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

Molecule X 1^Sigma+ State of HF. Basis AUG-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
78.782
35.
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
0.08
0.0023
Singularities of quadratic [1, 1, 0] approximant
2
0.0854
0.00238 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
-1.4344
0.323
Singularities of quadratic [1, 1, 1] approximant
2
11.5223
2.3
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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
-0.4049 + 0.4858 i
0.00814 + 0.00116 i
Singularities of quadratic [2, 1, 1] approximant
2
-0.4049 - 0.4858 i
0.00814 - 0.00116 i
3
0.713
0.00975
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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
-0.9833
0.0821
Singularities of quadratic [2, 2, 1] approximant
2
1.9261 + 1.8689 i
0.24 + 0.0763 i
3
1.9261 - 1.8689 i
0.24 - 0.0763 i
4
-6.5887
0.325 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
-0.0907
0.0000329
Singularities of quadratic [2, 2, 2] approximant
2
-0.0907
0.0000329 i
3
-0.8538
0.0263
4
1.8724
0.124
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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
-0.9423
0.0621
Singularities of quadratic [3, 2, 2] approximant
2
2.1859 + 1.8535 i
0.124 + 0.0895 i
3
2.1859 - 1.8535 i
0.124 - 0.0895 i
4
1.9762 + 3.9835 i
0.192 - 0.0303 i
5
1.9762 - 3.9835 i
0.192 + 0.0303 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
0.2299 + 0.e-4 i
0.000124 - 0.000124 i
Singularities of quadratic [3, 3, 2] approximant
2
0.2299 - 0.e-4 i
0.000124 + 0.000124 i
3
-0.9286
0.051
4
3.2589 + 1.0567 i
1.02 - 1.12 i
5
3.2589 - 1.0567 i
1.02 + 1.12 i
6
-13.384
0.706 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
-0.9777 + 0.004 i
7.04 + 0.565 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.9777 - 0.004 i
7.04 - 0.565 i
3
-1.4757 + 0.1716 i
0.67 + 0.518 i
4
-1.4757 - 0.1716 i
0.67 - 0.518 i
5
2.6201
0.988
6
-5.4176
3.9
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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
-0.8632
0.0269
Singularities of quadratic [4, 3, 3] approximant
2
-1.1513
0.0564 i
3
-1.4701
0.228
4
1.1612 + 1.9716 i
0.0471 - 0.00272 i
5
1.1612 - 1.9716 i
0.0471 + 0.00272 i
6
1.8883 + 1.7059 i
0.0267 + 0.0585 i
7
1.8883 - 1.7059 i
0.0267 - 0.0585 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
-0.8643
0.0278
Singularities of quadratic [4, 4, 3] approximant
2
-1.1435
0.0558 i
3
-1.4498
0.26
4
1.1185 + 2.1173 i
0.0545 + 0.00616 i
5
1.1185 - 2.1173 i
0.0545 - 0.00616 i
6
2.0954 + 1.8951 i
0.0208 + 0.0816 i
7
2.0954 - 1.8951 i
0.0208 - 0.0816 i
8
-66400.1306
485. 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
-0.8557
0.0267
Singularities of quadratic [4, 4, 4] approximant
2
-1.0631
0.0418 i
3
-1.3558
0.24
4
-1.0848 + 1.4395 i
0.00525 + 0.0431 i
5
-1.0848 - 1.4395 i
0.00525 - 0.0431 i
6
-1.1458 + 1.8599 i
0.0601 - 0.00818 i
7
-1.1458 - 1.8599 i
0.0601 + 0.00818 i
8
2.7429
2.27
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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
-0.8358
0.0296
Singularities of quadratic [5, 4, 4] approximant
2
-0.8986
0.0392 i
3
-1.0189
0.403
4
-1.4481
0.131 i
5
-1.9771
0.157
6
3.0474
652.
7
-3.3252
0.101 i
8
0.1488 + 3.5972 i
0.0205 + 0.174 i
9
0.1488 - 3.5972 i
0.0205 - 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
-0.8275 + 0.0272 i
0.00704 + 0.0114 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.8275 - 0.0272 i
0.00704 - 0.0114 i
3
-0.8939
0.0142
4
-2.1761
49.5 i
5
2.9704
18.5
6
0.3635 + 3.804 i
0.127 + 0.204 i
7
0.3635 - 3.804 i
0.127 - 0.204 i
8
-3.4599 + 1.9188 i
0.173 + 0.0763 i
9
-3.4599 - 1.9188 i
0.173 - 0.0763 i
10
56.9262
2.8 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
-0.8048 + 0.0269 i
0.00258 + 0.00446 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.8048 - 0.0269 i
0.00258 - 0.00446 i
3
-0.8461
0.00514
4
2.4891 + 0.0387 i
0.0766 - 0.0682 i
5
2.4891 - 0.0387 i
0.0766 + 0.0682 i
6
-3.1275
0.261 i
7
-1.2541 + 3.4607 i
0.361 + 0.0969 i
8
-1.2541 - 3.4607 i
0.361 - 0.0969 i
9
3.7859
0.966
10
-11.6994
0.551
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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
-0.7993 + 0.0241 i
0.00226 + 0.00383 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.7993 - 0.0241 i
0.00226 - 0.00383 i
3
-0.8389
0.00451
4
-1.0244 + 2.0626 i
0.00204 + 0.0499 i
5
-1.0244 - 2.0626 i
0.00204 - 0.0499 i
6
-1.2213 + 2.485 i
0.0489 - 0.00854 i
7
-1.2213 - 2.485 i
0.0489 + 0.00854 i
8
3.0043
33.2
9
0.6711 + 3.3519 i
0.0935 + 0.0798 i
10
0.6711 - 3.3519 i
0.0935 - 0.0798 i
11
-3.5092
0.145 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
-0.7865 + 0.0207 i
0.00129 + 0.00221 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.7865 - 0.0207 i
0.00129 - 0.00221 i
3
-0.8173
0.00256
4
1.7168
0.00924
5
1.7327
0.0094 i
6
-2.0795 + 0.8297 i
0.194 - 0.055 i
7
-2.0795 - 0.8297 i
0.194 + 0.055 i
8
-2.5924
0.192 i
9
-1.2732 + 3.4611 i
0.177 + 0.0829 i
10
-1.2732 - 3.4611 i
0.177 - 0.0829 i
11
3.9369
0.722
12
10.8764
3.2 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
-0.7861 + 0.0206 i
0.00127 + 0.00217 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.7861 - 0.0206 i
0.00127 - 0.00217 i
3
-0.8167
0.00252
4
1.6192
0.00655
5
1.6293
0.00663 i
6
-2.0788 + 0.829 i
0.193 - 0.06 i
7
-2.0788 - 0.829 i
0.193 + 0.06 i
8
-2.5815
0.195 i
9
-1.2285 + 3.4791 i
0.18 + 0.0703 i
10
-1.2285 - 3.4791 i
0.18 - 0.0703 i
11
3.8236
0.817
12
11.4128
3.55 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
-0.7825 + 0.0183 i
0.00119 + 0.00195 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.7825 - 0.0183 i
0.00119 - 0.00195 i
3
-0.8132
0.00237
4
-1.4407
1.68 i
5
-1.538
0.444
6
-2.533
0.333 i
7
2.8496
2.07
8
-0.9879 + 2.788 i
0.0725 + 0.0522 i
9
-0.9879 - 2.788 i
0.0725 - 0.0522 i
10
1.6716 + 2.8265 i
0.058 - 0.087 i
11
1.6716 - 2.8265 i
0.058 + 0.087 i
12
1.6783 + 4.5028 i
0.0396 + 0.11 i
13
1.6783 - 4.5028 i
0.0396 - 0.11 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
-0.7795 + 0.0163 i
0.00115 + 0.0018 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.7795 - 0.0163 i
0.00115 - 0.0018 i
3
-0.8107
0.00232
4
-1.1133
0.206 i
5
-1.1351
0.354
6
2.6046
0.32
7
-1.4014 + 2.805 i
0.0811 + 0.0958 i
8
-1.4014 - 2.805 i
0.0811 - 0.0958 i
9
-2.8892 + 1.2264 i
0.122 - 0.0956 i
10
-2.8892 - 1.2264 i
0.122 + 0.0956 i
11
3.4333
4.6 i
12
3.0934 + 3.2114 i
0.121 + 0.334 i
13
3.0934 - 3.2114 i
0.121 - 0.334 i
14
-9.633
0.989 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.