Singularities of Møller-Plesset series: example "o2"

Molecule Ne. Basis cc-pVTZ-(f). Structure ""

Content

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Unpublished reports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

-0.1605

0.00647

2

-0.1785

0.00683 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

-1.9937

0.584
Singularities of quadratic [1, 1, 1] approximant

2

7.8597

51.2

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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

-1.0786

0.0434
Singularities of quadratic [2, 1, 1] approximant

2

0.7163 + 1.4331 i

0.0594 + 0.00679 i

3

0.7163 - 1.4331 i

0.0594 - 0.00679 i

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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

-1.6704

0.31
Singularities of quadratic [2, 2, 1] approximant

2

5.3355 + 2.8566 i

1.13 + 1.17 i

3

5.3355 - 2.8566 i

1.13 - 1.17 i

4

-17.7119

1.07 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

0.5777 + 0.0009 i

0.00217 - 0.00217 i
Singularities of quadratic [2, 2, 2] approximant

2

0.5777 - 0.0009 i

0.00217 + 0.00217 i

3

-1.5528

0.168

4

3.3052

0.546

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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

-1.6335

0.334
Singularities of quadratic [3, 2, 2] approximant

2

3.2846

1.1

3

-2.0852 + 3.9416 i

0.237 - 0.494 i

4

-2.0852 - 3.9416 i

0.237 + 0.494 i

5

-5.2395

0.464 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. z_c c Location in the complex plane

1

-1.6705

0.46
Singularities of quadratic [3, 3, 2] approximant

2

3.2425

0.926

3

-3.4775

0.385 i

4

-2.692 + 3.3672 i

0.488 - 0.409 i

5

-2.692 - 3.3672 i

0.488 + 0.409 i

6

5259.0761

87.8 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. z_c c Location in the complex plane

1

-1.8616 + 0.1892 i

0.78 + 0.707 i
Singularities of quadratic [3, 3, 3] approximant

2

-1.8616 - 0.1892 i

0.78 - 0.707 i

3

3.2115

0.92

4

-3.2177

1.9

5

11.2609

2.68 i

6

-14.3388

1.14 i

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Table 11. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.3943

0.00131
Singularities of quadratic [4, 3, 3] approximant

2

0.3943

0.00131 i

3

-1.6763

0.494

4

3.3048

1.13

5

-3.5777

0.368 i

6

-2.4909 + 3.3804 i

0.397 - 0.386 i

7

-2.4909 - 3.3804 i

0.397 + 0.386 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. z_c c Location in the complex plane

1

-1.6316

0.365
Singularities of quadratic [4, 4, 3] approximant

2

-0.1141 + 1.8709 i

0.00797 - 0.0317 i

3

-0.1141 - 1.8709 i

0.00797 + 0.0317 i

4

0.0093 + 1.9606 i

0.0324 + 0.006 i

5

0.0093 - 1.9606 i

0.0324 - 0.006 i

6

2.4883

0.119

7

6.4486

1.47 i

8

-34.2029

49. 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. z_c c Location in the complex plane

1

-0.0212 + 1.1594 i

0.00345 - 0.00382 i
Singularities of quadratic [4, 4, 4] approximant

2

-0.0212 - 1.1594 i

0.00345 + 0.00382 i

3

-0.0092 + 1.1655 i

0.00387 + 0.00342 i

4

-0.0092 - 1.1655 i

0.00387 - 0.00342 i

5

-1.7271

1.81

6

2.6743

0.155

7

-2.8558

0.351 i

8

-5.018

1.87

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Table 14. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.0821

9.61e-8
Singularities of quadratic [5, 4, 4] approximant

2

0.0821

9.61e-8 i

3

-1.5803

0.169

4

2.0383

0.0271

5

-0.6984 + 2.2492 i

0.0591 + 0.0000469 i

6

-0.6984 - 2.2492 i

0.0591 - 0.0000469 i

7

-0.2307 + 2.8661 i

0.0286 + 0.0521 i

8

-0.2307 - 2.8661 i

0.0286 - 0.0521 i

9

2.946

0.066 i

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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	-0.1605	0.00647
2	-0.1785	0.00683 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	-1.9937	0.584
2	7.8597	51.2

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	-1.0786	0.0434
2	0.7163 + 1.4331 i	0.0594 + 0.00679 i
3	0.7163 - 1.4331 i	0.0594 - 0.00679 i

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	-1.6704	0.31
2	5.3355 + 2.8566 i	1.13 + 1.17 i
3	5.3355 - 2.8566 i	1.13 - 1.17 i
4	-17.7119	1.07 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	0.5777 + 0.0009 i	0.00217 - 0.00217 i
2	0.5777 - 0.0009 i	0.00217 + 0.00217 i
3	-1.5528	0.168
4	3.3052	0.546

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	-1.6335	0.334
2	3.2846	1.1
3	-2.0852 + 3.9416 i	0.237 - 0.494 i
4	-2.0852 - 3.9416 i	0.237 + 0.494 i
5	-5.2395	0.464 i

Table 9. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.6705	0.46
2	3.2425	0.926
3	-3.4775	0.385 i
4	-2.692 + 3.3672 i	0.488 - 0.409 i
5	-2.692 - 3.3672 i	0.488 + 0.409 i
6	5259.0761	87.8 i

Table 10. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.8616 + 0.1892 i	0.78 + 0.707 i
2	-1.8616 - 0.1892 i	0.78 - 0.707 i
3	3.2115	0.92
4	-3.2177	1.9
5	11.2609	2.68 i
6	-14.3388	1.14 i

Table 11. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.3943	0.00131
2	0.3943	0.00131 i
3	-1.6763	0.494
4	3.3048	1.13
5	-3.5777	0.368 i
6	-2.4909 + 3.3804 i	0.397 - 0.386 i
7	-2.4909 - 3.3804 i	0.397 + 0.386 i

Table 12. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-1.6316	0.365
2	-0.1141 + 1.8709 i	0.00797 - 0.0317 i
3	-0.1141 - 1.8709 i	0.00797 + 0.0317 i
4	0.0093 + 1.9606 i	0.0324 + 0.006 i
5	0.0093 - 1.9606 i	0.0324 - 0.006 i
6	2.4883	0.119
7	6.4486	1.47 i
8	-34.2029	49. i

Table 13. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-0.0212 + 1.1594 i	0.00345 - 0.00382 i
2	-0.0212 - 1.1594 i	0.00345 + 0.00382 i
3	-0.0092 + 1.1655 i	0.00387 + 0.00342 i
4	-0.0092 - 1.1655 i	0.00387 - 0.00342 i
5	-1.7271	1.81
6	2.6743	0.155
7	-2.8558	0.351 i
8	-5.018	1.87

Table 14. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.0821	9.61e-8
2	0.0821	9.61e-8 i
3	-1.5803	0.169
4	2.0383	0.0271
5	-0.6984 + 2.2492 i	0.0591 + 0.0000469 i
6	-0.6984 - 2.2492 i	0.0591 - 0.0000469 i
7	-0.2307 + 2.8661 i	0.0286 + 0.0521 i
8	-0.2307 - 2.8661 i	0.0286 - 0.0521 i
9	2.946	0.066 i

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