Singularities of Møller-Plesset series: example "a2"

Molecule BH. 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 [3, 3, 2]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
1
1.5257 + 0.4801 i
0.166 - 0.0301 i
Singularities of quadratic [3, 3, 2] approximant
2
1.5257 - 0.4801 i
0.166 + 0.0301 i
3
2.8473
3.37
4
-6.4736
0.0438
5
-9.4952
0.0509 i
6
21.9312
0.237 i
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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
1.5258 + 0.4568 i
0.179 - 0.0681 i
Singularities of quadratic [3, 3, 3] approximant
2
1.5258 - 0.4568 i
0.179 + 0.0681 i
3
2.7826
1.81
4
-5.734 + 0.526 i
0.0279 + 0.0367 i
5
-5.734 - 0.526 i
0.0279 - 0.0367 i
6
-109.5278
0.0208
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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
1.5261 + 0.4416 i
0.178 - 0.102 i
Singularities of quadratic [4, 3, 3] approximant
2
1.5261 - 0.4416 i
0.178 + 0.102 i
3
2.8706
2.12
4
-5.7987 + 2.978 i
0.00918 + 0.017 i
5
-5.7987 - 2.978 i
0.00918 - 0.017 i
6
-4.9795 + 8.4005 i
0.0258 - 0.00254 i
7
-4.9795 - 8.4005 i
0.0258 + 0.00254 i
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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
0.8872
0.00543
Singularities of quadratic [4, 4, 3] approximant
2
0.888
0.00545 i
3
1.5191 + 0.4597 i
0.148 - 0.0678 i
4
1.5191 - 0.4597 i
0.148 + 0.0678 i
5
2.9837
6.18
6
-5.6562
0.0296
7
-8.0078
0.0341 i
8
28.8515
0.208 i
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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
1.6588 + 0.1495 i
0.213 - 0.189 i
Singularities of quadratic [4, 4, 4] approximant
2
1.6588 - 0.1495 i
0.213 + 0.189 i
3
1.6752 + 0.4066 i
5.96 + 0.157 i
4
1.6752 - 0.4066 i
5.96 - 0.157 i
5
2.2505
0.469
6
-5.4088 + 1.1147 i
0.00764 + 0.00951 i
7
-5.4088 - 1.1147 i
0.00764 - 0.00951 i
8
-22.0942
0.0145
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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
1.4557 + 0.3185 i
0.056 + 0.0409 i
Singularities of quadratic [5, 4, 4] approximant
2
1.4557 - 0.3185 i
0.056 - 0.0409 i
3
1.8667
20.3
4
-2.0364
0.000299
5
-2.068
0.000296 i
6
1.9309 + 0.9203 i
0.0445 + 0.0709 i
7
1.9309 - 0.9203 i
0.0445 - 0.0709 i
8
-3.9059
0.00199
9
-10.1909
0.00645 i
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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
-0.6632
0.0000124
Singularities of quadratic [5, 5, 4] approximant
2
-0.6632
0.0000124 i
3
1.4547 + 0.261 i
0.0527 - 0.00204 i
4
1.4547 - 0.261 i
0.0527 + 0.00204 i
5
1.5661
0.0853
6
1.9399 + 0.7275 i
0.0934 + 0.0925 i
7
1.9399 - 0.7275 i
0.0934 - 0.0925 i
8
-4.6419
0.00602
9
-9.8258
0.0108 i
10
153.4361
0.121 i
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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
-1.2016 + 0.7873 i
8.7e-6 - 0.0000606 i
Singularities of quadratic [5, 5, 5] approximant
2
-1.2016 - 0.7873 i
8.7e-6 + 0.0000606 i
3
-1.2071 + 0.7883 i
0.0000605 + 8.98e-6 i
4
-1.2071 - 0.7883 i
0.0000605 - 8.98e-6 i
5
1.6615 + 0.4532 i
0.128 - 0.403 i
6
1.6615 - 0.4532 i
0.128 + 0.403 i
7
2.3269 + 0.2427 i
0.144 - 0.0854 i
8
2.3269 - 0.2427 i
0.144 + 0.0854 i
9
-4.0348
0.00149
10
4.655
2.9
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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
-0.9175 + 0.6593 i
9.69e-7 + 0.0000281 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.9175 - 0.6593 i
9.69e-7 - 0.0000281 i
3
-0.9185 + 0.6597 i
0.0000281 - 9.38e-7 i
4
-0.9185 - 0.6597 i
0.0000281 + 9.38e-7 i
5
1.6543 + 0.429 i
0.165 - 0.714 i
6
1.6543 - 0.429 i
0.165 + 0.714 i
7
2.3004 + 0.4493 i
0.202 - 0.0467 i
8
2.3004 - 0.4493 i
0.202 + 0.0467 i
9
3.9018
1.7
10
-4.0819
0.00187
11
-28.9909
0.0107 i
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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
-0.8853 + 0.457 i
4.95e-6 - 9.02e-6 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.8853 - 0.457 i
4.95e-6 + 9.02e-6 i
3
-0.8858 + 0.4565 i
9.01e-6 + 4.95e-6 i
4
-0.8858 - 0.4565 i
9.01e-6 - 4.95e-6 i
5
1.6751 + 0.4376 i
0.44 - 0.408 i
6
1.6751 - 0.4376 i
0.44 + 0.408 i
7
2.6201 + 0.2022 i
0.181 - 0.0689 i
8
2.6201 - 0.2022 i
0.181 + 0.0689 i
9
-3.7609
0.00116
10
9.1432
0.0847
11
18.8525
0.0226 i
12
-27.826
0.00958 i
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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
0.3857
2.48e-7
Singularities of quadratic [6, 6, 6] approximant
2
0.3857
2.48e-7 i
3
-0.8874 + 0.5691 i
5.01e-7 + 8.77e-6 i
4
-0.8874 - 0.5691 i
5.01e-7 - 8.77e-6 i
5
-0.8886 + 0.5689 i
8.77e-6 - 4.86e-7 i
6
-0.8886 - 0.5689 i
8.77e-6 + 4.86e-7 i
7
1.5704 + 0.4368 i
0.136 - 0.0324 i
8
1.5704 - 0.4368 i
0.136 + 0.0324 i
9
1.8612 + 0.3125 i
0.148 - 1.11 i
10
1.8612 - 0.3125 i
0.148 + 1.11 i
11
3.3356
2.38
12
-3.8191
0.00109
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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
0.6028
0.0000198
Singularities of quadratic [7, 6, 6] approximant
2
0.6028
0.0000198 i
3
-0.8783 + 0.6021 i
5.72e-7 + 0.0000177 i
4
-0.8783 - 0.6021 i
5.72e-7 - 0.0000177 i
5
-0.8791 + 0.6023 i
0.0000177 - 5.55e-7 i
6
-0.8791 - 0.6023 i
0.0000177 + 5.55e-7 i
7
1.6343 + 0.4285 i
0.225 + 0.555 i
8
1.6343 - 0.4285 i
0.225 - 0.555 i
9
2.1921 + 0.4677 i
0.203 - 0.0835 i
10
2.1921 - 0.4677 i
0.203 + 0.0835 i
11
3.558
1.5
12
-4.0246
0.00169
13
-31.172
0.011 i
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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
0.2425 + 0.e-5 i
9.4e-10 - 9.4e-10 i
Singularities of quadratic [7, 7, 6] approximant
2
0.2425 - 0.e-5 i
9.4e-10 + 9.4e-10 i
3
0.4473 + 0.e-5 i
6.42e-8 - 6.42e-8 i
4
0.4473 - 0.e-5 i
6.42e-8 + 6.42e-8 i
5
-0.9023 + 0.5607 i
8.51e-7 + 3.35e-6 i
6
-0.9023 - 0.5607 i
8.51e-7 - 3.35e-6 i
7
-0.9045 + 0.5596 i
3.35e-6 - 8.36e-7 i
8
-0.9045 - 0.5596 i
3.35e-6 + 8.36e-7 i
9
1.6716 + 0.2972 i
0.437 + 0.205 i
10
1.6716 - 0.2972 i
0.437 - 0.205 i
11
2.8576
5.16
12
-3.5732
0.000582
13
20.867
0.0651 i
14
-217.8922
0.0662 i
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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
-0.015
0
Singularities of quadratic [7, 7, 7] approximant
2
-0.015
0
3
-0.0476
0
4
-0.0476
0
5
0.8187 + 0.0003 i
7.75e-6 - 7.73e-6 i
6
0.8187 - 0.0003 i
7.75e-6 + 7.73e-6 i
7
-0.883 + 0.6036 i
2.98e-6 - 1.38e-6 i
8
-0.883 - 0.6036 i
2.98e-6 + 1.38e-6 i
9
-0.8846 + 0.6058 i
1.39e-6 + 3.e-6 i
10
-0.8846 - 0.6058 i
1.39e-6 - 3.e-6 i
11
1.6019 + 0.2334 i
0.0571 + 0.0892 i
12
1.6019 - 0.2334 i
0.0571 - 0.0892 i
13
3.0437
48.5
14
-3.9693
0.00166
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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
0.0442 + 0.781 i
2.05e-6 + 2.12e-7 i
Singularities of quadratic [8, 7, 7] approximant
2
0.0442 - 0.781 i
2.05e-6 - 2.12e-7 i
3
0.0441 + 0.781 i
2.12e-7 - 2.05e-6 i
4
0.0441 - 0.781 i
2.12e-7 + 2.05e-6 i
5
-0.9593 + 0.411 i
1.89e-6 + 1.21e-6 i
6
-0.9593 - 0.411 i
1.89e-6 - 1.21e-6 i
7
-0.959 + 0.4139 i
1.21e-6 - 1.9e-6 i
8
-0.959 - 0.4139 i
1.21e-6 + 1.9e-6 i
9
1.4206 + 0.0407 i
0.0147 - 0.0105 i
10
1.4206 - 0.0407 i
0.0147 + 0.0105 i
11
1.5627 + 0.3246 i
0.16 - 0.0678 i
12
1.5627 - 0.3246 i
0.16 + 0.0678 i
13
-3.2877
0.000341
14
3.6728
1.35
15
10.0286
0.287 i
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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
-0.0659 + 0.8661 i
8.25e-7 + 8.12e-8 i
Singularities of quadratic [8, 8, 7] approximant
2
-0.0659 - 0.8661 i
8.25e-7 - 8.12e-8 i
3
-0.0663 + 0.8664 i
8.22e-8 - 8.24e-7 i
4
-0.0663 - 0.8664 i
8.22e-8 + 8.24e-7 i
5
-0.8725 + 0.4409 i
2.92e-7 + 4.45e-7 i
6
-0.8725 - 0.4409 i
2.92e-7 - 4.45e-7 i
7
-0.8754 + 0.4428 i
4.48e-7 - 2.93e-7 i
8
-0.8754 - 0.4428 i
4.48e-7 + 2.93e-7 i
9
1.5291 + 0.5605 i
0.0111 + 0.0167 i
10
1.5291 - 0.5605 i
0.0111 - 0.0167 i
11
1.9675 + 0.6668 i
0.0118 - 0.0429 i
12
1.9675 - 0.6668 i
0.0118 + 0.0429 i
13
-2.9926 + 2.2542 i
2.71e-6 - 0.000276 i
14
-2.9926 - 2.2542 i
2.71e-6 + 0.000276 i
15
1.2817 + 7.5602 i
0.00154 + 0.00517 i
16
1.2817 - 7.5602 i
0.00154 - 0.00517 i
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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
-0.0722 + 0.8616 i
8.51e-7 + 5.87e-8 i
Singularities of quadratic [8, 8, 8] approximant
2
-0.0722 - 0.8616 i
8.51e-7 - 5.87e-8 i
3
-0.0726 + 0.8617 i
5.96e-8 - 8.51e-7 i
4
-0.0726 - 0.8617 i
5.96e-8 + 8.51e-7 i
5
-0.8784 + 0.4431 i
3.68e-7 + 5.15e-7 i
6
-0.8784 - 0.4431 i
3.68e-7 - 5.15e-7 i
7
-0.8813 + 0.4448 i
5.18e-7 - 3.68e-7 i
8
-0.8813 - 0.4448 i
5.18e-7 + 3.68e-7 i
9
1.5354 + 0.5634 i
0.0104 + 0.0182 i
10
1.5354 - 0.5634 i
0.0104 - 0.0182 i
11
2.0058 + 0.6956 i
0.0182 - 0.037 i
12
2.0058 - 0.6956 i
0.0182 + 0.037 i
13
-3.9119 + 3.0109 i
0.000246 + 0.000707 i
14
-3.9119 - 3.0109 i
0.000246 - 0.000707 i
15
1.6554 + 4.6796 i
0.00399 - 0.000321 i
16
1.6554 - 4.6796 i
0.00399 + 0.000321 i
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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
0.1096
0
Singularities of quadratic [9, 8, 8] approximant
2
0.1096
0
3
-0.0797 + 0.8462 i
2.43e-8 + 2.74e-7 i
4
-0.0797 - 0.8462 i
2.43e-8 - 2.74e-7 i
5
-0.08 + 0.8469 i
2.74e-7 - 2.37e-8 i
6
-0.08 - 0.8469 i
2.74e-7 + 2.37e-8 i
7
-0.889 + 0.4417 i
2.65e-7 - 2.66e-8 i
8
-0.889 - 0.4417 i
2.65e-7 + 2.66e-8 i
9
-0.887 + 0.4479 i
3.03e-8 + 2.67e-7 i
10
-0.887 - 0.4479 i
3.03e-8 - 2.67e-7 i
11
1.5298 + 0.6559 i
0.0044 - 0.00709 i
12
1.5298 - 0.6559 i
0.0044 + 0.00709 i
13
1.8359 + 1.0241 i
0.00846 - 0.00204 i
14
1.8359 - 1.0241 i
0.00846 + 0.00204 i
15
-3.9881 + 0.8327 i
0.000406 - 0.000382 i
16
-3.9881 - 0.8327 i
0.000406 + 0.000382 i
17
13.1965
0.0399
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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
-0.7507
5.54e-9
Singularities of quadratic [9, 9, 8] approximant
2
-0.7516
5.55e-9 i
3
-0.7849 + 0.5131 i
2.81e-8 - 2.46e-8 i
4
-0.7849 - 0.5131 i
2.81e-8 + 2.46e-8 i
5
-0.793 + 0.5219 i
2.5e-8 + 2.93e-8 i
6
-0.793 - 0.5219 i
2.5e-8 - 2.93e-8 i
7
-0.1227 + 0.9623 i
1.54e-7 + 1.08e-7 i
8
-0.1227 - 0.9623 i
1.54e-7 - 1.08e-7 i
9
-0.1188 + 0.9669 i
1.11e-7 - 1.57e-7 i
10
-0.1188 - 0.9669 i
1.11e-7 + 1.57e-7 i
11
1.2007
0.000288
12
1.2234
0.000348 i
13
2.083 + 0.3719 i
0.164 + 0.0358 i
14
2.083 - 0.3719 i
0.164 - 0.0358 i
15
-0.8809 + 2.2119 i
8.04e-6 + 0.000014 i
16
-0.8809 - 2.2119 i
8.04e-6 - 0.000014 i
17
0.3062 + 2.508 i
0.0000669 + 0.0000477 i
18
0.3062 - 2.508 i
0.0000669 - 0.0000477 i
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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
-0.5544 + 0.6949 i
2.21e-8 - 2.52e-8 i
Singularities of quadratic [9, 9, 9] approximant
2
-0.5544 - 0.6949 i
2.21e-8 + 2.52e-8 i
3
-0.5554 + 0.699 i
2.53e-8 + 2.24e-8 i
4
-0.5554 - 0.699 i
2.53e-8 - 2.24e-8 i
5
-0.0132 + 1.016 i
2.14e-7 - 3.16e-7 i
6
-0.0132 - 1.016 i
2.14e-7 + 3.16e-7 i
7
-0.0183 + 1.0205 i
3.06e-7 + 2.22e-7 i
8
-0.0183 - 1.0205 i
3.06e-7 - 2.22e-7 i
9
-1.1674 + 0.3666 i
9.23e-8 + 7.9e-8 i
10
-1.1674 - 0.3666 i
9.23e-8 - 7.9e-8 i
11
-1.2391 + 0.6009 i
1.02e-7 - 2.11e-7 i
12
-1.2391 - 0.6009 i
1.02e-7 + 2.11e-7 i
13
-1.3884
1.75e-7
14
1.4514
0.0105
15
1.5835 + 0.2469 i
0.0485 + 0.0359 i
16
1.5835 - 0.2469 i
0.0485 - 0.0359 i
17
1.9952
0.772 i
18
62.2625
0.00679
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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.