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

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
76.0136
37.
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
0.042
0.00113
Singularities of quadratic [1, 1, 0] approximant
2
0.044
0.00116 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
1.4777
0.669
Singularities of quadratic [1, 1, 1] approximant
2
-2.3274
3.44
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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
0.5509
0.0137
Singularities of quadratic [2, 1, 1] approximant
2
-0.3308 + 0.5581 i
0.0154 - 0.000271 i
3
-0.3308 - 0.5581 i
0.0154 + 0.000271 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
-1.3134
0.257
Singularities of quadratic [2, 2, 1] approximant
2
1.6066 + 0.7148 i
0.558 - 0.0144 i
3
1.6066 - 0.7148 i
0.558 + 0.0144 i
4
-25.707
1.62 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
-1.3176
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 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
-1.2435
0.171
Singularities of quadratic [3, 2, 2] approximant
2
1.1159 + 0.6018 i
0.12 - 0.0581 i
3
1.1159 - 0.6018 i
0.12 + 0.0581 i
4
2.2161
0.366
5
2.9644
6.51 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
1.0684
0.0725
Singularities of quadratic [3, 3, 2] approximant
2
1.1138 + 0.3439 i
0.0406 - 0.102 i
3
1.1138 - 0.3439 i
0.0406 + 0.102 i
4
-1.2647
0.192
5
2.5169
9.48 i
6
-102.5272
4.61 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
-1.129 + 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 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
1.1216 + 0.2343 i
0.0673 - 0.0126 i
Singularities of quadratic [4, 3, 3] approximant
2
1.1216 - 0.2343 i
0.0673 + 0.0126 i
3
0.0351 + 1.2436 i
0.0105 - 0.0237 i
4
0.0351 - 1.2436 i
0.0105 + 0.0237 i
5
0.2206 + 1.319 i
0.0258 + 0.00706 i
6
0.2206 - 1.319 i
0.0258 - 0.00706 i
7
-1.4087
0.994
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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
1.1917 + 0.5287 i
0.0974 - 0.276 i
Singularities of quadratic [4, 4, 3] approximant
2
1.1917 - 0.5287 i
0.0974 + 0.276 i
3
-1.5436
4.24
4
-2.3278
0.451 i
5
-0.9236 + 2.4465 i
0.334 + 0.013 i
6
-0.9236 - 2.4465 i
0.334 - 0.013 i
7
3.9086
0.35
8
9.9479
3.39 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
1.1574 + 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 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
-1.2095
0.0791
Singularities of quadratic [5, 4, 4] approximant
2
1.1661 + 0.5734 i
0.188 - 0.135 i
3
1.1661 - 0.5734 i
0.188 + 0.135 i
4
-1.5241
0.127 i
5
-1.5208 + 0.8452 i
0.0473 + 0.187 i
6
-1.5208 - 0.8452 i
0.0473 - 0.187 i
7
2.1853
0.288
8
-1.919 + 3.1576 i
0.367 - 0.502 i
9
-1.919 - 3.1576 i
0.367 + 0.502 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
-0.7122 + 0.364 i
0.000179 + 0.00139 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.7122 - 0.364 i
0.000179 - 0.00139 i
3
-0.7194 + 0.3572 i
0.0014 - 0.000155 i
4
-0.7194 - 0.3572 i
0.0014 + 0.000155 i
5
1.1318 + 0.49 i
0.031 + 0.118 i
6
1.1318 - 0.49 i
0.031 - 0.118 i
7
-1.7226 + 1.2192 i
0.0813 + 0.000878 i
8
-1.7226 - 1.2192 i
0.0813 - 0.000878 i
9
1.4642 + 2.2357 i
0.0217 - 0.154 i
10
1.4642 - 2.2357 i
0.0217 + 0.154 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
-1.0804 + 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 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
-1.1218 + 0.0325 i
0.0402 + 0.0364 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.1218 - 0.0325 i
0.0402 - 0.0364 i
3
1.163 + 0.5483 i
0.119 - 0.196 i
4
1.163 - 0.5483 i
0.119 + 0.196 i
5
-1.5279 + 0.3081 i
0.0341 + 0.318 i
6
-1.5279 - 0.3081 i
0.0341 - 0.318 i
7
-1.5219 + 1.8391 i
0.0553 + 0.27 i
8
-1.5219 - 1.8391 i
0.0553 - 0.27 i
9
2.9541
0.411
10
7.9992
38.7 i
11
-10.2438
0.832
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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
-1.0151 + 0.0765 i
0.000768 + 0.00426 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.0151 - 0.0765 i
0.000768 - 0.00426 i
3
-1.0747
0.00261
4
-1.1467
0.00461 i
5
1.1602 + 0.5504 i
0.118 - 0.182 i
6
1.1602 - 0.5504 i
0.118 + 0.182 i
7
-1.5997 + 1.3125 i
0.14 - 0.0448 i
8
-1.5997 - 1.3125 i
0.14 + 0.0448 i
9
2.5439
0.363
10
-0.7163 + 5.3288 i
0.684 + 0.222 i
11
-0.7163 - 5.3288 i
0.684 - 0.222 i
12
34.398
1.99 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
-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 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
-0.8466 + 0.0017 i
0.00144 + 0.00146 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.8466 - 0.0017 i
0.00144 - 0.00146 i
3
-1.0756
0.0129
4
-1.2245
0.0248 i
5
1.1627 + 0.5499 i
0.124 - 0.191 i
6
1.1627 - 0.5499 i
0.124 + 0.191 i
7
-1.7739 + 0.8677 i
0.169 + 0.181 i
8
-1.7739 - 0.8677 i
0.169 - 0.181 i
9
2.9581
0.388
10
-1.7112 + 2.6694 i
1.01 - 0.249 i
11
-1.7112 - 2.6694 i
1.01 + 0.249 i
12
4.4855
3.65 i
13
18.711
1.97
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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
-0.767 + 0.0005 i
0.000635 + 0.000637 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.767 - 0.0005 i
0.000635 - 0.000637 i
3
-1.0579
0.0114
4
-1.1899
0.0193 i
5
1.1625 + 0.5498 i
0.123 - 0.19 i
6
1.1625 - 0.5498 i
0.123 + 0.19 i
7
-1.7112 + 1.0033 i
0.145 + 0.0944 i
8
-1.7112 - 1.0033 i
0.145 - 0.0944 i
9
-1.4093 + 3.08 i
0.222 - 0.729 i
10
-1.4093 - 3.08 i
0.222 + 0.729 i
11
3.4304 + 0.0549 i
0.147 - 0.128 i
12
3.4304 - 0.0549 i
0.147 + 0.128 i
13
12.9331 + 13.7921 i
1.24 - 0.272 i
14
12.9331 - 13.7921 i
1.24 + 0.272 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

Designed by A. Sergeev.