To find the center of mass of an extended object like a bicycle, one may suspend this object under various angles, determine the plumb line in each case and preserve them like the yellow lines in the picture. The intersection of the plumb lines is the center of mass. Mathematically, this means finding the best intersection of a number of straight lines, a problem to be solved with least squares. Orthogonal plumb lines (being most independent) do the most accurate job.

We recall a little linear algebra to describe the fitting process in a simplified way. Let ${P}_{i}$ be one parameter of a vector $\mathbf{P}$ of parameters and ${E}_{k}$ one energy of a vector $\mathbf{E}$ of energies. All energy operators will be contained in the matrix A.

Orthogonal operators are normalized in batches of operators of the same type (2-body magnetic, 3-body electrostatic, etc.), which means that

${\left({A}^{T}A\right)}_{ii}={p}_{i}:{p}_{i}$ is the same for all operators in the batch. From the formula for the LSF error on the parameter

${P}_{i}$:

it can now be understood, why parameter errors are equal (and minimal) for all parameters in the same batch.

#### 3.1. Construction of an Orthogonal Set

First, let us make a subdivision of possible energy operators, to be able to survey the field:

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.

There are some useful properties of inner products that help defining a set of orthogonal operators. First, the inner product is independent of the coupling scheme.

Second, the behavior of the inner product as a function of the number of electrons in the shell is well-defined. If operators

${H}_{1}$ and

${H}_{2}$ occur together in the

${l}^{n}$ shell for the first time, then their inner product in the

${l}^{N}$ shell is closely related:

The coefficients

$\alpha $ and

$\beta $ only depend on

N and on the

$n-$particle characters of the two operators:

$\alpha $ and

$\beta $ are independent of the operators in question. In an orthogonal operator set, only the average energy operator has a non-zero trace. As a result, once operators are orthogonal in their parent configuration, i.e. the shell(s) where they first make their appearance, then they automatically remain orthogonal in all other configurations. This statement is equivalent to the below group theoretical result [

32]:

If ${H}_{1}$ and ${H}_{2}$ belong to different irreducible representations ${\mathsf{\Gamma}}_{1}$ and ${\mathsf{\Gamma}}_{2}$ (differing symmetries) of a group $\mathcal{G}$ ánd ${H}_{1}{H}_{2}$ does not contain the identity representation ${\mathsf{\Gamma}}_{0}$ of $\mathcal{G}$, then $\to {H}_{1}:{H}_{2}=0$.

This property has been used notably by Brian Judd to construct orthogonal operators based on Lie groups such as

$U(4\ell +2)$,

$Sp(4\ell +2)$,

$S0(2\ell +1)$ and

${G}_{2}$ [

33,

34].

Except building operators with well-defined group-theoretical properties, one may also start from elementary building blocks for inequivalent electrons that are orthogonal due to the well-known properties of

$9j$-symbols. To build a first orthogonal basis, we use annihilation and creation tensor operators [

31] with ranks

$\kappa $ and

k in spin- and orbital spaces, coupled to a total rank

t:

In the electrostatic

$t=0$ case, this simplifies to

$\kappa ={\kappa}^{\prime},k={k}^{\prime}$ yielding a number of

$(4{l}^{\prime}+2)$${e}_{\kappa k}$ basic orthogonal operators in

$l{l}^{\prime}({l}^{\prime}<l)$ [

35].

Such a first orthogonal basis of six electrostatic operators is given in

Table 9 for the dp configuration as an example:

In order to avoid square roots in the entries, the common normalization factors ${\eta}_{\kappa k}$ are given below each column: ${e}_{\kappa k}^{\prime}={\eta}_{\kappa k}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{e}_{\kappa k}$.

The next step is to find linear combinations of these operators to distinguish between first order direct and exchange

$({F}^{k},{G}^{k})$ Coulomb interactions and higher order effects. The Coulomb parameters are named

${C}_{i}$ and the distinct higher order parameters

${S}_{i}$ (Sack), respectively [

36,

37]. Operators are always written lower case to distinguish them from the corresponding parameters, e.g., the

${e}_{av}$ operator is associated with the parameter

${E}_{av}$.

The below example describes the route from the original orthogonal basis given in

Table 9 towards the final orthogonal operator set used for

${d}^{n}p$ configurations [

37] and may serve as a blueprint for any

$l{l}^{\prime}$ configuration.

Properties: ${f}^{k}\propto {e}_{0k}^{\prime}$ and $({e}_{0k}^{\prime}-{e}_{1k}^{\prime}):(3\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{e}_{0k}^{\prime}+{e}_{1k}^{\prime})=0$.

Immediate use: ${e}_{av}={e}_{00}^{\prime},\phantom{\rule{0.277778em}{0ex}}{c}_{1}={e}_{02}^{\prime}$ and ${s}_{1}=(3\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{e}_{01}^{\prime}+{e}_{11}^{\prime})$. Property: ${g}^{k}\propto ({e}_{0k}^{\prime}-{e}_{1k}^{\prime})$ applies to all exchange operators.

Use: we combine the remaining operators: ${e}_{10}^{\prime}$, ${e}_{12}^{\prime}$ and $({e}_{01}^{\prime}-{e}_{11}^{\prime})$ to: $(7{e}_{10}^{\prime}+{e}_{12}^{\prime})$ and its orthogonal counterpart $(4{e}_{10}^{\prime}-{e}_{12}^{\prime})$. To check the whole procedure afterwards, we verify the inner products:

$({e}_{01}^{\prime}-{e}_{11}^{\prime}):(3{e}_{01}^{\prime}+{e}_{11}^{\prime})=0$ and $(7{e}_{10}^{\prime}+{e}_{12}^{\prime}):(4{e}_{10}^{\prime}-{e}_{12}^{\prime})=0$.

Final results:

First order Coulomb: ${c}_{1}={e}_{02}^{\prime}$, ${c}_{2}=(7{e}_{10}^{\prime}+{e}_{12}^{\prime})$ and ${c}_{3}=11({e}_{01}^{\prime}-{e}_{11}^{\prime})-(4{e}_{10}^{\prime}-{e}_{12}^{\prime})$

Higher order: ${s}_{1}=(3{e}_{01}^{\prime}+{e}_{11}^{\prime}$), and ${s}_{2}=\frac{3}{2}({e}_{01}^{\prime}-{e}_{11}^{\prime})+2(4{e}_{10}^{\prime}-{e}_{12}^{\prime})$.

These final results are summarized in

Table 10.

All entries in a column are to be divided by the factors ${\eta}_{i}$ to ensure common normalization:

${e}_{av}:{e}_{av}={c}_{i}:{c}_{i}={s}_{i}:{s}_{i}=60$.

#### 3.2. Completeness

An issue sometimes raised in connection with orthogonal operators is the comparatively large number of parameters M versus the number of observed energy levels N. Usually N is equal or larger than M, and ideally maybe even much larger. As each orthogonal operator describes an independent physical effect, however, it may quickly be seen that with a small number of parameters, many effects are inevitably omitted and one can not hope to obtain a physically reliable fit. On the other hand, for each $\left(n\ell \right)-$configuration with one electron outside closed shells, we have two operators ${e}_{av}$ and ${\zeta}_{l}$ and therefore: $N=M=2$: for this case we seem to be used to a complete set already!

In fact, each complete set of operators may be shown to yield a unique joint solution to level energies and level compositions. Consequently, an operator set that consists of more operators than the number of levels in the configuration is actually not overcomplete. In principle there is, in addition to the level energies, sufficient physical information dependent on the level compositions (Landé g-factors, line strengths) to determine all parameter values unambiguously. In many cases the experimental information is far from complete, but theoretical or empirical knowledge of the parameters can readily be used to reduce the number of parameters to be varied. For a Hamiltonian consisting of angular operators and associated radial parameters to yield correct energies and level compositions, a complete operator set should be used as a fact of principle, even though the number of operators may exceed the number of fitted energy levels. Noble-gas configurations

${p}^{5}s$, using Landé g-factors for additional information on the level compositions, have been used to substantiate this point [

38]. The fit with

$M=5$ and

$N=4$ yielded physically realistic parameter values in line with

ab initio results. In practice, however, one can always neglect the smaller effects, or add them as non-variable quantities derived from empirical or theoretical knowledge.