Singularities of Møller-Plesset series: example "O2--aug-cc-pVDZ"

Molecule X 1^Sigma+ State of O2-. Basis AUG-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
-0.5863 + 0.0429 i
0.605 + 1.34 i
Singularities of quadratic [2, 2, 2] approximant
2
-0.5863 - 0.0429 i
0.605 - 1.34 i
3
-1.2334
0.77
4
2.4084
2.34
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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
-0.5652 + 0.0714 i
0.207 + 0.552 i
Singularities of quadratic [2, 2, 3] approximant
2
-0.5652 - 0.0714 i
0.207 - 0.552 i
3
-1.0352
0.396
4
2.9253
30.8
5
7.1608
1.32 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
-0.5867
17.3
Singularities of quadratic [2, 3, 3] approximant
2
-0.6228
3.16 i
3
1.9657
0.691
4
-2.0289
326.
5
-3.017
1.68 i
6
403.786
11.2 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
-0.5917 + 0.0384 i
1.16 + 1.44 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.5917 - 0.0384 i
1.16 - 1.44 i
3
0.8645
0.108
4
0.8811
0.109 i
5
-1.3193
1.1
6
2.7423
8.57
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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
-0.5808 + 0.0633 i
0.649 + 0.434 i
Singularities of quadratic [3, 3, 4] approximant
2
-0.5808 - 0.0633 i
0.649 - 0.434 i
3
-0.9319
0.624
4
-1.6087
0.354 i
5
1.7353
0.348
6
-0.9951 + 1.6883 i
0.367 - 0.182 i
7
-0.9951 - 1.6883 i
0.367 + 0.182 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
0.362
0.000328
Singularities of quadratic [3, 4, 4] approximant
2
0.3623
0.000328 i
3
-0.4648 + 0.0134 i
0.0166 + 0.013 i
4
-0.4648 - 0.0134 i
0.0166 - 0.013 i
5
-0.8439
0.0915
6
1.3718
0.0536
7
-2.2046
0.799 i
8
5.8892
3.94 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
-0.5504 + 0.1338 i
0.0687 - 0.0038 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.5504 - 0.1338 i
0.0687 + 0.0038 i
3
-0.629 + 0.1654 i
0.0253 + 0.0703 i
4
-0.629 - 0.1654 i
0.0253 - 0.0703 i
5
-1.4604
0.433
6
1.5615
0.183
7
3.7696
0.661 i
8
-4.9544
1.59 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
-0.5715 + 0.0931 i
0.239 + 0.0152 i
Singularities of quadratic [4, 4, 5] approximant
2
-0.5715 - 0.0931 i
0.239 - 0.0152 i
3
-0.7444 + 0.1046 i
0.184 + 0.232 i
4
-0.7444 - 0.1046 i
0.184 - 0.232 i
5
1.5379
0.201
6
-2.1686 + 0.9353 i
1.66 + 0.224 i
7
-2.1686 - 0.9353 i
1.66 - 0.224 i
8
2.3906
0.355 i
9
4.3953
4.93
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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
-0.5942
0.375
Singularities of quadratic [4, 5, 5] approximant
2
-0.592 + 0.0723 i
0.0451 + 0.327 i
3
-0.592 - 0.0723 i
0.0451 - 0.327 i
4
-0.4983 + 0.5501 i
0.00797 - 0.00501 i
5
-0.4983 - 0.5501 i
0.00797 + 0.00501 i
6
-0.4912 + 0.5781 i
0.0053 + 0.00794 i
7
-0.4912 - 0.5781 i
0.0053 - 0.00794 i
8
-1.4804
0.149 i
9
1.4925
0.106
10
7.7509
12.3 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
0.3452
0.0000641
Singularities of quadratic [5, 5, 5] approximant
2
0.3453
0.0000641 i
3
-0.4841 + 0.0164 i
0.0093 + 0.00639 i
4
-0.4841 - 0.0164 i
0.0093 - 0.00639 i
5
-0.5782 + 0.1969 i
0.0225 + 0.0126 i
6
-0.5782 - 0.1969 i
0.0225 - 0.0126 i
7
-0.6505
0.668
8
-0.8653
0.0528 i
9
1.414
0.0546
10
-25.9719
0.457
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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
-0.5687 + 0.0692 i
0.377 + 0.417 i
Singularities of quadratic [5, 5, 6] approximant
2
-0.5687 - 0.0692 i
0.377 - 0.417 i
3
-0.8341
2.45
4
-0.8763
19.5 i
5
-1.4684 + 1.0037 i
0.307 + 0.668 i
6
-1.4684 - 1.0037 i
0.307 - 0.668 i
7
1.9305 + 0.5516 i
0.277 + 0.0516 i
8
1.9305 - 0.5516 i
0.277 - 0.0516 i
9
1.5205 + 1.3753 i
0.0955 - 0.205 i
10
1.5205 - 1.3753 i
0.0955 + 0.205 i
11
3.5009
0.393
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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
-0.5696 + 0.0698 i
0.435 + 0.386 i
Singularities of quadratic [5, 6, 6] approximant
2
-0.5696 - 0.0698 i
0.435 - 0.386 i
3
-0.7769
0.652
4
-0.9631
0.56 i
5
1.5441 + 0.4115 i
0.0904 - 0.105 i
6
1.5441 - 0.4115 i
0.0904 + 0.105 i
7
-1.3204 + 1.2207 i
0.445 + 0.107 i
8
-1.3204 - 1.2207 i
0.445 - 0.107 i
9
2.0508 + 1.018 i
0.171 + 0.144 i
10
2.0508 - 1.018 i
0.171 - 0.144 i
11
-1.3832 + 6.1748 i
0.638 + 0.473 i
12
-1.3832 - 6.1748 i
0.638 - 0.473 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
-0.5734 + 0.0706 i
0.671 + 0.209 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.5734 - 0.0706 i
0.671 - 0.209 i
3
-0.7177
0.335
4
1.2516
0.0116
5
-1.2964
0.116 i
6
1.2896 + 0.6064 i
0.0106 + 0.0255 i
7
1.2896 - 0.6064 i
0.0106 - 0.0255 i
8
-1.6548
0.176
9
1.6722
0.0089 i
10
1.767
0.0116
11
-1.2203 + 1.3043 i
0.156 - 0.223 i
12
-1.2203 - 1.3043 i
0.156 + 0.223 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
-0.5665 + 0.0695 i
0.25 + 0.407 i
Singularities of quadratic [6, 6, 7] approximant
2
-0.5665 - 0.0695 i
0.25 - 0.407 i
3
-0.7731 + 0.1038 i
0.027 - 0.25 i
4
-0.7731 - 0.1038 i
0.027 + 0.25 i
5
-0.9073
0.188
6
-1.2048
0.597 i
7
-1.3706 + 0.8677 i
0.253 - 0.577 i
8
-1.3706 - 0.8677 i
0.253 + 0.577 i
9
1.5913 + 0.4808 i
0.116 - 0.0438 i
10
1.5913 - 0.4808 i
0.116 + 0.0438 i
11
1.7511 + 1.3545 i
0.0429 - 0.185 i
12
1.7511 - 1.3545 i
0.0429 + 0.185 i
13
52.4936
6.22
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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
-0.5716 + 0.0739 i
0.457 + 0.138 i
Singularities of quadratic [6, 7, 7] approximant
2
-0.5716 - 0.0739 i
0.457 - 0.138 i
3
-0.6343
0.624
4
-0.6832
0.514 i
5
-0.8671 + 0.1302 i
0.288 + 0.167 i
6
-0.8671 - 0.1302 i
0.288 - 0.167 i
7
1.6233 + 0.4488 i
0.139 - 0.122 i
8
1.6233 - 0.4488 i
0.139 + 0.122 i
9
-1.3749 + 1.0848 i
0.345 + 0.32 i
10
-1.3749 - 1.0848 i
0.345 - 0.32 i
11
1.8232 + 1.1543 i
0.115 + 0.207 i
12
1.8232 - 1.1543 i
0.115 - 0.207 i
13
4.4191 + 10.3127 i
0.969 + 0.115 i
14
4.4191 - 10.3127 i
0.969 - 0.115 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.