Singularities of Møller-Plesset series: example "HF-cc-pVDZ-1.5Re"

Molecule X 1^Sigma+ State of HF. Basis 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
-1.4389 + 0.3127 i
0.115 + 0.0929 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.4389 - 0.3127 i
0.115 - 0.0929 i
3
1.629
0.303
4
-2.9076
0.458
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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
-1.1156 + 0.2101 i
0.00722 + 0.0228 i
Singularities of quadratic [2, 2, 3] approximant
2
-1.1156 - 0.2101 i
0.00722 - 0.0228 i
3
-1.1999
0.0182
4
1.6266
0.263
5
9.05
1.66 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
-1.2453
0.0667
Singularities of quadratic [2, 3, 3] approximant
2
1.1664 + 0.5654 i
0.0277 - 0.0135 i
3
1.1664 - 0.5654 i
0.0277 + 0.0135 i
4
1.6135 + 0.9509 i
0.0221 + 0.056 i
5
1.6135 - 0.9509 i
0.0221 - 0.056 i
6
-32.0753
360. 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
0.7716 + 0.0033 i
0.0107 - 0.0105 i
Singularities of quadratic [3, 3, 3] approximant
2
0.7716 - 0.0033 i
0.0107 + 0.0105 i
3
-1.2252 + 0.3397 i
0.0337 + 0.0334 i
4
-1.2252 - 0.3397 i
0.0337 - 0.0334 i
5
1.6621
0.454
6
-1.8126
0.0683
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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
-1.3074
0.109
Singularities of quadratic [3, 3, 4] approximant
2
1.8654 + 0.6307 i
0.453 - 0.093 i
3
1.8654 - 0.6307 i
0.453 + 0.093 i
4
0.4047 + 2.8715 i
0.17 + 0.225 i
5
0.4047 - 2.8715 i
0.17 - 0.225 i
6
4.8006
0.721
7
-15.0649
0.855 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
0.2981 + 0.e-5 i
0.0000699 - 0.0000699 i
Singularities of quadratic [3, 4, 4] approximant
2
0.2981 - 0.e-5 i
0.0000699 + 0.0000699 i
3
-1.3154
0.127
4
0.4889 + 2.1539 i
0.0565 + 0.0504 i
5
0.4889 - 2.1539 i
0.0565 - 0.0504 i
6
2.086 + 1.3133 i
0.00942 + 0.125 i
7
2.086 - 1.3133 i
0.00942 - 0.125 i
8
-10.6956
1.36 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
-1.2193
0.0508
Singularities of quadratic [4, 4, 4] approximant
2
1.6194 + 0.6238 i
0.168 - 0.0456 i
3
1.6194 - 0.6238 i
0.168 + 0.0456 i
4
-1.4498 + 1.1203 i
0.0399 + 0.0784 i
5
-1.4498 - 1.1203 i
0.0399 - 0.0784 i
6
-2.4409 + 1.5588 i
0.302 - 0.0507 i
7
-2.4409 - 1.5588 i
0.302 + 0.0507 i
8
3.0699
0.693
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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
-1.1731
0.0458
Singularities of quadratic [4, 4, 5] approximant
2
-1.4037
0.0547 i
3
1.6577 + 0.4876 i
0.202 - 0.283 i
4
1.6577 - 0.4876 i
0.202 + 0.283 i
5
-1.8353 + 0.6173 i
0.0288 - 0.159 i
6
-1.8353 - 0.6173 i
0.0288 + 0.159 i
7
2.2663
0.277
8
-0.9516 + 3.3833 i
0.169 - 0.311 i
9
-0.9516 - 3.3833 i
0.169 + 0.311 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
-1.14
0.0216
Singularities of quadratic [4, 5, 5] approximant
2
-1.4803
0.045 i
3
1.3889 + 0.618 i
0.032 - 0.00518 i
4
1.3889 - 0.618 i
0.032 + 0.00518 i
5
2.0427 + 0.5825 i
0.0263 + 0.105 i
6
2.0427 - 0.5825 i
0.0263 - 0.105 i
7
-1.896 + 1.0754 i
0.0658 + 0.0395 i
8
-1.896 - 1.0754 i
0.0658 - 0.0395 i
9
0.7908 + 2.6659 i
0.0781 - 0.0208 i
10
0.7908 - 2.6659 i
0.0781 + 0.0208 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
-1.1265 + 0.0237 i
0.0338 + 0.0388 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.1265 - 0.0237 i
0.0338 - 0.0388 i
3
-1.3726
0.0957
4
1.6473 + 0.5058 i
0.161 - 0.249 i
5
1.6473 - 0.5058 i
0.161 + 0.249 i
6
2.5289
0.311
7
-1.701 + 2.5556 i
0.975 + 0.158 i
8
-1.701 - 2.5556 i
0.975 - 0.158 i
9
-2.7172 + 1.6934 i
0.108 - 0.434 i
10
-2.7172 - 1.6934 i
0.108 + 0.434 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
-1.0653
0.00981
Singularities of quadratic [5, 5, 6] approximant
2
-1.1964
0.0118 i
3
-1.5208 + 0.4379 i
0.00833 + 0.0334 i
4
-1.5208 - 0.4379 i
0.00833 - 0.0334 i
5
1.638 + 0.5513 i
0.219 - 0.0856 i
6
1.638 - 0.5513 i
0.219 + 0.0856 i
7
-1.5014 + 0.8694 i
0.0186 - 0.032 i
8
-1.5014 - 0.8694 i
0.0186 + 0.032 i
9
2.205
0.268
10
-0.9702 + 2.9381 i
0.11 - 0.124 i
11
-0.9702 - 2.9381 i
0.11 + 0.124 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
-1.0667
0.00769
Singularities of quadratic [5, 6, 6] approximant
2
-1.3004
0.013 i
3
-1.2825 + 0.5732 i
0.00406 + 0.0101 i
4
-1.2825 - 0.5732 i
0.00406 - 0.0101 i
5
-1.3067 + 0.7437 i
0.0109 - 0.00816 i
6
-1.3067 - 0.7437 i
0.0109 + 0.00816 i
7
1.6409 + 0.556 i
0.226 - 0.0658 i
8
1.6409 - 0.556 i
0.226 + 0.0658 i
9
2.176
0.26
10
-1.0147 + 2.7562 i
0.114 - 0.0681 i
11
-1.0147 - 2.7562 i
0.114 + 0.0681 i
12
684.7159
20.2 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
-0.9916 + 0.0168 i
0.00181 + 0.00197 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.9916 - 0.0168 i
0.00181 - 0.00197 i
3
-1.1262
0.00626
4
1.6109 + 0.4935 i
0.0286 - 0.2 i
5
1.6109 - 0.4935 i
0.0286 + 0.2 i
6
-1.9518
1.07 i
7
-1.5973 + 2.0649 i
0.0181 - 0.127 i
8
-1.5973 - 2.0649 i
0.0181 + 0.127 i
9
2.864 + 0.9524 i
0.166 + 0.33 i
10
2.864 - 0.9524 i
0.166 - 0.33 i
11
2.6686 + 3.8801 i
0.0879 + 0.295 i
12
2.6686 - 3.8801 i
0.0879 - 0.295 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
-0.991 + 0.0167 i
0.0018 + 0.00196 i
Singularities of quadratic [6, 6, 7] approximant
2
-0.991 - 0.0167 i
0.0018 - 0.00196 i
3
-1.1257
0.00624
4
1.6111 + 0.4945 i
0.0321 - 0.2 i
5
1.6111 - 0.4945 i
0.0321 + 0.2 i
6
-1.9491
1.04 i
7
-1.5904 + 2.062 i
0.0175 - 0.125 i
8
-1.5904 - 2.062 i
0.0175 + 0.125 i
9
2.9371 + 0.9383 i
0.173 + 0.368 i
10
2.9371 - 0.9383 i
0.173 - 0.368 i
11
2.5091 + 3.8304 i
0.0642 + 0.281 i
12
2.5091 - 3.8304 i
0.0642 - 0.281 i
13
268.6152
4.25
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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
-0.9902 + 0.0169 i
0.00172 + 0.00187 i
Singularities of quadratic [6, 7, 7] approximant
2
-0.9902 - 0.0169 i
0.00172 - 0.00187 i
3
-1.1237
0.00598
4
1.607 + 0.4783 i
0.0217 + 0.184 i
5
1.607 - 0.4783 i
0.0217 - 0.184 i
6
-1.9671
1.3 i
7
2.2985 + 1.1843 i
0.108 + 0.151 i
8
2.2985 - 1.1843 i
0.108 - 0.151 i
9
-1.583 + 2.099 i
0.00158 - 0.133 i
10
-1.583 - 2.099 i
0.00158 + 0.133 i
11
2.9504 + 2.0635 i
0.251 + 0.103 i
12
2.9504 - 2.0635 i
0.251 - 0.103 i
13
4.1233
1.57
14
8.1813
2.83 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.