Improvements to second-order lifetime calculations within the parametric potential model: application to the Al iv case

P. Ceyzeriat and M. Aymar

Author Affiliations

P. Ceyzeriat^{1} and M. Aymar^{2}

^{1}Laboratoire de Spectrométrie lonique et Moléculaire, Centre National de la Recherche Scientifique, Laboratoire 171, Université Lyon I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cédex,
France

^{2}Laboratoire Aimé Cotton, Bât. 505, Centre National de la Recherche Scientifique, II Campus, 91405 Orsay,
France

P. Ceyzeriat and M. Aymar, "Improvements to second-order lifetime calculations within the parametric potential model: application to the Al iv case," J. Opt. Soc. Am. 72, 116-125 (1982)

Dipolar electric radiative lifetimes are calculated for Al iv. The semiempirical method that associates the parametric procedure, the optimization of a central parametric potential, and a second-order calculation of the radial matrix elements of the transition operator is shown to be related in a natural way to the effective-operator formalism and to perturbation theory. In contrast to previous applications of this method, all second-order contributions are taken into account. Two types of improvement are introduced, namely, the contributions arising from the inner shells and p^{5} shell excitations and the contributions of the continuum. They are shown to lead to a significant decrease in the discrepancies between the dipole-length and dipole-velocity results. Furthermore, their sizes in the second-order corrections are far from negligible in most cases and can even contribute in an essential way.

J. E. Hansen J. Opt. Soc. Am. 59(6) 722-726 (1969)

References

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See text, Section 3.B.
Deviation Δτ/τ between the length and velocity results expressed in percentages.
Complete second-order correction.

Table 2

Relative Length–Velocity Discrepancy Mean Value and Standard Deviation over Subsets of Results and Over the Whole Set of Results within the Different Levels of Approximation in Columns a–e

Mean-Value Standard Deviation

First-Order Result

Second-Order Results with No Improvement

Second-Order Results with One Improvement

Final Results

a

b

c

d

e

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3s)$

10.22

6.93

0.635

1.335

2.575

σ(3s)

0.51

0.5

0.505

0.535

0.505

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3p)$

7.369

4.515

4.258

2.435

2.861

σ(3p)

9.661

5.765

1.494

0.430

3.864

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3d)$

4.008

2.853

2.07

3.805

1.114

σ(3d)

1.900

0.603

1.322

5.484

0.619

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4s)$

2.337

5.96

1.98

1.385

0.355

σ(4s)

1.916

5.603

1.025

0.281

0.305

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4p)$

2.388

6.548

6.244

1.901

0.8959

σ(4p)

2.373

2.463

6.842

0.655

0.567

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4d)$

3.319

2.316

0.5875

3.982

0.786

σ(4d)

1.929

0.789

0.463

5.573

0.389

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4f)$

3.217

1.772

1.706

1.6425

1.499

σ(4f)

1.428

0.513

1.030

0.417

0.504

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(5s)$

1.8

7.737

3.162

1.460

0.225

σ(5s)

1.676

7.445

2.275

0.075

0.102

whole set of results

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}$

3.956

3.978

2.715

2.584

1.300

σ

4.587

3.933

3.445

3.511

1.760

Table 3

Probabilities (in nanoseconds) of Resonance Transitions from Low-Lying Excited States Calculated Using Different Methods^{a}

L, length; V, velocity.
a, b, and e are at the levels of approximation as noted in a, b, and e columns in Tables 1 and 2. DHF, RRPA, RPA (Ref. 27) (oscillator strength results have been converted into transition probabilities, HF (Ref. 26), and HF + second order (Ref. 28) are defined in Section 3.C.

Tables (3)

Table 1

Lifetime Results (in nanoseconds) in Five Cases Corresponding to Different Degrees of Approximation

See text, Section 3.B.
Deviation Δτ/τ between the length and velocity results expressed in percentages.
Complete second-order correction.

Table 2

Relative Length–Velocity Discrepancy Mean Value and Standard Deviation over Subsets of Results and Over the Whole Set of Results within the Different Levels of Approximation in Columns a–e

Mean-Value Standard Deviation

First-Order Result

Second-Order Results with No Improvement

Second-Order Results with One Improvement

Final Results

a

b

c

d

e

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3s)$

10.22

6.93

0.635

1.335

2.575

σ(3s)

0.51

0.5

0.505

0.535

0.505

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3p)$

7.369

4.515

4.258

2.435

2.861

σ(3p)

9.661

5.765

1.494

0.430

3.864

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(3d)$

4.008

2.853

2.07

3.805

1.114

σ(3d)

1.900

0.603

1.322

5.484

0.619

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4s)$

2.337

5.96

1.98

1.385

0.355

σ(4s)

1.916

5.603

1.025

0.281

0.305

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4p)$

2.388

6.548

6.244

1.901

0.8959

σ(4p)

2.373

2.463

6.842

0.655

0.567

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4d)$

3.319

2.316

0.5875

3.982

0.786

σ(4d)

1.929

0.789

0.463

5.573

0.389

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(4f)$

3.217

1.772

1.706

1.6425

1.499

σ(4f)

1.428

0.513

1.030

0.417

0.504

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}(5s)$

1.8

7.737

3.162

1.460

0.225

σ(5s)

1.676

7.445

2.275

0.075

0.102

whole set of results

$\frac{\overline{\mathrm{\Delta}\tau}}{\tau}$

3.956

3.978

2.715

2.584

1.300

σ

4.587

3.933

3.445

3.511

1.760

Table 3

Probabilities (in nanoseconds) of Resonance Transitions from Low-Lying Excited States Calculated Using Different Methods^{a}

L, length; V, velocity.
a, b, and e are at the levels of approximation as noted in a, b, and e columns in Tables 1 and 2. DHF, RRPA, RPA (Ref. 27) (oscillator strength results have been converted into transition probabilities, HF (Ref. 26), and HF + second order (Ref. 28) are defined in Section 3.C.