Singularities of Møller-Plesset series: example "Ne cc-pVDZ"

Molecule Ne. Basis cc-pVDZ. Structure ""

Content


ExamplesAr cc-pVDZBH aug-cc-pVQZ 0.9r_eBH aug-cc-pVQZ 1.0r_eBH aug-cc-pVQZ 1.1r_eBH aug-cc-pVQZ 1.2r_eBH aug-cc-pVQZ 1.3r_eBH aug-cc-pVQZ 1.4r_eBH aug-cc-pVQZ 1.5r_eBH aug-cc-pVQZ 1.6r_eBH aug-cc-pVQZ 1.7r_eBH aug-cc-pVQZ 1.8r_eBH aug-cc-pVQZ 1.9r_eBH aug-cc-pVQZ 2.0r_eBH aug-cc-pVQZ 2.1r_eBH aug-cc-pVQZ 2.2r_eBH 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 aug-cc-pVDZ 1.5r_eHF aug-cc-pVDZ 2.0r_eHF aug-cc-pVDZ r_eHF cc-pVDZ 1.5ReHF cc-pVDZ 2ReHF cc-pVDZ Rena-pl aug-cc-pvdzNe cc-pVDZO2- aug-cc-pVDZ
MoleculeArX 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 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 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 HFX 1^Sigma+ State of HFX 1^Sigma+ State of HFNa+NeX 1^Sigma+ State of O2-
Basiscc-pVDZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZAUG-CC-PVDZAUG-CC-PVDZAUG-CC-PVDZCC-PVDZCC-PVDZCC-PVDZAUG-CC-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 [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
19.6646
7.3
Singularities of quadratic [1, 0, 0] approximant
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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.8062
0.0559
Singularities of quadratic [1, 1, 0] approximant
2
1.2676
0.07 i
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Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.7826
0.556
Singularities of quadratic [1, 1, 1] approximant
2
10.8617
16.8
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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.4643 + 1.1398 i
0.0581 + 0.017 i
Singularities of quadratic [2, 1, 1] approximant
2
-1.4643 - 1.1398 i
0.0581 - 0.017 i
3
1.9252
0.0756
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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.145 + 0.9165 i
0.181 + 0.075 i
Singularities of quadratic [2, 2, 1] approximant
2
-2.145 - 0.9165 i
0.181 - 0.075 i
3
2.4936
0.218
4
49.4782
2.65 i
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Table 6. 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
-1.5372 + 0.8804 i
0.0492 + 0.0297 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.5372 - 0.8804 i
0.0492 - 0.0297 i
3
2.4143
0.145
4
-5.1013
0.183
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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.953 + 0.9686 i
0.126 + 0.0776 i
Singularities of quadratic [3, 2, 2] approximant
2
-1.953 - 0.9686 i
0.126 - 0.0776 i
3
2.4377 + 1.0462 i
0.0553 - 0.238 i
4
2.4377 - 1.0462 i
0.0553 + 0.238 i
5
5.12
0.239
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Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.7213
0.0343
Singularities of quadratic [3, 3, 2] approximant
2
-2.0715
0.0425 i
3
-1.8145 + 1.7814 i
0.019 + 0.118 i
4
-1.8145 - 1.7814 i
0.019 - 0.118 i
5
4.4232 + 1.1067 i
0.569 + 0.858 i
6
4.4232 - 1.1067 i
0.569 - 0.858 i
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Table 9. 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
-2.6401 + 1.202 i
0.378 - 0.00435 i
Singularities of quadratic [3, 3, 3] approximant
2
-2.6401 - 1.202 i
0.378 + 0.00435 i
3
2.9183
0.359
4
-0.2282 + 4.8803 i
0.534 + 0.22 i
5
-0.2282 - 4.8803 i
0.534 - 0.22 i
6
-8.1308
0.865
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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.5819 + 0.8233 i
0.243 + 0.634 i
Singularities of quadratic [4, 3, 3] approximant
2
-2.5819 - 0.8233 i
0.243 - 0.634 i
3
3.0776
0.695
4
-1.7433 + 3.6791 i
0.481 + 0.0899 i
5
-1.7433 - 3.6791 i
0.481 - 0.0899 i
6
1.0111 + 5.8127 i
0.0496 - 0.543 i
7
1.0111 - 5.8127 i
0.0496 + 0.543 i
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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.4452 + 1.1779 i
0.243 + 0.00391 i
Singularities of quadratic [4, 4, 3] approximant
2
-2.4452 - 1.1779 i
0.243 - 0.00391 i
3
3.0221
0.465
4
-0.5642 + 3.7351 i
0.158 + 0.272 i
5
-0.5642 - 3.7351 i
0.158 - 0.272 i
6
-3.5399 + 5.327 i
0.395 + 0.13 i
7
-3.5399 - 5.327 i
0.395 - 0.13 i
8
13.7514
1.41 i
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Table 12. 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
-2.6059 + 0.9048 i
0.375 + 0.474 i
Singularities of quadratic [4, 4, 4] approximant
2
-2.6059 - 0.9048 i
0.375 - 0.474 i
3
3.5156 + 0.5871 i
1.01 - 0.178 i
4
3.5156 - 0.5871 i
1.01 + 0.178 i
5
-1.2265 + 4.1923 i
0.354 - 0.439 i
6
-1.2265 - 4.1923 i
0.354 + 0.439 i
7
5.7503 + 4.6669 i
1.1 + 0.0731 i
8
5.7503 - 4.6669 i
1.1 - 0.0731 i
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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.605 + 0.9113 i
0.392 + 0.455 i
Singularities of quadratic [5, 4, 4] approximant
2
-2.605 - 0.9113 i
0.392 - 0.455 i
3
3.3866 + 0.6055 i
0.751 - 0.396 i
4
3.3866 - 0.6055 i
0.751 + 0.396 i
5
-1.2593 + 4.2408 i
0.411 - 0.497 i
6
-1.2593 - 4.2408 i
0.411 + 0.497 i
7
6.6596
10.
8
5.2844 + 10.3478 i
1.11 + 0.598 i
9
5.2844 - 10.3478 i
1.11 - 0.598 i
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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
2.0254 + 0.0095 i
0.0158 - 0.0156 i
Singularities of quadratic [5, 5, 4] approximant
2
2.0254 - 0.0095 i
0.0158 + 0.0156 i
3
-2.6195 + 0.947 i
0.576 + 0.321 i
4
-2.6195 - 0.947 i
0.576 - 0.321 i
5
3.2998
9.14
6
-2.173 + 3.7953 i
0.335 + 0.442 i
7
-2.173 - 3.7953 i
0.335 - 0.442 i
8
0.8909 + 4.5759 i
0.163 - 0.267 i
9
0.8909 - 4.5759 i
0.163 + 0.267 i
10
212.9098
93.6 i
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Table 15. 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
-0.7039
0.000082
Singularities of quadratic [5, 5, 5] approximant
2
-0.7039
0.000082 i
3
-2.5824 + 0.878 i
0.203 + 0.442 i
4
-2.5824 - 0.878 i
0.203 - 0.442 i
5
3.1124 + 0.4988 i
0.0584 + 0.614 i
6
3.1124 - 0.4988 i
0.0584 - 0.614 i
7
3.9952
0.477
8
-0.7905 + 4.2716 i
0.0763 + 0.452 i
9
-0.7905 - 4.2716 i
0.0763 - 0.452 i
10
-38.0357
0.768
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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.1741
4.51e-8
Singularities of quadratic [6, 5, 5] approximant
2
-0.1741
4.51e-8 i
3
-2.588 + 0.8726 i
0.197 + 0.47 i
4
-2.588 - 0.8726 i
0.197 - 0.47 i
5
3.1422 + 0.5085 i
0.0618 + 0.616 i
6
3.1422 - 0.5085 i
0.0618 - 0.616 i
7
-0.7875 + 4.1655 i
0.0264 + 0.375 i
8
-0.7875 - 4.1655 i
0.0264 - 0.375 i
9
4.3677
0.554
10
-13.0403
1.56
11
14.4482
5.1 i
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Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.6246 + 0.898 i
0.432 + 0.515 i
Singularities of quadratic [6, 6, 5] approximant
2
-2.6246 - 0.898 i
0.432 - 0.515 i
3
3.1352 + 0.6042 i
0.324 - 0.381 i
4
3.1352 - 0.6042 i
0.324 + 0.381 i
5
4.2105
0.641
6
-1.0446 + 4.4698 i
0.356 - 0.937 i
7
-1.0446 - 4.4698 i
0.356 + 0.937 i
8
-2.4394 + 8.8738 i
1.07 - 0.0622 i
9
-2.4394 - 8.8738 i
1.07 + 0.0622 i
10
-8.2016 + 4.8537 i
0.727 + 0.784 i
11
-8.2016 - 4.8537 i
0.727 - 0.784 i
12
18.9919
1.71 i
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Table 18. 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
-2.6238 + 0.9024 i
0.439 + 0.489 i
Singularities of quadratic [6, 6, 6] approximant
2
-2.6238 - 0.9024 i
0.439 - 0.489 i
3
3.1174 + 0.6484 i
0.301 - 0.243 i
4
3.1174 - 0.6484 i
0.301 + 0.243 i
5
-0.9669 + 4.3617 i
0.151 - 0.679 i
6
-0.9669 - 4.3617 i
0.151 + 0.679 i
7
4.9178 + 0.0089 i
0.347 - 0.325 i
8
4.9178 - 0.0089 i
0.347 + 0.325 i
9
-5.7052
0.594
10
-6.4109
0.607 i
11
9.3662
8.48
12
-50.7405
0.48
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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.6246 + 0.8995 i
0.437 + 0.507 i
Singularities of quadratic [7, 6, 6] approximant
2
-2.6246 - 0.8995 i
0.437 - 0.507 i
3
3.1332 + 0.6108 i
0.325 - 0.356 i
4
3.1332 - 0.6108 i
0.325 + 0.356 i
5
4.2286
0.664
6
-1.0134 + 4.4297 i
0.261 - 0.838 i
7
-1.0134 - 4.4297 i
0.261 + 0.838 i
8
-8.9294 + 2.6968 i
0.338 + 0.506 i
9
-8.9294 - 2.6968 i
0.338 - 0.506 i
10
12.866
1.96 i
11
-2.6274 + 14.1364 i
0.738 + 0.227 i
12
-2.6274 - 14.1364 i
0.738 - 0.227 i
13
-67.1679
0.955
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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-2.6247 + 0.8994 i
0.437 + 0.508 i
Singularities of quadratic [7, 7, 6] approximant
2
-2.6247 - 0.8994 i
0.437 - 0.508 i
3
3.1329 + 0.6113 i
0.324 - 0.354 i
4
3.1329 - 0.6113 i
0.324 + 0.354 i
5
4.2312
0.667
6
-1.0111 + 4.4317 i
0.256 - 0.843 i
7
-1.0111 - 4.4317 i
0.256 + 0.843 i
8
-8.8019 + 2.9507 i
0.372 + 0.541 i
9
-8.8019 - 2.9507 i
0.372 - 0.541 i
10
11.7184
2.16 i
11
-4.636 + 13.1486 i
0.726 + 0.0774 i
12
-4.636 - 13.1486 i
0.726 - 0.0774 i
13
42.9551 + 26.4481 i
0.982 - 1.8 i
14
42.9551 - 26.4481 i
0.982 + 1.8 i
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ExamplesAr cc-pVDZBH aug-cc-pVQZ 0.9r_eBH aug-cc-pVQZ 1.0r_eBH aug-cc-pVQZ 1.1r_eBH aug-cc-pVQZ 1.2r_eBH aug-cc-pVQZ 1.3r_eBH aug-cc-pVQZ 1.4r_eBH aug-cc-pVQZ 1.5r_eBH aug-cc-pVQZ 1.6r_eBH aug-cc-pVQZ 1.7r_eBH aug-cc-pVQZ 1.8r_eBH aug-cc-pVQZ 1.9r_eBH aug-cc-pVQZ 2.0r_eBH aug-cc-pVQZ 2.1r_eBH aug-cc-pVQZ 2.2r_eBH 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 aug-cc-pVDZ 1.5r_eHF aug-cc-pVDZ 2.0r_eHF aug-cc-pVDZ r_eHF cc-pVDZ 1.5ReHF cc-pVDZ 2ReHF cc-pVDZ Rena-pl aug-cc-pvdzNe cc-pVDZO2- aug-cc-pVDZ
MoleculeArX 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 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 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 HFX 1^Sigma+ State of HFX 1^Sigma+ State of HFNa+NeX 1^Sigma+ State of O2-
Basiscc-pVDZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZAUG-CC-PVQZCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZAUG-CC-PVDZAUG-CC-PVDZAUG-CC-PVDZCC-PVDZCC-PVDZCC-PVDZAUG-CC-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

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