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

Molecule X 1^Sigma+ State of BH. Basis 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
0.7324
0.106
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
0.95
0.184
Singularities of quadratic [1, 1, 0] approximant
2
49.2127
1.32 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
1.4318
3.14
Singularities of quadratic [1, 1, 1] approximant
2
-2.9953
0.122
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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
1.6439
9.03
Singularities of quadratic [2, 1, 1] approximant
2
2.5977
1.12 i
3
-14.3916
0.714
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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
-0.4252
0.00159
Singularities of quadratic [2, 2, 1] approximant
2
-0.433
0.0016 i
3
4.0331 + 1.6193 i
0.056 - 0.0694 i
4
4.0331 - 1.6193 i
0.056 + 0.0694 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
1.3157 + 0.3983 i
0.186 - 0.987 i
Singularities of quadratic [2, 2, 2] approximant
2
1.3157 - 0.3983 i
0.186 + 0.987 i
3
1.6397
0.439
4
-3.8802
0.168
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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
1.2771 + 0.5524 i
0.321 - 0.131 i
Singularities of quadratic [3, 2, 2] approximant
2
1.2771 - 0.5524 i
0.321 + 0.131 i
3
1.7386
0.395
4
-3.9437
0.141
5
13.1327
0.222 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
1.2935 + 0.3668 i
0.4 + 0.34 i
Singularities of quadratic [3, 3, 2] approximant
2
1.2935 - 0.3668 i
0.4 - 0.34 i
3
2.425
0.359
4
-5.5007 + 3.531 i
0.351 + 0.139 i
5
-5.5007 - 3.531 i
0.351 - 0.139 i
6
8.6513
3.8 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
1.3175 + 0.454 i
0.365 - 0.645 i
Singularities of quadratic [3, 3, 3] approximant
2
1.3175 - 0.454 i
0.365 + 0.645 i
3
1.758
0.419
4
-2.8057
0.0903
5
-2.9621 + 0.7466 i
0.0533 - 0.0852 i
6
-2.9621 - 0.7466 i
0.0533 + 0.0852 i
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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
-0.4635
0.0000408
Singularities of quadratic [4, 3, 3] approximant
2
-0.4639
0.0000408 i
3
1.063
0.0198
4
1.0307 + 0.7645 i
0.0109 + 0.0137 i
5
1.0307 - 0.7645 i
0.0109 - 0.0137 i
6
-1.4316 + 1.9893 i
0.00689 - 0.00631 i
7
-1.4316 - 1.9893 i
0.00689 + 0.00631 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
1.164 + 0.3736 i
0.105 + 0.0412 i
Singularities of quadratic [4, 4, 3] approximant
2
1.164 - 0.3736 i
0.105 - 0.0412 i
3
1.7894 + 0.8373 i
0.125 + 0.0315 i
4
1.7894 - 0.8373 i
0.125 - 0.0315 i
5
2.8489
0.185
6
-2.7812 + 2.9405 i
0.075 + 0.0148 i
7
-2.7812 - 2.9405 i
0.075 - 0.0148 i
8
10.4607
5.75 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
1.2502
0.0432
Singularities of quadratic [4, 4, 4] approximant
2
1.1618 + 0.4721 i
0.0508 - 0.0941 i
3
1.1618 - 0.4721 i
0.0508 + 0.0941 i
4
1.3085
0.0532 i
5
3.565
1.1
6
-3.5389 + 2.2055 i
0.00178 + 0.18 i
7
-3.5389 - 2.2055 i
0.00178 - 0.18 i
8
-27.6148
0.0746
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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
0.4945
0.000095
Singularities of quadratic [5, 4, 4] approximant
2
0.4946
0.0000951 i
3
1.1379 + 0.4285 i
0.0164 + 0.0564 i
4
1.1379 - 0.4285 i
0.0164 - 0.0564 i
5
-2.0586 + 0.0624 i
0.00226 + 0.00214 i
6
-2.0586 - 0.0624 i
0.00226 - 0.00214 i
7
2.4202
0.746
8
-1.5227 + 3.6752 i
0.00838 + 0.0402 i
9
-1.5227 - 3.6752 i
0.00838 - 0.0402 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
1.1612 + 0.471 i
0.0433 - 0.105 i
Singularities of quadratic [5, 5, 4] approximant
2
1.1612 - 0.471 i
0.0433 + 0.105 i
3
1.2893
0.0779
4
1.4261
0.131 i
5
3.105 + 2.3237 i
0.0883 - 0.236 i
6
3.105 - 2.3237 i
0.0883 + 0.236 i
7
-2.9863 + 2.4893 i
0.064 + 0.0572 i
8
-2.9863 - 2.4893 i
0.064 - 0.0572 i
9
5.3341 + 18.011 i
0.198 - 0.227 i
10
5.3341 - 18.011 i
0.198 + 0.227 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
1.1899 + 0.45 i
0.0843 + 0.175 i
Singularities of quadratic [5, 5, 5] approximant
2
1.1899 - 0.45 i
0.0843 - 0.175 i
3
1.3683 + 0.6412 i
0.179 + 0.00106 i
4
1.3683 - 0.6412 i
0.179 - 0.00106 i
5
1.6392 + 0.6696 i
0.303 - 0.379 i
6
1.6392 - 0.6696 i
0.303 + 0.379 i
7
1.9415
0.435
8
-3.2089 + 2.5267 i
0.092 + 0.083 i
9
-3.2089 - 2.5267 i
0.092 - 0.083 i
10
-21.7813
0.11
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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
1.1911 + 0.4348 i
0.122 + 0.104 i
Singularities of quadratic [6, 5, 5] approximant
2
1.1911 - 0.4348 i
0.122 - 0.104 i
3
1.2771 + 0.6296 i
0.137 - 0.0853 i
4
1.2771 - 0.6296 i
0.137 + 0.0853 i
5
1.4481 + 0.582 i
0.396 + 0.218 i
6
1.4481 - 0.582 i
0.396 - 0.218 i
7
2.3125
0.414
8
-3.1672 + 2.5206 i
0.0876 + 0.0748 i
9
-3.1672 - 2.5206 i
0.0876 - 0.0748 i
10
-18.2277
0.134
11
33.5154
0.259 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
1.2051 + 0.4502 i
0.157 + 0.22 i
Singularities of quadratic [6, 6, 5] approximant
2
1.2051 - 0.4502 i
0.157 - 0.22 i
3
1.312 + 0.5831 i
0.289 - 0.11 i
4
1.312 - 0.5831 i
0.289 + 0.11 i
5
1.4929 + 0.5375 i
1.44 + 0.249 i
6
1.4929 - 0.5375 i
1.44 - 0.249 i
7
2.3856
0.431
8
-3.1994 + 2.5018 i
0.084 + 0.0883 i
9
-3.1994 - 2.5018 i
0.084 - 0.0883 i
10
-10.8049 + 24.8143 i
0.162 + 0.0435 i
11
-10.8049 - 24.8143 i
0.162 - 0.0435 i
12
30.6162
0.225 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
-0.779
0.0000749
Singularities of quadratic [6, 6, 6] approximant
2
-0.779
0.0000749 i
3
1.2021 + 0.4834 i
0.172 - 0.261 i
4
1.2021 - 0.4834 i
0.172 + 0.261 i
5
1.4136 + 0.4964 i
0.359 + 0.0796 i
6
1.4136 - 0.4964 i
0.359 - 0.0796 i
7
1.6744 + 0.5068 i
5.21 + 0.507 i
8
1.6744 - 0.5068 i
5.21 - 0.507 i
9
2.0649
0.39
10
-3.1992 + 2.5779 i
0.11 + 0.0687 i
11
-3.1992 - 2.5779 i
0.11 - 0.0687 i
12
-16.6362
0.111
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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
1.1889 + 0.49 i
0.138 - 0.142 i
Singularities of quadratic [7, 6, 6] approximant
2
1.1889 - 0.49 i
0.138 + 0.142 i
3
1.5433 + 0.4146 i
0.326 + 0.00532 i
4
1.5433 - 0.4146 i
0.326 - 0.00532 i
5
1.8092
0.311
6
1.8667 + 0.5392 i
1.28 + 0.757 i
7
1.8667 - 0.5392 i
1.28 - 0.757 i
8
-3.9129
0.0747
9
-3.8595 + 2.519 i
0.795 + 2.73 i
10
-3.8595 - 2.519 i
0.795 - 2.73 i
11
-5.2322
0.053 i
12
-8.8283 + 2.0454 i
0.0459 + 0.033 i
13
-8.8283 - 2.0454 i
0.0459 - 0.033 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
1.1919 + 0.4753 i
0.0151 - 0.213 i
Singularities of quadratic [7, 7, 6] approximant
2
1.1919 - 0.4753 i
0.0151 + 0.213 i
3
1.4315 + 0.4202 i
0.827 - 1.41 i
4
1.4315 - 0.4202 i
0.827 + 1.41 i
5
1.4114 + 0.5328 i
0.18 + 0.228 i
6
1.4114 - 0.5328 i
0.18 - 0.228 i
7
-1.8107 + 0.0006 i
0.000713 + 0.000713 i
8
-1.8107 - 0.0006 i
0.000713 - 0.000713 i
9
3.2171 + 1.6403 i
0.147 - 0.252 i
10
3.2171 - 1.6403 i
0.147 + 0.252 i
11
-3.1669 + 2.2514 i
0.00851 + 0.0694 i
12
-3.1669 - 2.2514 i
0.00851 - 0.0694 i
13
3.3057 + 6.5986 i
0.0553 - 0.138 i
14
3.3057 - 6.5986 i
0.0553 + 0.138 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.