Singularities of Møller-Plesset series: example "BH-cc-pVQZ-2Re"

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
1.3609 + 0.6111 i
0.336 - 0.0504 i
Singularities of quadratic [2, 2, 2] approximant
2
1.3609 - 0.6111 i
0.336 + 0.0504 i
3
-2.1331
0.129
4
2.4636
0.83
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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
0.8721 + 0.046 i
0.0151 - 0.0129 i
Singularities of quadratic [2, 2, 3] approximant
2
0.8721 - 0.046 i
0.0151 + 0.0129 i
3
-1.863 + 1.164 i
0.0585 + 0.0153 i
4
-1.863 - 1.164 i
0.0585 - 0.0153 i
5
3.8212
0.188
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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
1.3214 + 0.3009 i
0.71 + 0.407 i
Singularities of quadratic [2, 3, 3] approximant
2
1.3214 - 0.3009 i
0.71 - 0.407 i
3
1.7917
0.44
4
-2.2674
0.234
5
-7.6377
0.204 i
6
165.2231
1.13 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
1.3129 + 0.3446 i
0.397 + 0.452 i
Singularities of quadratic [3, 3, 3] approximant
2
1.3129 - 0.3446 i
0.397 - 0.452 i
3
1.9386
0.393
4
-1.8922 + 0.5598 i
0.0202 + 0.0646 i
5
-1.8922 - 0.5598 i
0.0202 - 0.0646 i
6
-2.291
0.0509
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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
1.302 + 0.4236 i
0.128 - 0.507 i
Singularities of quadratic [3, 3, 4] approximant
2
1.302 - 0.4236 i
0.128 + 0.507 i
3
1.794
0.385
4
-2.7722
3.65
5
-3.1795
2.39 i
6
4.3762
1.8 i
7
37.5955
3.13
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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
-0.2199 + 0.e-5 i
8.22e-6 + 8.22e-6 i
Singularities of quadratic [3, 4, 4] approximant
2
-0.2199 - 0.e-5 i
8.22e-6 - 8.22e-6 i
3
1.3745
0.177
4
1.2925 + 0.5483 i
0.17 + 0.0555 i
5
1.2925 - 0.5483 i
0.17 - 0.0555 i
6
-2.8806
3.1
7
-4.062
0.162 i
8
19.0849
0.296 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
1.2293 + 0.3693 i
0.236 + 0.224 i
Singularities of quadratic [4, 4, 4] approximant
2
1.2293 - 0.3693 i
0.236 - 0.224 i
3
1.5348 + 0.2849 i
0.344 - 0.0904 i
4
1.5348 - 0.2849 i
0.344 + 0.0904 i
5
1.7967
0.893
6
-2.5131 + 0.642 i
0.201 + 0.184 i
7
-2.5131 - 0.642 i
0.201 - 0.184 i
8
-5.7011
0.356
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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
1.1929 + 0.4479 i
0.0654 - 0.108 i
Singularities of quadratic [4, 4, 5] approximant
2
1.1929 - 0.4479 i
0.0654 + 0.108 i
3
1.3846 + 0.0275 i
0.0524 - 0.0415 i
4
1.3846 - 0.0275 i
0.0524 + 0.0415 i
5
-2.5311 + 0.5498 i
0.0797 + 0.339 i
6
-2.5311 - 0.5498 i
0.0797 - 0.339 i
7
3.0338
0.806
8
-6.8035
0.256
9
7.6544
0.922 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
1.1783 + 0.4516 i
0.0521 - 0.0814 i
Singularities of quadratic [4, 5, 5] approximant
2
1.1783 - 0.4516 i
0.0521 + 0.0814 i
3
1.2732
0.05
4
1.3624
0.0715 i
5
-2.4981 + 0.5002 i
0.0684 - 0.33 i
6
-2.4981 - 0.5002 i
0.0684 + 0.33 i
7
3.2322
1.36
8
-5.9468
0.181
9
6.5528
0.923 i
10
-321.6057
1.56 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
1.1796 + 0.4513 i
0.0527 - 0.086 i
Singularities of quadratic [5, 5, 5] approximant
2
1.1796 - 0.4513 i
0.0527 + 0.086 i
3
1.2905
0.0589
4
1.3986
0.0918 i
5
-2.5154 + 0.5455 i
0.0576 + 0.331 i
6
-2.5154 - 0.5455 i
0.0576 - 0.331 i
7
3.4429 + 1.1421 i
0.0167 - 0.876 i
8
3.4429 - 1.1421 i
0.0167 + 0.876 i
9
-6.0682
0.244
10
11.5789
0.923
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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
1.1893 + 0.4488 i
0.0551 - 0.113 i
Singularities of quadratic [5, 5, 6] approximant
2
1.1893 - 0.4488 i
0.0551 + 0.113 i
3
1.3777
0.105
4
1.556
0.238 i
5
-2.5214 + 0.7078 i
0.179 + 0.0875 i
6
-2.5214 - 0.7078 i
0.179 - 0.0875 i
7
2.6159 + 1.6433 i
0.0602 + 0.263 i
8
2.6159 - 1.6433 i
0.0602 - 0.263 i
9
-5.4934 + 1.8217 i
0.195 + 0.289 i
10
-5.4934 - 1.8217 i
0.195 - 0.289 i
11
15.6132
0.436
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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
1.2053 + 0.4441 i
0.0464 - 0.182 i
Singularities of quadratic [5, 6, 6] approximant
2
1.2053 - 0.4441 i
0.0464 + 0.182 i
3
1.5672
0.281
4
1.6058 + 0.5728 i
0.221 + 0.107 i
5
1.6058 - 0.5728 i
0.221 - 0.107 i
6
1.7984 + 0.8906 i
0.265 - 0.0845 i
7
1.7984 - 0.8906 i
0.265 + 0.0845 i
8
-2.5376 + 0.6034 i
0.196 + 0.256 i
9
-2.5376 - 0.6034 i
0.196 - 0.256 i
10
-5.7674
0.287
11
28.7043
0.355 i
12
-117.831
0.642 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
1.2491 + 0.0271 i
0.0626 - 0.0427 i
Singularities of quadratic [6, 6, 6] approximant
2
1.2491 - 0.0271 i
0.0626 + 0.0427 i
3
1.1829 + 0.4562 i
0.0516 - 0.0885 i
4
1.1829 - 0.4562 i
0.0516 + 0.0885 i
5
1.4361
0.506
6
2.1816 + 0.3841 i
0.427 - 0.33 i
7
2.1816 - 0.3841 i
0.427 + 0.33 i
8
2.0549 + 1.3166 i
0.12 + 0.145 i
9
2.0549 - 1.3166 i
0.12 - 0.145 i
10
-2.5339 + 0.6026 i
0.177 + 0.253 i
11
-2.5339 - 0.6026 i
0.177 - 0.253 i
12
-6.3062
0.362
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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
1.2175 + 0.4509 i
0.1 - 0.251 i
Singularities of quadratic [6, 6, 7] approximant
2
1.2175 - 0.4509 i
0.1 + 0.251 i
3
1.4181 + 0.4734 i
0.336 + 0.108 i
4
1.4181 - 0.4734 i
0.336 - 0.108 i
5
1.6022 + 0.4412 i
32.6 + 146. i
6
1.6022 - 0.4412 i
32.6 - 146. i
7
2.1694
0.418
8
-2.5357 + 0.6169 i
0.198 + 0.226 i
9
-2.5357 - 0.6169 i
0.198 - 0.226 i
10
5.7464
2.13 i
11
-5.9498
0.391
12
-17.722
0.642 i
13
20.9485
1.5
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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
-1.0918
0.000494
Singularities of quadratic [6, 7, 7] approximant
2
-1.0918
0.000494 i
3
1.2057 + 0.4603 i
0.119 - 0.131 i
4
1.2057 - 0.4603 i
0.119 + 0.131 i
5
1.6263
0.216
6
1.6172 + 0.3753 i
0.312 - 0.0547 i
7
1.6172 - 0.3753 i
0.312 + 0.0547 i
8
1.8831 + 0.6584 i
0.864 - 0.254 i
9
1.8831 - 0.6584 i
0.864 + 0.254 i
10
-2.5555 + 0.6181 i
0.293 + 0.232 i
11
-2.5555 - 0.6181 i
0.293 - 0.232 i
12
-5.0771
0.227
13
16.2706
0.311 i
14
-50.0076
0.428 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

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Designed by A. Sergeev.