Two-center resonant photoionization-excitation driven by combined intra- and interatomic electron correlations

Ionization-excitation of an atom induced by the absorption of a single photon in the presence of a neighboring atom is studied. The latter is, first, resonantly photoexcited and, afterwards, transfers the excitation energy radiationlessly to the other atom, leading to its ionization with simultaneous excitation. The process relies on the combined effects of interatomic and intraatomic electron correlations. Under suitable conditions, it can dominate by several orders of magnitude over direct photoionization-excitation and even over direct photoionization. In addition, we briefly discuss another kind of two-center resonant photoionization with excitation where the ionization and residual excitation in the final state are located at different atomic sites.


I. INTRODUCTION
Absorption of a single photon by an atom (or molecule) typically leads to excitation or ionization of a single electron.From a theoretical point of view this is in line with the fact that photoabsorption is induced by a one-body operator.However, due to intraatomic electron correlations, the absorption of a single photon may also lead to the simultaneous transition of two or more electrons in an atom.This is distinctly exemplified by the processes of single-photon double ionization [1,2] as well as photoionization accompanied by excitation [3][4][5][6][7][8][9][10][11][12], which would not exist in the absence of electron correlations.
Electron correlations also play a prominent role in resonant photoionization.Here, the absorption of a single photon leads to the population of an autoionizing state in the atom, which afterwards deexcites via Auger decay, releasing an electron into the continuum.Resonant photoionization may also occur in two-center atomic systems, where it is driven by interatomic correlations.Here, an atom B is first resonantly excited by photoabsorption and, afterwards, deexcites by transfering the excess energy radiationlessly to a neighboring atom A of different species, causing its ionization.This two-center resonant photoionization (2CPI) has been studied both theoretically [13][14][15][16][17] and experimentally [18,19] in recent years.
We note that the second step in 2CPI -i.e., the radiationless decay of an excited atomic system by energy transfer to a neighboring atomic system upon which the latter emits an electron -is called interatomic Auger decay [20] or interatomic Coulombic decay (ICD) [21].The study [21] has triggered extensive investigations of such decays in various systems such as noble gas dimers and clusters, both theoretically and experimentally [22][23][24].
Recent experiments have observed two-electron transitions after single-photon absorption in dimers and clusters due to interatomic electron correlations.Photofragmentation of the 4 He 2 dimer into He + ions, which proceeds via photoejection of an electron from one of the helium atoms followed by an (e, 2e) reaction at the other atom, was studied experimentally [25] and theoretically [26].Collisional mechanisms leading to double ionization have also been observed in Ne clusters after resonant inner-valence photoexcitation [27].Double ionization of magnesium in Mg-He clusters was found to be largely enhanced due to electron-transfer-mediated decay [28,29]: after photoionization, the resulting He + ion was neutralized via electron transfer from a Mg atom and a second electron was simultaneously ejected from Mg to keep the energy balance.Double ICD has been predicted to occur in endohedral fullerenes such as Mg@C 60 [30,31].A related experiment on alkali dimers attached to helium droplets observed double ionization followed by dissociation of the dimer due to energy transfer from excited helium atoms [32].Single-photon double ionization due to combined intra-and interatomic correlations has been studied theoretically in diatomic systems [33].
In the present paper, we study resonant photoionization with simultaneous excitation in two-center atomic systems.The process relies on the joint action of intraatomic and interatomic electron correlations.In The excitation energy is sufficiently large to lead -upon radiationless energy transfer to a neighbouring atom A of different atomic speciesto ionization-excitation of the latter (see Fig. 1).The correlation-driven process may be termed two-center resonant photoionization-excitation (2CPIE).We will show that, under suitable conditions, the cross section for 2CPIE can largely exceed the cross sections for direct photoionization-excitation and even for direct photoionization of an isolated atom A. In addition, we will discuss a variant of 2CPI where atom B deexcites only partially, leading to ionization of atom A, while atom B ends up in an excited state.
From photoionization studies of single atoms it is known that photoionization-excitation is a relevant process which leads to satellite lines in the photoionization cross section [4].The cross section for photoionization with excitation can be comparable or even larger than the cross section for single-photon double ionization [11].
Our paper is organized as follows.In Sec.II we present our theoretical approach to 2CPIE and discuss its relation with one-center photoionization-excitation.In Sec.III we apply our formalism to various two-center atomic systems and demonstrate the relevance of 2CPIE.In Sec.IV the related process of 2CPI of an atom A with residual excitation in a neighbour atom B is introduced and discussed.Conclusions are given in Sec.V.
Atomic units (a.u.) are used throughout unless explicitly stated otherwise.

A. General considerations
Let us consider a system consisting of two atoms, A and B, separated by a sufficiently large distance R such that their individuality is basically preserved.The atoms, which are initially in their ground states, are exposed to a resonant electromagnetic field.The latter will be treated as a classical electromagnetic wave of linear polarization, whose electric field component reads Here ω = ck and k are the angular frequency and wave vector, and F 0 = F 0 e z denotes the field strength vector which is chosen to define the z direction.
Assuming the atoms to be at rest, we take the position of the nucleus of atom A as the origin and denote the coordinates of the nucleus of atom B, the two (active) electrons of atom A and that of atom B by R, r j (j ∈ {1, 2}) and r 3 = R + ξ, respectively, where ξ is the position of the electron of atom B with respect to its nucleus.Let atom B have an excited state χ e reachable from the ground state χ g by a dipole-allowed transition.
The total Hamiltonian describing the two atoms in the external electromagnetic field reads where Ĥ0 is the sum of the Hamiltonians for the noninteracting atoms A and B, VAB the interaction between the atoms and Ŵ = ŴA + ŴB the interaction of the atoms with the electromagnetic field.Within the dipole approximation and length gauge, the interaction Ŵ reads Ŵ = j=1,2,3 The two terms with j ∈ {1, 2} compose the interaction ŴA with the electrons in atom A, whereas the term with j = 3 the interaction ŴB with the electron in atom B.
For electrons undergoing electric dipole transitions, the interatomic interaction reads It is assumed that ω ge R/c ≪ 1, where ω ge = ϵ e − ϵ g is the atomic transition frequency and c the speed of light, such that retardation effects can be neglected.
In the process of 2CPIE one has essentially three different basic three-electron configurations, which are schematically illustrated in Fig. 1: (I) Ψ g;g = Φ g (r 1 , r 2 )χ g (ξ) with total energy E g;g = ε g + ϵ g , where both atoms are in their corresponding ground states Φ g and χ g ; (II) Ψ g;e = Φ g (r 1 , r 2 )χ e (ξ) with total energy E g;e = ε g + ϵ e , in which atom A is in the ground state while atom B is in the excited state χ e ; (III) Ψ p,e;g = Φ p,e (r 1 , r 2 )χ g (ξ) with total energy E p,e;g = ε p,e + ϵ g and ε p,e = ε p + ε e , where one electron of atom A has been emitted into the continuum with asymptotic momentum p and energy ε p = p 2 2 , whereas the other one has been excited to a bound state of energy ε e , while the electron of atom B has returned to the ground state.
Within the second order of time-dependent perturbation theory, the probability amplitude for 2CPIE can be written as By performing the inner time integral, we obtain e −i(Eg;g+ω−Ep,e;g)t .(6) Here we have kept only the term with the resonant denominator and inserted the total width Γ = Γ rad + Γ ICD of the excited state χ e in atom B. It accounts for the finite lifetime of this state and consists of the radiative width and the ICD width where the integral is taken over the emission angles of the ICD electron that is ejected from atom A, and Φ p ′ ,g denotes the state of atom A where one electron has been emitted with asymptotic momentum p ′ and energy ε p ′ = ε g + ω ge − ε + g , while the other electron is in the ground state with energy ε + g of the resulting singly charged ion.We note that, in the considered scenario, there is an additional contribution to Γ ICD associated with the decay channel where deexcitation of atom B leads to photoionization with excitation of atom A. This contribution, however, is usually small and may be neglected.
Taking also the outer time integral, we arrive at where the detuning from the resonance ∆ = ω − ω ge has been introduced.The delta function in Eq. ( 9) displays the energy conservation in the process.In this relation, the energies of atom B have dropped out, in accordance with its role as catalyzer.
From the transition amplitude we can obtain the fully differential ionization cross section in the usual way by taking the absolute square and dividing it by the interaction time τ and the incident flux j = The factor (2π) −3 arises from the fact that the continuum states in our calculations are normalized to a quantization volume of unity.Performing the integration over ε p with the help of the δ-function in Eq. ( 9), we obtain the angle-differential cross section 2 p=pc (11) where we have introduced the unit vector e R = R/R along the internuclear separation and the angle θ R between R and the field direction.Besides, p c denotes the momentum value of the electron emitted into the continuum, as determined by the δ-function in Eq. ( 9).

B. Relation to one-center processes
We can draw a comparison with the direct photoionization-excitation of atom A by the electromagnetic field.The corresponding probability amplitude in the first order of perturbation theory is given by S (1)  p,e = −i For the special cases, when the separation vector R between the atoms A and B is oriented either along the field direction or perpendicular to it, we can cast Eq. ( 9) into the form S (1)   p,e which translates into the relation between the corresponding cross sections.Here, α = −2 for R ∥ F 0 and α = 1 for R ⊥ F 0 .When the field is exactly resonant with the transition χ g → χ e in atom B and the interatomic distance is sufficiently large, so that Γ ICD ≪ Γ rad , this expression becomes Here we have used formula (7) for the radiative width.Eq. (15) shows that the cross section for 2CPIE can be largely enhanced by a factor [c/(ωR)] 6 ≫ 1 as compared with the usual one-center process of photoionizationexcitation, where the neighboring atom B is not involved.For example, assuming ω ≈ 10 eV (corresponding to the first excitation energy in hydrogen) and R = 10 Å, an enormous enhancement by 8 orders of magnitude results.This implies further that 2CPIE can even very strongly exceed the direct photoionization of atom A. Because the ratio between photoionization-excitation and photoionization (without excitation) is typically of the order of few percent [7][8][9][10][11][12], an enhancement by six orders remains.We point out that, nevertheless, 2CPIE is usually not the dominant ionization channel because 2CPI is much stronger, being enhanced over one-center photoionization by a factor [c/(ωR)] 6 as well [13].The ratio of 2CPIEto-2CPI is therefore of similar size as the ratio between the corresponding one-center processes.In special cases, however, 2CPIE can be comparable or even larger than 2CPI (just as single-center PIE can sometimes dominate over single-center photoionization [4]); see Sec.III.
The processes of 2CPIE and direct one-center photoionization-excitation lead to the same final state, since atom B eventually returns to its ground state and thus serves as a catalyzer.Therefore, the corresponding probability amplitudes ( 9) and ( 12) are generally subject to quantum interference.However, for parameters where the two-center channel strongly dominates, the interference is of minor importance and may be neglected.
The extremely high efficiency of 2CPIE arises from its resonant nature, whereas one-center photoionizationexcitation is a nonresonant process, in general.We note, however, that under special circumstances also onecenter PIE can proceed in a resonant way: photoabsorption by an atom may lead to the population of an intraatomic autoionizing state, which is able to decay, upon electron emission, not only to the ground state but also to an excited state of the resulting ion.In the latter case, the resonant photoexcitation of the autoionizing state leads to photoionization-excitation in an isolated atom.Corresponding resonance structures in PIE have been predicted to occur in He between about 70 eV to 73 eV, stemming from doubly excited states [6].Another example is PIE of Ca through excitation of the 3p → 3d resonance at about 31 eV [4].

III. NUMERICAL EXAMPLES AND DISCUSSION
In this section, we illustrate characteristic properties of 2CPIE by way of some concrete examples.
We first consider a two-center system composed of a neutral He atom as center A and a Li + ion as center B. Our consideration is motivated by the fact that helium constitutes a benchmark for photoionization-excitation studies [6][7][8][9][10][11][12].The threshold energy amounts to 65.4 eV, when the electron in the created He + ion occupies an n = 2 state.If a singly charged Li + ion is located in close vicinity to a He atom and is subject to an external field resonant to its 1s 2 → 1s3p transition at ω ge ≈ 69.65 eV [34], photoionization-excitation of helium via 2CPIE may occur.Its cross section is given by Eq. ( 14) as assuming for definiteness that the internuclear axis lies along the field direction.The radiative and ICD decay widths are given by Γ rad ≈ 7.76 × 10 9 s −1 [34] and Γ ICD ≈ 3c 4 2πω 4 ge R 6 Γ rad σ PI (ω ge ) (see, e.g., [35]), where σ PI (ω ge ) ≈ 1 Mb [11] denotes the single-center photoionization cross section of helium.Accordingly, Γ ICD /Γ rad ≈ (7.2/R [a.u.]) 6 .If the resonance condition is exactly met (∆ = 0) and Γ rad ≫ Γ ICD , the enhancement of 2CPIE over one-center PIE amounts to σ 2CPIE /σ PIE ∼ 2 × 10 11 /(R [a.u.]) 6 .For instance, at R = 10 a.u., the 2CPI cross section is larger than σ PIE ≈ 0.1 Mb [11,12] by five orders of magnitude.Due to of this huge difference, the quantum interference that both processes are subject to, is immaterial for the considered parameters.
The very large enhancement results from our assumption of exact resonance.In an experiment, however, the applied photon beam will not be perfectly monochromatic but have a certain frequency width ∆ω.For a typical value ∆ω ∼ 1 meV (see, e.g., [18]) this beam width is much larger than Γ rad ≈ 5 µeV in our example.While the cross section for nonresonant single-center photoionization-excitation would be practically constant over the bandwidth of the photon beam, 2CPIE would only proceed efficiently for those beam frequencies that are very close to the resonant value ω ge .As a consequence, if a He-Li + system is exposed to a photon beam whose frequency range encompasses the resonant frequency ω ge , the ratio of cross sections given above will be reduced approximately by a factor Γ rad /∆ω ∼ 5 × 10 −3 .At R = 10 a.u., the beam-averaged cross section for 2CPIE will still exceed the one-center cross section σ PIE by three orders of magnitude.
Apart from He-Li + , there is also a number of conceivable systems composed of two neutral atoms, wherein 2CPIE can proceed.As relation (15) shows, it is advantageous for 2CPIE if low interatomic energy transfers are sufficient to induce the process in atom A. (i) As a first diatomic system, let us consider the combination of Ca as atom A with H as atom B. By an external field of frequency ω ≈ 10.2 eV the 1s → 2p transition in H is resonantly excited, whose energy can be transfered to a neighbouring Ca(4s 2 ), causing its ionization with 4s → 4p excitation.The enhancement of 2CPIE over single-center photoionization-excitation would be of order ∼ (365/R [a.u.]) 6 which considerably exceeds unity up to interatomic distances of several nanometers.Also in absolute terms the 2CPIE cross section can be very large, given that σ PIE ∼ 0.1 Mb at ω ≈ 10 eV [4,5].
Interestingly, 2CPIE could even compete with 2CPI in this case, because for an isolated Ca atom at this energy one has σ PIE ≳ σ PI [4], since σ PI runs through a Cooper minimum there [36].
(ii) A diatomic system with very similar properties would be Mg as atom A and H as atom B, where the latter is resonantly excited from 1s → 3p at ω ≈ 12.09 eV, which can be transfered to Mg(3s 2 ), causing its ionization with 3s → 3p excitation very close to threshold.
(iii) Finally, one can imagine any of the atoms listed in Table I in the vicinity of He as atom B. This could be realized, for instance, by attachment to He droplets, similarly to the recent experiments [29,32].Exposing the system to an external field of frequency ω ≈ 21.2 eV, the 1s 2 → 1s2p transition in He would be resonantly excited, with subsequent interatomic energy transfer to, e.g, Mg(3s 2 ), causing its ionization with simultaneous 3s → 3p (at 4.42 eV) or 3s → 4s (at 8.65 eV) or 3s → 4p (at 10.0 eV) excitation of the Mg + ion.Here, the respective enhancements of 2CPIE over single-center PIE would amount to ∼ (176/R [a.u.]) 6 .
It is interesting to note that an early PIE experiment relied on a rather similar setup.In [3], the resonance radiation from He at 21.2 eV was used to study photoionization-excitation of Ca atoms.

IV. 2CPI WITH RESIDUAL EXCITATION
Before moving on to the conclusion we briefly describe a variant of 2CPI that is related to 2CPIE.In the standard form of 2CPI, an atom B is resonantly photoexcited and, afterwards, deexcites back into its initial state (typically the ground state), transfering the energy release to a neighbouring atom A which gets ionized [13][14][15][16].However, as Fig. 2 shows, it is also conceivable that the deexcitation of atom B does not proceed fully down to the initial state, but instead to another excited state of intermediate energy.If the corresponding energy difference ϵ e − ϵ ′ e is larger than the ionization potential of atom A, then the latter could still be ionized via interatomic energy transfer from atom B. Accordingly, in this process -which could be termed 2CPI with residual excitationthe two-center system ends up in an ionized-excited state, but in contrast to 2CPIE the ionization and excitation are located at two different centers.
The probability amplitude for 2CPI with residual excitation is given by Here, Γ rad and Γ (ge) rad denote the rates for the radiative decays χ e → χ ′ e and χ e → χ g , respectively.Note that the cross section for ordinary 2CPI [13] can be obtained from Eq. ( 18) by formally setting χ ′ e ≡ χ g and ϵ ′ e ≡ ϵ g therein.
Under suitable conditions, 2CPI with residual excitation can have a larger cross section than usual 2CPI [37].This is because smaller energy transfers are generally beneficial for the efficiency of interatomic processes, which can overcompensate a possibly smaller transition matrix element (or decay rate) between the states χ e and χ ′ e in atom B as compared with χ e and χ g .As an example, let us consider a system composed of a neutral Ca atom as center A and a singly charged Li + ion as center B. The system is irradiated by an electromagnetic field of frequency ω ≈ 69.65 eV, resonant with the 1s 2 → 1s3p transition in Li + .After resonant photoexcitation, the 1s3p state may either decay back into the ground state or, alternatively, into the intermediate 1s2s excited state, releasing an energy of ω e ′ e ≈ 8.73 eV.In the first case, 2CPI may happen, whereas in the second case, 2CPI with residual excitation can arise.According to Eq. (19), the ratio between the corresponding cross sections amounts to σ (e ′ ) 2CPI /σ 2CPI ∼ 10 3 , where we have used the decay rates Γ (e ′ e) rad ≈ 2.83 × 10 8 s −1 , Γ (ge) rad ≈ 7.76 × 10 9 s −1 [34] and σ PI (ω e ′ e ) ≳ σ PI (ω ge ) ∼ 0.1 Mb [4].Thus, for the considered Ca-Li + system and resonant field frequency, 2CPI with residual excitation can strongly outperform ordinary 2CPI.
We finally note that the 1s2s state reached in Li + after 2CPI with residual excitation is metastable.The energy stored in this state might therefore induce subsequent reactions, such as Penning ionization, for example.

V. CONCLUSION
The process of two-center resonant photoionizationexcitation has been studied where the resonant photoexcitation of an atom B leads to ionization with simultaneous excitation of a neighbouring atom A via the combined action of interatomic and intra-atomic electron-electron correlations.It was shown that, due to its resonant character, 2CPIE can largely dominate over the ordinary one-center photoionization-excitation of an isolated atom A. The enhancement may persist for interatomic distances up to a several nanometers, as was demonstrated by considering various two-center systems.2CPIE could in principle be observed in an experimental setup similar to those in [29,32].
In addition, we briefly discussed the related process of 2CPI with residual excitation, where the excitation energy of a resonantly photoexcited atom B is only partially transfered to a neighbour atom A, leading to its ionization, while the rest of the energy remains in atom B. It was shown that this partial interatomic energy transfer may lead to very efficient ionization.Under suitable conditions, it can largely dominate over the ordinary 2CPI where the full excitation energy is transfered.

FIG. 1 :
FIG.1: Scheme of two-center resonant photoionizationexcitation (2CPIE).First, atom B is resonantly photoexcited.Afterwards, upon radiationless energy transfer to atom A, the latter is singly ionized and simultaneously excited.The process involves three active electrons and relies on both interatomic and intraatomic electron correlations, as indicated by the dashed arrows.

FIG. 2 :ge ω e ′ e 5 σ
FIG.2: 2CPI with residual excitation.As in Fig.1, atom B is first resonantly photoexcited, but afterwards deexcites not fully down to its initial, but to an intermediate excited state whose energy ϵ ′ e lies in between ϵg and ϵe.The energy set free still suffices to singly ionize the neighbouring atom A. The process involves two active electrons and relies solely on interatomic electron correlations.

TABLE I :
Table I lists a few examples of atoms with rather low ionizationexcitation energies, which would represent suitable candidates.Examples of candidates of atoms A and transition energies in A + , which are suitable for the 2CPIE process.