Singularities of Møller-Plesset series: example "BH-cc-pVTZ-1.5Re"

Molecule X 1^Sigma+ State of BH. Basis CC-PVTZ. Structure ""

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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

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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
1.6083 + 0.6144 i
0.374 - 0.0803 i
Singularities of quadratic [2, 2, 2] approximant
2
1.6083 - 0.6144 i
0.374 + 0.0803 i
3
2.5562
0.898
4
-3.7243
0.139
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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
1.5678 + 0.5103 i
0.402 - 0.308 i
Singularities of quadratic [2, 2, 3] approximant
2
1.5678 - 0.5103 i
0.402 + 0.308 i
3
2.0623
0.464
4
-3.7131
0.159
5
-23.7455
0.203 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
1.5697 + 0.4571 i
0.352 - 0.573 i
Singularities of quadratic [2, 3, 3] approximant
2
1.5697 - 0.4571 i
0.352 + 0.573 i
3
2.0015
0.45
4
-3.8895
0.205
5
-17.2548
0.236 i
6
439.7192
1.26 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
1.5698 + 0.4577 i
0.362 - 0.565 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5698 - 0.4577 i
0.362 + 0.565 i
3
1.9945
0.45
4
-3.8627
0.195
5
-22.9793
0.199 i
6
62.1794
14. 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
1.57 + 0.4555 i
0.349 - 0.583 i
Singularities of quadratic [3, 3, 4] approximant
2
1.57 - 0.4555 i
0.349 + 0.583 i
3
2.001
0.451
4
-3.882
0.203
5
-17.953
0.225 i
6
51.9427
2.37 i
7
113.3485
1.
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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
0.0088
4.47e-10 - 4.47e-10 i
Singularities of quadratic [3, 4, 4] approximant
2
0.0088
4.47e-10 + 4.47e-10 i
3
1.569 + 0.4541 i
0.326 - 0.591 i
4
1.569 - 0.4541 i
0.326 + 0.591 i
5
2.0091
0.452
6
-3.8769
0.199
7
-17.7487
0.237 i
8
427.8969
1.23 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
0.9977 + 0.0015 i
0.0089 - 0.00887 i
Singularities of quadratic [4, 4, 4] approximant
2
0.9977 - 0.0015 i
0.0089 + 0.00887 i
3
1.5278 + 0.4186 i
0.0624 + 0.367 i
4
1.5278 - 0.4186 i
0.0624 - 0.367 i
5
2.2372
0.503
6
-4.1898
0.299
7
-5.532
83.7 i
8
-8.446
0.25
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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
0.9489 + 1.0046 i
0.011 + 0.00254 i
Singularities of quadratic [4, 4, 5] approximant
2
0.9489 - 1.0046 i
0.011 - 0.00254 i
3
0.9448 + 1.0144 i
0.00265 - 0.011 i
4
0.9448 - 1.0144 i
0.00265 + 0.011 i
5
1.4892 + 0.4362 i
0.0638 + 0.198 i
6
1.4892 - 0.4362 i
0.0638 - 0.198 i
7
2.3631
0.635
8
-4.0395
0.324
9
-14.0228
0.199 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
1.0086
0.00184
Singularities of quadratic [4, 5, 5] approximant
2
1.0115
0.00185 i
3
1.409 + 0.4355 i
0.0125 + 0.0693 i
4
1.409 - 0.4355 i
0.0125 - 0.0693 i
5
-2.8942
0.0162
6
-3.238
0.019 i
7
3.2811
2.82
8
-7.0787 + 3.0878 i
0.0463 + 0.0751 i
9
-7.0787 - 3.0878 i
0.0463 - 0.0751 i
10
24.1824
0.243 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
1.0634
0.0027
Singularities of quadratic [5, 5, 5] approximant
2
1.0678
0.00272 i
3
1.4107 + 0.4391 i
0.00806 + 0.0716 i
4
1.4107 - 0.4391 i
0.00806 - 0.0716 i
5
-3.1512
0.0272
6
3.2161
2.22
7
-3.9459
0.0409 i
8
-5.3866 + 4.1494 i
0.0525 + 0.0763 i
9
-5.3866 - 4.1494 i
0.0525 - 0.0763 i
10
91.0823
0.156 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
1.4217
0.0333
Singularities of quadratic [5, 5, 6] approximant
2
1.4769
0.0398 i
3
1.4273 + 0.4706 i
0.044 - 0.0917 i
4
1.4273 - 0.4706 i
0.044 + 0.0917 i
5
-2.4695
0.016
6
-2.5179
0.0163 i
7
3.2478 + 1.0416 i
0.577 + 0.832 i
8
3.2478 - 1.0416 i
0.577 - 0.832 i
9
-4.5648
6.3
10
5.8612
0.603
11
-9.7269
0.204 i
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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
1.4609 + 0.5172 i
0.146 - 0.0348 i
Singularities of quadratic [5, 6, 6] approximant
2
1.4609 - 0.5172 i
0.146 + 0.0348 i
3
1.6687 + 0.2848 i
0.282 + 0.0582 i
4
1.6687 - 0.2848 i
0.282 - 0.0582 i
5
1.8458 + 0.5573 i
0.536 - 0.544 i
6
1.8458 - 0.5573 i
0.536 + 0.544 i
7
-2.6789
0.0148
8
-2.8129
0.0155 i
9
2.9183
2.72
10
-5.7136
0.298
11
-9.0497
0.104 i
12
78.8297
0.441 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
1.4582 + 0.4875 i
0.129 - 0.128 i
Singularities of quadratic [6, 6, 6] approximant
2
1.4582 - 0.4875 i
0.129 + 0.128 i
3
1.8178 + 0.0952 i
0.144 - 0.0767 i
4
1.8178 - 0.0952 i
0.144 + 0.0767 i
5
2.3614
2.68
6
2.2598 + 0.7262 i
1.09 - 0.407 i
7
2.2598 - 0.7262 i
1.09 + 0.407 i
8
-2.7458
0.0221
9
-2.8788
0.0228 i
10
-5.8522 + 0.9418 i
0.0107 + 0.281 i
11
-5.8522 - 0.9418 i
0.0107 - 0.281 i
12
-25.4217
8.56
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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
1.4589 + 0.4851 i
0.127 - 0.141 i
Singularities of quadratic [6, 6, 7] approximant
2
1.4589 - 0.4851 i
0.127 + 0.141 i
3
1.8812 + 0.019 i
0.0537 - 0.0488 i
4
1.8812 - 0.019 i
0.0537 + 0.0488 i
5
2.1825
0.767
6
2.2639 + 0.7657 i
0.946 - 0.211 i
7
2.2639 - 0.7657 i
0.946 + 0.211 i
8
-2.7465
0.0254
9
-2.8653
0.0261 i
10
-5.7581 + 0.7237 i
0.000339 + 0.385 i
11
-5.7581 - 0.7237 i
0.000339 - 0.385 i
12
-31.3405
2.99
13
56.5769
0.402 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
1.4609 + 0.4811 i
0.119 - 0.167 i
Singularities of quadratic [6, 7, 7] approximant
2
1.4609 - 0.4811 i
0.119 + 0.167 i
3
1.8483
0.211
4
1.991 + 0.3299 i
0.437 - 0.217 i
5
1.991 - 0.3299 i
0.437 + 0.217 i
6
2.2351 + 0.7581 i
0.924 + 0.00751 i
7
2.2351 - 0.7581 i
0.924 - 0.00751 i
8
-2.8111 + 0.0449 i
0.0285 + 0.0307 i
9
-2.8111 - 0.0449 i
0.0285 - 0.0307 i
10
-4.4526 + 0.3603 i
0.164 + 0.0336 i
11
-4.4526 - 0.3603 i
0.164 - 0.0336 i
12
-10.1087
0.12
13
-27.5529
0.154 i
14
46.0536
0.327 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.