Singularities of Møller-Plesset series: example "o1"

Molecule Ne. Basis cc-pVDZ. Structure ""

Content


Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)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 [0, 0, 0]
The most stable singularity is highlighted.
No. zcc Location in the complex plane
Singularities of quadratic [0, 0, 0] approximant
Top of Page  Top of the page    

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
Top of Page  Top of the page    

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
0.8061
0.0558
Singularities of quadratic [1, 1, 0] approximant
2
1.2672
0.07 i
Top of Page  Top of the page    

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
-2.7839
0.557
Singularities of quadratic [1, 1, 1] approximant
2
10.8432
17.
Top of Page  Top of the page    

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
-1.4628 + 1.1392 i
0.0579 + 0.017 i
Singularities of quadratic [2, 1, 1] approximant
2
-1.4628 - 1.1392 i
0.0579 - 0.017 i
3
1.9226
0.0753
Top of Page  Top of the page    

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
-2.1582 + 0.9046 i
0.186 + 0.0777 i
Singularities of quadratic [2, 2, 1] approximant
2
-2.1582 - 0.9046 i
0.186 - 0.0777 i
3
2.5034
0.223
4
47.3754
2.55 i
Top of Page  Top of the page    

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
-1.5464 + 0.8795 i
0.0501 + 0.0301 i
Singularities of quadratic [2, 2, 2] approximant
2
-1.5464 - 0.8795 i
0.0501 - 0.0301 i
3
2.4197
0.147
4
-5.1819
0.191
Top of Page  Top of the page    

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
-1.9502 + 0.9635 i
0.125 + 0.0763 i
Singularities of quadratic [3, 2, 2] approximant
2
-1.9502 - 0.9635 i
0.125 - 0.0763 i
3
2.5557 + 1.0539 i
0.0533 - 0.286 i
4
2.5557 - 1.0539 i
0.0533 + 0.286 i
5
5.8211
0.291
Top of Page  Top of the page    


Exampleso1o2o3o4o5o6o7o8o9
MoleculeNeNeF-HFH2OCH2CH2C2N2
Basiscc-pVDZcc-pVTZ-(f)cc-pVTZ-(f)cc-pVTZ-(f/d)cc-pVDZ(+)aug-cc-pVDZcc-pVTZ-(f/d)cc-pVDZ(+)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.