Singularities of Møller-Plesset series: example "a83"

Molecule HCl. Basis 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 [3, 3, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.7844
0.00101
Singularities of quadratic [3, 3, 2] approximant
2
0.7933
0.00102 i
3
-2.0367 + 0.7328 i
0.0154 + 0.00647 i
4
-2.0367 - 0.7328 i
0.0154 - 0.00647 i
5
2.103 + 1.8386 i
0.0362 - 0.0717 i
6
2.103 - 1.8386 i
0.0362 + 0.0717 i
Top of Page  Top of the page    

Table 2. 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
-1.7104
0.00828
Singularities of quadratic [3, 3, 3] approximant
2
-1.2378 + 2.61 i
0.00702 + 0.013 i
3
-1.2378 - 2.61 i
0.00702 - 0.013 i
4
1.775 + 2.4598 i
0.00237 - 0.0412 i
5
1.775 - 2.4598 i
0.00237 + 0.0412 i
6
3.7629
0.299
Top of Page  Top of the page    

Table 3. 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
-1.8115 + 0.322 i
0.00614 + 0.00803 i
Singularities of quadratic [4, 3, 3] approximant
2
-1.8115 - 0.322 i
0.00614 - 0.00803 i
3
2.5281
0.873
4
-2.7489
0.0136
5
-0.9185 + 3.8385 i
0.098 - 0.021 i
6
-0.9185 - 3.8385 i
0.098 + 0.021 i
7
4.5356
1.44 i
Top of Page  Top of the page    

Table 4. 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
1.1683 + 1.0809 i
0.0025 - 0.00101 i
Singularities of quadratic [4, 4, 3] approximant
2
1.1683 - 1.0809 i
0.0025 + 0.00101 i
3
1.2397 + 1.0335 i
0.0013 + 0.00256 i
4
1.2397 - 1.0335 i
0.0013 - 0.00256 i
5
-1.679
0.00619
6
-0.6793 + 2.912 i
0.0154 - 0.000603 i
7
-0.6793 - 2.912 i
0.0154 + 0.000603 i
8
-289.8786
2.41 i
Top of Page  Top of the page    

Table 5. 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
-1.5887
0.00311
Singularities of quadratic [4, 4, 4] approximant
2
0.0913 + 1.669 i
0.000481 + 0.000603 i
3
0.0913 - 1.669 i
0.000481 - 0.000603 i
4
-0.0184 + 1.8631 i
0.000561 - 0.000587 i
5
-0.0184 - 1.8631 i
0.000561 + 0.000587 i
6
0.3441 + 1.9746 i
0.00136 + 0.0000108 i
7
0.3441 - 1.9746 i
0.00136 - 0.0000108 i
8
-3.224
0.0584 i
Top of Page  Top of the page    

Table 6. 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
-1.5315
0.00209
Singularities of quadratic [5, 4, 4] approximant
2
0.0987 + 1.7337 i
0.000413 + 0.000744 i
3
0.0987 - 1.7337 i
0.000413 - 0.000744 i
4
-0.2402 + 1.8719 i
0.00101 - 0.000416 i
5
-0.2402 - 1.8719 i
0.00101 + 0.000416 i
6
0.4543 + 1.8409 i
0.00136 - 0.000716 i
7
0.4543 - 1.8409 i
0.00136 + 0.000716 i
8
-2.298
0.00838 i
9
-7.0949
0.0211
Top of Page  Top of the page    

Table 7. 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
-1.2074 + 0.0054 i
0.000249 + 0.000251 i
Singularities of quadratic [5, 5, 4] approximant
2
-1.2074 - 0.0054 i
0.000249 - 0.000251 i
3
-1.5984
0.00241
4
2.3929 + 0.571 i
0.0534 - 0.227 i
5
2.3929 - 0.571 i
0.0534 + 0.227 i
6
2.9661 + 1.5259 i
0.0201 - 0.184 i
7
2.9661 - 1.5259 i
0.0201 + 0.184 i
8
0.2311 + 3.5339 i
0.0288 - 0.0444 i
9
0.2311 - 3.5339 i
0.0288 + 0.0444 i
10
-11.6853
0.355 i
Top of Page  Top of the page    

Table 8. 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
-1.0329 + 0.0013 i
0.0000954 + 0.0000955 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.0329 - 0.0013 i
0.0000954 - 0.0000955 i
3
-1.5887
0.00251
4
2.0245 + 1.1174 i
0.0287 + 0.00901 i
5
2.0245 - 1.1174 i
0.0287 - 0.00901 i
6
1.8545 + 1.3967 i
0.0106 - 0.0195 i
7
1.8545 - 1.3967 i
0.0106 + 0.0195 i
8
0.0901 + 3.0406 i
0.00895 - 0.0192 i
9
0.0901 - 3.0406 i
0.00895 + 0.0192 i
10
-5.314
8.76 i
Top of Page  Top of the page    

Table 9. 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
-1.4764 + 0.0389 i
0.00079 + 0.00103 i
Singularities of quadratic [6, 5, 5] approximant
2
-1.4764 - 0.0389 i
0.00079 - 0.00103 i
3
-1.6257
0.00181
4
2.4762 + 0.5816 i
0.383 - 0.29 i
5
2.4762 - 0.5816 i
0.383 + 0.29 i
6
3.7381
1.05
7
-0.6092 + 4.3072 i
0.0846 + 0.048 i
8
-0.6092 - 4.3072 i
0.0846 - 0.048 i
9
2.7836 + 3.5692 i
0.313 + 0.513 i
10
2.7836 - 3.5692 i
0.313 - 0.513 i
11
-10.8656
0.103 i
Top of Page  Top of the page    

Table 10. 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
-1.5355 + 0.0685 i
0.00111 + 0.00159 i
Singularities of quadratic [6, 6, 5] approximant
2
-1.5355 - 0.0685 i
0.00111 - 0.00159 i
3
-1.6973
0.00232
4
2.4282 + 0.6585 i
0.216 - 0.159 i
5
2.4282 - 0.6585 i
0.216 + 0.159 i
6
-2.7463
0.145 i
7
-3.4008
0.0556
8
3.0953 + 1.6365 i
0.109 - 0.234 i
9
3.0953 - 1.6365 i
0.109 + 0.234 i
10
0.3563 + 3.8354 i
0.0524 - 0.0815 i
11
0.3563 - 3.8354 i
0.0524 + 0.0815 i
12
-10.5342
0.573 i
Top of Page  Top of the page    

Table 11. 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
-1.5164 + 0.0475 i
0.00121 + 0.0017 i
Singularities of quadratic [6, 6, 6] approximant
2
-1.5164 - 0.0475 i
0.00121 - 0.0017 i
3
-1.674
0.00258
4
2.2363 + 0.7676 i
0.0965 - 0.0182 i
5
2.2363 - 0.7676 i
0.0965 + 0.0182 i
6
2.6172 + 1.0304 i
0.00726 - 0.134 i
7
2.6172 - 1.0304 i
0.00726 + 0.134 i
8
1.3993 + 3.5479 i
0.307 - 0.0666 i
9
1.3993 - 3.5479 i
0.307 + 0.0666 i
10
-1.4259 + 4.0714 i
0.0221 + 0.0514 i
11
-1.4259 - 4.0714 i
0.0221 - 0.0514 i
12
16.0583
1.93
Top of Page  Top of the page    

Table 12. 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
-1.5192 + 0.0463 i
0.00132 + 0.00186 i
Singularities of quadratic [7, 6, 6] approximant
2
-1.5192 - 0.0463 i
0.00132 - 0.00186 i
3
-1.6817
0.00281
4
2.2251 + 0.778 i
0.0869 - 0.00387 i
5
2.2251 - 0.778 i
0.0869 + 0.00387 i
6
2.6538 + 1.0272 i
0.0299 - 0.132 i
7
2.6538 - 1.0272 i
0.0299 + 0.132 i
8
1.4954 + 3.5304 i
0.278 - 0.249 i
9
1.4954 - 3.5304 i
0.278 + 0.249 i
10
-1.301 + 3.9891 i
0.0242 + 0.041 i
11
-1.301 - 3.9891 i
0.0242 - 0.041 i
12
3.1388 + 10.4519 i
0.209 + 0.0475 i
13
3.1388 - 10.4519 i
0.209 - 0.0475 i
Top of Page  Top of the page    

Table 13. 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
-1.4918 + 0.054 i
0.000593 + 0.000883 i
Singularities of quadratic [7, 7, 6] approximant
2
-1.4918 - 0.054 i
0.000593 - 0.000883 i
3
-1.6066
0.00121
4
1.9657
0.013
5
2.1804 + 0.0947 i
0.00636 + 0.0175 i
6
2.1804 - 0.0947 i
0.00636 - 0.0175 i
7
2.4458 + 0.8194 i
0.0477 - 0.124 i
8
2.4458 - 0.8194 i
0.0477 + 0.124 i
9
1.065 + 3.7595 i
0.0959 + 0.0943 i
10
1.065 - 3.7595 i
0.0959 - 0.0943 i
11
-4.0566
0.0588 i
12
-5.3041 + 4.8052 i
0.335 + 0.0646 i
13
-5.3041 - 4.8052 i
0.335 - 0.0646 i
14
21.7421
1.52 i
Top of Page  Top of the page    

Table 14. 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
-1.4873 + 0.0538 i
0.000526 + 0.000805 i
Singularities of quadratic [7, 7, 7] approximant
2
-1.4873 - 0.0538 i
0.000526 - 0.000805 i
3
-1.5944
0.00107
4
2.1816
0.0384
5
2.1749 + 0.3538 i
0.0362 - 0.0253 i
6
2.1749 - 0.3538 i
0.0362 + 0.0253 i
7
2.4682 + 0.7979 i
0.0716 - 0.0999 i
8
2.4682 - 0.7979 i
0.0716 + 0.0999 i
9
-3.452
0.104 i
10
1.0858 + 3.8392 i
0.0875 + 0.119 i
11
1.0858 - 3.8392 i
0.0875 - 0.119 i
12
-4.9385
0.132
13
-0.6911 + 8.7055 i
0.549 - 1.47 i
14
-0.6911 - 8.7055 i
0.549 + 1.47 i
Top of Page  Top of the page    

Table 15. 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
-1.4838 + 0.0543 i
0.000472 + 0.000739 i
Singularities of quadratic [8, 7, 7] approximant
2
-1.4838 - 0.0543 i
0.000472 - 0.000739 i
3
-1.5845
0.00095
4
2.0158
0.0129
5
2.148 + 0.227 i
0.0066 - 0.0208 i
6
2.148 - 0.227 i
0.0066 + 0.0208 i
7
2.3915 + 0.8686 i
0.0361 - 0.0733 i
8
2.3915 - 0.8686 i
0.0361 + 0.0733 i
9
-3.183
0.162 i
10
1.1918 + 3.7109 i
0.107 + 0.0665 i
11
1.1918 - 3.7109 i
0.107 - 0.0665 i
12
-3.983
0.075
13
-1.6669 + 7.2606 i
0.681 - 0.623 i
14
-1.6669 - 7.2606 i
0.681 + 0.623 i
15
-11.7319
0.213 i
Top of Page  Top of the page    

Table 16. 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
-1.485 + 0.0552 i
0.000485 + 0.000745 i
Singularities of quadratic [8, 8, 7] approximant
2
-1.485 - 0.0552 i
0.000485 - 0.000745 i
3
1.4927
0.000512
4
1.4961
0.000513 i
5
-1.5883
0.000978
6
2.1359 + 0.7971 i
0.0172 + 0.0191 i
7
2.1359 - 0.7971 i
0.0172 - 0.0191 i
8
2.5318 + 0.222 i
0.0586 + 0.0222 i
9
2.5318 - 0.222 i
0.0586 - 0.0222 i
10
-3.7019
0.065 i
11
1.23 + 4.1166 i
0.32 + 0.14 i
12
1.23 - 4.1166 i
0.32 - 0.14 i
13
4.177 + 3.5185 i
0.449 + 0.45 i
14
4.177 - 3.5185 i
0.449 - 0.45 i
15
-6.2332 + 3.305 i
0.322 + 0.203 i
16
-6.2332 - 3.305 i
0.322 - 0.203 i
Top of Page  Top of the page    

Table 17. 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
-0.5377
2.1e-7
Singularities of quadratic [8, 8, 8] approximant
2
-0.5377
2.1e-7 i
3
-1.4953 + 0.0411 i
0.000789 + 0.00136 i
4
-1.4953 - 0.0411 i
0.000789 - 0.00136 i
5
-1.6153
0.00161
6
2.0079
0.00782
7
2.0766 + 0.2661 i
0.00666 - 0.00992 i
8
2.0766 - 0.2661 i
0.00666 + 0.00992 i
9
2.3311 + 0.9602 i
0.0288 - 0.0307 i
10
2.3311 - 0.9602 i
0.0288 + 0.0307 i
11
-2.6495
0.353 i
12
-2.9917
0.0267
13
1.065 + 3.6384 i
0.0424 + 0.0647 i
14
1.065 - 3.6384 i
0.0424 - 0.0647 i
15
-0.2003 + 5.7737 i
0.499 - 0.388 i
16
-0.2003 - 5.7737 i
0.499 + 0.388 i
Top of Page  Top of the page    

Table 18. 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
0.0934
0
Singularities of quadratic [9, 8, 8] approximant
2
0.0934
0
3
-0.2933
1.52e-10
4
-0.2933
1.52e-10 i
5
-1.4505
0.000481
6
1.5482 + 0.0058 i
0.000133 - 0.000129 i
7
1.5482 - 0.0058 i
0.000133 + 0.000129 i
8
-1.7945 + 0.4859 i
0.00079 + 0.000604 i
9
-1.7945 - 0.4859 i
0.00079 - 0.000604 i
10
-1.8458 + 0.4171 i
0.00109 - 0.000735 i
11
-1.8458 - 0.4171 i
0.00109 + 0.000735 i
12
2.06 + 0.6775 i
0.00691 - 0.0047 i
13
2.06 - 0.6775 i
0.00691 + 0.0047 i
14
1.3771 + 3.2028 i
0.0151 - 0.019 i
15
1.3771 - 3.2028 i
0.0151 + 0.019 i
16
3.1414 + 2.822 i
0.0277 - 0.0877 i
17
3.1414 - 2.822 i
0.0277 + 0.0877 i
Top of Page  Top of the page    

Table 19. Singularities with their weights for the quadratic approximant [9, 9, 8]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
0.0473 + 1.164 i
9.24e-6 - 3.32e-6 i
Singularities of quadratic [9, 9, 8] approximant
2
0.0473 - 1.164 i
9.24e-6 + 3.32e-6 i
3
0.0474 + 1.164 i
3.32e-6 + 9.24e-6 i
4
0.0474 - 1.164 i
3.32e-6 - 9.24e-6 i
5
-1.4607 + 0.0682 i
0.00022 + 0.000367 i
6
-1.4607 - 0.0682 i
0.00022 - 0.000367 i
7
-1.5283
0.000414
8
2.0114 + 0.485 i
0.00685 + 0.00397 i
9
2.0114 - 0.485 i
0.00685 - 0.00397 i
10
2.3614 + 1.2516 i
0.0127 + 0.00896 i
11
2.3614 - 1.2516 i
0.0127 - 0.00896 i
12
2.9527
10.8
13
-1.5073 + 3.3624 i
0.00602 + 0.00779 i
14
-1.5073 - 3.3624 i
0.00602 - 0.00779 i
15
-0.5189 + 3.8243 i
0.000668 - 0.0151 i
16
-0.5189 - 3.8243 i
0.000668 + 0.0151 i
17
5.2785
0.529 i
18
-12.825
0.0877 i
Top of Page  Top of the page    

Table 20. Singularities with their weights for the quadratic approximant [9, 9, 9]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
-0.6616
1.94e-8
Singularities of quadratic [9, 9, 9] approximant
2
-0.6616
1.94e-8 i
3
-1.1348 + 0.7382 i
6.61e-6 - 9.68e-6 i
4
-1.1348 - 0.7382 i
6.61e-6 + 9.68e-6 i
5
-1.1373 + 0.7391 i
9.76e-6 + 6.69e-6 i
6
-1.1373 - 0.7391 i
9.76e-6 - 6.69e-6 i
7
-1.4334
0.000266
8
0.3437 + 1.4016 i
6.26e-6 + 8.99e-6 i
9
0.3437 - 1.4016 i
6.26e-6 - 8.99e-6 i
10
0.3449 + 1.4024 i
9.e-6 - 6.29e-6 i
11
0.3449 - 1.4024 i
9.e-6 + 6.29e-6 i
12
1.9434 + 0.5537 i
0.00101 + 0.00303 i
13
1.9434 - 0.5537 i
0.00101 - 0.00303 i
14
1.8665 + 1.6823 i
0.00186 - 0.00119 i
15
1.8665 - 1.6823 i
0.00186 + 0.00119 i
16
2.1079 + 2.2366 i
0.00122 - 0.00581 i
17
2.1079 - 2.2366 i
0.00122 + 0.00581 i
18
-11.7487
0.434 i
Top of Page  Top of the page    


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.