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

Molecule X 1^Sigma+ State of HF. Basis CC-PVDZ. Structure ""

Content


ExamplesBH-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-cc-pVDZ-1.5ReHF-cc-pVDZ-2ReHF-cc-pVDZ-ReO2--aug-cc-pVDZ
MoleculeX 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 O2-
BasisCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZCC-PVDZCC-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 [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 2. Singularities with their weights for the quadratic approximant [2, 2, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.0497 + 0.e-4 i
9.32e-7 - 9.32e-7 i
Singularities of quadratic [2, 2, 3] approximant
2
0.0497 - 0.e-4 i
9.32e-7 + 9.32e-7 i
3
-1.4168
0.038
4
2.1869
0.115
5
4.1308
2.42 i
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Table 3. Singularities with their weights for the quadratic approximant [2, 3, 3]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.6674
0.183
Singularities of quadratic [2, 3, 3] approximant
2
2.4018
0.574
3
-3.5917
0.207 i
4
-0.0381 + 4.7216 i
0.176 - 0.381 i
5
-0.0381 - 4.7216 i
0.176 + 0.381 i
6
71.6813
92.3 i
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Table 4. 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 5. Singularities with their weights for the quadratic approximant [3, 3, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.5082
0.0623
Singularities of quadratic [3, 3, 4] approximant
2
2.545
1.19
3
-1.8594 + 1.7986 i
0.0448 + 0.0833 i
4
-1.8594 - 1.7986 i
0.0448 - 0.0833 i
5
-0.7929 + 3.4832 i
0.0818 - 0.146 i
6
-0.7929 - 3.4832 i
0.0818 + 0.146 i
7
-4.1492
0.952 i
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Table 6. Singularities with their weights for the quadratic approximant [3, 4, 4]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.4023
0.0334
Singularities of quadratic [3, 4, 4] approximant
2
-1.7432
0.0467 i
3
2.647
2.8
4
-2.5132 + 0.9888 i
0.0358 + 0.0997 i
5
-2.5132 - 0.9888 i
0.0358 - 0.0997 i
6
0.0673 + 3.5199 i
0.0011 + 0.159 i
7
0.0673 - 3.5199 i
0.0011 - 0.159 i
8
48727.1801
50.6 i
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Table 7. 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 8. Singularities with their weights for the quadratic approximant [4, 4, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.371
0.0561
Singularities of quadratic [4, 4, 5] approximant
2
1.3734
0.0567 i
3
-1.4328
0.0393
4
-1.951
0.0662 i
5
2.5753
1.49
6
-2.2977 + 1.2315 i
0.032 + 0.0953 i
7
-2.2977 - 1.2315 i
0.032 - 0.0953 i
8
-0.0899 + 3.6685 i
0.0168 - 0.176 i
9
-0.0899 - 3.6685 i
0.0168 + 0.176 i
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Table 9. Singularities with their weights for the quadratic approximant [4, 5, 5]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.303 + 1.2977 i
0.000884 - 0.0000599 i
Singularities of quadratic [4, 5, 5] approximant
2
0.303 - 1.2977 i
0.000884 + 0.0000599 i
3
0.3089 + 1.3019 i
0.0000624 + 0.000888 i
4
0.3089 - 1.3019 i
0.0000624 - 0.000888 i
5
-1.3755
0.0157
6
-2.1414
0.125 i
7
-1.4287 + 2.4223 i
0.014 - 0.0365 i
8
-1.4287 - 2.4223 i
0.014 + 0.0365 i
9
3.7964 + 1.0061 i
0.168 - 0.173 i
10
3.7964 - 1.0061 i
0.168 + 0.173 i
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Table 10. 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 11. Singularities with their weights for the quadratic approximant [5, 5, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.3706
0.017
Singularities of quadratic [5, 5, 6] approximant
2
-1.8698
0.0503 i
3
-1.5221 + 1.6219 i
0.0142 + 0.0154 i
4
-1.5221 - 1.6219 i
0.0142 - 0.0154 i
5
2.5777 + 0.714 i
0.215 + 0.0955 i
6
2.5777 - 0.714 i
0.215 - 0.0955 i
7
-2.303 + 1.6594 i
0.0317 - 0.0208 i
8
-2.303 - 1.6594 i
0.0317 + 0.0208 i
9
-0.2328 + 3.2604 i
0.0121 - 0.0582 i
10
-0.2328 - 3.2604 i
0.0121 + 0.0582 i
11
3.7736
4.8
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Table 12. Singularities with their weights for the quadratic approximant [5, 6, 6]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.2369 + 0.0097 i
0.00113 + 0.00123 i
Singularities of quadratic [5, 6, 6] approximant
2
-1.2369 - 0.0097 i
0.00113 - 0.00123 i
3
-1.3298
0.00391
4
2.5099 + 0.3973 i
0.138 + 0.6 i
5
2.5099 - 0.3973 i
0.138 - 0.6 i
6
-2.6719
2.66 i
7
3.1153
0.516
8
-1.8069 + 2.6526 i
0.0821 - 0.125 i
9
-1.8069 - 2.6526 i
0.0821 + 0.125 i
10
0.9158 + 5.7184 i
0.456 - 0.106 i
11
0.9158 - 5.7184 i
0.456 + 0.106 i
12
14.0242
1.26 i
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Table 13. 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 14. Singularities with their weights for the quadratic approximant [6, 6, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.4511
1.93e-6
Singularities of quadratic [6, 6, 7] approximant
2
0.4511
1.93e-6 i
3
-1.2724
0.00106
4
-1.288 + 0.0246 i
0.000352 - 0.00136 i
5
-1.288 - 0.0246 i
0.000352 + 0.00136 i
6
2.5235 + 0.3758 i
0.19 - 1.05 i
7
2.5235 - 0.3758 i
0.19 + 1.05 i
8
2.7833
0.494
9
-3.1586
28.6 i
10
-1.9982 + 3.2615 i
0.149 + 0.466 i
11
-1.9982 - 3.2615 i
0.149 - 0.466 i
12
-3.6285 + 4.9255 i
6.31 + 2.52 i
13
-3.6285 - 4.9255 i
6.31 - 2.52 i
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Table 15. Singularities with their weights for the quadratic approximant [6, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-1.3019
0.0019
Singularities of quadratic [6, 7, 7] approximant
2
-1.3022 + 0.0439 i
0.000103 + 0.00268 i
3
-1.3022 - 0.0439 i
0.000103 - 0.00268 i
4
2.4505 + 0.314 i
0.252 + 0.0665 i
5
2.4505 - 0.314 i
0.252 - 0.0665 i
6
3.1119 + 0.856 i
0.297 + 0.0313 i
7
3.1119 - 0.856 i
0.297 - 0.0313 i
8
-1.6236 + 3.1011 i
0.0853 + 0.187 i
9
-1.6236 - 3.1011 i
0.0853 - 0.187 i
10
-3.8269 + 0.6813 i
2.14 - 2.09 i
11
-3.8269 - 0.6813 i
2.14 + 2.09 i
12
4.8758
0.513
13
-5.5228
238. i
14
13.1502
4.67 i
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ExamplesBH-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-cc-pVDZ-1.5ReHF-cc-pVDZ-2ReHF-cc-pVDZ-ReO2--aug-cc-pVDZ
MoleculeX 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 O2-
BasisCC-PVDZCC-PVDZCC-PVDZCC-PVQZCC-PVQZCC-PVQZCC-PVTZCC-PVTZCC-PVTZAUG-CC-PV5ZAUG-CC-PVQZCC-PVDZCC-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.