Singularities of Møller-Plesset series: example "a75"

Molecule HF. 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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [0, 0, 0] approximant
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Table 2. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [1, 0, 0] approximant
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Table 3. 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
0.069
0.00184
Singularities of quadratic [1, 1, 0] approximant
2
0.0729
0.00189 i
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Table 4. 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
-1.4286
0.323
Singularities of quadratic [1, 1, 1] approximant
2
11.3835
2.34
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Table 5. 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
-0.3793 + 0.4656 i
0.00702 + 0.000941 i
Singularities of quadratic [2, 1, 1] approximant
2
-0.3793 - 0.4656 i
0.00702 - 0.000941 i
3
0.6751
0.00835
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Table 6. 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
-0.9789
0.0818
Singularities of quadratic [2, 2, 1] approximant
2
1.9076 + 1.8554 i
0.237 + 0.074 i
3
1.9076 - 1.8554 i
0.237 - 0.074 i
4
-6.6761
0.33 i
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Table 7. 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
-0.1101
0.0000544
Singularities of quadratic [2, 2, 2] approximant
2
-0.1101
0.0000544 i
3
-0.8482
0.0258
4
1.8738
0.127
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Table 8. 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
-0.9386
0.0621
Singularities of quadratic [3, 2, 2] approximant
2
2.1946 + 1.8639 i
0.121 + 0.0912 i
3
2.1946 - 1.8639 i
0.121 - 0.0912 i
4
1.9943 + 3.922 i
0.188 - 0.0311 i
5
1.9943 - 3.922 i
0.188 + 0.0311 i
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Table 9. 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
0.2561 + 0.e-4 i
0.000171 - 0.000171 i
Singularities of quadratic [3, 3, 2] approximant
2
0.2561 - 0.e-4 i
0.000171 + 0.000171 i
3
-0.925
0.0512
4
3.2146 + 1.064 i
0.917 - 1.1 i
5
3.2146 - 1.064 i
0.917 + 1.1 i
6
-13.7888
0.728 i
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Table 10. 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
-0.9744 + 0.0021 i
12.5 - 6.53 i
Singularities of quadratic [3, 3, 3] approximant
2
-0.9744 - 0.0021 i
12.5 + 6.53 i
3
-1.4677 + 0.1736 i
0.651 + 0.502 i
4
-1.4677 - 0.1736 i
0.651 - 0.502 i
5
2.6101
0.985
6
-5.3355
3.77
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Table 11. 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
-0.8621
0.0286
Singularities of quadratic [4, 3, 3] approximant
2
-1.1321
0.0557 i
3
-1.4244
0.296
4
1.0705 + 2.235 i
0.0602 + 0.0152 i
5
1.0705 - 2.235 i
0.0602 - 0.0152 i
6
2.3259 + 2.0662 i
0.0101 + 0.106 i
7
2.3259 - 2.0662 i
0.0101 - 0.106 i
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Table 12. 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
-0.8615
0.0281
Singularities of quadratic [4, 4, 3] approximant
2
-1.1363
0.0559 i
3
-1.435
0.273
4
1.0954 + 2.156 i
0.0566 + 0.00922 i
5
1.0954 - 2.156 i
0.0566 - 0.00922 i
6
2.1629 + 1.9536 i
0.0177 + 0.0898 i
7
2.1629 - 1.9536 i
0.0177 - 0.0898 i
8
-201657.1219
986. i
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Table 13. 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
-0.8529
0.0268
Singularities of quadratic [4, 4, 4] approximant
2
-1.0622
0.0423 i
3
-1.3543
0.235
4
-1.0717 + 1.4443 i
0.00633 + 0.0425 i
5
-1.0717 - 1.4443 i
0.00633 - 0.0425 i
6
-1.1155 + 1.8574 i
0.0587 - 0.0102 i
7
-1.1155 - 1.8574 i
0.0587 + 0.0102 i
8
2.7369
2.33
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Table 14. 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
-0.8317
0.0251
Singularities of quadratic [5, 4, 4] approximant
2
-0.9099
0.0316 i
3
-1.049
6.01
4
-1.3528
0.107 i
5
-1.8225
0.23
6
3.0456
2.01e3
7
-3.3374
0.108 i
8
0.1266 + 3.5687 i
0.0163 + 0.172 i
9
0.1266 - 3.5687 i
0.0163 - 0.172 i
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Table 15. 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
-0.8258 + 0.0269 i
0.00734 + 0.0119 i
Singularities of quadratic [5, 5, 4] approximant
2
-0.8258 - 0.0269 i
0.00734 - 0.0119 i
3
-0.8926
0.0148
4
-2.1463
146. i
5
2.9681
21.3
6
0.3481 + 3.7761 i
0.122 + 0.201 i
7
0.3481 - 3.7761 i
0.122 - 0.201 i
8
-3.4422 + 1.7955 i
0.164 + 0.0778 i
9
-3.4422 - 1.7955 i
0.164 - 0.0778 i
10
57.6228
2.83 i
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Table 16. 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
-0.8026 + 0.0268 i
0.00261 + 0.00452 i
Singularities of quadratic [5, 5, 5] approximant
2
-0.8026 - 0.0268 i
0.00261 - 0.00452 i
3
-0.8436
0.00519
4
2.5259 + 0.0339 i
0.0792 - 0.0712 i
5
2.5259 - 0.0339 i
0.0792 + 0.0712 i
6
-3.0894
0.265 i
7
-1.2685 + 3.4551 i
0.37 + 0.102 i
8
-1.2685 - 3.4551 i
0.37 - 0.102 i
9
3.863
0.919
10
-12.3298
0.579
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Table 17. 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
-0.797 + 0.024 i
0.00227 + 0.00386 i
Singularities of quadratic [6, 5, 5] approximant
2
-0.797 - 0.024 i
0.00227 - 0.00386 i
3
-0.8361
0.00453
4
-0.9998 + 2.0323 i
0.00148 + 0.0474 i
5
-0.9998 - 2.0323 i
0.00148 - 0.0474 i
6
-1.1524 + 2.4196 i
0.0463 - 0.00731 i
7
-1.1524 - 2.4196 i
0.0463 + 0.00731 i
8
3.0034
43.2
9
0.6193 + 3.3242 i
0.0863 + 0.0821 i
10
0.6193 - 3.3242 i
0.0863 - 0.0821 i
11
-3.4451
0.148 i
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Table 18. 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
-0.7847 + 0.0206 i
0.00133 + 0.00228 i
Singularities of quadratic [6, 6, 5] approximant
2
-0.7847 - 0.0206 i
0.00133 - 0.00228 i
3
-0.8155
0.00264
4
1.9064
0.0184
5
1.9449
0.0192 i
6
-2.0725 + 0.8387 i
0.192 - 0.0469 i
7
-2.0725 - 0.8387 i
0.192 + 0.0469 i
8
-2.6511
0.186 i
9
-1.3744 + 3.4073 i
0.166 + 0.107 i
10
-1.3744 - 3.4073 i
0.166 - 0.107 i
11
4.3878
0.532
12
8.7687
1.65 i
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Table 19. 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
-0.7837 + 0.0203 i
0.00127 + 0.00218 i
Singularities of quadratic [6, 6, 6] approximant
2
-0.7837 - 0.0203 i
0.00127 - 0.00218 i
3
-0.8139
0.00252
4
1.6653
0.00797
5
1.6787
0.00808 i
6
-2.0684 + 0.8295 i
0.19 - 0.058 i
7
-2.0684 - 0.8295 i
0.19 + 0.058 i
8
-2.5948
0.193 i
9
-1.2579 + 3.4582 i
0.179 + 0.0758 i
10
-1.2579 - 3.4582 i
0.179 - 0.0758 i
11
3.9189
0.722
12
10.4397
2.67 i
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Table 20. 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
-0.7801 + 0.0182 i
0.00118 + 0.00194 i
Singularities of quadratic [7, 6, 6] approximant
2
-0.7801 - 0.0182 i
0.00118 - 0.00194 i
3
-0.8102
0.00235
4
-1.4891
0.881 i
5
-1.6157
0.738
6
-2.438
0.371 i
7
2.8344
1.93
8
-0.9584 + 2.7968 i
0.0747 + 0.0458 i
9
-0.9584 - 2.7968 i
0.0747 - 0.0458 i
10
1.6517 + 2.7365 i
0.046 - 0.0834 i
11
1.6517 - 2.7365 i
0.046 + 0.0834 i
12
1.7276 + 4.1286 i
0.05 + 0.0938 i
13
1.7276 - 4.1286 i
0.05 - 0.0938 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.