State-Selective Double Photoionization of Atomic Carbon and Neon

Abstract

Double photoionization (DPI) allows for a sensitive and direct probe of electron correlation, which governs the structure of all matter. For atoms, much of the work in theory and experiment that informs our fullest understanding of this process has been conducted on helium, and efforts continue to explore many-electron targets with the same level of detail to understand the angular distributions of the ejected electrons in full dimensionality. Expanding on previous results, we consider here the double photoionization of two $2p$ valence electrons of atomic carbon and neon and explore the possible continuum states that are connected by dipole selection rules to the coupling of the outgoing electrons in ³P, ¹D, and ¹S initial states of the target atoms. Carbon and neon share these possible symmetries for the coupling of their valence electrons. Results are presented for the energy-sharing single differential cross section (SDCS) and triple differential cross section (TDCS), further elucidating the impact of the initial state symmetry in determining the angular distributions that are impacted by the correlation that drives the DPI process.

1. Introduction

Double photoionization (DPI) studies on atomic and molecular targets serve as a means to directly probe the consequences of electron correlation, a fundamental topic that governs the organization and structure of all matter. Total cross sections for DPI from targets continue to see new developments [1,2]. Beyond the total cross section, the use of sophisticated coincidence measurements that account for the angular distributions of the outgoing electrons removed by the action of a single photoabsorption reveals a great deal about the consequences of the initial and final state symmetries accessible from selection rules and also the considerations of energy sharing of the excess energy above the double ionization threshold. From the theory side, it is a challenge to accurately represent the correlation both in the initial and final states even for purely two-electron targets. However, good agreement between theory and experiment currently exists for the simplest two-electron systems, atomic helium [3,4,5,6,7,8,9,10] and molecular hydrogen [3,9,11,12,13,14,15], and the challenge remains being able to describe multi-electron systems with the same level of detail to determine the angular body-frame distribution of the outgoing electrons and the fully differential cross sections that contain the most detail observable in DPI events.

In fact, this highest level of detail for DPI studies is the triple differential cross section (TDCS), describing the angular distributions of both electrons and their energy sharing above the double ionization threshold. Thus far, progress beyond purely two-electron targets has largely focused on single-photon DPI from $n{s}^2$ helium-like targets [16,17,18,19,20,21,22,23,24,25,26,27]. What these systems have in common is the transition removing two electrons from the $n{s}^2$ configurations in an overall transition as in atomic helium; specifically, this is from a ¹S initial state to a ¹P final state double continuum. As we might expect, the richness of multi-electron atomic targets allows for the consideration of other initial state symmetries and, therefore, other possibilities for the final state continua accessed, and indeed, there are fewer theoretical studies that accurately describe alternative continua symmetries.

Recently, we considered valence double ionization of the lowest-energy configuration of ground state atomic carbon [28], where DPI of the pair of $2p^2$ electrons coupled into the ³P term made up the initial state. This allowed for exploration of the DPI cross sections and angular distributions into final states notably distinct from the helium-like atoms that many theories have previously considered. Indeed, the final states accessible in single-photon DPI consisted of the odd-parity terms ³P + ³D. We expanded on the previous results of Carter and Kelly [29] for this lowest-energy term of carbon to report the TDCS results and compare our energy-sharing single differential cross section (SDCS) and total DPI cross sections with those existing many-body perturbation theory (MBPT) results. However, a more expansive treatment of DPI of the $2p^2$ electrons of carbon should also consider the other $LS$ coupling possibilities for this atom, namely the ¹S and ¹D initial-state couplings in addition to the ³P ground state previously considered. Here, we complete the study and report single-photon double ionization TDCS and SDCS results for all of the $LS$ states of carbon.

Our results here rely on a “two-active electron” treatment of the target atom, whereby the electron correlation between the outgoing electrons is described exactly in a full configuration interaction (CI) expansion, but the other four electrons beyond those valence electrons being removed, namely the $1s^{2}2s^2$ electrons, are held fixed in all CI expansion terms; this is also called a “frozen-core approximation”, and here, we utilize it for all but the doubly ionized electrons. The nature of this accounting is that those electrons held and fixed in the CI expansion essentially provide a closed-shell Coulomb and exchange interaction with the valence $2p^2$ electrons that are actually acted on by the photon (and each other through the correlation), resulting in their double ionization into the continuum. Without relaxing the core electrons in this approximation, their representation as atomic orbitals is used throughout to provide the Coulomb screening and exchange interaction that impacts the ionized electrons of interest.

It is also interesting, as we did in the purely ³P study, to compare the state-selective DPI results of carbon with those of neon, since the vacancy of two electrons after photoionization of two of its valence electrons leads to the same possible final-state couplings as carbon possesses. We provide the same comparison between carbon and neon, albeit with the caveat that neon leaves behind an open-shell target that complicates the results and may not rigorously justify the validity of our frozen-core approximation, since closed forms of the average interactions of the remaining $2p^4$ electrons in the open valence shell are not simply given by spherically-symmetric $2J-K$ Coulomb and exchange operators. Nevertheless, we discuss the apparent success of treating the open-shell problem in an average way that has achieved previous accuracy in the TDCS results from both neon and argon [30].

In what follows, we overview the method in Section 2, focusing on the the determination of the double ionization amplitudes from a two-electron scattering solution and also a description of the frozen electrons’ impact on the active electrons. Section 3 presents the results of the differential cross sections, both the energy-sharing SDCS and the full TDCS in the state-selective framework that reveals the impacts of symmetry considerations from the initial-state coupling that distinguishes each case. As before, we compare the results of both carbon and neon in each instance.

2. Theory

We provide a brief overview of the key aspects and refer the reader to the application of these ideas to carbon and neon in Ref. [28]. In addition, further details of the radial orbital discrete variable representation (orbital DVR) and its application in time-independent single-photon double ionization [27,31] and also with a time-dependent formalism [32] will amplify the following theoretical overview for the reader. Atomic units are assumed below unless otherwise stated.

2.1. Single-Photon Double Photoionization (DPI) Amplitudes and Cross Sections

The triple differential cross section (TDCS) for single-photon double ionization (in the velocity gauge) is given by

$${\displaystyle \frac{d^3\sigma }{d{E}_{1}d{\Omega }_{1}d{\Omega }_2}}={\displaystyle \frac{4{\pi }^2}{\omega c}}{k}_1{k}_2|f({\mathbf{k}}_1,{\mathbf{k}}_2){|}^2,$$

(1)

where $f({\mathbf{k}}_1,{\mathbf{k}}_2)$ represents the double ionization amplitude that carries the outgoing electrons with momenta ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$, respectively, at the total excess energy $E=(k_1^2+k_2^2)/2$ above the double ionization potential. The double ionization amplitudes are determined by solving a driven Schrödinger equation

$$(E-H){\Psi }_{\mathrm{sc}}^{+}({\mathbf{r}}_1,{\mathbf{r}}_2)=(\overrightarrow{ϵ}·[-i\overrightarrow{{\nabla }_1}-i\overrightarrow{{\nabla }_2}]){\Psi }_0({\mathbf{r}}_1,{\mathbf{r}}_2),$$

(2)

where $E=E_0+\omega$ is the total available energy, $\omega$ is the photon energy, and $E_0$ is the double ionization potential (subtracting the frozen-core energy from the total atomic energy, since the former does not change as a result of the photoabsorption in the frozen-core approximation). The dipole operator of each electron is shown in the velocity gauge. The continuum electron dynamics are contained in ${\Psi }_{\mathrm{sc}}^{+}({\mathbf{r}}_1,{\mathbf{r}}_2)$, which is state-selected by specifying the $LS$ coupling connected by one-photon selection rules for to the two-electron coupling of the initial state ${\Psi }_0({\mathbf{r}}_1,{\mathbf{r}}_2)$. This initial state, ${\Psi }_0$, has the electrons coupled into either the ³P, ¹D or ¹S terms (all of even parity, we omit those labels for simplicity going forward). The initial state appears as part of the driving term with the action of the photoabsorption in the dipole approximation (using linear polarization for the photon throughout). Thus, the symmetry of the final state continua for the outgoing electrons is determined (see Equation (10) below). This driven equation is solved for the scattered wave solution ${\Psi }_{\mathrm{sc}}^{+}({\mathbf{r}}_1,{\mathbf{r}}_2)$, which contains the continuum electron dynamics at total energy E, with exterior complex scaling (ECS) being used to impose the outgoing-wave boundary conditions (see Ref. [33] for details).

Explicitly, the double ionization amplitudes are computed in a partial-wave expansion that takes full advantage of the atomic symmetry for the two-active electrons, giving

$$f({\mathbf{k}}_1,{\mathbf{k}}_2)=\sum _{l_1,l_2}\left({\displaystyle \frac{2}{\pi }}\right)i^{-(l_1+l_2)}{e}^{i{\eta }_{l_1}\left(k_1\right)+i{\eta }_{l_2}\left(k_2\right)}\times \left[{\mathcal{F}}_{l_1,l_2}(k_1,k_2){\mathcal{Y}}_{l_1,l_2}^{LM}({\widehat{\mathbf{k}}}_1,{\widehat{\mathbf{k}}}_2)\right],$$

(3)

where coupled-spherical harmonics ${\mathcal{Y}}_{l_1,l_2}^{LM}({\widehat{\mathbf{k}}}_1,{\widehat{\mathbf{k}}}_2)$ describe the angular coordinates of the outgoing electrons, and radial amplitudes ${\mathcal{F}}_{l_1,l_2}(k_1,k_2)$ can be computed using a surface integral formulation for each of the partial waves by integrating along a hyper-spherical arc with appropriate one-electron ”testing functions”. These testing functions are continuum states of the individual one-body Hamiltonian h of the residual dication (see Equation (6) below). This testing function formalism works to isolate the double ionization continuum from any single-ionization channels by relying on the orthogonality of the testing function to these two-body channels. More details on the testing function formalism can also be in Refs. [31,33] (for when core electrons are also present). We note that as long as the corresponding one-body Hamiltonian terms for each active electron are used in defining the full two-electron Hamiltonian of Equation (5), this orthogonality is assured, and the method isolates the double ionization channels from any contributions in the energetically open two-body channels (e.g., excitation-ionization).

Integrating the TDCS over the angular coordinates of the electrons ${\Omega }_1$ and ${\Omega }_2$ gives the single differential cross section (SDCS), which describes the energy sharing of the outgoing electrons for a fixed total energy above the double ionization threshold,

$${\displaystyle \frac{d\sigma }{d{E}_1}}=\int \int {\displaystyle \frac{d^3\sigma }{d{E}_{1}d{\Omega }_{1}d{\Omega }_2}}d{\Omega }_{1}d{\Omega }_2.$$

(4)

2.2. Describing the Ionized Electrons in the Presence of Other Frozen Electrons

We briefly overview the frozen-core approximation for the DPI of carbon and mention the modifications necessary to treat the DPI of two of the $2p^6$ electrons of neon at the end of this subsection. In the case of carbon, this approximation treats the DPI as a quasi-two-electron problem. That is, we approximate the initial state of the target as a full CI of the $2p^2$ valence electrons of the neutral target that experiences the influence of the remaining $1s^{2}2s^2$ core electrons. These core electrons are otherwise held fixed in the same (neutral) orbitals in all expansion configurations. Thus, the effective Hamiltonian for the outgoing electrons is

$$H=h\left(1\right)+h\left(2\right)+{\displaystyle \frac{1}{r_{12}}},$$

(5)

where the electron repulsion term $1/r_{12}=1/|{\mathbf{r}}_1-{\mathbf{r}}_2|$ correlates the valence electrons, and the interaction of the frozen electrons on each of those to be ionized is included in the one-body operator $h\left(i\right)$,

$$h\left(i\right)=T-{\displaystyle \frac{Z}{r_i}}+\sum _{occ}\left(2J_{occ}-K_{occ}\right),$$

(6)

for electrons $i=1$ or 2. In the one-body Hamiltonian h of each electron, T is the one-electron kinetic energy, $-Z/r$ is the nuclear attraction, and the terms in the sum over occupied orbitals, $2J_{occ}$ and $K_{occ}$, account for the direct and exchange interactions felt by the $2p^2$ valence electrons from the $1s^2$ and $2s^2$ core orbitals of carbon. Because of the closed-shell nature of these core electrons, the description of their interaction on the valence electrons as $2J_{occ}-K_{occ}$ is appropriate, where the direct operator is given by

$$J_{nl}\left(\mathbf{r}\right)=\int {\displaystyle \frac{{\left|{\phi }_{nl}\left({\mathbf{r}}^{\prime }\right)\right|}^2}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}d{\mathbf{r}}^{\prime },$$

(7)

and the exchange interaction is such that when operating on the orbital $\chi \left(\mathbf{r}\right)$, it is given by

$$K_{nl}\left(\mathbf{r}\right)\chi \left(\mathbf{r}\right)={\phi }_{nl}\left(\mathbf{r}\right)\int {\displaystyle \frac{{\phi }_{nl}^{*}\left({\mathbf{r}}^{\prime }\right)\chi \left({\mathbf{r}}^{\prime }\right)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}d{\mathbf{r}}^{\prime }.$$

(8)

Thus, the effect of the doubly-occupied $1s$ and $2s$ orbitals provides the Coulombic screening and non-local exchange interaction seen by the fully active pair of $2p$ electrons for atomic carbon that are to be removed by the photon. The double ionization potential is the total energy minus the energy of the frozen-core electrons; this defines $E_0$, the ground state energy of the two-active electron Hamiltonian in Equation (5) for a given $LS$ coupling in our state-selective treatment. In particular, the two-electron bound and continuum states are written as radial wave functions using a two-electron partial wave expansion in terms of coupled spherical harmonics,

$$\Psi ({\mathbf{r}}_1,{\mathbf{r}}_2)=\sum _{l_1{l}_2}{\displaystyle \frac{1}{r_1{r}_2}}{\psi }_{l_1,l_2}(r_1,r_2){\mathcal{Y}}_{l_1,l_2}^{LM}({\widehat{\mathbf{r}}}_1,{\widehat{\mathbf{r}}}_2).$$

(9)

The symmetry of the initial states is determined by the total angular momentum quantum numbers, L and M, that characterize the coupled spherical harmonics ${\mathcal{Y}}_{l_1,l_2}^{LM}({\widehat{\mathbf{r}}}_1,{\widehat{\mathbf{r}}}_2)$. In the case of either carbon or neon, diagonalizing the full Hamiltonian in Equation (5) with the appropriate L and M values determines the initial-state symmetry as either ³P, ¹S or ¹D, always chosen with even parity. In the case of neon, this implies a similar coupling of the remaining $2p^4$ valence electrons that remain on the dication, so that the overall symmetry of the target (active plus frozen-open-valence electrons) is initially the ¹S state of the neutral neon atom. In either case, however, the selection rules for one-photon absorption in the dipole approximation results in the following possibilities for the continua that originate from the bound states of each term:

$$\begin{array}{cc}\hfill {}^{1}S& \to {}^{1}P\hfill \\ \hfill {}^{1}D& \to {}^{1}P+{}^{1}D+{}^{1}F\hfill \\ \hfill {}^{3}P& \to {}^{3}P+{}^{3}D,\hfill \end{array}$$

(10)

where the initial states on the left are all even parity, and the final state continua in each case all possess odd parity.

It remains to describe the frozen-core electron interactions and to prohibit their duplicate occupation in the expansion of Equation (9). The radial profile of the core atomic orbitals held fixed in the CI can be determined in a number of ways, including standard atomic orbital codes [34,35]. Once determined, these atomic orbitals, namely the $1s$ and $2s$ radial orbitals of carbon, are numerically resolved on a radial grid that is constructed using a finite element method with discrete variable representation (FEM-DVR) [36]. Full details of this transformation to an atomic orbital basis over a subset of the primitive FEM-DVR basis are given in Ref. [31]. We highlight here that the transformation of the underlying FEM-DVR grid basis into an atomic orbital basis provides coverage that spans the radial extent of the bound atomic orbitals. These transformed (radial) atomic orbitals, $\phi \left(r\right)$, are thus linear combinations of the first M primitive FEM-DVR functions,

$${\phi }_{\alpha }\left(r\right)=\sum _{j=1}^M{U}_{\alpha j}{\chi }_j\left(r\right).$$

(11)

Thus, the atomic orbitals ${\phi }_{\alpha }\left(r\right)$, each constructed from the underlying FEM-DVR functions ${\chi }_j\left(r\right)$, with $j=1\dots M$, are utilized as the atomic orbital basis in the CI expansion of both active and frozen electrons. This also facilitates the removal of the frozen-occupation orbitals from the active set using projection methods to prevent violations of the Pauli exclusion principle in the expansion of Equation (9). This atomic orbital transformation over a subset of the radial grid leads to notable efficiency in the transformation of the two-electron matrix elements, resulting from the diagonal properties of the radial coordinate in the underlying FEM-DVR basis. In particular, the evaluation of electronic repulsion terms is never a full four-index transformation but rather a much simpler two-index transformation [31].

The procedure above is appropriate when accounting for the interaction with the core electrons, as well as providing for their removal from the CI active space when the core orbitals are fully occupied. However, in the case of neon, the dication that remains is open-shell, and while the angular momentum coupling of the remaining four $2p$ valence electrons is entirely determined by our initial coupling of the outgoing $2p$ electrons in this state-selective approach, their interaction with the active electrons cannot be written in a closed-form based on the Coulomb and exchange operators of Equations (7) and (8) without introducing energy-dependent terms into the Hamiltonian. We have previously used a state-averaged Hamiltonian that approximates this open-shell interaction in the simplest fashion as a direct interaction term $4J_{2p}\left(\mathbf{r}\right)$. This represents the Coulomb screening of the remaining four $2p$ valence electrons of the open-shell dication and ignores the exchange interaction. Thus, the effective one-electron Hamiltonian in the case of neon, following Equation (6) is (for $i=1$ or 2)

$$h\left(i\right)=T-{\displaystyle \frac{Z}{r_i}}+\sum _{1s,2s}\left(2J_{occ}-K_{occ}\right)+4J_{2p},$$

(12)

where $Z=10$. This approximation renders the influence of the frozen-valence electrons in the open shell as consisting of a Coulomb screening by the remaining four $2p$ electrons, thus leading to the correct asymptotic charge of the dication as $+2$ seen by the outgoing electrons. We note that, despite the simplicity of this treatment, the computed TDCS results show excellent agreement in both magnitude and angular distributions with experimental results [30] for both neon [37] and argon [38]. From our previous studies, the angular distributions for the different $LS$ states are determined by their initial state coupling and are not otherwise distinguished by the (spherically averaged) one-body Hamiltonian in Equation (12) above.

2.3. Computational Details

In the case of carbon, the radial grid for expanding the atomic orbitals as well as the primitive FEM-DVR grid that describes the continuum consisted of finite element boundaries from the nucleus at $r=1.0$, 9.0 and increased by $8.0$ bohr until $r=R_0=57.0$, where the ECS tail consisted of two additional finite elements up to $r_{\max }=90.0$ with an ECS rotation angle of $\theta ={30}^{\circ }$. The atomic orbitals are transformed on the subset of the grid within the first 3 finite elements (up to $r=17.0$), with 18-th order DVR in each individual finite element. The results that follow appear converged in terms of the radial parameters of the FEM-DVR grid. For the angular coordinates represented via the coupled spherical harmonics, sufficient convergence of the results was achieved with up to $l_{\mathrm{lmax}}=5$ for each electron in the partial wave expansion for ¹D and ³P couplings. The ¹S term, however, required higher partial waves to converge, and up to $l_{\mathrm{lmax}}=7$ was used. For neon, 18-th order DVR in each finite element was also used, with the right boundaries of the first few FEMs at $r=2.0,6.0,13.0$ and $20.0$ and then increased by $8.0$ bohr until $r=R_0=60.0$, with an ECS tail extending to $r_{\max }=90.0$. Similar partial wave angular momenta $l_{\mathrm{lmax}}$ for each electron were used for each term of neon as in carbon, leading to sufficient convergence in the angular distributions of the TDCS we observed.

The DPI results that follow have been computed using a velocity gauge form of the dipole operator. We note that small differences in the angular patterns and the magnitude of the cross section results were observed in comparing length and velocity gauge results, but this is expected given the inexactness of the atomic wave functions with frozen electrons. We refer the reader to our previous analysis of neon DPI, Ref. [30] (and the Supplemental Material published therein) for a detailed comparison of the TDCS results based on the gauge form. In all cases observed here, these differences for in converged results are of the order of a few percent and do not substantially change the analysis that follows.

With these grid parameters, the ground state energy of each initial state coupled into $LS$ terms represents the double ionization potential $|E_0|$; these values are tabulated in

Table 1. Initial state energies of each $LS$ term for the atomic targets carbon and neon (in the two-active electron approximation). Values for the ³P coupling are shown as the ground-state energy $E_0$, i.e., the negative double ionization potential, while the values shown for the ¹D and ¹S columns show the energy splitting relative to the ³P state for each atom. Literature values are also shown directly below the present results. All values shown are in eV units.

	Carbon			Neon
	3P Double IP	1D	1S	3P Double IP	1D	1S
present results	$E_0=-35.2$	1.30	2.91	$E_0=-65.9$	2.38	5.50
NIST values [39]	$E_0=-35.6$	1.26	2.68	$E_0=-62.5$	3.20	6.94

. We show the value of the two-active electron ground state energy $E_0$ for the ³P state of each target (in eV), followed by the term-splitting energy of the ¹D and ¹S states relative to the ³P state. We also compare the present results with reference values from the NIST Atomic Spectral Database [39]. Generally, we see excellent agreement for the carbon atom energies in the frozen-core approximation and larger differences with the literature values in the case of neon. Again, we expect to see more deviation in the latter target given the lack of exchange interaction between the outgoing electrons and the frozen-valence (open-shell) $2p^4$ electrons. Still, the agreement with accepted values is fairly accurate for neon given the approximate nature of the interactions with the remaining electrons that stay bound on the target dication.

3. Results

To focus the results presented, we selected a single photon energy (adjusted for each state-selected target) to examine. Since our previous study on the ³P state of carbon explored TDCS results at $E=10$ eV excess energy, rather than reproduce the results at $E=10$ eV that we have published earlier, we present results here computed such that each $LS$ coupling of the carbon atom target has $E=15$ eV excess energy to share between the outgoing electrons. In order to provide a comparison to neon, the DPI results for neon were conducted at a photon energy where the excess energy that results has the same ratio to the double ionization potential as does 15 eV excess energy divided by the double IP of the analogous state of carbon. In other words, we have scaled the photon energy to provide the same excess energy to the ground state of each target, as shown in Table 1. This provides consistent excess energy relative to the initial state energy $E_0$ and makes for a comparison of the results at the same fraction of total kinetic energy of the outgoing electrons compared to the energetic threshold of that particular double continua.

3.1. Single Differential Cross Sections

We begin with a comparison of the single differential cross section (SDCS), which exhibits the energy sharing of the outgoing electrons in each case. Figure 1 shows the SDCS results for each term of the target carbon (black solid lines) and neon (blue dashed lines) at comparable excess energies (∼1.4 times the double IP for each target state). We see a characteristically smooth “smile-like” energy-sharing cross section for each target in their ³P (top panel), ¹D (middle panel) and ¹S (bottom panel), similar to helium and other $n{s}^2$ targets. We note the neon results are generally about half the magnitude of the carbon SDCS results, perhaps reflecting the smaller orbital radial extent of the $2p$ valence electrons in the initial state in neon. Also, the ³P results for both carbon and neon are smaller in magnitude than the ¹D and ¹S SDCS, while the ¹D and ¹S energy-sharing cross sections are roughly comparable in value.

In order to further examine the trends of the energy-sharing cross section, Figure 2 presents the SDCS normalized to its maximum value in Figure 1 for each of the initial-state terms of carbon (black solid curves) and neon (blue dashed curves). Each state of the atomic targets is labeled by symbols along the curves as shown. Plotting the SDCS in this relative fashion shows the variation in the depths of the smile, for which we see that the ³P results show a greater variation in the cross section as the energy sharing is changed, while the other couplings result in flatter, less-responsive cross sections along the energy sharing, with the ¹D results changing the least to the energy sharing for both atomic targets.

3.2. State-Selective Triple Differential Cross Sections

In examining the triple differential cross section (TDCS) for the cases that follow, we examine the angular distributions in the co-planar geometry, meaning ${\mathbf{k}}_1$, ${\mathbf{k}}_2$ and the photon (linear) polarization $\overrightarrow{ϵ}$ lie in the same plane. Also, the photon polarization always lies in the horizontal direction of each panel polar plot that follows. Figure 3 shows the results for the DPI of ³P carbon (black solid curves) and neon (red dashed curves) with 15 eV of excess energy shared equally between the electrons ($E_1/E=50\%$). Each panel in Figure 3 diplays the TDCS as the fixed electron direction (labeled as electron 1 and indicated by the blue arrow) is changed relative to the linear polarization of the photon from values of ${\theta }_1=0^{\circ }$, ${30}^{\circ }$, ${60}^{\circ }$ and ${90}^{\circ }$. To provide a better comparison between the carbon and neon results, the neon TDCS magnitudes have been scaled by a factor of 2 (in all of the TDCS figures that follow). In all cases that follow, as well, the (purple) numbers shown in the upper-right of each panel indicate the magnitude of the TDCS as the radii size of each polar plot (in units of b/(eV sr²). Similar to our results at 10 eV excess energy [28] for the ³P-coupled targets, the cross sections between neon and carbon show similar angular distributions overall, with the neon major lobes slightly more oriented towards the fixed electron direction than what carbon TDCS generally exhibit. As the angle of ${\theta }_1$ is rotated away from the polarization direction, we see a growth in the minor lobe that points in the opposite direction to ${\mathbf{k}}_1$, with neon showing this back-to-back lobe (permissible in this symmetry, in contrast to the ¹S initial state, which blocks the back-to-back emission at equal energy sharing due to parity conservation [40]) as more comparable in magnitude to the major lobes that flank it.

For this initial-state coupling, we also consider how the TDCS results in the same excess energy but is compared at unequal sharings. Figure 4 and Figure 5 show the TDCS for ³P carbon and neon at 20% and 80% energy sharings, respectively. For the fixed electron at 20% energy sharing, we see the appearance of secondary lobes for both carbon and neon towards the backward direction for ${\theta }_1=0^{\circ }$, and these secondary structures appear larger than for the equal energy-sharing cases for the other intermediate fixed-electron directions; the angular distributions do look like equal energy sharing when ${\theta }_1={90}^{\circ }$, however. It also appears that the magnitudes of the neon to carbon cross sections are similar to the equal energy sharing case, but when the fixed electron carries a larger share of the energy in the 80% of Figure 5, the neon cross section becomes larger. Also for the slower electron plotted at this energy sharing, the primary lobes are pushed further back away from the fixed (faster) electron.

Although done at a different excess photon energy than our previous study focusing on ³P symmetry carbon and neon [28], similar angular patterns emerge due to the symmetry of this initial state coupling. We now consider the TDCS results for the ¹D term. Figure 6 shows the carbon and neon data in similar co-planar plots as the fixed electron is changed relative to the polarization and for equal energy sharing. We can see that, generally, the angular distributions that result are largely dominated by two lobes split along the backward direction from the fixed electron, again as a consequence of the re-emergence of the parity selection rule that prohibits the back-to-back ejection of the electrons at equal energy sharing [40]. The secondary lobe is slightly attenuated at the intermediate fixed-electron directions of ${\theta }_1={30}^{\circ }$ and ${60}^{\circ }$, but the angular patterns are symmetric about the fixed electron in the other geometries, as expected. Furthermore, the TDCS from the ¹D ground state have generally more rounded and less angularly narrow primary lobes compared to the ³P cases seen above. The magnitude of the cross sections is also larger than the ³P results, which is consistent with the SDCS results reported above.

Moving to unequal energy sharings for the ¹D initial state coupling at these excess energies, we examine the TDCS results at 20% and 80% of the excess energy E carried by the fixed electron in Figure 7 and Figure 8, respectively. We see that the prohibition against back-to-back ejection of the photoelectrons relaxes when $E_1\ne E_2$, and this becomes the most favored direction for the plotted electron when ${\theta }_1=0^{\circ }$ (along the polarization direction), and the angular patterns become modified as the fixed electron direction at ${\theta }_1={90}^{\circ }$ (perpendicular to the polarization) leads to the two-lobe patterns backward from ${\mathbf{k}}_1$, similar to the equal energy sharing cases. In fact, it is noteworthy that this term leads to angular distributions that, generally, look rather helium-like for the variations of the fixed-electron’s energy and direction considered here, despite the fact that the helium DPI transition arises from the $1s^2$ ¹S initial state.

Observing the carbon and neon results from the ¹D term ionizing two $2p$ electrons above, overlapping largely in the resulting angular patterns with the ground-state helium-like DPI results, it is thus interesting to next consider the case of the initial state coupling that actually mirrors the $n{s}^2$ DPI term state itself. Figure 9, Figure 10 and Figure 11 show the DPI results of carbon and neon for transitions that result from the active electrons initially coupled in the ¹S state, again at 50%, 20%, and 80%, energy sharings, respectively.

We find that for equal energy sharing, the TDCS results still prohibit back-to-back ejection, but now have additional lobes that are most prominent closer towards the fixed electron for ${\theta }_1=0^{\circ }$, and more oriented towards the fixed electron direction for the neon case than carbon. Moving the fixed electron away from the polarization in the other panels shows these additional lobes become significantly diminished, and the primary structures appear (as before) directed more away from the fixed electron, but the appearance of more complicated secondary structures in the angular patterns persists. Although the initial state coupling is ¹S and would appear to be most aligned with ground-state helium DPI, we note that the valence electrons coupled into this same term originate in the $2p^2$ configuration, not the helium $1s^2$ configuration. Thus, the initial state angular momentum of each electron largely impacts these results, producing more complicated angular distributions than the helium-like counterparts; this is true both in the treatment of the electrons themselves for carbon and their vacancies in the neon case. We also see a large change in the magnitude of the TDCS results in moving the fixed electron away from the polarization direction, and this is generally more dramatic for the atomic targets here in this symmetry than what is observed for DPI from the helium-like targets.

Now considering the unequal energy-sharing cases for the ¹S initial state coupling, we see in Figure 10 the presence of a smaller structure and generally more complicated angular patterns, which are the same for helium-like targets. Additionally, the primary lobes themselves appear more narrowly peaked than what is observed for helium. This is also the case in examining plots of the slow electron with the fixed electron at $E_1/E=80\%$ in Figure 11. In all three of the ¹S symmetry results presented, the scaled neon results also tend to track the angular patterns of carbon fairly well, with slight differences in the directions of the primary peaks. We also see that unequal energy-sharing cases show the magnitude of the cross section is less sensitive to changes in the fixed electron direction when compared to $E_1/E=50\%$ energy sharing.

3.3. Examining the Component M States That Constitute the TDCS

Having presented the TDCS results for the three $LS$ terms of the initial states, we generally see signatures of the coupling that manifest themselves in the distinct continua that are dipole-connected to the bound electrons, as in Equation (10) for both carbon and neon. In computing these TDCS results presented above, we had to average the initial states and sum the final states of the total magnetic quantum number $M=m_1+m_2$. That is, using linear polarization aligned along (and thus defining) the body-frame z-axis, the total M results remain preserved from the initial state that makes up the right-hand-side of Equation (2) into the double continua. For each of the terms considered, there are $2L+1$ total M states that must be considered in calculating the TDCS. It is instructive to examine these M-specific angular distributions.

In fact, this is greatly simplified for all but the ¹D initial state coupling. Considering first the ¹S possibilities, since a total of $L=0$, there is only one possibility, and the results shown in Figure 9, Figure 10 and Figure 11 are the only cases to be considered since $M=0$ is the only oriented initial state to be averaged. In examining the contributions to the TDCS results for ³P symmetry, contributions from three states need to be considered: $M=0$ and $M=\pm 1$. However, for the co-planar geometries that we have chosen in these studies, the $M=0$ cross sections result in an angular node in this plane, and since the $M=+1$ and $M=-1$ angular distributions are the same, the TDCS results presented in Figure 3, Figure 4 and Figure 5 have the same angular patterns as $|M|=1$ alone. That is, the actual TDCS results in the co-planar geometry are just the pure $M=1$ (or identical $M=-1$) values weighted by the averaging factor $2/3$. We specify that the amplitudes themselves for $M=0$ in the ³P symmetry are not zero-valued but that they possess phases that eliminate the TDCS when in the co-planar geometry ${\varphi }_1={\varphi }_2=0^{\circ }$. Examining the results out of plane would require a more non-trivial averaging with $M=0$ contributing $1/3$ of the result to the TDCS. It is also the case that the SDCS results presented above in Section 3.1 have been averaged over the M states possible in all ³P ¹D and ¹S terms we considered.

This leaves only ¹D symmetry to examine by M values, which we present below. Figure 12 shows the unique component M TDCS results for both carbon (black solid curve) and neon (red dashed curve, and scaled by factor of 2) in the co-planar geometry. Different energy sharings of the fixed electron along the polarization direction (${\theta }_1=0^{\circ }$) are indicated in the rows of panels, as shown. We see that the individual M components are distinguished from the others in terms of their angular distributions. Each of the M components prohibits back-to-back ejection of the photoelectrons at equal energy sharing (middle row results), and in fact, this is also the case at unequal energy sharing for the $M=\pm 1$ $M=\pm 2$ angular distributions. We can see that the back-to-back ejection is dominant in the $M=0$ TDCS when $E_1\ne E_2$. It is also observed that the $M=0$ TDCS results are substantially larger in magnitude compared to the other total M components, and their effect on the overall TDCS results can be seen in the overall TDCS presented above, particularly at unequal energy sharing. Overall, the $M=\pm 1$ angular distributions exhibit a four-lobe structure where the minor lobes in the backwards direction are smallest relative to the perpendicular major lobes at equal energy sharing, and as the energy sharing is made more unequal, the primary and secondary lobes become more comparable in magnitude to each other. It is the $M=\pm 2$ results that appear to trend larger in magnitude as the plotted electron is given less of the available energy, but the angular distribution appears to not change in the number lobes with their direction moving more away from the fixed electron as the plotted electron is made slower.

If we now examine the contributions to the ¹D TDCS for individual M components with the fixed electron perpendicular to the polarization direction, we arrive at the results shown in Figure 13. Again, the equal energy sharing results exhibit the parity conservation effect to prohibit back-to-back ejection, but now in contrast to the fixed electron along the polarization direction, the $M=0$ results tend to not have back-to-back ejection even for cases where $E_1\ne E_2$. In fact, in this geometry, the $|M|=1$ unequal energy-sharing cases now contain a small cross section feature in the back-to-back direction. The four-fold angular pattern is now most prominent in $M=\pm 2$ when the fixed electron at ${\theta }_1={90}^{\circ }$, while the largest magnitude contribution to the physical TDCS now appears to come from $M=\pm 1$. In all of the panels, as before, the carbon and neon results display generally similar angular distributions as each other in all panels. The net effect of these M-valued components when properly averaged is the relatively simple two-lobed structure backward from the fixed electron seen in the lower-right panels of Figure 6, Figure 7 and Figure 8.

4. Conclusions

We have amplified the previous study on double photoionization of two $2p$ electrons in carbon and neon, adding new results beyond the ³P coupling of the initial state to also now consider the other possibilities for the $LS$ coupling of these outgoing valence electrons from ¹D and ¹S bound states, leaving the ionized electrons in the dipole-connected double continua of known symmetries, as shown in Equation (10). Comparing carbon and neon results throughout makes use of the overlapping symmetries for the valence electrons that can be state-selectively placed into the continuum by photoabsorption. Generally, the cross section features and TDCS results for these two atomic targets are similar, with the features of the angular distributions and the relative magnitudes of the TDCS results largely dictated by the coupling symmetry of the outgoing electrons.

The present results also expand the observed angular distributions from atomic targets that are notably distinct from helium and helium-like $n{s}^2$ systems whose symmetry considerations have been extensively explored by both theory and experimental measurements. In fact, we have observed differences in the present results from the ¹S → ¹P state-selected double ionization that most align with the ground-state helium DPI process, noting that the character of the bound state orbitals ($2p$ vs. $1s$) is not the same, though their overall $LS$ coupling leads to the same term states in the initial and final states for the outgoing electrons. The results presented here also rely on a frozen-electron approximation that renders the double photoionization event as a two-active electron CI with the influence of the other electrons that remain bound to the target on the ejected electrons taken into account in a mean-field approach using standard Coulomb and exchange interactions. The validity of this model has been tested for both neon and argon and seems to give surprisingly good agreement with experimental measurements. We note that the correlation that actually drives the action at hand in single-photon DPI and which, therefore, must be accurately accounted for, exists between the outgoing electrons themselves and our treatment of the DPI events and represents this correlation accurately in the CI expansion.

While many-electron atomic systems are certainly of interest to consider further in DPI studies, it is also clear that molecular targets allow for more richness in studying electron correlation, primarily because the body-frame orientation of the molecule impacts the results and thus becomes another consideration of how the photon polarization is incident relative to the molecule. Numerous studies on H₂ from both theory and experiment [3,9,11,12,14,15] reflect this added dimensionality even for this simplest molecular target, and other work to investigate more complicated molecular targets that also undergo DPI with Coulomb explosion (i.e., dissociation into charged fragment channels) remains an active area of research [41]. To that end, the results here have informed and inspired recent work to develop an orbital basis in the molecular frame [42] so that many-electron studies in molecular targets can also be considered using these theoretical considerations.