Singularities of Møller-Plesset series: example "o1"

Molecule Ne. Basis cc-pVDZ. Structure ""

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Definition of quadratic approximants

Examples	o1	o2	o3	o4	o5	o6	o7	o8	o9
Molecule	Ne	Ne	F-	HF	H2O	CH₂	CH₂	C₂	N₂
Basis	cc-pVDZ	cc-pVTZ-(f)	cc-pVTZ-(f)	cc-pVTZ-(f/d)	cc-pVDZ(+)	aug-cc-pVDZ	cc-pVTZ-(f/d)	cc-pVDZ(+)	cc-pVDZ

Plot of singularities

List of examples

Mathematica programs

Work in UMassD

Unpublished reports

Quadratic approximants

[n₁, n₂, n₃] 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 z_c is defined so that
f ~ c(1 - z/z_c)^1/2 at z -> z_c.
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 = z_c.

Table 1. Singularities with their weights for the quadratic approximant [0, 0, 0]
The most stable singularity is highlighted.

No. z_c c 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. z_c c 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. z_c c Location in the complex plane

1

0.8061

0.0558
Singularities of quadratic [1, 1, 0] approximant

2

1.2672

0.07 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. z_c c Location in the complex plane

1

-2.7839

0.557
Singularities of quadratic [1, 1, 1] approximant

2

10.8432

17.

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Table 5. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.

No. z_c c 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

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Table 6. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.

No. z_c c 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

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Table 7. Singularities with their weights for the quadratic approximant [2, 2, 2]
The most stable singularity is highlighted.

No. z_c c 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

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Table 8. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.

No. z_c c 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

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Table 1. Singularities with their weights for the quadratic approximant [0, 0, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane

Table 2. Singularities with their weights for the quadratic approximant [1, 0, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane

Table 3. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.8061	0.0558
2	1.2672	0.07 i

Table 4. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-2.7839	0.557
2	10.8432	17.

Table 5. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.4628 + 1.1392 i	0.0579 + 0.017 i
2	-1.4628 - 1.1392 i	0.0579 - 0.017 i
3	1.9226	0.0753

Table 6. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-2.1582 + 0.9046 i	0.186 + 0.0777 i
2	-2.1582 - 0.9046 i	0.186 - 0.0777 i
3	2.5034	0.223
4	47.3754	2.55 i

Table 7. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.5464 + 0.8795 i	0.0501 + 0.0301 i
2	-1.5464 - 0.8795 i	0.0501 - 0.0301 i
3	2.4197	0.147
4	-5.1819	0.191

Table 8. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.9502 + 0.9635 i	0.125 + 0.0763 i
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

Examples	o1	o2	o3	o4	o5	o6	o7	o8	o9
Molecule	Ne	Ne	F-	HF	H2O	CH₂	CH₂	C₂	N₂
Basis	cc-pVDZ	cc-pVTZ-(f)	cc-pVTZ-(f)	cc-pVTZ-(f/d)	cc-pVDZ(+)	aug-cc-pVDZ	cc-pVTZ-(f/d)	cc-pVDZ(+)	cc-pVDZ

Plot of singularities

List of examples

Mathematica programs

Work in UMassD

Unpublished reports