Singularities of Møller-Plesset series: example "H--cc-pV5Z"

Molecule H- ion. Basis AUG-CC-PV5Z. 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
Plot of singularities Blank Molecule - icon for Allen-dataList of examples Blank Mathematica programs Blank Work in UMassD Blank Waste iconUnpublished reports

Quadratic approximants

[n1, n2, n3] 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
 
1.3368
 
0.0434
 Singularities of quadratic [2, 2, 2] approximant 
2
 
2.1034
 
0.0871 i
 
3
 
10.0647
 
0.0755
 
4
 
-10.7919
 
2.73
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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
 
1.3605
 
0.0516
 Singularities of quadratic [2, 2, 3] approximant 
2
 
2.1407
 
0.0961 i
 
3
 
-6.5591
 
0.568
 
4
 
6.0711 + 5.0139 i
 
0.0648 - 0.0736 i
 
5
 
6.0711 - 5.0139 i
 
0.0648 + 0.0736 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
 
0.664
 
0.000618
 Singularities of quadratic [2, 3, 3] approximant 
2
 
0.6705
 
0.000618 i
 
3
 
1.1971
 
0.00917
 
4
 
3.0455
 
0.741 i
 
5
 
-5.4311
 
0.068
 
6
 
-15.1764
 
0.822 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
 
-0.0973 + 0.e-5 i
 
1.7e-7 + 1.7e-7 i
 Singularities of quadratic [3, 3, 3] approximant 
2
 
-0.0973 - 0.e-5 i
 
1.7e-7 - 1.7e-7 i
 
3
 
1.1673 + 0.3612 i
 
0.00506 - 0.00132 i
 
4
 
1.1673 - 0.3612 i
 
0.00506 + 0.00132 i
 
5
 
1.7743
 
0.0199
 
6
 
32.989
 
0.0303 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
 
1.2991
 
0.0106
 Singularities of quadratic [3, 3, 4] approximant 
2
 
1.3956 + 0.6519 i
 
0.00943 - 0.015 i
 
3
 
1.3956 - 0.6519 i
 
0.00943 + 0.015 i
 
4
 
1.774
 
0.0364 i
 
5
 
2.9478
 
0.467
 
6
 
-3.4791 + 6.0367 i
 
0.0645 - 0.0435 i
 
7
 
-3.4791 - 6.0367 i
 
0.0645 + 0.0435 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
 
1.4096 + 0.371 i
 
0.02 + 0.000945 i
 Singularities of quadratic [3, 4, 4] approximant 
2
 
1.4096 - 0.371 i
 
0.02 - 0.000945 i
 
3
 
1.5742 + 1.7988 i
 
0.0148 + 0.00581 i
 
4
 
1.5742 - 1.7988 i
 
0.0148 - 0.00581 i
 
5
 
1.3609 + 2.9873 i
 
0.0151 - 0.00846 i
 
6
 
1.3609 - 2.9873 i
 
0.0151 + 0.00846 i
 
7
 
-4.6373 + 0.5599 i
 
0.0119 + 0.00778 i
 
8
 
-4.6373 - 0.5599 i
 
0.0119 - 0.00778 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
 
1.4489 + 0.281 i
 
0.0272 - 0.0439 i
 Singularities of quadratic [4, 4, 4] approximant 
2
 
1.4489 - 0.281 i
 
0.0272 + 0.0439 i
 
3
 
1.9501 + 0.9484 i
 
0.0238 + 0.0476 i
 
4
 
1.9501 - 0.9484 i
 
0.0238 - 0.0476 i
 
5
 
1.7157 + 2.8947 i
 
0.0272 + 0.0161 i
 
6
 
1.7157 - 2.8947 i
 
0.0272 - 0.0161 i
 
7
 
-0.0112 + 6.6159 i
 
0.0281 - 0.0231 i
 
8
 
-0.0112 - 6.6159 i
 
0.0281 + 0.0231 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
 
0.1016
 
5.39e-9 - 5.39e-9 i
 Singularities of quadratic [4, 4, 5] approximant 
2
 
0.1016
 
5.39e-9 + 5.39e-9 i
 
3
 
1.4142 + 0.4346 i
 
0.0112 + 0.00838 i
 
4
 
1.4142 - 0.4346 i
 
0.0112 - 0.00838 i
 
5
 
1.5689 + 1.9677 i
 
0.00186 + 0.0144 i
 
6
 
1.5689 - 1.9677 i
 
0.00186 - 0.0144 i
 
7
 
1.6725 + 5.3755 i
 
0.0113 + 0.023 i
 
8
 
1.6725 - 5.3755 i
 
0.0113 - 0.023 i
 
9
 
10.329
 
0.0803
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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
 
1.4438 + 0.2602 i
 
0.00993 - 0.0522 i
 Singularities of quadratic [4, 5, 5] approximant 
2
 
1.4438 - 0.2602 i
 
0.00993 + 0.0522 i
 
3
 
1.8665 + 0.9321 i
 
0.0235 + 0.03 i
 
4
 
1.8665 - 0.9321 i
 
0.0235 - 0.03 i
 
5
 
1.6335 + 2.7804 i
 
0.0193 + 0.00888 i
 
6
 
1.6335 - 2.7804 i
 
0.0193 - 0.00888 i
 
7
 
-0.6951 + 5.5948 i
 
0.018 - 0.0164 i
 
8
 
-0.6951 - 5.5948 i
 
0.018 + 0.0164 i
 
9
 
12.9401 + 14.9992 i
 
0.0578 - 0.00181 i
 
10
 
12.9401 - 14.9992 i
 
0.0578 + 0.00181 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
 
1.3793 + 0.3146 i
 
0.013 - 0.0126 i
 Singularities of quadratic [5, 5, 5] approximant 
2
 
1.3793 - 0.3146 i
 
0.013 + 0.0126 i
 
3
 
1.6896 + 0.2609 i
 
0.0154 + 0.0305 i
 
4
 
1.6896 - 0.2609 i
 
0.0154 - 0.0305 i
 
5
 
1.7107 + 0.9299 i
 
0.0142 + 0.0201 i
 
6
 
1.7107 - 0.9299 i
 
0.0142 - 0.0201 i
 
7
 
1.6481 + 2.9057 i
 
0.0233 + 0.0113 i
 
8
 
1.6481 - 2.9057 i
 
0.0233 - 0.0113 i
 
9
 
-0.0018 + 6.1414 i
 
0.0245 - 0.0212 i
 
10
 
-0.0018 - 6.1414 i
 
0.0245 + 0.0212 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
 
1.4399 + 0.2821 i
 
0.0224 - 0.039 i
 Singularities of quadratic [5, 5, 6] approximant 
2
 
1.4399 - 0.2821 i
 
0.0224 + 0.039 i
 
3
 
1.8801 + 1.0814 i
 
0.0035 + 0.0364 i
 
4
 
1.8801 - 1.0814 i
 
0.0035 - 0.0364 i
 
5
 
1.5257 + 3.2079 i
 
0.00498 + 0.0186 i
 
6
 
1.5257 - 3.2079 i
 
0.00498 - 0.0186 i
 
7
 
-2.5468 + 3.9706 i
 
0.00574 + 0.00595 i
 
8
 
-2.5468 - 3.9706 i
 
0.00574 - 0.00595 i
 
9
 
-3.5578 + 3.5523 i
 
0.0081 - 0.0043 i
 
10
 
-3.5578 - 3.5523 i
 
0.0081 + 0.0043 i
 
11
 
-12.8551
 
0.384
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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
 
1.4396 + 0.3151 i
 
0.0258 - 0.00663 i
 Singularities of quadratic [5, 6, 6] approximant 
2
 
1.4396 - 0.3151 i
 
0.0258 + 0.00663 i
 
3
 
1.501
 
0.0391
 
4
 
1.6535
 
3.24 i
 
5
 
2.136 + 1.4428 i
 
0.0483 + 0.0314 i
 
6
 
2.136 - 1.4428 i
 
0.0483 - 0.0314 i
 
7
 
2.7204
 
0.311
 
8
 
-3.5237 + 0.035 i
 
0.00168 + 0.00165 i
 
9
 
-3.5237 - 0.035 i
 
0.00168 - 0.00165 i
 
10
 
0.6449 + 4.3066 i
 
0.0212 + 0.0126 i
 
11
 
0.6449 - 4.3066 i
 
0.0212 - 0.0126 i
 
12
 
8.0551
 
0.086 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
 
1.4559 + 0.3036 i
 
0.0435 - 0.00935 i
 Singularities of quadratic [6, 6, 6] approximant 
2
 
1.4559 - 0.3036 i
 
0.0435 + 0.00935 i
 
3
 
1.7119
 
0.0621
 
4
 
2.0313 + 0.42 i
 
0.0498 + 0.125 i
 
5
 
2.0313 - 0.42 i
 
0.0498 - 0.125 i
 
6
 
2.1325 + 1.5777 i
 
0.0267 + 0.0394 i
 
7
 
2.1325 - 1.5777 i
 
0.0267 - 0.0394 i
 
8
 
-3.8379 + 0.0543 i
 
0.00236 + 0.00227 i
 
9
 
-3.8379 - 0.0543 i
 
0.00236 - 0.00227 i
 
10
 
0.5927 + 4.2771 i
 
0.02 + 0.0123 i
 
11
 
0.5927 - 4.2771 i
 
0.02 - 0.0123 i
 
12
 
9.0224
 
0.0761 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
 
1.4554 + 0.2996 i
 
0.043 - 0.016 i
 Singularities of quadratic [6, 6, 7] approximant 
2
 
1.4554 - 0.2996 i
 
0.043 + 0.016 i
 
3
 
1.6764
 
0.0712
 
4
 
1.9388
 
0.199 i
 
5
 
2.239 + 1.3475 i
 
0.108 + 0.0637 i
 
6
 
2.239 - 1.3475 i
 
0.108 - 0.0637 i
 
7
 
3.0428
 
0.24
 
8
 
0.7335 + 4.5262 i
 
0.0323 + 0.0189 i
 
9
 
0.7335 - 4.5262 i
 
0.0323 - 0.0189 i
 
10
 
-5.1375 + 0.2493 i
 
0.00971 + 0.00837 i
 
11
 
-5.1375 - 0.2493 i
 
0.00971 - 0.00837 i
 
12
 
19.6945
 
0.142 i
 
13
 
77.4099
 
2.2
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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
 
-0.5573
 
7.4e-7
 Singularities of quadratic [6, 7, 7] approximant 
2
 
-0.5573
 
7.4e-7 i
 
3
 
1.4462 + 0.3171 i
 
0.0297 - 0.00226 i
 
4
 
1.4462 - 0.3171 i
 
0.0297 + 0.00226 i
 
5
 
1.5741
 
0.0496
 
6
 
2.0159 + 0.1787 i
 
0.122 + 0.0914 i
 
7
 
2.0159 - 0.1787 i
 
0.122 - 0.0914 i
 
8
 
2.1287 + 1.563 i
 
0.0265 + 0.0357 i
 
9
 
2.1287 - 1.563 i
 
0.0265 - 0.0357 i
 
10
 
-3.8948 + 0.0535 i
 
0.0029 + 0.00282 i
 
11
 
-3.8948 - 0.0535 i
 
0.0029 - 0.00282 i
 
12
 
0.5632 + 4.3231 i
 
0.0185 + 0.0146 i
 
13
 
0.5632 - 4.3231 i
 
0.0185 - 0.0146 i
 
14
 
7.7334
 
0.0796 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.