Singularities of Møller-Plesset series: example "a45"

Molecule C2. 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

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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
-2.2575
1.41
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
-0.3276
0.0614
Singularities of quadratic [1, 1, 0] approximant
2
-0.855
0.0993 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
-0.7732
0.482
Singularities of quadratic [1, 1, 1] approximant
2
1.6974
3.14
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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
-0.9385
1.72
Singularities of quadratic [2, 1, 1] approximant
2
1.4547
0.844
3
-2.7429
0.833 i
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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
1.2488
0.457
Singularities of quadratic [2, 2, 1] approximant
2
-1.2595 + 0.2706 i
1.65 - 1.03 i
3
-1.2595 - 0.2706 i
1.65 + 1.03 i
4
131.6219
4.86 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
-1.1965 + 0.3195 i
1.62 - 0.359 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.1965 - 0.3195 i
1.62 + 0.359 i
3
1.2757
0.499
4
-19.4413
3.56e3
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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
-0.6639
0.082
Singularities of quadratic [3, 2, 2] approximant
2
-0.7987 + 0.0618 i
0.042 - 0.0952 i
3
-0.7987 - 0.0618 i
0.042 + 0.0952 i
4
1.2029
0.341
5
-1.3708
699. 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
-0.8297 + 0.4026 i
0.183 + 0.0479 i
Singularities of quadratic [3, 3, 2] approximant
2
-0.8297 - 0.4026 i
0.183 - 0.0479 i
3
1.1669
0.277
4
-1.2035 + 0.4556 i
0.0417 - 0.376 i
5
-1.2035 - 0.4556 i
0.0417 + 0.376 i
6
115.5956
11.7 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
-0.5061 + 0.1173 i
0.00556 + 0.00312 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.5061 - 0.1173 i
0.00556 - 0.00312 i
3
-0.5063 + 0.1547 i
0.00305 - 0.00667 i
4
-0.5063 - 0.1547 i
0.00305 + 0.00667 i
5
1.1235
0.182
6
-3.9631
0.511
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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
-0.9014 + 0.1456 i
0.356 + 0.0481 i
Singularities of quadratic [4, 3, 3] approximant
2
-0.9014 - 0.1456 i
0.356 - 0.0481 i
3
1.2576
0.643
4
-0.1345 + 1.3902 i
0.0492 + 0.122 i
5
-0.1345 - 1.3902 i
0.0492 - 0.122 i
6
-0.4801 + 1.5301 i
0.139 - 0.0342 i
7
-0.4801 - 1.5301 i
0.139 + 0.0342 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
-0.9626 + 0.3154 i
0.526 + 0.791 i
Singularities of quadratic [4, 4, 3] approximant
2
-0.9626 - 0.3154 i
0.526 - 0.791 i
3
1.1948 + 0.3122 i
0.16 - 0.235 i
4
1.1948 - 0.3122 i
0.16 + 0.235 i
5
1.7604
0.316
6
-1.8951
1.43
7
-3.0676
9.28 i
8
14.2401
10.9 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
-0.9127 + 0.2611 i
0.29 - 0.233 i
Singularities of quadratic [4, 4, 4] approximant
2
-0.9127 - 0.2611 i
0.29 + 0.233 i
3
1.2313 + 0.3763 i
0.223 - 0.0876 i
4
1.2313 - 0.3763 i
0.223 + 0.0876 i
5
-1.1708 + 0.6464 i
0.255 - 0.0129 i
6
-1.1708 - 0.6464 i
0.255 + 0.0129 i
7
-1.4972
1.21
8
2.7567
2.68
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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
-0.8854 + 0.2175 i
0.203 + 0.00228 i
Singularities of quadratic [5, 4, 4] approximant
2
-0.8854 - 0.2175 i
0.203 - 0.00228 i
3
-1.0938
1.38
4
-1.0844 + 0.5813 i
0.213 + 0.106 i
5
-1.0844 - 0.5813 i
0.213 - 0.106 i
6
1.2304 + 0.3813 i
0.219 - 0.0733 i
7
1.2304 - 0.3813 i
0.219 + 0.0733 i
8
2.7544
2.99
9
-34.7857
0.542 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
-0.9549 + 0.3315 i
0.597 + 0.532 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.9549 - 0.3315 i
0.597 - 0.532 i
3
1.2329 + 0.3181 i
0.198 - 0.292 i
4
1.2329 - 0.3181 i
0.198 + 0.292 i
5
-1.5857
1.4
6
2.1365
0.447
7
-2.4589
2.93 i
8
5.16
22.4 i
9
-5.4255 + 3.783 i
1.3 + 0.897 i
10
-5.4255 - 3.783 i
1.3 - 0.897 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
-0.955 + 0.3342 i
0.612 + 0.48 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.955 - 0.3342 i
0.612 - 0.48 i
3
1.2313 + 0.323 i
0.201 - 0.269 i
4
1.2313 - 0.323 i
0.201 + 0.269 i
5
-1.5067
1.25
6
2.1916
0.506
7
-3.0299
0.747 i
8
-2.8051 + 3.5101 i
0.411 + 0.406 i
9
-2.8051 - 3.5101 i
0.411 - 0.406 i
10
9.4584
3.76 i
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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
-0.9539 + 0.3289 i
0.545 + 0.568 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.9539 - 0.3289 i
0.545 - 0.568 i
3
1.2216 + 0.3448 i
0.206 - 0.17 i
4
1.2216 - 0.3448 i
0.206 + 0.17 i
5
-1.6333
1.44
6
2.2225
0.776
7
0.8593 + 2.5653 i
0.165 - 0.276 i
8
0.8593 - 2.5653 i
0.165 + 0.276 i
9
0.3064 + 2.9198 i
0.243 + 0.204 i
10
0.3064 - 2.9198 i
0.243 - 0.204 i
11
-3.0106
1.45 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
-0.954 + 0.3294 i
0.554 + 0.56 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.954 - 0.3294 i
0.554 - 0.56 i
3
1.2172 + 0.3488 i
0.195 - 0.154 i
4
1.2172 - 0.3488 i
0.195 + 0.154 i
5
-1.6222
1.42
6
2.3307
1.17
7
1.3788 + 2.1767 i
0.012 - 0.326 i
8
1.3788 - 2.1767 i
0.012 + 0.326 i
9
-2.8981
1.58 i
10
1.581 + 3.4057 i
0.361 + 0.107 i
11
1.581 - 3.4057 i
0.361 - 0.107 i
12
62.3914
1.91 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
-0.9591 + 0.3285 i
0.695 + 0.626 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.9591 - 0.3285 i
0.695 - 0.626 i
3
1.2166 + 0.3354 i
0.127 - 0.214 i
4
1.2166 - 0.3354 i
0.127 + 0.214 i
5
1.1167 + 1.0374 i
0.0617 - 0.0472 i
6
1.1167 - 1.0374 i
0.0617 + 0.0472 i
7
1.1665 + 0.9898 i
0.0485 + 0.059 i
8
1.1665 - 0.9898 i
0.0485 - 0.059 i
9
-1.5281 + 0.3455 i
0.86 - 0.493 i
10
-1.5281 - 0.3455 i
0.86 + 0.493 i
11
-2.1428
0.666
12
3.7492
152.
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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
-0.9562 + 0.3284 i
0.604 + 0.604 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.9562 - 0.3284 i
0.604 - 0.604 i
3
1.2684
0.276
4
1.2969 + 0.2219 i
0.0502 - 0.441 i
5
1.2969 - 0.2219 i
0.0502 + 0.441 i
6
1.2753 + 0.3482 i
0.878 + 1.01 i
7
1.2753 - 0.3482 i
0.878 - 1.01 i
8
-1.8068 + 0.2518 i
1.56 - 1.26 i
9
-1.8068 - 0.2518 i
1.56 + 1.26 i
10
2.3937
0.715 i
11
-5.0215
0.837
12
2.9629 + 4.1011 i
0.411 + 1.15 i
13
2.9629 - 4.1011 i
0.411 - 1.15 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
-0.956 + 0.3284 i
0.598 + 0.602 i
Singularities of quadratic [7, 7, 6] approximant
2
-0.956 - 0.3284 i
0.598 - 0.602 i
3
1.2051 + 0.327 i
0.0953 - 0.274 i
4
1.2051 - 0.327 i
0.0953 + 0.274 i
5
1.4982 + 0.3994 i
0.246 + 0.0779 i
6
1.4982 - 0.3994 i
0.246 - 0.0779 i
7
1.7851
0.5
8
-1.7995 + 0.2582 i
1.54 - 1.17 i
9
-1.7995 - 0.2582 i
1.54 + 1.17 i
10
2.3162
0.338 i
11
-4.6111
0.667
12
1.6999 + 5.3312 i
0.213 - 0.798 i
13
1.6999 - 5.3312 i
0.213 + 0.798 i
14
-40.2633
1.47 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.