Inner Products of Spherical Tensor Operators: A Late Chapter of Racah Algebra

Abstract

Inner products of spherical tensor operators have been used since the early eighties to define orthogonal operators. However, the basic theory and properties are largely missing in the literature. An inner product in any configuration is directly proportional to the inner product taken in the most basic configuration in which it can occur. The formula for the proportionality factor in question is presented for the first time. This allows the inner products in and between arbitrary configurations to be calculated in advance. In addition, inner products are shown to be independent of the coupling scheme used to construct the state functions. Applications such as the orthogonal operator method and projections of ab initio calculations check for the completeness of the used basis of operators and, importantly, check the matrix elements in any arbitrary configuration, as discussed and illustrated with examples. Closed formulae for the inner products of the well-known Slater and spin–orbit operators are given.

1. Introduction

Spherical tensor operators are widely used in the fields of atomic, molecular, and nuclear physics; see, e.g., [1]. The inner product of these trace-class operators, defined as the double contraction or double-dot product of their matrices, was not considered until half a century later [2,3]; only then was its added value seen. The purpose of the present work is to explain and illustrate the theory and applications of the Hilbert–Schmidt inner product in the current context. In this paper, the operator inner product is shown to be a property of operators rather than of their accidental matrix elements; the inner product of two operators remains the same regardless of the bra and ket configurations, except for a simple proportionality factor p. The derivation of a closed expression for p, referred to as the ‘parent factor’ hereinafter, is one of the key results of the present work. Vanishing inner products or orthogonality ensure the least correlation between the energy operators, and this increased stability appears to be a powerful tool in reducing the deviations between calculated and experimental energy values in complex spectra $(Z>20)$. In addition, the linear algebra of operators can fruitfully be used to project a variety of contributions onto the orthogonal operator set, both algebraically and numerically, using a second key result explained in this work. This procedure supersedes the often-used but incomplete direct proportionality, such as the $\Delta -$factors in hyperfine structures. It was actually in this way that the Blume and Watson theory for the spin–orbit interaction [4] was generalized and corrected [5] and other contributions than $s\to d$ excitations to the Trees operator were found.

In what follows, operators are all factorized as a product of a spin–angular operator (lower case) and a radial factor (upper case): $H_i=p_i{P}_i$. In the orthogonal operator method, the model space is spanned by a set of orthogonal operators $H_i=p_i{P}_i$ with inner products $p_i:p_j=0(i\ne j)$, where the angular operators form an orthogonal set $\left\{p_i\right\}$ and the radial factors $P_i$ are treated as parameters.

Operators may be expressed as double tensors with ranks $k=0,1,2$ in separate spin and orbital spaces [6]. Only if the two double tensors in an inner product are diagonal in all tensor ranks can they form a scalar number. There are three energy subspaces of operators that are orthogonal by their tensorial character: $T^{\left(00\right)0}\to$ electrostatic, $T^{\left(11\right)0}\to$ spin–orbit, and $T^{\left(22\right)0}\to$ spin–spin. Operators acting on different electrons belong to different orthogonal subspaces as well.

The layout of this article is as follows. In Section 2, some relevant linear algebra and group theoretical results are reviewed in the present context of inner products of spherical tensor operators. The formulae are later necessary for the elaboration of the projection method in Section 4. In Section 3, an expression for the parent factor p is developed. This parent factor is the proportionality factor of the inner product in any arbitrary configuration and the inner product in the (most elementary) parent configuration. The formula and its complete derivation are given here for the first time. The parent factor p not only applies to intra-configuration but also to inter-configuration inner products. This original result is elaborated in Section 3.1. Combined with the closed formulae for the parent inner products, this provides an important check for the matrices of arbitrary configuration interaction operators. For completeness, closed formulae for the inner products of spin–orbit fine structure operators are given in Section 3.2. In Section 4, it is shown how radial parameters can be calculated with a projection method using inner products. The $N-$dependence of the projections is a newly derived result following directly from a ratio of parent factors. The method is shortly exemplified in Section 4.1 and Section 4.2 by means of an ab initio calculation of an electrostatic and a magnetic higher-order parameter in the $d^{N}s$ configurations coming from the existing literature.

2. An Operator Hilbert Space

The inner product of two operators u and v is defined as

$$\begin{array}{c}\hfill u:v=\sum _{\Psi ,{\Psi }^{\prime }}⟨\Psi \mid u\mid {\Psi }^{\prime }⟩⟨{\Psi }^{\prime }\mid v\mid \Psi ⟩\end{array}$$

(1)

Equating $u:v$ to zero defines a set of orthogonal operators to be used in a least-squares fit (LSF) procedure. However, the use of the concept of the operator inner product does not stop there. The resulting linear algebra allows the definition of an operator projection, and this opens up new possibilities that can fruitfully be exploited in ab initio calculations.

Let the operator space be spanned by a set of orthogonal operators $H_i=p_i{P}_i$ with $p_i:p_j=0(i\ne j)$, where the spin–angular operators form an orthogonal set $\left\{p_i\right\}$, and the radial factors $P_i$ are treated as parameters. Any arbitrary operator $U=vV$, written as a product of a spin–angular operator v and a radial factor V, can now be expressed in terms of the complete basis set of orthogonal operators:

$$\begin{array}{c}\hfill v=\sum _i{\alpha }_i·p_i=\sum _i{\displaystyle \frac{v:p_i}{p_i:p_i}}·p_i\end{array}$$

(2)

By the reverse, the contribution of U to a particular parameter $P_i$ is given by

$$\begin{array}{c}\hfill \Delta P_i={\alpha }_i·V={\displaystyle \frac{v:p_i}{p_i:p_i}}·V\end{array}$$

(3)

Applications of Equation (3) and its dependence on the number of electrons N are further elaborated upon in Section 4.

In addition, the projection of any physical operator $U=vV$ on a finite (and possibly incomplete) basis $\left\{p_i\right\}$ is complete if and only if the magnitude of the angular operator equals the sum of the magnitudes of its projections:

$$\begin{array}{c}\hfill v:v=\sum _i{\displaystyle \frac{{\left(v:p_i\right)}^2}{p_i:p_i}}\end{array}$$

(4)

Equation (4) can be used to find the percentages by which a given operator is represented in subsets of a particular type, such as the first, second, or third order, or of a particular n-particle character.

Alternatively, any ‘new’ operator t describing effects that are not yet completely covered by the original orthogonal operator set will naturally not be completely represented by $\left\{p_i\right\}$. It may, however, be orthogonalized straightforwardly:

$$\begin{array}{c}\hfill t_{\perp }=t-\sum _i{\alpha }_i·p_i=t-\sum _i{\displaystyle \frac{t:p_i}{p_i:p_i}}·p_i\end{array}$$

(5)

A group-theoretical proof of the conserved orthogonality as a function of N was given by [7]. Consider the two operators $H_1$ and $H_2$ that belong to different irreducible representations ${\Gamma }_1$ and ${\Gamma }_2$ of a group $\mathcal{G}$. If the collection of states $|{\psi }_n⟩$ forms a basis for a representation (not necessarily irreducible) of the group $\mathcal{G}$, then the equation

$$\begin{array}{c}\hfill \sum _{n,m}⟨{\psi }_n\left|H_1\right|{\psi }_m⟩⟨{\psi }_m\left|H_2\right|{\psi }_n⟩=0\end{array}$$

(6)

follows from the fact that $H_1{H}_2$ does not contain the identity representation ${\Gamma }_0$ of $\mathcal{G}$. Any two operators of differing symmetries ${\Gamma }_1$ and ${\Gamma }_2$ (i.e., for which ${\Gamma }_1{\Gamma }_2$ does not contain the identity representation) are automatically orthogonal. 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. Notably, Brian Judd used this property to construct orthogonal operators based on Lie groups such as U$(4\ell +2)$, Sp$(4\ell +2)$, SO$(2\ell +1)$, and ${\mathrm{G}}_2$ [8].

3. The Parent Factor: $\mathit{N}$-Dependence and the $\mathit{n}$-Particle Character

Denoting the zero-particle unit operator as $\mathbb{1}$, the trace Tr $\left(H\right)$ of an operator H will be $H:\mathbb{1}$ in the notation of inner products. In particular, $\mathbb{1}:\mathbb{1}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)$ is the number of states in the $l^N$ configuration, the general case given by

$$\begin{array}{c}\hfill \mathbb{1}:\mathbb{1}=\prod _{\mathrm{shell}l}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)\end{array}$$

(7)

Let $H_n^{\left(N\right)}$ be an $n-$particle operator in the $l^N$ shell; n refers to the number of electrons that $H_n^{\left(N\right)}$ operates on and N is the number of electrons in the shell involved, so $l^n$ is the parent shell and $N\ge n$.

Its contribution to the average energy equals the trace divided by the number of states in the shell:

$$\begin{array}{c}\hfill \mathrm{Tr}{H}_n^{\left(N\right)}{\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)}^{-1}=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)·\mathrm{Tr}{H}_n^{\left(n\right)}{\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{n}}\right)}^{-1}\end{array}$$

(8)

From the useful identity

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n}{N-n}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)}^{-1}=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{n}}\right)}^{-1}\end{array}$$

(9)

it follows immediately for any pure n-particle operator in $l^N$ that

$$\begin{array}{c}\hfill \mathrm{Tr}{H}_n^{\left(N\right)}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n}{N-n}}\right)\mathrm{Tr}{H}_n^{\left(n\right)}\end{array}$$

(10)

Given that $\mathbb{1}:\mathbb{1}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)$, one may conclude that the unit matrix $e_{\mathrm{av}}=\mathbb{1}$, i.e., the angular part of $E_{\mathrm{av}}$, is a zero-particle operator. The trace of an arbitrary operator yields its average energy contribution. Expressed in double tensors $T^{\left(\kappa k\right)t}$, one finds that only electrostatic operators $H=T^{\left(00\right)0}$ may have a trace. From the above, it also directly follows that the condition for ’no shift’ of any electrostatic operator H implies Tr $H=0$.

Next, consider two intra-configuration energy operators $H_1$ and $H_2$, with $n-$particle characters in the $l^N$ shell of $n_1$ and $n_2$, respectively. The inner product $H_1:H_2$ is the trace of the operator product $H_1{H}_2$:

$$\begin{array}{c}\hfill H_1:H_2^{\left(N\right)}=H_1{H}_2:{\mathbb{1}}^{\left(N\right)}=\mathrm{Tr}{\left(H_1{H}_2\right)}^{\left(N\right)}\end{array}$$

(11)

The superscript $\left(N\right)$ in the above refers to the shell $l^N$ in which the inner product is taken. Inner products are properties of the operators, not just of their matrix elements in a particular configuration. Therefore, the inner product of two operators in any configuration is closely related to the inner product in their parent configuration, i.e., the configuration where they first appear together:

$$\begin{array}{c}\hfill H_1:H_2^{\left(N\right)}=\alpha \mathrm{Tr}{H}_1^{\left(n\right)}·\mathrm{Tr}{H}_2^{\left(n\right)}+\beta H_1:H_2^{\left(n\right)}\end{array}$$

(12)

The coefficients $\alpha$ and $\beta$ only depend on N, $n_1$, and $n_2$ and are independent of the operators in question. In practice, $\alpha$ is hardly important, as electrostatic operators other than the average energy are usually defined to be traceless. Ref. [9] defined his operators $f^2$ and $f^4$ (associated with the Slater integrals $F^2$ and $F^4$) as early as 1960 to be traceless by extracting the average energy contributions $-2/63·(f^2+f^4)$ and, thus, orthogonalized them to the average energy operator avant la lettre; however, they still remain non-orthogonal to one another. The application of Equation (5) gives

$$\begin{array}{c}\hfill f_{\perp }^4=f^4+{\displaystyle \frac{10}{53}}{f}^2\end{array}$$

(13)

To find an explicit expression for $\beta$, consider two intra-configuration energy operators $H_1$ and $H_2$, with $n-$particle characters in the $l^N$ shell of $n_1$ and $n_2$, respectively. Assume that $n_2\ge n_1$ for the sake of the discussion. For the operator product $H_1{H}_2$ to exist in the $l^N$ shell, it follows that $n_2\le N\le 4l+2-n_1$. To understand what happens to the $n-$particle character of a product $H_1{H}_2$, let us consider the product of a two- and three-particle operator schematically in a second quantization: $\left({\mathbf{a}}^{†}{\mathbf{a}}^{†}\mathbf{a}\mathbf{a}\right)\left({\mathbf{a}}^{†}{\mathbf{a}}^{†}{\mathbf{a}}^{†}\mathbf{a}\mathbf{a}\mathbf{a}\right)$. To obtain pure $n-$particle operators, all of the creation operators have to be moved to the left. However, when a creation and an annihilation operator of the same shell are interchanged, Equation (14) applies:

$$\begin{array}{c}\hfill {\left(\mathbf{a}{\mathbf{a}}^{†}\right)}^{\kappa k}={(-1)}^{\kappa +k}{\left({\mathbf{a}}^{†}\mathbf{a}\right)}^{\kappa k}+\delta (\kappa ,0)\delta (k,0){[{\displaystyle \frac{1}{2}},l]}^{{\displaystyle \frac{1}{2}}}\end{array}$$

(14)

This gives a branching into an n-particle and an $(n-1)$-particle operator, as the second term has one ${\mathbf{a}}^{†}\mathbf{a}$ pair less. In the above example, one finds a three-, four-, and five-particle operator. The $n-$particle character of $H_1{H}_2$, therefore, ranges from $n_2$ to $n_1+n_2$. A series of pure $n-$particle operators now emerges; multiplied with the required Pauli phase and statistical weighting factors, one finds the following with the use of Equation (10):

$$\begin{array}{cc}\hfill & H_1:H_2^{\left(N\right)}=\mathrm{Tr}{\left(H_1{H}_2\right)}^{\left(N\right)}\hfill \\ \hfill & =\prod _{\mathrm{shell}l}\sum _{n=n_2}^{n_1+n_2}{(-1)}^{n-n_2}\left({\displaystyle \genfrac{}{}{0pt}{}{n_1}{n-n_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n}{N-n}}\right)H_1:H_2^{\left(n_2\right)}\hfill \end{array}$$

(15)

where $n_1$ and $n_2$ are the $n-$particle characters of the operators $H_1$ and $H_2$ in the shell characterized by orbital angular momentum l. The summation over n can be carried out explicitly with the following mathematical formula, to be proved with Zeilberger’s algorithm after a transformation to hypergeometric functions [10]:

$$\begin{array}{c}\hfill \sum _{k=0}^n{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{a-k}{b-k}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{a-n}{b}}\right)\end{array}$$

(16)

Equation (16), valid for $a\ge n\ge 0$ and $a\ge b\ge 0$, is also given by [11] in Equations (A1.1) and (A1.2). With the substitutions $a\to 4l+2-n_2,b\to N-n_2,n\to n_1$, and $k\to n-n_2$, Equation (16) gives

$$\begin{array}{c}\hfill \sum _{n=n_2}^{n_1+n_2}{(-1)}^{n-n_2}\left({\displaystyle \genfrac{}{}{0pt}{}{n_1}{n-n_2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n}{N-n}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_1-n_2}{N-n_2}}\right)\end{array}$$

(17)

The expression for the coefficient $\beta$ introduced in Equation (12) and the parent factor $p={\prod }_{\left(\mathrm{shell}l\right)}\beta$ thus becomes:

$$\begin{array}{cc}\hfill & H_1:H_2^{\left(N\right)}=\prod _{\mathrm{shell}l}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_1-n_2}{N-n_2}}\right)H_1:H_2^{\left(n_2\right)}\to \hfill \\ \hfill & p=\prod _{\mathrm{shell}l}\beta (n_1,n_2,N)=\prod _{\mathrm{shell}l}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_1-n_2}{N-n_2}}\right)\hfill \end{array}$$

(18)

The coefficient $\alpha$ is of no importance when dealing with orthogonal or configuration interaction operators, as all of them are traceless except the $e_{av}$ operator. For the sake of completeness, the expression is given as follows:

$$\begin{array}{c}\hfill \alpha (n_1,n_2,N)={\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{n_2}}\right)}^{-1}\left[\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n_1}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{n_2}{n_1}}\right)}^{-1}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_2}{N-n_2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_1-n_2}{N-n_2}}\right)\right]\end{array}$$

(19)

The N-dependence of the magnitude $H:H$ of an intra-configuration operator H constitutes a special case with $n_1=n_2$:

$$\begin{array}{c}\hfill H:H^{\left(N\right)}=\mathrm{Tr}{\left(HH\right)}^{\left(N\right)}=\prod _{\mathrm{shell}l}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-2n}{N-n}}\right)H:H^{\left(n\right)}\end{array}$$

(20)

Equation (20) allows the prediction of the inner products of traceless intra-configuration operators in any configuration without the need to calculate the matrix elements explicitly.

This special case $\beta (n,n,N)$ is also given by [8], derived from group-theoretical arguments.

One can verify that $\beta (n,n,N)$ is invariant under conjugation, as it should indicate that

$$\begin{array}{c}\hfill \beta (n,n,N)=\beta (n,n,4l+2-N)\end{array}$$

(21)

An operator may have different $n-$particle characters in different shells; for example, the three-particle operator $t_{\mathrm{dds}}$ in $d^{N}s$ configurations has $n=2$ in the d shell and $n=1$ in the s shell. In such cases, the total factor p is simply found as the product of the $\beta$’s for each individual shell. The result for $t_{\mathrm{dds}}$ is then $p=\left({\displaystyle \genfrac{}{}{0pt}{}{10-4}{N-2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{2-2}{1-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{6}{N-2}}\right)$. Given that ${(t_{\mathrm{dds}}:t_{\mathrm{dds}})}^{d^{2}s}=90$, this yields ${(t_{\mathrm{dds}}:t_{\mathrm{dds}})}^{d^{5}s}=\left({\displaystyle \genfrac{}{}{0pt}{}{6}{3}}\right)·90=1800$.

Given the N-dependence of the inner products, the full matrix of inner products for any particular configuration can now be predicted in advance and, thus, serves as a rather strict check on the operator calculation.

3.1. The Parent Factor of Inter-Configuration Operators

Equation (18) applies to operators H between different configurations as well. Such inter-configuration operators are intrinsically traceless, so only the parent factor $p={\prod }_{\left(\mathrm{shell}l\right)}\beta (n_1,n_2,N)$ is needed to find the N-dependence of $H_1:H_2$.

Here, the shell occupations of bra and ket states are different, and so are, potentially, the $n-$particle characters $n_1$ and $n_2$ of H in the relevant bra- and ket-shells; note that the bra–ket symmetry (Hermiticity) is retained, as $N-n_2=N_{\mathrm{bra}}-n_{\mathrm{bra}}=N_{\mathrm{ket}}-n_{\mathrm{ket}}$.

For passive spectator shells ${\overline{l}}^M$, $n_2=n_1=0$, and an additional factor $\left({\displaystyle \genfrac{}{}{0pt}{}{4\overline{l}+2}{M}}\right)$ appears.

In the vast majority of cases, H concerns the Coulomb interaction C.

Even with complex bra and ket configurations, the inner products of the operators associated with any of the $R^k$ and $R^{k^{\prime }}$ integrals turn out to be directly proportional to those of the underlying two-particle (parent) interaction.

The integer proportionality coefficient (hereinafter referred to as parent factor p) is equal for any combination of tensor ranks k and $k^{\prime }$ and, like the inner products themselves, independent of the coupling scheme. The occurring inner products may, therefore, be predicted prior to the actual calculation of the matrix elements and can serve as a strict check on the results of the computer program used. The parent factors p of the Coulomb interactions $l^n\leftrightarrow l^{n-1}{l}^{\prime }$ and $l^n{l}^{\prime }\leftrightarrow l^n{l}^{″}$ and some other Coulomb inter-configuration operators are exemplified below.

For $⟨l^N\left|C\right|l^{N-1}{l}^{\prime }⟩$, e.g., one obtains $n_2=2$ (bra) and $n_1=1$ (ket), yielding

$$\begin{array}{c}\hfill C:C^{l^N\leftrightarrow l^{N-1}{l}^{\prime }}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-1-2}{N-2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }+2-1-0}{0-0}}\right)C:C^{l^2\leftrightarrow l{l}^{\prime }}\\ \hfill =\left({\displaystyle \genfrac{}{}{0pt}{}{4l-1}{N-2}}\right)C:C^{l^2\leftrightarrow l{l}^{\prime }}\to p=\left({\displaystyle \genfrac{}{}{0pt}{}{4l-1}{N-2}}\right)\end{array}$$

(22a)

Equation (18) holds irrespective of possible Brillouin corrections coming from the off-diagonal potential.

A more general example is the following:

$$\begin{array}{c}\hfill C:C^{l^{\prime N}{l}^M\leftrightarrow l^{\prime N-1}{l}^{M+1}}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }-1}{N-2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l+1}{M}}\right)C:C^{l^{\prime 2}\leftrightarrow l^{\prime }l}\to p=\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }-1}{N-2}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l+1}{M}}\right)\\ \hfill \mathrm{or},\mathrm{interchanging}{l}^{\prime }\leftrightarrow l\mathrm{and}N-1\leftrightarrow M,\\ \hfill \left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }+1}{N-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l-1}{M-1}}\right)C:C^{l^{\prime }l\leftrightarrow l^2}p=\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }+1}{N-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l-1}{M-1}}\right)\end{array}$$

(22b)

Similarly, for the Coulomb interaction $⟨l^N{l}^{\prime }\left|C\right|l^N{l}^{″}⟩$, one obtains $n_2=n_1=1$, yielding

$$\begin{array}{c}\hfill C:C^{l^N{l}^{\prime }\leftrightarrow l^N{l}^{″}}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-1-1}{N-1}}\right)C:C^{l{l}^{\prime }\leftrightarrow l{l}^{″}}\to p=\left({\displaystyle \genfrac{}{}{0pt}{}{4l}{N-1}}\right)\end{array}$$

(22c)

A concrete example yielding $p=78$ is the following:

$$\begin{array}{ccc}\hfill C:C^{p^6{f}^{2}d\leftrightarrow p^5{f}^{3}d}& =& \left({\displaystyle \genfrac{}{}{0pt}{}{6-0-1}{6-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{14-0-1}{3-1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{10-1-1}{1-1}}\right)C:C^{pd\leftrightarrow fd}\hfill \\ & =& 78·C:C^{pd\leftrightarrow fd}\hfill \end{array}$$

(23)

The parent inner products $C:C^{l_a{l}_b\leftrightarrow l_c{l}_d}$ occurring at the upper right-hand side can be calculated not only numerically but also algebraically using straightforward Racah algebra. Recall that the results will be the same for any coupling scheme.

The spin–angular part of the Coulomb interaction can be written as $C={\sum }_{k{k}^{\prime }}(D^k+E^{k^{\prime }})$. The parent inner products of the operators $D^k$ and $E^{k^{\prime }}$, associated with the direct and exchange $R^k$-integrals, respectively, become the following for non-equivalent electrons:

$$\begin{array}{cc}\hfill & D^k:D^{k^{\prime }}=8·\delta \left(k{k}^{\prime }\right)·{⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩}^2{⟨l_b\Vert C^{\left(k\right)}\Vert l_d⟩}^2·{\left[k\right]}^{-1}\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill & E^k:E^{k^{\prime }}=8·\delta \left(k{k}^{\prime }\right)·{⟨l_a\Vert C^{\left(k\right)}\Vert l_d⟩}^2{⟨l_b\Vert C^{\left(k\right)}\Vert l_c⟩}^2·{\left[k\right]}^{-1}\hfill \end{array}$$

(24b)

$$\begin{array}{cc}\hfill & D^k:E^{k^{\prime }}=-4·⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩⟨l_b\Vert C^{\left(k\right)}\Vert l_d⟩\hfill \\ \hfill & ·⟨l_a\Vert C^{\left(k^{\prime }\right)}\Vert l_d⟩⟨l_b\Vert C^{\left(k^{\prime }\right)}\Vert l_c⟩·{(-1)}^{l_c+l_d}·\left\{\begin{array}{ccc}{l}_c& l_b& k^{\prime }\\ l_d& l_a& k\end{array}\right\}\hfill \end{array}$$

(24c)

When both bra and ket contain equivalent electrons, $l_a\equiv l_b$ and $l_c\equiv l_d$, one finds

$$\begin{array}{cc}\hfill & D^k:D^{k^{\prime }}=4·\delta \left(k{k}^{\prime }\right)·{⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩}^4·{\left[k\right]}^{-1}\hfill \\ \hfill & -2·{⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩}^2{⟨l_a\Vert C^{\left(k^{\prime }\right)}\Vert l_c⟩}^2·\left\{\begin{array}{ccc}{l}_a& l_c& k^{\prime }\\ l_a& l_c& k\end{array}\right\}\hfill \end{array}$$

(24d)

Finally, if either bra or ket contains equivalent electrons, $l_a\equiv l_b$, this yields

$$\begin{array}{cc}\hfill & D^k:D^{k^{\prime }}=8·\delta \left(k{k}^{\prime }\right)·{⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩}^2·{⟨l_a\Vert C^{\left(k\right)}\Vert l_d⟩}^2·{\left[k\right]}^{-1}\hfill \\ \hfill & -4·⟨l_a\Vert C^{\left(k\right)}\Vert l_c⟩⟨l_a\Vert C^{\left(k\right)}\Vert l_d⟩⟨l_a\Vert C^{\left(k^{\prime }\right)}\Vert l_c⟩⟨l_a\Vert C^{\left(k^{\prime }\right)}\Vert l_d⟩·\left\{\begin{array}{ccc}{l}_c& l_a& k^{\prime }\\ l_d& l_a& k\end{array}\right\}\hfill \end{array}$$

(24e)

In the above, the role of the Pauli exclusion principle for equivalent electrons is readily recognized as similar to the role of the exchange for non-equivalent electrons.

3.2. Fine Structure

The most important fine structure operator is the spin–orbit coupling $Z\left(a\right)=z\left(a\right)·\zeta \left(a\right)$, where $z\left(a\right)$ is the spin–angular operator ${\sum }_i{\overrightarrow{s}}_i·{\overrightarrow{l}}_i=\sqrt{l(l+1)(2l+1)/2}{\left({\mathbf{a}}^{†}\mathbf{a}\right)}^{\left(11\right)0}$, and the spin–orbit coupling constant $\zeta \left(a\right)$ is the corresponding radial parameter. To find its relativistic form, we consider the $T^{\left(11\right)0}$ term of the nuclear attraction after a $jj\to SL$ recoupling:

$$\begin{array}{c}\hfill -\sqrt{{\displaystyle \frac{l(l+1)(2l+1)}{2}}}·\zeta \left(a\right)=\sum _{j}3(2j+1)\left\{\begin{array}{ccc}{\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& 1\\ l& l& 1\\ j& j& 0\end{array}\right\}·\left(a_j\Vert -Z/r\Vert a_j\right)\end{array}$$

(25)

and calculate the reduced matrix element of the operator $-Z/r$:

$$\begin{array}{c}\hfill \left(a_j\Vert -Z/r\Vert a_j\right)=\sqrt{2j+1}{\int }_0^{\infty }\left(F_j^2+G_j^2\right){\displaystyle \frac{-Z}{r}}\mathrm{d}r\end{array}$$

(26)

where $F_j$ and $G_j$ are the large and small radial components of the wavefunction $a_j$. After simplifying the 9j-symbol, the final result becomes

$$\begin{array}{c}\hfill \zeta \left(a\right)=\sum _j(2j+1)·{\displaystyle \frac{l(l+1)+\frac{3}{4}-j(j+1)}{l(l+1)(2l+1)}}{\int }_0^{\infty }\left(F_j^2+G_j^2\right){\displaystyle \frac{Z}{r}}\mathrm{d}r\end{array}$$

(27a)

Only j-dependent terms in the integral will survive summation over j.

In the Pauli limit, this means that only $G_j\approx {\displaystyle \frac{1}{2}}\alpha \left(F_j^{\prime }+\kappa ·F_j/r\right)$ contributes.

After integration by parts, the well-known non-relativistic limit appears, which is now derived without any explicit reference to magnetic effects:

$$\begin{array}{c}\hfill \zeta \left(a\right)={\displaystyle \frac{1}{2}}{\alpha }^{2}Z{\int }_0^{\infty }{a}^2\left(r\right)r^{-3}\mathrm{d}r\end{array}$$

(27b)

Inner products can be deployed to calculate additional single- and two-particle contributions to the basic expression (27) of $\zeta \left(a\right)$. Taking the N-dependence of the inner product into account, we obtain in a configuration $l^N{l}^{\prime M}$ the following explicit expressions for the inner products:

$$\begin{array}{cc}\hfill & z\left(l\right):z{\left(l\right)}^{\left(1\right)}=l(l+1)(2l+1)/2\hfill \end{array}$$

(28a)

$$\begin{array}{cc}\hfill & z\left(l\right):z{\left(l\right)}^{l^N{l}^{\prime M}}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }+2}{M}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l}{N-1}}\right)l(l+1)(2l+1)/2\hfill \end{array}$$

(28b)

$$\begin{array}{cc}\hfill & z\left(l^{\prime }\right):z{\left(l^{\prime }\right)}^{l^N{l}^{\prime M}}=\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2}{N}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{4l^{\prime }}{M-1}}\right)l^{\prime }(l^{\prime }+1)(2l^{\prime }+1)/2\hfill \end{array}$$

(28c)

$$\begin{array}{cc}\hfill & z\left(l\right):z{\left(l^{\prime }\right)}^{l^N{l}^{\prime M}}=0\hfill \end{array}$$

(28d)

4. Use in Ab Initio Calculations

The progression of the N-dependence may be calculated straightforwardly. For a single shell $l^N$, the contribution of an arbitrary $n-$electron operator $U=vV$ to an orthogonal $m-$electron parameter $P_i(m\le n)$ is given by Equation (3):

$$\begin{array}{c}\hfill \Delta P_i={\displaystyle \frac{v:p_i^{\left(N\right)}}{p_i:p_i^{\left(N\right)}}}\end{array}$$

(29)

where the inner products are calculated in $l^N$. The use of inner products from the parent configuration $l^n$ yields a prefactor Q to be calculated from Equation (18) again:

$$\begin{array}{c}\hfill Q={\displaystyle \frac{\beta (m,n,N)·\beta (m,m,n)}{\beta (m,m,N)}}=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{N}{m}}\right)}^{-1}\end{array}$$

(30)

This dependence Q on the number of electrons is actually the normalized ratio of weighting factors and could have been seen in advance. In summary,

$$\begin{array}{c}\hfill \Delta P_i=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{N}{m}}\right)}^{-1}{\displaystyle \frac{v:p_i^{\left(n\right)}}{p_i:p_i^{\left(n\right)}}}V\end{array}$$

(31)

Obviously, $Q=1$ for $n=m$ or $N=n$.

In the case of direct proportionality, $v=a·p_i$, and Equation (31) reduces to $\Delta P_i=a·V$.

A complete ab initio calculation of a parameter $P_i={\sum }_i\Delta P_i$ requires the summation over all possible contributing operators U.

Formula (31) can be applied both numerically and algebraically. Examples of both of these cases will be briefly explained below.

4.1. Second-Order Calculation of Electrostatic Parameters

Equation (31) is readily programmed to allow numerical projections, e.g., of Slater–Condon integrals [12], or perturbative effects onto the orthogonal operator set to calculate the contributions to any parameter of choice.

Earlier ab initio calculations of parameters, whether analytical or numerical, could only take into account contributions that are directly proportional to the operator concerned. Equation (31), on the other hand, allows us to calculate all ab initio contributions with a non-zero inner product.

An example of this can be found in an ab initio calculation of the Trees operator $T_1$ in Fe VI (Table 6 of [13]). Nominally, $T_1$ ‘only’ accounts for $1s,2s,3s\to 3d$, and $3d\to ns/\epsilon s$ excitations. Using Equation (31), one finds that there are important non-zero contributions (in particular, $3d\to ng/\epsilon g$ excitations) with a different spin–angular character as well.

Second order ab initio contributions may also be derived algebraically from Equation (31). An example is the physical content of the three-body (or, rather, (2+1)-body) electrostatic $d^{N}s$ parameters $T_{\mathrm{dds}}$; sample results can be found in Table 1, taken from [14].

Table 1. Calculated contributions to $T_{\mathrm{dds}}$ in Fe VI and Ga V compared to experiments.

${\mathit{T}}_{\mathbf{dds}}$	Fe VI (3${\mathrm{d}}^2$4s)	Ga V (3${\mathrm{d}}^8$4s)
$3d\to d^{\prime }$	27.8	37.3
$4s\to d^{\prime }$	−118.9	−99.6
$4s\to g$	3.0	2.3
Total calc.	−88.1	−60.1
Fitted value	−91.2	−58.6

4.2. First- and Second-Order Calculation of Magnetic Parameters

Since the days of [4] it has been realized that the mutual spin–orbit interaction (MSO) contributes appreciably to the value of the spin–orbit parameter $\zeta$. In addition, the $\zeta$-contribution of the electrostatic spin–orbit interaction EL-SO turns out to be even more important for lower ionization stages.

However, only the simpler case of direct proportionality was considered. Equation (31) can be used to find the projections on $\zeta$ of all operators of magnetic origin (such as MSO and EL-SO operators or Dirac–Breit operators in spherical tensor form), not just the proportional ones. This generalizes the Blume and Watson theory to open shells and correct intrashell MSO contributions. The MSO $W^k$- and $N^k$ integrals used below are defined by [15].

The dominant MSO contribution is the intershell case with $V=N^0(ab;ab)$. To apply Equation (31) either numerically or algebraically in a configuration $l^N{l}^{\prime M}$, one needs the following:

$$\begin{array}{c}\hfill \mathrm{MSO}:z\left(l\right)=-(4l^{\prime }+2)l(l+1)(2l+1)N^0(ab;ab)\end{array}$$

(32a)

$$\begin{array}{c}\hfill z\left(l\right):z{\left(l\right)}^{l^N{l}^{\prime M}}=(4l^{\prime }+2)l(l+1)(2l+1)/2\end{array}$$

(32b)

$$\begin{array}{c}\hfill Q=\left[\left({\displaystyle \genfrac{}{}{0pt}{}{N}{1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{N}{1}}\right)}^{-1}\right]\left[\left({\displaystyle \genfrac{}{}{0pt}{}{M}{1}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{1}{0}}\right){\left({\displaystyle \genfrac{}{}{0pt}{}{M}{0}}\right)}^{-1}\right]=M\end{array}$$

(32c)

The final result can thus be written as follows:

$$\begin{array}{c}\hfill \Delta \zeta \left(a\right)=-2M\sum _b{N}^0(ab;ab)\end{array}$$

(33a)

With Equation (33a) being the most important part of $\Delta \zeta$, it may be interesting to trace its relativistic origin. The contribution comes from the spin–own–orbit part of MSO and, thus, the relativistic Coulomb interaction. Using a procedure to find the $T^{\left(11\right)0}$ term that is similar to that in Section 3.2, the result is found to be

$$\begin{array}{ccc}\hfill \Delta \zeta \left(a\right)& =& -\sum _b\left({\displaystyle \frac{M}{4l^{\prime }+2}}\right)\sum _{j,j^{\prime }}(2j+1)(2j^{\prime }+1){\displaystyle \frac{l(l+1)+\frac{3}{4}-j(j+1)}{l(l+1)(2l+1)}}\hfill \\ & \times & {\int }_0^{\infty }{\int }_0^{\infty }{(F_j^2+G_j^2)}_1{\left(r_{>}\right)}^{-1}{(F_{j^{\prime }}^2+G_{j^{\prime }}^2)}_2\mathrm{d}{r}_1\mathrm{d}{r}_2\hfill \end{array}$$

(33b)

In the Pauli limit, again, only $G_j$ depends on j, and therefore, the integral ${\int }_0^{\infty }{\int }_0^{\infty }{\left(G_j^2\right)}_1{\left(r_{>}\right)}^{-1}{\left(F_{j^{\prime }}^2\right)}_2\mathrm{d}{r}_1\mathrm{d}{r}_2$ is the leading term surviving the summation over j. Integrating by parts to remove $\mathrm{d}F/\mathrm{d}r$ and carrying out the summations over j and $j^{\prime }$, one obtains exactly Expression (33a) as the Pauli limit of Equation (33b).

In the second order, mixed electrostatic–magnetic contributions called EL-SO appear on the scene. As a consequence of the properties of spin–orbit interaction, EL-SO only connects configurations A and B that differ by a single electron substitution from an occupied orbital a to a virtual orbital v, conserving the orbital angular momentum: $nl\to n^{\prime }l$. Consider a configuration $l{l}^{\prime }$ occupied by orbitals $ab$, respectively. The reduced EL-SO matrix elements needed to apply Equation (31) are given below:

$$\begin{array}{c}\hfill ⟨l{l}^{\prime }\left(SL\right)\Vert \mathrm{EL}-\mathrm{SO}\Vert l{l}^{\prime }\left(S^{\prime }{L}^{\prime }\right)⟩=-⟨l{l}^{\prime }\left(SL\right)\Vert z_l\Vert l{l}^{\prime }\left(S^{\prime }{L}^{\prime }\right)⟩\\ \hfill ·{\displaystyle \frac{1}{2}}\left[⟨l{l}^{\prime }\left(SL\right)\left|C\right|l{l}^{\prime }\left(SL\right)⟩+⟨l{l}^{\prime }\left(S^{\prime }{L}^{\prime }\right)\left|C\right|l{l}^{\prime }\left(S^{\prime }{L}^{\prime }\right)⟩\right]·\sum _{a\to v}{\displaystyle \frac{2\zeta \left(av\right)R^k(ab;bv)}{\Delta E}}\end{array}$$

(34)

The virtual orbitals $v=n^{\prime }l/\epsilon l$ are calculated, e.g., with B-splines in the potential of configuration A and summed over. With high ionization and/or nuclear charge, the potential should be fully relativistic. It is essential to include the off-diagonal potential in the calculation as well, among other things, to cancel the unrealistically large contributions from direct $R^0$ Slater integrals.

An illustrative example of first- and second-order contributions calculated algebraically from Equation (31) is the second most important magnetic effect in $d^{N}s$ configurations described by the two-body parameter $A_{\mathrm{mso}}$. The result of all projections is

$$\begin{array}{ccc}\hfill A_{\mathrm{mso}}& =& -{\displaystyle \frac{6}{5}}{W}^1(ds;sd)+4N^0(ds;ds)-{\displaystyle \frac{1}{10}}\sum _{d\to d^{\prime }}{\displaystyle \frac{2\zeta \left(d{d}^{\prime }\right)R^2(ds;s{d}^{\prime })}{E_{d{d}^{\prime }}}}\hfill \end{array}$$

(35)

Numerical values of Equation (35) for Fe VI and Ga V are given in Table 2, taken from [16].

Table 2. Calculated contributions to $A_{\mathrm{mso}}$ in Fe VI and Ga V in comparison with experiments.

${\mathit{A}}_{\mathbf{mso}}$	Fe VI (3${\mathrm{d}}^2$4s)	Ga V (3${\mathrm{d}}^8$4s)
$-6/5·W^1$	−0.18	−0.26
$4·N^0$	1.78	2.24
$3d\to d^{\prime }$	1.60	4.06
Total calc.	3.20	6.04
Experiment	2.98	6.59

The high-order parameters $A_{\mathrm{mso}}$ and $T_{\mathrm{dds}}$ turn out to improve the fit of $5d^{N-1}6s$ configurations to the experimental levels considerably ([17]).

5. Summary and Discussion

It may be helpful to summarize below some basic properties of inner products in the present context:

• Commutative: $u:v=v:u$.
• Distributive: $(u+v):w=u:w+v:w$.
• Multiplicative: $(\alpha ·u):v=\alpha ·(u:v)$.
• Positive definite: $u:u=0\leftrightarrow u=0$.
• If $u\ne 0$ and $v\ne 0$, $u:v=0\leftrightarrow u\perp v$.
• Diagonal in tensor ranks: $u^{\left(\kappa k\right)t}:v^{\left({\kappa }^{\prime }{k}^{\prime }\right)t^{\prime }}=\delta (t,t^{\prime })\delta (\kappa ,{\kappa }^{\prime })\delta (k,k^{\prime })·(u:v)$. As the only way to form a scalar number, this condition is obvious.
• For energy operators $H^{\left(kk\right)0}$, it implies that electrostatic operators ($k=0$), spin–orbit operators ($k=1$) and spin–spin operators ($k=2$) belong to disjoint orthogonal subspaces.
• If u and v are operators with n-particle ranks $n_1$ and $n_2$ in shell l, the explicit N-dependence of $u:v$ is given by the parent factor $p={\prod }_{\mathrm{shell}l}\left({\displaystyle \genfrac{}{}{0pt}{}{4l+2-n_1-n_2}{N-n_2}}\right)$.
• $u:v$ is independent of the coupling scheme used.

The last proposition may require some further explanation. The most frequently used coupling schemes in atomic theory are $SL$, $jK$, $j_{1}J$, and $jJ$ coupling. Physical states are a linear combination of the pure basis states in a particular coupling scheme. A recoupling matrix, i.e., a transition matrix $R=⟨{\Psi }_a|{\Psi }_b⟩$ between coupling schemes a and b, is, in fact, a rotation matrix between two bases with the property $R^T=R^{-1}$. Therefore, the fourfold interposition of R, $R^T$, $R^{\prime }$, and $R^{\prime T}$ in Equation (1) just yields a twofold $\mathbb{1}$ and no change.

Inner products of spherical tensor operators prove to be a useful extra tool in the description of complex atoms. They allow the construction of orthogonal operator sets, which have the advantage of being more complete and stable in a fitting procedure. The dependence on the number N of electrons in arbitrary configurations is contained in a simple parent factor p. Therefore, the inner products are already known prior to the actual matrix calculation and may serve as an almost (except for overall phase factors) sufficient check of the results, including configuration interaction operators. Another application of inner products is the possibility to calculate radial parameters by projection; this calculation can be done either algebraically or numerically. The ’translation’ of traditional Slater–Condon parameters from Cowan’s program suite to orthogonal parameters can be used as a starting point. In addition, ab initio parameter calculations of whatever origin (relativistic variational or perturbative) are made possible by a basic projection using inner products. This is concretely illustrated by an electrostatic and a magnetic example. These results are clearly somewhat older, and it is hoped that the newly derived tools from Equations (18) and (31), key results of this work, will breathe new life into the field of atomic structure calculations.