# Orthogonal Operators: Applications, Origin and Outlook

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Applications

#### 2.1. Oscillator Strengths Involving Close Lying Levels

#### 2.2. Interplay with ab initio Calculations

#### 2.3. Configuration Interaction

#### 2.4. Transition Probabilities Improved

- Use of core-polarization to account for the induced dipole moment, which is particularly important in the case of large, loosely bound (lower ionization stages) ionic cores.This usually decreases the E1-integral by 5-10%: $\overrightarrow{d}\to \overrightarrow{d}\left(1-\frac{{\alpha}_{d}}{{r}^{3}}\right)$ where the dipole polarizability ${\alpha}_{d}$ (in terms of ${a}_{0}^{3}$) is either taken from literature or calculated ab initio. A cutoff radius is introduced here to avoid divergence at $r=0$.For E2 transitions, the electric quadrupole polarization ${\alpha}_{q}$ is used.
- Use of MCDHF calculated transition integrals.
- Inclusion of essential configurations in the model space for full diagonalization.
- Use of perturbation theory: let $\Psi $ and ${\Psi}^{\prime}$ refer to the full odd and even states of the system, to be approximated by the model states $\alpha $ and ${\alpha}^{\prime}$ respectively, and $\beta ,\gamma ..$ far-lying configurations to be summed over. The first order expression $\u2329{\alpha}^{\prime}\left|\phantom{\rule{0.166667em}{0ex}}\mathbf{r}\phantom{\rule{0.166667em}{0ex}}\right|\alpha \u232a$ of the dipole operator $\mathbf{r}$ can be corrected to second order by linking the virtual configurations $\beta ,\gamma $ to the model configurations $\alpha ,{\alpha}^{\prime}$ with the Coulomb operator V:$$\begin{array}{c}\hfill \u2329{\Psi}^{\prime}\left|\phantom{\rule{0.166667em}{0ex}}\mathbf{r}\phantom{\rule{0.166667em}{0ex}}\right|\Psi \u232a=\u2329{\alpha}^{\prime}\left|\phantom{\rule{0.166667em}{0ex}}\mathbf{r}\phantom{\rule{0.166667em}{0ex}}\right|\alpha \u232a+\sum _{\beta}\frac{\u2329{\alpha}^{\prime}\left|\phantom{\rule{0.166667em}{0ex}}\mathbf{r}\phantom{\rule{0.166667em}{0ex}}\right|\beta \u232a\u2329\beta \left|V\right|\alpha \u232a}{{E}_{\alpha}-{E}_{\beta}}+\sum _{\gamma}\frac{\u2329{\alpha}^{\prime}\left|V\right|\gamma \u232a\u2329\gamma \left|\phantom{\rule{0.166667em}{0ex}}\mathbf{r}\phantom{\rule{0.166667em}{0ex}}\right|\alpha \u232a}{{E}_{{\alpha}^{\prime}}-{E}_{\gamma}}\end{array}$$

## 3. Origin

#### 3.1. Construction of an Orthogonal Set

- There are three subspaces of operators that are orthogonal by their tensorial character: expressed as double tensors with ranks $k=0,1,2$ in separate spin- and orbital spaces [31], one distinguishes: ${T}^{\left(00\right)0}\to $ electrostatic, ${T}^{\left(11\right)0}\to $ spin-orbit and ${T}^{\left(22\right)0}\to $ spin-spin.
- Operators acting on different electrons belong in different orthogonal subspaces as well. The $\ell -\ell $ and $\ell -{\ell}^{\prime}$ interactions are described, for example, by separate orthogonal operators.
- In addition, each operator has a unique $n-$particle character, i.e., the number of electrons it acts on (only the average energy is a 0-particle operator). We distinguish $n=2,3,4$ in the electrostatic space, $n=1,2,\left(3\right)$ in the spin-orbit space and $n=2$ in the spin-spin space. An operator may have different $n-$particle characters in different shells: the Trees operator ${T}_{1}$ has a three-particle character in the d-shell, while the ${T}_{\mathrm{dds}}$ operator has a two-particle character in the d-shell and a 1-particle character in the s-shell.
- A further classification is the order of perturbation theory: preferably, we describe first- and second (or higher) order effects by different operators. In line with the previous point: n-body operators occur in the $(n-1)$ order of perturbation.

#### 3.2. Completeness

## 4. Outlook and Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The values of the ${E}_{av}$ parameter in a number of LSF to the 3d${}^{3}$ configuration in Cr IV.

**b**) The values of the ${E}_{av}$ parameter using the same procedure as in (

**a**), but using a set of orthogonal equivalents.

Conventional | Orthogonal Operators | |
---|---|---|

# configurations | large | limited |

parameter interdependence | yes | no |

2-body electrostatic | yes | yes |

3-body electrostatic | only ${T}_{1}$ and ${T}_{2}$ | yes |

4-body electrostatic | no | yes |

1-body magnetic | yes | yes |

2-body magnetic | no | yes |

mean error | medium | small |

initial preparation | small | medium |

transition probabilities | generally sufficient | close to experiment |

use | automated | more case to case |

**Table 2.**Two close lying levels in the $J=5$ matrix of the 3d${}^{5}$4p configuration of Fe III calculated by the conventional and the orthogonal method.

Conventional Method, Overall Mean Error $\mathit{\sigma}=139$ cm${}^{-1}\phantom{\rule{0.277778em}{0ex}}$ | ||||
---|---|---|---|---|

Exp | Calc | Diff. | Eigenvector composition | |

139509.2 | 139407.4 | 101.8 | 49% | ${{(}^{2}H)}^{3}I+21\%{{(}^{4}F)}^{3}G$ |

139463.0 | 139378.4 | 84.7 | 49% | ${{(}^{2}H)}^{3}I+32\%{{(}^{4}F)}^{3}G$ |

Orthogonal method, overall mean error $\sigma =12$ cm${}^{-1}\phantom{\rule{0.277778em}{0ex}}$ | ||||

Exp | Calc | Diff. | Eigenvector composition | |

139509.2 | 139504.1 | 5.0 | 83% | ${{(}^{2}H)}^{3}I+\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}6\%{{(}^{2}H)}^{3}H$ |

139463.0 | 139476.0 | −13.0 | 44% | ${{(}^{4}F)}^{3}G+29\%{{(}^{2}G)}^{3}G$ |

**Table 3.**Transition probabilities ($gA$) in Fe III calculated by the conventional (Conv.) and the orthogonal (Orth.) method, compared to experiment [14]: B. are estimates of photographic blackening on a logarithmic scale, Int. are scaled intensities calculated from B.

The 3d${}^{5}$4s-3d${}^{5}$4p Array | ||||||||
---|---|---|---|---|---|---|---|---|

$\lambda \phantom{\rule{0.277778em}{0ex}}$(Å) | B. | Int. | Conv | $\left({10}^{8}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}\right)$ | Orth | $\left({10}^{8}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}\right)$ | Transition | |

2041.203 | 14 | 22 | 3 | 18 | 3 | 26 | 3 | ${{(}^{4}F)}^{3}{F}_{4}-139463.0$ |

2039.283 | 11 | 2 | 8 | 17 | 2 | 4 | 40 | ${{(}^{4}F)}^{3}{F}_{4}-139509.2$ |

2012.901 | 10 | 1 | 4 | 6 | 95 | 2 | 58 | ${{(}^{2}G)}^{3}{G}_{4}-139463.0$ |

2011.034 | 13 | 11 | 2 | 6 | 07 | 11 | 0 | ${{(}^{2}G)}^{3}{G}_{4}-139509.2$ |

Percentages | ||||
---|---|---|---|---|

Level↓ | (4p)${}^{4}$G | (5p)${}^{6}$F | (5p)${}^{4}$F | |

Raassen and Uylings [17] | 90040.5 | 16 | 36 | 41 |

90072.7 | 76 | 9 | 8 | |

Corrégé and Hibbert [15] | 90042.8 | 43 | 29 | 13 |

90067.4 | 41 | 27 | 16 | |

Uylings and Raassen * | 90042.7 | 44 | 26 | 22 |

90067.4 | 48 | 23 | 22 |

Source | 2507.552 | 2509.097 | Sum |
---|---|---|---|

Kurucz [18] | 0.001 | 0.297 | 0.298 |

Raassen and Uylings [17] | 0.237 | 0.045 | 0.282 |

Corrégé and Hibbert [15] | 0.138 | 0.136 | 0.274 |

Uylings and Raassen${}^{*}$ | 0.148 | 0.134 | 0.282 |

**Table 6.**Second-order contributions $\Delta {T}_{1}\propto {R}^{k}{R}^{{k}^{\prime}}/\Delta E$ to the three-particle Trees parameter in Fe VI (3d${}^{3}$).

Exc. (${\mathit{kk}}^{\prime}$) | 22 | 24 | 44 |
---|---|---|---|

$s\to 3d$ | −12.067 | - | - |

$3d\to s$ | 0.209 | - | - |

$3d\to {d}^{\prime}$ | −0.198 | 0.405 | −0.129 |

$3d\to g$ | 2.391 | 0.710 | −1.107 |

$3d\to i$ | - | - | 0.037 |

Total calc. | −9.727 | ||

Fitted value | −8.452 |

**Table 7.**Calculated contributions to ${T}_{\mathrm{dds}}$ and ${A}_{\mathrm{mso}}$ in Fe VI (3d${}^{2}$4s) compared with experiment.

${\mathit{T}}_{\mathbf{dds}}$ | ${\mathit{A}}_{\mathbf{mso}}$ | ||
---|---|---|---|

$3d\to {d}^{\prime}$ | 27.8 | $3d\to {d}^{\prime}$ | 1.60 |

$4s\to {d}^{\prime}$ | −118.9 | −$\frac{6}{5}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{W}^{1}$ | −0.18 |

$4s\to g$ | 3.0 | $4\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{N}^{0}$ | 1.78 |

Total calc. | −88.1 | 3.20 | |

Fitted value | −91.2 | 2.98 |

Fit(1) | DF | DF + Breit | B-splines | Fit(2) | |
---|---|---|---|---|---|

${\zeta}_{d}$ | 578.63 | 636.97 | 579.56 | 598.09 | 594.52 |

A${}_{c}$ | 0 | 4.43 | 2.95 | 3.16 | 2.84 |

A${}_{3}$ | 0 | 0.18 | 2.07 | 1.97 | 2.41 |

A${}_{4}$ | 0 | 4.39 | 4.37 | 4.31 | 3.86 |

A${}_{5}$ | 0 | 1.64 | 7.18 | 7.05 | 6.86 |

A${}_{6}$ | 0 | 2.33 | −9.22 | −9.08 | −9.85 |

A${}_{1}$ | 0 | −0.12 | 0.41 | 0.88 | 0.90 |

A${}_{2}$ | 0 | 0.12 | −2.31 | −2.73 | −2.90 |

$\sigma $ | 28.3 | 73.4 | 14.2 | 5.8 | 1.9 |

${\mathit{e}}_{00}^{\prime}$ | ${\mathit{e}}_{10}^{\prime}$ | ${\mathit{e}}_{01}^{\prime}$ | ${\mathit{e}}_{11}^{\prime}$ | ${\mathit{e}}_{02}^{\prime}$ | ${\mathit{e}}_{12}^{\prime}$ | ||
---|---|---|---|---|---|---|---|

${}^{1}P$ | 1 | 3 | 3 | 9 | 7 | 21 | |

${}^{1}D$ | 1 | 3 | 1 | 3 | −7 | −21 | |

${}^{1}F$ | 1 | 3 | −2 | −6 | 2 | 6 | |

${}^{3}P$ | 1 | −1 | 3 | -3 | 7 | -7 | |

${}^{3}D$ | 1 | −1 | 1 | −1 | −7 | 7 | |

${}^{3}F$ | 1 | −1 | −2 | 2 | 2 | −2 | |

${\eta}_{\kappa k}$ | 1 | $\sqrt{3}$ | 2 | $2\sqrt{3}$ | $2\sqrt{7}$ | $2\sqrt{21}$ |

${\mathit{e}}_{\mathbf{av}}$ | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | ||
---|---|---|---|---|---|---|---|

${}^{1}P$ | 1 | 7 | 42 | −57 | 9 | −27 | |

${}^{1}D$ | 1 | −7 | 0 | −55 | 3 | 63 | |

${}^{1}F$ | 1 | 2 | 27 | 38 | −6 | 18 | |

${}^{3}P$ | 1 | 7 | −14 | 63 | 3 | 15 | |

${}^{3}D$ | 1 | −7 | 0 | 33 | 1 | −19 | |

${}^{3}F$ | 1 | 2 | −9 | −42 | −2 | −10 | |

${\eta}_{i}$ | 1 | $2\sqrt{7}$ | $\sqrt{231}$ | $2\sqrt{517}$ | $2\sqrt{3}$ | $2\sqrt{141}$ |

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Uylings, P.; Raassen, T. Orthogonal Operators: Applications, Origin and Outlook. *Atoms* **2019**, *7*, 102.
https://doi.org/10.3390/atoms7040102

**AMA Style**

Uylings P, Raassen T. Orthogonal Operators: Applications, Origin and Outlook. *Atoms*. 2019; 7(4):102.
https://doi.org/10.3390/atoms7040102

**Chicago/Turabian Style**

Uylings, Peter, and Ton Raassen. 2019. "Orthogonal Operators: Applications, Origin and Outlook" *Atoms* 7, no. 4: 102.
https://doi.org/10.3390/atoms7040102