Singularities of Møller-Plesset series: example "HF cc-pVDZ 1.5Re"

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
127.6372
56.3
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
0.0469
0.00103
Singularities of quadratic [1, 1, 0] approximant
2
0.0488
0.00105 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
2.247
1.14
Singularities of quadratic [1, 1, 1] approximant
2
-2.7447
2.23
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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
0.6063
0.00807
Singularities of quadratic [2, 1, 1] approximant
2
-0.3489 + 0.5673 i
0.00856 + 0.0000572 i
3
-0.3489 - 0.5673 i
0.00856 - 0.0000572 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
-1.5164
0.219
Singularities of quadratic [2, 2, 1] approximant
2
2.056 + 0.9651 i
0.51 - 0.0055 i
3
2.056 - 0.9651 i
0.51 + 0.0055 i
4
-15.2423
0.974 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
-1.4389 + 0.3127 i
0.115 + 0.0929 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.4389 - 0.3127 i
0.115 - 0.0929 i
3
1.629
0.303
4
-2.9076
0.458
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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
-1.0976 + 0.1954 i
0.00481 + 0.0203 i
Singularities of quadratic [3, 2, 2] approximant
2
-1.0976 - 0.1954 i
0.00481 - 0.0203 i
3
-1.1472
0.0152
4
1.6222
0.256
5
8.7645
1.74 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
-1.2449
0.0665
Singularities of quadratic [3, 3, 2] approximant
2
1.1651 + 0.5692 i
0.0278 - 0.0132 i
3
1.1651 - 0.5692 i
0.0278 + 0.0132 i
4
1.5946 + 0.9599 i
0.0202 + 0.0555 i
5
1.5946 - 0.9599 i
0.0202 - 0.0555 i
6
-36.1078
5.54e4 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
0.7716 + 0.0033 i
0.0107 - 0.0105 i
Singularities of quadratic [3, 3, 3] approximant
2
0.7716 - 0.0033 i
0.0107 + 0.0105 i
3
-1.2252 + 0.3397 i
0.0337 + 0.0334 i
4
-1.2252 - 0.3397 i
0.0337 - 0.0334 i
5
1.6621
0.454
6
-1.8126
0.0683
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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
-1.3075
0.109
Singularities of quadratic [4, 3, 3] approximant
2
1.8651 + 0.6169 i
0.477 - 0.125 i
3
1.8651 - 0.6169 i
0.477 + 0.125 i
4
0.3847 + 2.846 i
0.156 + 0.222 i
5
0.3847 - 2.846 i
0.156 - 0.222 i
6
4.4325
0.625
7
-15.8256
0.925 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
0.318 + 0.e-5 i
0.0000891 - 0.0000891 i
Singularities of quadratic [4, 4, 3] approximant
2
0.318 - 0.e-5 i
0.0000891 + 0.0000891 i
3
-1.3148
0.126
4
0.4736 + 2.1458 i
0.055 + 0.0509 i
5
0.4736 - 2.1458 i
0.055 - 0.0509 i
6
2.1194 + 1.3394 i
0.00415 + 0.125 i
7
2.1194 - 1.3394 i
0.00415 - 0.125 i
8
-11.7132
1.66 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
-1.2193
0.0508
Singularities of quadratic [4, 4, 4] approximant
2
1.6194 + 0.6238 i
0.168 - 0.0456 i
3
1.6194 - 0.6238 i
0.168 + 0.0456 i
4
-1.4498 + 1.1203 i
0.0399 + 0.0784 i
5
-1.4498 - 1.1203 i
0.0399 - 0.0784 i
6
-2.4409 + 1.5588 i
0.302 - 0.0507 i
7
-2.4409 - 1.5588 i
0.302 + 0.0507 i
8
3.0699
0.693
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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
-1.1722
0.0448
Singularities of quadratic [5, 4, 4] approximant
2
-1.406
0.0541 i
3
1.6568 + 0.4905 i
0.207 - 0.271 i
4
1.6568 - 0.4905 i
0.207 + 0.271 i
5
-1.839 + 0.625 i
0.0229 - 0.157 i
6
-1.839 - 0.625 i
0.0229 + 0.157 i
7
2.2579
0.276
8
-0.9158 + 3.3868 i
0.15 - 0.311 i
9
-0.9158 - 3.3868 i
0.15 + 0.311 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
-1.1398
0.0218
Singularities of quadratic [5, 5, 4] approximant
2
-1.4732
0.0442 i
3
1.4094 + 0.6247 i
0.0367 - 0.00438 i
4
1.4094 - 0.6247 i
0.0367 + 0.00438 i
5
-1.9068 + 1.0629 i
0.0653 + 0.0414 i
6
-1.9068 - 1.0629 i
0.0653 - 0.0414 i
7
2.19 + 0.5611 i
0.0311 + 0.16 i
8
2.19 - 0.5611 i
0.0311 - 0.16 i
9
0.7158 + 2.6785 i
0.0798 - 0.0129 i
10
0.7158 - 2.6785 i
0.0798 + 0.0129 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
-1.1265 + 0.0237 i
0.0338 + 0.0388 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.1265 - 0.0237 i
0.0338 - 0.0388 i
3
-1.3726
0.0957
4
1.6473 + 0.5058 i
0.161 - 0.249 i
5
1.6473 - 0.5058 i
0.161 + 0.249 i
6
2.5289
0.311
7
-1.701 + 2.5556 i
0.975 + 0.158 i
8
-1.701 - 2.5556 i
0.975 - 0.158 i
9
-2.7172 + 1.6934 i
0.108 - 0.434 i
10
-2.7172 - 1.6934 i
0.108 + 0.434 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
-1.0651
0.0104
Singularities of quadratic [6, 5, 5] approximant
2
-1.1832
0.0121 i
3
-1.5653 + 0.3897 i
0.00987 + 0.0425 i
4
-1.5653 - 0.3897 i
0.00987 - 0.0425 i
5
1.6375 + 0.5504 i
0.217 - 0.0891 i
6
1.6375 - 0.5504 i
0.217 + 0.0891 i
7
-1.5349 + 0.8927 i
0.0185 - 0.0391 i
8
-1.5349 - 0.8927 i
0.0185 + 0.0391 i
9
2.2106
0.269
10
-0.9623 + 2.9772 i
0.107 - 0.136 i
11
-0.9623 - 2.9772 i
0.107 + 0.136 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
-1.067
0.00753
Singularities of quadratic [6, 6, 5] approximant
2
-1.3167
0.0137 i
3
-1.2571 + 0.5817 i
0.00378 + 0.00888 i
4
-1.2571 - 0.5817 i
0.00378 - 0.00888 i
5
-1.2816 + 0.7322 i
0.00971 - 0.00691 i
6
-1.2816 - 0.7322 i
0.00971 + 0.00691 i
7
1.6412 + 0.5567 i
0.226 - 0.0628 i
8
1.6412 - 0.5567 i
0.226 + 0.0628 i
9
2.1717
0.259
10
-1.0188 + 2.7384 i
0.114 - 0.0628 i
11
-1.0188 - 2.7384 i
0.114 + 0.0628 i
12
396.2356
28.4 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
-0.9916 + 0.0168 i
0.00181 + 0.00197 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.9916 - 0.0168 i
0.00181 - 0.00197 i
3
-1.1262
0.00626
4
1.6109 + 0.4935 i
0.0286 - 0.2 i
5
1.6109 - 0.4935 i
0.0286 + 0.2 i
6
-1.9518
1.07 i
7
-1.5973 + 2.0649 i
0.0181 - 0.127 i
8
-1.5973 - 2.0649 i
0.0181 + 0.127 i
9
2.864 + 0.9524 i
0.166 + 0.33 i
10
2.864 - 0.9524 i
0.166 - 0.33 i
11
2.6686 + 3.8801 i
0.0879 + 0.295 i
12
2.6686 - 3.8801 i
0.0879 - 0.295 i
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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
-0.991 + 0.0167 i
0.0018 + 0.00196 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.991 - 0.0167 i
0.0018 - 0.00196 i
3
-1.1257
0.00624
4
1.6111 + 0.4945 i
0.0321 - 0.2 i
5
1.6111 - 0.4945 i
0.0321 + 0.2 i
6
-1.949
1.04 i
7
-1.5904 + 2.062 i
0.0175 - 0.125 i
8
-1.5904 - 2.062 i
0.0175 + 0.125 i
9
2.9371 + 0.9381 i
0.173 + 0.368 i
10
2.9371 - 0.9381 i
0.173 - 0.368 i
11
2.5094 + 3.8315 i
0.0642 + 0.281 i
12
2.5094 - 3.8315 i
0.0642 - 0.281 i
13
297.3509
4.29
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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
-0.9902 + 0.0169 i
0.00172 + 0.00187 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.9902 - 0.0169 i
0.00172 - 0.00187 i
3
-1.1238
0.00599
4
1.6073 + 0.4763 i
0.028 + 0.184 i
5
1.6073 - 0.4763 i
0.028 - 0.184 i
6
-1.9667
1.29 i
7
2.2598 + 1.121 i
0.121 + 0.142 i
8
2.2598 - 1.121 i
0.121 - 0.142 i
9
-1.5844 + 2.0983 i
0.00224 - 0.133 i
10
-1.5844 - 2.0983 i
0.00224 + 0.133 i
11
3.2553 + 1.842 i
0.278 + 0.12 i
12
3.2553 - 1.842 i
0.278 - 0.12 i
13
4.2746
0.642
14
5.9686
0.72 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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