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

Molecule X 1^Sigma+ State of HF. 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
-1.3176
0.254
Singularities of quadratic [2, 2, 2] approximant
2
1.4885 + 0.7347 i
0.434 - 0.0542 i
3
1.4885 - 0.7347 i
0.434 + 0.0542 i
4
142.5848
2.36
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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
-1.2447
0.172
Singularities of quadratic [2, 2, 3] approximant
2
1.1141 + 0.5923 i
0.117 - 0.0606 i
3
1.1141 - 0.5923 i
0.117 + 0.0606 i
4
2.0624
0.308
5
3.0245
21.7 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
1.0656
0.0718
Singularities of quadratic [2, 3, 3] approximant
2
1.1137 + 0.3423 i
0.0397 - 0.101 i
3
1.1137 - 0.3423 i
0.0397 + 0.101 i
4
-1.2645
0.192
5
2.4971
10.5 i
6
-122.4242
4.62 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
-1.129 + 0.3127 i
0.0383 + 0.0693 i
Singularities of quadratic [3, 3, 3] approximant
2
-1.129 - 0.3127 i
0.0383 - 0.0693 i
3
1.1958 + 0.4603 i
0.00824 - 0.327 i
4
1.1958 - 0.4603 i
0.00824 + 0.327 i
5
-1.3769
0.0702
6
1.9545
0.262
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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
1.152 + 0.267 i
0.0924 - 0.0069 i
Singularities of quadratic [3, 3, 4] approximant
2
1.152 - 0.267 i
0.0924 + 0.0069 i
3
-0.0181 + 1.342 i
0.0181 - 0.0294 i
4
-0.0181 - 1.342 i
0.0181 + 0.0294 i
5
-1.4151
1.04
6
0.2181 + 1.4944 i
0.0343 + 0.0137 i
7
0.2181 - 1.4944 i
0.0343 - 0.0137 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
1.1914 + 0.5291 i
0.0988 - 0.275 i
Singularities of quadratic [3, 4, 4] approximant
2
1.1914 - 0.5291 i
0.0988 + 0.275 i
3
-1.5421
4.09
4
-2.3394
0.447 i
5
-0.929 + 2.4397 i
0.335 + 0.0156 i
6
-0.929 - 2.4397 i
0.335 - 0.0156 i
7
3.8073
0.351
8
11.1326
5.41 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
1.1574 + 0.5691 i
0.165 - 0.139 i
Singularities of quadratic [4, 4, 4] approximant
2
1.1574 - 0.5691 i
0.165 + 0.139 i
3
-1.359
0.307
4
-1.5143 + 1.2094 i
0.0658 + 0.805 i
5
-1.5143 - 1.2094 i
0.0658 - 0.805 i
6
-1.5375 + 1.4425 i
1.47 + 0.888 i
7
-1.5375 - 1.4425 i
1.47 - 0.888 i
8
2.2304
0.305
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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
-1.2108
0.0804
Singularities of quadratic [4, 4, 5] approximant
2
1.1662 + 0.573 i
0.188 - 0.137 i
3
1.1662 - 0.573 i
0.188 + 0.137 i
4
-1.5298
0.129 i
5
-1.52 + 0.8424 i
0.0454 + 0.19 i
6
-1.52 - 0.8424 i
0.0454 - 0.19 i
7
2.1943
0.289
8
-1.9516 + 3.144 i
0.392 - 0.503 i
9
-1.9516 - 3.144 i
0.392 + 0.503 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
-0.7359 + 0.3711 i
0.000211 + 0.00166 i
Singularities of quadratic [4, 5, 5] approximant
2
-0.7359 - 0.3711 i
0.000211 - 0.00166 i
3
-0.7444 + 0.3627 i
0.00166 - 0.000177 i
4
-0.7444 - 0.3627 i
0.00166 + 0.000177 i
5
1.1346 + 0.4931 i
0.0286 + 0.125 i
6
1.1346 - 0.4931 i
0.0286 - 0.125 i
7
-1.7125 + 1.2239 i
0.0816 - 0.0013 i
8
-1.7125 - 1.2239 i
0.0816 + 0.0013 i
9
1.5349 + 2.271 i
0.0214 - 0.167 i
10
1.5349 - 2.271 i
0.0214 + 0.167 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
-1.0804 + 0.0513 i
0.0096 + 0.0109 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.0804 - 0.0513 i
0.0096 - 0.0109 i
3
1.162 + 0.5473 i
0.113 - 0.193 i
4
1.162 - 0.5473 i
0.113 + 0.193 i
5
-1.3545 + 0.1126 i
0.0394 + 0.00295 i
6
-1.3545 - 0.1126 i
0.0394 - 0.00295 i
7
-1.5825 + 1.5339 i
0.119 - 0.173 i
8
-1.5825 - 1.5339 i
0.119 + 0.173 i
9
2.8527
0.401
10
-12.7017
0.656
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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
-1.1216 + 0.0335 i
0.0383 + 0.0349 i
Singularities of quadratic [5, 5, 6] approximant
2
-1.1216 - 0.0335 i
0.0383 - 0.0349 i
3
1.163 + 0.5483 i
0.119 - 0.195 i
4
1.163 - 0.5483 i
0.119 + 0.195 i
5
-1.5229 + 0.3025 i
0.01 + 0.306 i
6
-1.5229 - 0.3025 i
0.01 - 0.306 i
7
-1.5264 + 1.83 i
0.0471 + 0.271 i
8
-1.5264 - 1.83 i
0.0471 - 0.271 i
9
2.9411
0.409
10
8.3057
25.5 i
11
-10.4251
0.83
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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
-1.0159 + 0.0769 i
0.000809 + 0.00431 i
Singularities of quadratic [5, 6, 6] approximant
2
-1.0159 - 0.0769 i
0.000809 - 0.00431 i
3
-1.0789
0.00258
4
-1.1451
0.00436 i
5
1.1602 + 0.5504 i
0.118 - 0.182 i
6
1.1602 - 0.5504 i
0.118 + 0.182 i
7
-1.5991 + 1.314 i
0.14 - 0.0455 i
8
-1.5991 - 1.314 i
0.14 + 0.0455 i
9
2.5426
0.363
10
-0.688 + 5.3531 i
0.685 + 0.218 i
11
-0.688 - 5.3531 i
0.685 - 0.218 i
12
31.6072
1.97 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
-0.4586 + 0.e-5 i
0.0000467 + 0.0000467 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.4586 - 0.e-5 i
0.0000467 - 0.0000467 i
3
-1.0633
0.0199
4
-1.1521
0.025 i
5
1.1609 + 0.5497 i
0.118 - 0.185 i
6
1.1609 - 0.5497 i
0.118 + 0.185 i
7
-1.9096 + 0.716 i
0.139 + 0.266 i
8
-1.9096 - 0.716 i
0.139 - 0.266 i
9
2.6741
0.371
10
-2.1102 + 2.0528 i
0.26 - 4.66 i
11
-2.1102 - 2.0528 i
0.26 + 4.66 i
12
-5.0786
0.542
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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
-0.8515 + 0.0018 i
0.00153 + 0.00154 i
Singularities of quadratic [6, 6, 7] approximant
2
-0.8515 - 0.0018 i
0.00153 - 0.00154 i
3
-1.0774
0.013
4
-1.2276
0.0254 i
5
1.1627 + 0.5499 i
0.124 - 0.191 i
6
1.1627 - 0.5499 i
0.124 + 0.191 i
7
-1.7764 + 0.8556 i
0.17 + 0.19 i
8
-1.7764 - 0.8556 i
0.17 - 0.19 i
9
2.9492
0.389
10
-1.7203 + 2.6395 i
1.02 - 0.166 i
11
-1.7203 - 2.6395 i
1.02 + 0.166 i
12
4.5288
4.09 i
13
18.6813
2.08
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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
-0.7662 + 0.0005 i
0.000629 + 0.00063 i
Singularities of quadratic [6, 7, 7] approximant
2
-0.7662 - 0.0005 i
0.000629 - 0.00063 i
3
-1.0577
0.0113
4
-1.1896
0.0192 i
5
1.1625 + 0.5498 i
0.123 - 0.19 i
6
1.1625 - 0.5498 i
0.123 + 0.19 i
7
-1.7098 + 1.0048 i
0.145 + 0.0935 i
8
-1.7098 - 1.0048 i
0.145 - 0.0935 i
9
3.3413
0.235
10
-1.399 + 3.0839 i
0.213 - 0.723 i
11
-1.399 - 3.0839 i
0.213 + 0.723 i
12
3.5441
0.301 i
13
10.6928 + 13.6194 i
1.28 - 0.17 i
14
10.6928 - 13.6194 i
1.28 + 0.17 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.