Singularities of Møller-Plesset series: example "a1"

Molecule Ar. Basis aug-cc-pVDZ. Structure "mpn_Rfci"

Content


Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-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
3.0375
0.938
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
3.9562
1.64
Singularities of quadratic [1, 1, 0] approximant
2
198.2823
11.6 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
3.0922
0.68
Singularities of quadratic [1, 1, 1] approximant
2
13.2156
3.46 i
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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
0.5998
0.0176
Singularities of quadratic [2, 1, 1] approximant
2
0.6174
0.0179 i
3
5.9286
1.19e6
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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
-1.195 + 0.0923 i
0.0108 + 0.00973 i
Singularities of quadratic [2, 2, 1] approximant
2
-1.195 - 0.0923 i
0.0108 - 0.00973 i
3
1.8229
0.0712
4
6.8976
0.619 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
-1.2759
0.00512
Singularities of quadratic [2, 2, 2] approximant
2
1.157 + 0.9604 i
0.0107 + 0.000432 i
3
1.157 - 0.9604 i
0.0107 - 0.000432 i
4
-3.4477
0.011 i
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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
-1.9294
0.0294
Singularities of quadratic [3, 2, 2] approximant
2
-1.866 + 2.0303 i
0.0219 + 0.0456 i
3
-1.866 - 2.0303 i
0.0219 - 0.0456 i
4
2.855 + 1.6465 i
0.195 + 0.148 i
5
2.855 - 1.6465 i
0.195 - 0.148 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
-1.6512 + 0.6966 i
0.0106 + 0.00482 i
Singularities of quadratic [3, 3, 2] approximant
2
-1.6512 - 0.6966 i
0.0106 - 0.00482 i
3
1.9108
0.0281
4
0.4957 + 2.3486 i
0.0241 + 0.0145 i
5
0.4957 - 2.3486 i
0.0241 - 0.0145 i
6
4.2424
0.368 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
-1.6298
0.0117
Singularities of quadratic [3, 3, 3] approximant
2
-1.3782 + 1.929 i
0.00825 + 0.0174 i
3
-1.3782 - 1.929 i
0.00825 - 0.0174 i
4
2.686
1.27
5
1.3465 + 2.6048 i
0.0118 - 0.0619 i
6
1.3465 - 2.6048 i
0.0118 + 0.0619 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
-1.6889
0.0166
Singularities of quadratic [4, 3, 3] approximant
2
-1.5927 + 2.0658 i
0.00337 + 0.0279 i
3
-1.5927 - 2.0658 i
0.00337 - 0.0279 i
4
2.6961
0.688
5
1.0702 + 3.1952 i
0.0712 - 0.0787 i
6
1.0702 - 3.1952 i
0.0712 + 0.0787 i
7
-6.6937
0.0655 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
-1.0018 + 0.0048 i
0.00045 + 0.00045 i
Singularities of quadratic [4, 4, 3] approximant
2
-1.0018 - 0.0048 i
0.00045 - 0.00045 i
3
-1.6577
0.00736
4
2.4299
0.122
5
-0.5261 + 3.0016 i
0.0296 - 0.053 i
6
-0.5261 - 3.0016 i
0.0296 + 0.053 i
7
-5.2065
0.892 i
8
7.7061
4.59 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
-1.7104 + 0.1328 i
0.0198 + 0.00729 i
Singularities of quadratic [4, 4, 4] approximant
2
-1.7104 - 0.1328 i
0.0198 - 0.00729 i
3
-2.031 + 0.9011 i
0.0152 + 0.0162 i
4
-2.031 - 0.9011 i
0.0152 - 0.0162 i
5
2.4462
0.126
6
-0.1145 + 2.8777 i
0.0137 + 0.0494 i
7
-0.1145 - 2.8777 i
0.0137 - 0.0494 i
8
5.4292
124. i
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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
-1.4637
0.0152
Singularities of quadratic [5, 4, 4] approximant
2
-1.5446
0.0154 i
3
-2.3643
1.94e3
4
2.6486
0.412
5
-2.9789
0.0766 i
6
0.4367 + 3.7301 i
0.225 - 0.0243 i
7
0.4367 - 3.7301 i
0.225 + 0.0243 i
8
-2.4335 + 2.9224 i
0.0791 - 0.0621 i
9
-2.4335 - 2.9224 i
0.0791 + 0.0621 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
-1.397
0.00234
Singularities of quadratic [5, 5, 4] approximant
2
-1.5744 + 0.6958 i
0.0023 - 0.00201 i
3
-1.5744 - 0.6958 i
0.0023 + 0.00201 i
4
-1.894 + 0.6655 i
0.000636 + 0.00347 i
5
-1.894 - 0.6655 i
0.000636 - 0.00347 i
6
-2.089
0.00406 i
7
2.6244
0.364
8
0.1774 + 3.3018 i
0.0901 + 0.0489 i
9
0.1774 - 3.3018 i
0.0901 - 0.0489 i
10
34.1213
1.98 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
0.0661
3.7e-10
Singularities of quadratic [5, 5, 5] approximant
2
0.0661
3.7e-10 i
3
-1.2827 + 0.0346 i
0.000523 + 0.000557 i
4
-1.2827 - 0.0346 i
0.000523 - 0.000557 i
5
-1.5293
0.00186
6
2.6562
0.495
7
0.0645 + 3.2991 i
0.0583 + 0.0659 i
8
0.0645 - 3.2991 i
0.0583 - 0.0659 i
9
-4.9233
7.72 i
10
-24.2892
1.75
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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
-1.3135 + 0.0367 i
0.000816 + 0.000889 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.3135 - 0.0367 i
0.000816 - 0.000889 i
3
1.5158
0.00215
4
1.5199
0.00217 i
5
-1.5784
0.00276
6
2.7664
2.55
7
-0.0123 + 3.2403 i
0.0296 + 0.0665 i
8
-0.0123 - 3.2403 i
0.0296 - 0.0665 i
9
-7.7849
2.87 i
10
-9.787
3.73
11
13.7243
5.99 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
-1.2869 + 0.0302 i
0.000666 + 0.00073 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.2869 - 0.0302 i
0.000666 - 0.00073 i
3
-1.5415
0.00229
4
-0.4454 + 2.1561 i
0.00528 - 0.00375 i
5
-0.4454 - 2.1561 i
0.00528 + 0.00375 i
6
-0.4189 + 2.2157 i
0.00382 + 0.00559 i
7
-0.4189 - 2.2157 i
0.00382 - 0.00559 i
8
2.6073
0.305
9
0.3891 + 3.2607 i
0.0839 - 0.0242 i
10
0.3891 - 3.2607 i
0.0839 + 0.0242 i
11
-3.9611
1.74 i
12
39.2207
2.12 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
-1.2441 + 0.0143 i
0.000842 + 0.000888 i
Singularities of quadratic [6, 6, 6] approximant
2
-1.2441 - 0.0143 i
0.000842 - 0.000888 i
3
-1.5143
0.00353
4
-1.8233
0.0114 i
5
-2.1607
0.331
6
2.5747
0.232
7
0.0033 + 3.6511 i
0.0582 + 0.268 i
8
0.0033 - 3.6511 i
0.0582 - 0.268 i
9
-3.0654 + 3.5536 i
0.0123 + 0.172 i
10
-3.0654 - 3.5536 i
0.0123 - 0.172 i
11
5.7093
3.79 i
12
18.9284
2.46
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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
-1.2357 + 0.0212 i
0.000382 + 0.000416 i
Singularities of quadratic [7, 6, 6] approximant
2
-1.2357 - 0.0212 i
0.000382 - 0.000416 i
3
-1.4586
0.00137
4
-2.201
0.0157 i
5
2.5765
0.227
6
-3.0595
0.0462
7
-0.5928 + 3.0397 i
0.0299 - 0.00379 i
8
-0.5928 - 3.0397 i
0.0299 + 0.00379 i
9
-1.219 + 3.241 i
0.0145 - 0.0311 i
10
-1.219 - 3.241 i
0.0145 + 0.0311 i
11
1.154 + 3.9779 i
0.173 + 0.154 i
12
1.154 - 3.9779 i
0.173 - 0.154 i
13
114.4447
11.1 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
-0.9711
0.0000275
Singularities of quadratic [7, 7, 6] approximant
2
-0.9722
0.0000274 i
3
-1.2431 + 0.0486 i
0.000143 + 0.000127 i
4
-1.2431 - 0.0486 i
0.000143 - 0.000127 i
5
-1.4058
0.000435
6
2.5997 + 0.1291 i
0.114 - 0.2 i
7
2.5997 - 0.1291 i
0.114 + 0.2 i
8
2.712
0.196
9
0.3373 + 3.5701 i
0.147 + 0.00636 i
10
0.3373 - 3.5701 i
0.147 - 0.00636 i
11
-3.6341
7.98 i
12
-3.4095 + 5.8373 i
0.32 + 0.586 i
13
-3.4095 - 5.8373 i
0.32 - 0.586 i
14
66.4601
19.5 i
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Examplesa1a2a8a16a22a30a38a44a45a51a62a69a75a83a84a85a86a87a88a90a91
MoleculeArBHBHBHBHBHBHBO+C2CN+N2HFHFHClHClF-Cl-Cl-NeOH-SH-
Basisaug-cc-pVDZcc-pVDZcc-pVTZcc-pVQZaug-cc-pVDZaug-cc-pVTZaug-cc-pVQZcc-pVDZcc-pVDZcc-pVDZcc-pVDZcc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZcc-pVDZaug-cc-pVDZaug-cc-pVDZaug-cc-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.