 - 0.0417 i
 
5
 
1.7312 + 2.6282 i
 
0.00402 + 0.0276 i
 
6
 
1.7312 - 2.6282 i
 
0.00402 - 0.0276 i
 
7
 
-3.1516 + 1.71 i
 
0.000384 + 0.00169 i
 
8
 
-3.1516 - 1.71 i
 
0.000384 - 0.00169 i
 
9
 
-3.3377 + 1.6603 i
 
0.00174 - 0.000263 i
 
10
 
-3.3377 - 1.6603 i
 
0.00174 + 0.000263 i
 
11
 
3.7252 + 2.7096 i
 
0.135 + 0.102 i
 
12
 
3.7252 - 2.7096 i
 
0.135 - 0.102 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
 
1.5016 + 0.3139 i
 
0.0532 - 0.0539 i
 Singularities of quadratic [6, 6, 6] approximant 
2
 
1.5016 - 0.3139 i
 
0.0532 + 0.0539 i
 
3
 
1.8306 + 0.8659 i
 
0.0337 - 0.00116 i
 
4
 
1.8306 - 0.8659 i
 
0.0337 + 0.00116 i
 
5
 
1.649 + 1.8667 i
 
0.00424 - 0.01 i
 
6
 
1.649 - 1.8667 i
 
0.00424 + 0.01 i
 
7
 
0.822 + 3.5013 i
 
0.00402 + 0.00476 i
 
8
 
0.822 - 3.5013 i
 
0.00402 - 0.00476 i
 
9
 
-6.1426 + 2.1283 i
 
0.0102 + 0.00144 i
 
10
 
-6.1426 - 2.1283 i
 
0.0102 - 0.00144 i
 
11
 
-0.9194 + 9.9324 i
 
0.011 - 0.00232 i
 
12
 
-0.9194 - 9.9324 i
 
0.011 + 0.00232 i
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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
 
1.501 + 0.3068 i
 
0.0278 - 0.0603 i
 Singularities of quadratic [6, 6, 7] approximant 
2
 
1.501 - 0.3068 i
 
0.0278 + 0.0603 i
 
3
 
1.814 + 0.0616 i
 
0.154 + 0.208 i
 
4
 
1.814 - 0.0616 i
 
0.154 - 0.208 i
 
5
 
1.9027 + 1.0782 i
 
0.0146 + 0.0397 i
 
6
 
1.9027 - 1.0782 i
 
0.0146 - 0.0397 i
 
7
 
1.8998 + 2.3339 i
 
0.0258 + 0.012 i
 
8
 
1.8998 - 2.3339 i
 
0.0258 - 0.012 i
 
9
 
1.529 + 5.0743 i
 
0.0308 - 0.0179 i
 
10
 
1.529 - 5.0743 i
 
0.0308 + 0.0179 i
 
11
 
-6.8288 + 6.5917 i
 
0.117 - 0.0902 i
 
12
 
-6.8288 - 6.5917 i
 
0.117 + 0.0902 i
 
13
 
-15.7944
 
0.193
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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
 
1.504 + 0.3198 i
 
0.0568 - 0.0393 i
 Singularities of quadratic [6, 7, 7] approximant 
2
 
1.504 - 0.3198 i
 
0.0568 + 0.0393 i
 
3
 
1.9598 + 0.7135 i
 
0.0581 - 0.0602 i
 
4
 
1.9598 - 0.7135 i
 
0.0581 + 0.0602 i
 
5
 
2.2441
 
0.0989
 
6
 
1.9699 + 1.5926 i
 
0.0225 + 0.0312 i
 
7
 
1.9699 - 1.5926 i
 
0.0225 - 0.0312 i
 
8
 
1.675 + 3.2409 i
 
0.0277 - 0.00306 i
 
9
 
1.675 - 3.2409 i
 
0.0277 + 0.00306 i
 
10
 
-5.0831 + 4.1086 i
 
0.0098 + 0.0367 i
 
11
 
-5.0831 - 4.1086 i
 
0.0098 - 0.0367 i
 
12
 
-5.5154 + 6.4195 i
 
0.032 - 0.0213 i
 
13
 
-5.5154 - 6.4195 i
 
0.032 + 0.0213 i
 
14
 
10.6968
 
0.0894 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.