Singularities of Møller-Plesset series: example "a90"

Molecule OH-. 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

Plot of singularities Blank Molecule - icon for Allen-dataList of examples Blank Mathematica programs Blank Work in UMassD Blank Waste iconUnpublished reports

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
-0.4456
0.0047
Singularities of quadratic [2, 2, 2] approximant
2
-0.4992 + 0.064 i
0.00172 - 0.00597 i
3
-0.4992 - 0.064 i
0.00172 + 0.00597 i
4
1.9049
0.316
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Table 2. 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
-0.6642
0.0565
Singularities of quadratic [3, 2, 2] approximant
2
0.7553 + 1.8063 i
0.111 + 0.0504 i
3
0.7553 - 1.8063 i
0.111 - 0.0504 i
4
2.1002
22.3
5
3.1396
0.274 i
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Table 3. 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
-0.6775
0.0711
Singularities of quadratic [3, 3, 2] approximant
2
-2.4382
0.18 i
3
2.7694
3.57e4
4
-0.7364 + 2.9562 i
0.0674 + 0.231 i
5
-0.7364 - 2.9562 i
0.0674 - 0.231 i
6
14.6157
0.907 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
-0.6828
0.0924
Singularities of quadratic [3, 3, 3] approximant
2
-0.9616 + 0.5508 i
0.0499 - 0.0517 i
3
-0.9616 - 0.5508 i
0.0499 + 0.0517 i
4
-1.1169 + 0.6652 i
0.0481 + 0.0722 i
5
-1.1169 - 0.6652 i
0.0481 - 0.0722 i
6
2.2951
1.06
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Table 5. 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
-0.6821
0.0894
Singularities of quadratic [4, 3, 3] approximant
2
-0.9695 + 0.5853 i
0.0441 - 0.0558 i
3
-0.9695 - 0.5853 i
0.0441 + 0.0558 i
4
-1.1053 + 0.7221 i
0.0584 + 0.066 i
5
-1.1053 - 0.7221 i
0.0584 - 0.066 i
6
2.2991
1.08
7
535.1893
1.35 i
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Table 6. 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
-0.6675
0.0849
Singularities of quadratic [4, 4, 3] approximant
2
-0.7001
0.349 i
3
-0.7206
0.18
4
-1.9567
0.153 i
5
2.686
32.9
6
-1.3377 + 3.0196 i
0.0303 + 0.302 i
7
-1.3377 - 3.0196 i
0.0303 - 0.302 i
8
17.1222
1.04 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
-0.6604
0.0843
Singularities of quadratic [4, 4, 4] approximant
2
-0.683
0.259 i
3
-0.7081
0.149
4
-2.1904
0.152 i
5
2.602
9.97
6
-1.1915 + 2.607 i
0.0106 - 0.255 i
7
-1.1915 - 2.607 i
0.0106 + 0.255 i
8
-20.2966
76.
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Table 8. 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
-0.5584 + 0.0012 i
0.0081 + 0.00831 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.5584 - 0.0012 i
0.0081 - 0.00831 i
3
-0.6687
0.0474
4
-1.8007
0.163 i
5
2.6051
6.94
6
-1.8543 + 2.8684 i
0.0997 - 0.474 i
7
-1.8543 - 2.8684 i
0.0997 + 0.474 i
8
6.2307
1.46 i
9
18.3346
53.4
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Table 9. 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
0.4221
0.000111
Singularities of quadratic [5, 5, 4] approximant
2
0.4221
0.000111 i
3
-0.5485 + 0.0011 i
0.00575 + 0.00583 i
4
-0.5485 - 0.0011 i
0.00575 - 0.00583 i
5
-0.6674
0.0449
6
-1.824
0.155 i
7
2.691
52.7
8
-1.5444 + 2.8719 i
0.0455 - 0.325 i
9
-1.5444 - 2.8719 i
0.0455 + 0.325 i
10
22.4239
1.26 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
-0.0577
2.33e-8 + 2.33e-8 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.0577
2.33e-8 - 2.33e-8 i
3
-0.5412 + 0.0011 i
0.00439 + 0.00443 i
4
-0.5412 - 0.0011 i
0.00439 - 0.00443 i
5
-0.6666
0.0432
6
-1.8349
0.153 i
7
2.6768
32.9
8
-1.5031 + 2.899 i
0.026 - 0.317 i
9
-1.5031 - 2.899 i
0.026 + 0.317 i
10
21.4157
1.2 i
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Table 11. 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
-0.5851
0.198
Singularities of quadratic [6, 5, 5] approximant
2
-0.5859
0.3 i
3
-0.6746
0.0628
4
1.0719 + 0.1397 i
0.0000535 + 0.00112 i
5
1.0719 - 0.1397 i
0.0000535 - 0.00112 i
6
1.0786 + 0.1372 i
0.00111 - 0.0000834 i
7
1.0786 - 0.1372 i
0.00111 + 0.0000834 i
8
-1.7209
0.193 i
9
-2.2578 + 2.2834 i
0.877 - 0.316 i
10
-2.2578 - 2.2834 i
0.877 + 0.316 i
11
3.404
1.19
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Table 12. 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
-0.5834 + 0.0005 i
0.0969 + 0.133 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.5834 - 0.0005 i
0.0969 - 0.133 i
3
-0.6739
0.0607
4
0.7771 + 0.5115 i
0.000638 - 0.00143 i
5
0.7771 - 0.5115 i
0.000638 + 0.00143 i
6
0.7781 + 0.5115 i
0.00143 + 0.000636 i
7
0.7781 - 0.5115 i
0.00143 - 0.000636 i
8
-1.74
0.183 i
9
2.9912
4.28
10
-2.063 + 2.4273 i
0.519 - 0.367 i
11
-2.063 - 2.4273 i
0.519 + 0.367 i
12
195.6441
6.63 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
-0.5778 + 0.0016 i
0.0129 + 0.0136 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.5778 - 0.0016 i
0.0129 - 0.0136 i
3
-0.6707
0.0501
4
2.0489
0.884
5
2.1693
0.594 i
6
-2.2979 + 0.1721 i
0.0707 - 0.0957 i
7
-2.2979 - 0.1721 i
0.0707 + 0.0957 i
8
2.8525 + 1.032 i
0.712 + 0.621 i
9
2.8525 - 1.032 i
0.712 - 0.621 i
10
-2.2848 + 2.0031 i
0.22 - 0.765 i
11
-2.2848 - 2.0031 i
0.22 + 0.765 i
12
9.8341
7.95
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Table 14. 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
-0.5764 + 0.0016 i
0.0117 + 0.0123 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.5764 - 0.0016 i
0.0117 - 0.0123 i
3
-0.6699
0.0482
4
-1.3626
0.269 i
5
-1.4727 + 0.1802 i
0.0319 - 0.272 i
6
-1.4727 - 0.1802 i
0.0319 + 0.272 i
7
1.4004 + 0.7144 i
0.00358 - 0.0201 i
8
1.4004 - 0.7144 i
0.00358 + 0.0201 i
9
1.4223 + 0.6986 i
0.0202 + 0.00306 i
10
1.4223 - 0.6986 i
0.0202 - 0.00306 i
11
-2.3208 + 2.2486 i
0.772 - 0.311 i
12
-2.3208 - 2.2486 i
0.772 + 0.311 i
13
3.2638
1.41
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Table 15. 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
-0.5748 + 0.0023 i
0.00647 + 0.00659 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.5748 - 0.0023 i
0.00647 - 0.00659 i
3
-0.6678
0.0408
4
-0.3044 + 1.4816 i
0.00494 - 0.00759 i
5
-0.3044 - 1.4816 i
0.00494 + 0.00759 i
6
-0.2922 + 1.5145 i
0.00773 + 0.00489 i
7
-0.2922 - 1.5145 i
0.00773 - 0.00489 i
8
-1.6524
4.3 i
9
-1.7772 + 0.5243 i
0.0952 + 0.11 i
10
-1.7772 - 0.5243 i
0.0952 - 0.11 i
11
1.8881
0.0593
12
2.1767
0.0958 i
13
2.5539 + 4.209 i
0.0813 + 0.301 i
14
2.5539 - 4.209 i
0.0813 - 0.301 i
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Table 16. Singularities with their weights for the quadratic approximant [7, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.5477 + 0.0004 i
0.00691 + 0.00703 i
Singularities of quadratic [7, 7, 7] approximant
2
-0.5477 - 0.0004 i
0.00691 - 0.00703 i
3
-0.5938
0.0548
4
-0.5959
0.0771 i
5
-0.6735
0.0583
6
-1.8443
0.173 i
7
1.8625 + 0.9937 i
0.0726 - 0.0652 i
8
1.8625 - 0.9937 i
0.0726 + 0.0652 i
9
2.057 + 0.9006 i
0.083 + 0.0699 i
10
2.057 - 0.9006 i
0.083 - 0.0699 i
11
-2.2935 + 2.0905 i
1.07 - 1.21 i
12
-2.2935 - 2.0905 i
1.07 + 1.21 i
13
3.1603
1.11
14
-5.6758
0.762
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Table 17. Singularities with their weights for the quadratic approximant [8, 7, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.569 + 0.003 i
0.00288 + 0.00294 i
Singularities of quadratic [8, 7, 7] approximant
2
-0.569 - 0.003 i
0.00288 - 0.00294 i
3
-0.6551
0.0193
4
-0.7763
1.19 i
5
-0.8078
0.149
6
-1.2033
0.203 i
7
-1.3772 + 0.1945 i
0.361 + 0.0275 i
8
-1.3772 - 0.1945 i
0.361 - 0.0275 i
9
1.8206 + 0.9756 i
0.0709 + 0.102 i
10
1.8206 - 0.9756 i
0.0709 - 0.102 i
11
1.9024 + 1.0738 i
0.116 - 0.0581 i
12
1.9024 - 1.0738 i
0.116 + 0.0581 i
13
2.5726
9.
14
-2.1114 + 1.8723 i
0.412 + 0.214 i
15
-2.1114 - 1.8723 i
0.412 - 0.214 i
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Table 18. Singularities with their weights for the quadratic approximant [8, 8, 7]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.5662 + 0.0023 i
0.00328 + 0.00342 i
Singularities of quadratic [8, 8, 7] approximant
2
-0.5662 - 0.0023 i
0.00328 - 0.00342 i
3
-0.6373
0.0134
4
-0.6601
0.0234 i
5
-0.6931
0.544
6
1.7765 + 0.8806 i
0.0119 - 0.0999 i
7
1.7765 - 0.8806 i
0.0119 + 0.0999 i
8
-1.9805 + 0.4101 i
0.0719 - 0.112 i
9
-1.9805 - 0.4101 i
0.0719 + 0.112 i
10
1.8766 + 0.8481 i
0.102 + 0.00996 i
11
1.8766 - 0.8481 i
0.102 - 0.00996 i
12
-1.9376 + 1.7989 i
0.333 + 0.0542 i
13
-1.9376 - 1.7989 i
0.333 - 0.0542 i
14
-2.8741
0.519 i
15
3.0925
1.7
16
192.2089
19.8 i
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Table 19. Singularities with their weights for the quadratic approximant [8, 8, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.417
0.0000183
Singularities of quadratic [8, 8, 8] approximant
2
-0.417
0.0000183 i
3
-0.5669 + 0.0046 i
0.00133 + 0.00129 i
4
-0.5669 - 0.0046 i
0.00133 - 0.00129 i
5
-0.6415
0.00905
6
-0.6913
0.035 i
7
-0.7181
0.688
8
1.7742 + 0.8709 i
0.0266 - 0.0317 i
9
1.7742 - 0.8709 i
0.0266 + 0.0317 i
10
1.8659 + 0.7213 i
0.0379 + 0.02 i
11
1.8659 - 0.7213 i
0.0379 - 0.02 i
12
-2.1087
0.135 i
13
-3.0842
0.273
14
-2.5219 + 2.187 i
0.691 + 0.937 i
15
-2.5219 - 2.187 i
0.691 - 0.937 i
16
4.5383
0.992
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Table 20. Singularities with their weights for the quadratic approximant [9, 8, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.2154
6.24e-8
Singularities of quadratic [9, 8, 8] approximant
2
-0.2154
6.24e-8 i
3
-0.5667 + 0.0036 i
0.00181 + 0.00184 i
4
-0.5667 - 0.0036 i
0.00181 - 0.00184 i
5
-0.6404
0.0101
6
-0.6803
0.0274 i
7
-0.709
4.42
8
1.7735 + 0.8735 i
0.0272 - 0.0289 i
9
1.7735 - 0.8735 i
0.0272 + 0.0289 i
10
1.8625 + 0.7181 i
0.0355 + 0.0203 i
11
1.8625 - 0.7181 i
0.0355 - 0.0203 i
12
-2.2418
0.0983 i
13
-2.5982
0.133
14
-2.3878 + 2.2717 i
0.287 + 0.695 i
15
-2.3878 - 2.2717 i
0.287 - 0.695 i
16
4.5134
1.02
17
-41.3403
1.17 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.