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

Molecule X 1^Sigma+ State of BH. Basis CC-PVQZ. 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
1.5137
0.499
Singularities of quadratic [2, 2, 2] approximant
2
3.1207
0.616 i
3
4.4818
10.7
4
-5.443
0.36
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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
1.3496
0.0979
Singularities of quadratic [2, 2, 3] approximant
2
1.3339 + 0.2174 i
0.00204 - 0.141 i
3
1.3339 - 0.2174 i
0.00204 + 0.141 i
4
-5.0169
0.335
5
-20.1982
0.327 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
1.5719 + 0.314 i
0.28 - 0.222 i
Singularities of quadratic [2, 3, 3] approximant
2
1.5719 - 0.314 i
0.28 + 0.222 i
3
2.1468
0.506
4
-4.2318
0.157
5
-17.9777
0.22 i
6
42.9622
0.475 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
1.599 + 0.3238 i
0.322 - 0.174 i
Singularities of quadratic [3, 3, 3] approximant
2
1.599 - 0.3238 i
0.322 + 0.174 i
3
2.3355
0.834
4
-4.2686
0.152
5
11.4329
0.49 i
6
-224.3151
0.111 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
1.6051 + 0.3334 i
0.32 - 0.13 i
Singularities of quadratic [3, 3, 4] approximant
2
1.6051 - 0.3334 i
0.32 + 0.13 i
3
2.4038
1.11
4
-4.1648
0.128
5
13.4634
0.45 i
6
-0.2111 + 17.9753 i
0.07 + 0.279 i
7
-0.2111 - 17.9753 i
0.07 - 0.279 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
1.6046 + 0.3392 i
0.315 - 0.0996 i
Singularities of quadratic [3, 4, 4] approximant
2
1.6046 - 0.3392 i
0.315 + 0.0996 i
3
2.365
1.07
4
-3.0738
0.0273
5
-3.3638
0.0286 i
6
-6.6791 + 2.0373 i
0.0594 - 0.224 i
7
-6.6791 - 2.0373 i
0.0594 + 0.224 i
8
58.4353
0.6 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
1.5999 + 0.3101 i
0.314 - 0.256 i
Singularities of quadratic [4, 4, 4] approximant
2
1.5999 - 0.3101 i
0.314 + 0.256 i
3
-1.8672 + 0.0011 i
0.00341 + 0.00342 i
4
-1.8672 - 0.0011 i
0.00341 - 0.00342 i
5
2.3937
0.811
6
-4.0805
0.0956
7
5.9291
0.818 i
8
13.8429
8.52
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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
-0.2473 + 0.e-5 i
2.7e-6 + 2.7e-6 i
Singularities of quadratic [4, 4, 5] approximant
2
-0.2473 - 0.e-5 i
2.7e-6 - 2.7e-6 i
3
1.5973 + 0.3062 i
0.285 - 0.279 i
4
1.5973 - 0.3062 i
0.285 + 0.279 i
5
2.4282
0.883
6
-4.0874
0.105
7
6.5316
0.578 i
8
12.2942 + 16.0297 i
0.137 + 0.446 i
9
12.2942 - 16.0297 i
0.137 - 0.446 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
1.2258 + 0.0048 i
0.0213 - 0.0215 i
Singularities of quadratic [4, 5, 5] approximant
2
1.2258 - 0.0048 i
0.0213 + 0.0215 i
3
1.6422 + 0.3741 i
0.322 + 0.12 i
4
1.6422 - 0.3741 i
0.322 - 0.12 i
5
2.3724
1.51
6
-3.1434
0.043
7
-3.3221
0.0445 i
8
-5.4841
27.9
9
-10.1229
0.219 i
10
75.7234
0.695 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
1.2942 + 0.0094 i
0.0303 - 0.0307 i
Singularities of quadratic [5, 5, 5] approximant
2
1.2942 - 0.0094 i
0.0303 + 0.0307 i
3
1.6568 + 0.3847 i
0.293 + 0.182 i
4
1.6568 - 0.3847 i
0.293 - 0.182 i
5
2.3818
1.85
6
-3.5478
0.0492
7
-4.7808
0.0541 i
8
-4.9641 + 3.2827 i
0.0704 - 0.111 i
9
-4.9641 - 3.2827 i
0.0704 + 0.111 i
10
-22.0044
4.01
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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
1.3932 + 0.0365 i
0.0338 - 0.0298 i
Singularities of quadratic [5, 5, 6] approximant
2
1.3932 - 0.0365 i
0.0338 + 0.0298 i
3
1.7338 + 0.4579 i
0.0817 - 0.28 i
4
1.7338 - 0.4579 i
0.0817 + 0.28 i
5
2.0833
1.31
6
-3.0502 + 0.0722 i
0.0161 + 0.0163 i
7
-3.0502 - 0.0722 i
0.0161 - 0.0163 i
8
-4.371
0.0908
9
4.9312
86.2 i
10
7.6856
0.644
11
-11.1518
0.499 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
1.4418 + 0.0661 i
0.0502 - 0.038 i
Singularities of quadratic [5, 6, 6] approximant
2
1.4418 - 0.0661 i
0.0502 + 0.038 i
3
1.6875 + 0.5343 i
0.0302 - 0.147 i
4
1.6875 - 0.5343 i
0.0302 + 0.147 i
5
2.0476 + 0.2382 i
0.64 - 0.00592 i
6
2.0476 - 0.2382 i
0.64 + 0.00592 i
7
3.0082
582.
8
-3.2416
0.174
9
-3.3216
0.172 i
10
-5.1783
0.986
11
-10.0571
0.284 i
12
133.3694
1.05 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
1.5739 + 0.1599 i
1.42 - 3.9 i
Singularities of quadratic [6, 6, 6] approximant
2
1.5739 - 0.1599 i
1.42 + 3.9 i
3
1.577 + 0.436 i
0.159 - 0.0536 i
4
1.577 - 0.436 i
0.159 + 0.0536 i
5
1.6681 + 0.1793 i
0.0138 - 0.205 i
6
1.6681 - 0.1793 i
0.0138 + 0.205 i
7
1.9698
0.184
8
-3.1442
0.0333
9
-3.3514
0.0359 i
10
-5.2287
5.81
11
17.7283
0.469 i
12
-17.7717
0.176 i
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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
1.466
0.0527
Singularities of quadratic [6, 6, 7] approximant
2
1.4752 + 0.3131 i
0.0292 + 0.00507 i
3
1.4752 - 0.3131 i
0.0292 - 0.00507 i
4
1.4705 + 0.4241 i
0.014 - 0.0237 i
5
1.4705 - 0.4241 i
0.014 + 0.0237 i
6
1.9415 + 0.5142 i
0.0807 + 0.124 i
7
1.9415 - 0.5142 i
0.0807 - 0.124 i
8
-3.1118
0.0144
9
-3.5455
0.0171 i
10
-7.5919
0.108
11
-4.4657 + 9.1025 i
0.00285 + 0.078 i
12
-4.4657 - 9.1025 i
0.00285 - 0.078 i
13
-32.3506
0.2 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
1.3979 + 0.195 i
0.0564 + 0.0726 i
Singularities of quadratic [6, 7, 7] approximant
2
1.3979 - 0.195 i
0.0564 - 0.0726 i
3
1.4384 + 0.2008 i
0.0676 - 0.0508 i
4
1.4384 - 0.2008 i
0.0676 + 0.0508 i
5
1.6765 + 0.4204 i
0.108 + 0.38 i
6
1.6765 - 0.4204 i
0.108 - 0.38 i
7
2.0722
0.648
8
-3.215
0.25
9
-3.2693
0.263 i
10
4.3451
11.5 i
11
-4.9945
0.545
12
5.3458
0.827
13
-10.7331
0.296 i
14
224.8864
1.54 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.