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

Molecule X 1^Sigma+ State of HF. 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
17.3215
6.98
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
0.5779
0.0407
Singularities of quadratic [1, 1, 0] approximant
2
0.8651
0.0497 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
-3.317
1.41
Singularities of quadratic [1, 1, 1] approximant
2
3.6642
3.02
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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
1.5043
0.0767
Singularities of quadratic [2, 1, 1] approximant
2
-1.3062 + 1.3546 i
0.0813 + 0.0119 i
3
-1.3062 - 1.3546 i
0.0813 - 0.0119 i
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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
-1.7172
0.143
Singularities of quadratic [2, 2, 1] approximant
2
2.3931 + 1.7039 i
0.344 + 0.0388 i
3
2.3931 - 1.7039 i
0.344 - 0.0388 i
4
-9.6748
0.498 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
-1.8622 + 0.2673 i
0.256 + 0.2 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.8622 - 0.2673 i
0.256 - 0.2 i
3
2.4813
0.577
4
-5.8362
2.55
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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
-0.4151
0.000477
Singularities of quadratic [3, 2, 2] approximant
2
-0.4162
0.000477 i
3
-1.3611
0.0272
4
2.232
0.161
5
5.0244
25. i
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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
-1.6684
0.184
Singularities of quadratic [3, 3, 2] approximant
2
2.4011
0.573
3
-3.5631
0.208 i
4
-0.026 + 4.7462 i
0.177 - 0.384 i
5
-0.026 - 4.7462 i
0.177 + 0.384 i
6
58.7247
361. 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
-1.5415
0.0786
Singularities of quadratic [3, 3, 3] approximant
2
2.4699
0.717
3
-1.4596 + 2.1472 i
0.0481 + 0.0783 i
4
-1.4596 - 2.1472 i
0.0481 - 0.0783 i
5
-0.8267 + 2.639 i
0.0796 - 0.0876 i
6
-0.8267 - 2.639 i
0.0796 + 0.0876 i
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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
-1.555
0.0875
Singularities of quadratic [4, 3, 3] approximant
2
2.3925
0.462
3
-0.6166 + 2.3518 i
0.0664 - 0.0588 i
4
-0.6166 - 2.3518 i
0.0664 + 0.0588 i
5
-1.1561 + 2.2695 i
0.0405 + 0.0685 i
6
-1.1561 - 2.2695 i
0.0405 - 0.0685 i
7
11.122
4.82 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
-1.4023
0.0334
Singularities of quadratic [4, 4, 3] approximant
2
-1.7435
0.0467 i
3
2.647
2.8
4
-2.5125 + 0.9894 i
0.0357 + 0.0997 i
5
-2.5125 - 0.9894 i
0.0357 - 0.0997 i
6
0.0675 + 3.5202 i
0.00112 + 0.159 i
7
0.0675 - 3.5202 i
0.00112 - 0.159 i
8
6291.4684
177. 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
-1.45
0.0427
Singularities of quadratic [4, 4, 4] approximant
2
-2.2837
0.098 i
3
-2.0327 + 1.3351 i
0.0301 + 0.0896 i
4
-2.0327 - 1.3351 i
0.0301 - 0.0896 i
5
2.7746
16.
6
0.3996 + 4.2325 i
0.0305 + 0.218 i
7
0.3996 - 4.2325 i
0.0305 - 0.218 i
8
6.1602
0.59 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
1.3968
0.0317
Singularities of quadratic [5, 4, 4] approximant
2
1.4011
0.032 i
3
-1.4307
0.0379
4
-1.9622
0.0669 i
5
2.556
1.2
6
-2.2997 + 1.2697 i
0.0369 + 0.0899 i
7
-2.2997 - 1.2697 i
0.0369 - 0.0899 i
8
-0.0425 + 3.6646 i
0.0019 - 0.174 i
9
-0.0425 - 3.6646 i
0.0019 + 0.174 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
-1.3764
0.016
Singularities of quadratic [5, 5, 4] approximant
2
0.2773 + 1.4523 i
0.00137 + 0.000175 i
3
0.2773 - 1.4523 i
0.00137 - 0.000175 i
4
0.2867 + 1.4633 i
0.000175 - 0.00139 i
5
0.2867 - 1.4633 i
0.000175 + 0.00139 i
6
-2.1232
0.116 i
7
-1.4268 + 2.396 i
0.0149 - 0.033 i
8
-1.4268 - 2.396 i
0.0149 + 0.033 i
9
3.6636 + 1.0528 i
0.153 - 0.181 i
10
3.6636 - 1.0528 i
0.153 + 0.181 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
-1.1546 + 0.0029 i
0.000583 + 0.000589 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.1546 - 0.0029 i
0.000583 - 0.000589 i
3
-1.3336
0.00551
4
2.0992 + 0.0735 i
0.0362 - 0.0288 i
5
2.0992 - 0.0735 i
0.0362 + 0.0288 i
6
-2.6957
5.88 i
7
-1.5533 + 2.7765 i
0.0101 + 0.127 i
8
-1.5533 - 2.7765 i
0.0101 - 0.127 i
9
3.1926
0.653
10
111.4405
0.976 i
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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
-1.3712
0.0174
Singularities of quadratic [6, 5, 5] approximant
2
-1.8463
0.0472 i
3
-1.5706 + 1.6235 i
0.0151 + 0.0184 i
4
-1.5706 - 1.6235 i
0.0151 - 0.0184 i
5
2.576 + 0.7034 i
0.227 + 0.0891 i
6
2.576 - 0.7034 i
0.227 - 0.0891 i
7
-2.4512 + 1.5245 i
0.0416 - 0.0187 i
8
-2.4512 - 1.5245 i
0.0416 + 0.0187 i
9
-0.2317 + 3.3195 i
0.0111 - 0.0648 i
10
-0.2317 - 3.3195 i
0.0111 + 0.0648 i
11
3.7334
3.84
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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
-1.2402 + 0.01 i
0.00113 + 0.00125 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.2402 - 0.01 i
0.00113 - 0.00125 i
3
-1.3289
0.00378
4
2.509 + 0.3912 i
0.171 + 0.6 i
5
2.509 - 0.3912 i
0.171 - 0.6 i
6
-2.6747
2.68 i
7
3.0991
0.513
8
-1.8165 + 2.6524 i
0.0853 - 0.126 i
9
-1.8165 - 2.6524 i
0.0853 + 0.126 i
10
0.7904 + 5.7242 i
0.471 - 0.085 i
11
0.7904 - 5.7242 i
0.471 + 0.085 i
12
16.0796
1.29 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
-1.3049 + 0.0369 i
0.000366 + 0.00267 i
Singularities of quadratic [6, 6, 6] approximant
2
-1.3049 - 0.0369 i
0.000366 - 0.00267 i
3
-1.3135
0.00194
4
2.4861 + 0.3113 i
0.526 + 0.325 i
5
2.4861 - 0.3113 i
0.526 - 0.325 i
6
-2.742
3.2 i
7
2.9698
0.546
8
-2.0836 + 2.6771 i
0.19 - 0.131 i
9
-2.0836 - 2.6771 i
0.19 + 0.131 i
10
-1.1489 + 5.2214 i
0.754 + 0.594 i
11
-1.1489 - 5.2214 i
0.754 - 0.594 i
12
-13.7663
3.62
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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
0.369
6.23e-7
Singularities of quadratic [7, 6, 6] approximant
2
0.369
6.23e-7 i
3
-1.2707
0.001
4
-1.287 + 0.0225 i
0.000368 - 0.00124 i
5
-1.287 - 0.0225 i
0.000368 + 0.00124 i
6
2.5219 + 0.3735 i
0.112 - 1.04 i
7
2.5219 - 0.3735 i
0.112 + 1.04 i
8
2.7942
0.499
9
-3.1476
31.6 i
10
-2.0203 + 3.2433 i
0.125 + 0.486 i
11
-2.0203 - 3.2433 i
0.125 - 0.486 i
12
-3.4761 + 5.0417 i
5.14 + 3.09 i
13
-3.4761 - 5.0417 i
5.14 - 3.09 i
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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
-1.3006
0.00187
Singularities of quadratic [7, 7, 6] approximant
2
-1.3017 + 0.0435 i
0.0000807 + 0.00264 i
3
-1.3017 - 0.0435 i
0.0000807 - 0.00264 i
4
2.4477 + 0.3103 i
0.243 + 0.0562 i
5
2.4477 - 0.3103 i
0.243 - 0.0562 i
6
3.0906 + 0.8683 i
0.286 + 0.0247 i
7
3.0906 - 0.8683 i
0.286 - 0.0247 i
8
-1.6168 + 3.1027 i
0.0874 + 0.183 i
9
-1.6168 - 3.1027 i
0.0874 - 0.183 i
10
-3.7889 + 0.7136 i
2.09 - 1.83 i
11
-3.7889 - 0.7136 i
2.09 + 1.83 i
12
4.8135
0.508
13
-5.7375
146. i
14
12.5348
3.75 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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