Hartree-Fock calculation of the differential photoionization cross sections of small Li clusters S. A. Galitskiy, A. N. Artemyev, K. Jänkälä, B. M. Lagutin, and Ph. V. Demekhin Citation: The Journal of Chemical Physics 142, 034306 (2015); doi: 10.1063/1.4905722 View online: http://dx.doi.org/10.1063/1.4905722 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron reemission processes following photoelectron recapture due to post-collision interaction in innershell photoionization of water molecules J. Chem. Phys. 138, 214308 (2013); 10.1063/1.4808028 Probing the structures of neutral boron clusters using infrared/vacuum ultraviolet two color ionization: B11, B16, and B17 J. Chem. Phys. 137, 014317 (2012); 10.1063/1.4732308 Extreme ionization of Xe clusters driven by ultraintense laser fields J. Chem. Phys. 127, 074305 (2007); 10.1063/1.2762217 Photoionization of small krypton clusters in the Kr 3 d regime: Evidence for site-specific photoemission J. Chem. Phys. 123, 154304 (2005); 10.1063/1.2060709 Electron propagator theory calculations of molecular photoionization cross sections: The first-row hydrides J. Chem. Phys. 121, 4143 (2004); 10.1063/1.1773135

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THE JOURNAL OF CHEMICAL PHYSICS 142, 034306 (2015)

Hartree-Fock calculation of the differential photoionization cross sections of small Li clusters S. A. Galitskiy,1 A. N. Artemyev,1 K. Jänkälä,2 B. M. Lagutin,3 and Ph. V. Demekhin1,3,a)

1

Institut für Physik, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany Department of Physics, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland 3 Research Institute of Physics, Southern Federal University, Stachki Ave. 194, 344090 Rostov-on-Don, Russia 2

(Received 23 October 2014; accepted 29 December 2014; published online 20 January 2015) Cross sections and angular distribution parameters for the single-photon ionization of all electron orbitals of Li2−8 are systematically computed in a broad interval of the photoelectron kinetic energies for the energetically most stable geometry of each cluster. Calculations of the partial photoelectron continuum waves in clusters are carried out by the single center method within the Hartree-Fock approximation. We study photoionization cross sections per one electron and analyze in some details general trends in the photoionization of inner and outer shells with respect to the size and geometry of a cluster. The present differential cross sections computed for Li2 are in a good agreement with the available theoretical data, whereas those computed for Li3−8 clusters can be considered as theoretical predictions. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4905722]

I. INTRODUCTION

Neutral size-varied alkali-metal clusters have attracted considerable attention from both theoretical and experimental sides due to their nontrivial structural, reactional, and spectroscopic properties.1–4 Small Lin clusters play a central role in these studies5–9 for the following reasons. Lithium is the simplest metallic element consisting of a localized 1s2 core and a single delocalized valence 2s-electron. Therefore, it is an ideal prototypical case to study fundamental physical properties of simple metallic systems in the bordering region between atomic, molecular, and solid state physics. Also, Lin clusters are still obedient to the existing quantum chemistry methodologies, although only to rather nonstandard and sophisticated ones. Therefore, it is a good starting point for theoretical investigation of metals, and such ab-initio studies can provide benchmarks for comparison with less rigorous theoretical approaches. Plenty of comprehensive theoretical studies of lithium clusters focusing on their structural and electronic properties have already been published.10–18 The ground (optimal) and a few of the first excited geometries of small Lin clusters are therefore well known and characterized.5–9 However, the structural and electronic properties of clusters cannot be directly determined in experiments. Nevertheless, they can be accessed indirectly via different spectroscopic observables. In particular, photoabsorption spectra of small Lin clusters are found to be extremely size and geometry selective.19,20 Being accompanied with the accurate theoretical interpretation,7 such experimental data may serve for unique identification of clusters. Complementary information can be obtained by a photoelectron spectroscopy, i.e., addressing the electron continuum spectrum of clusters. However, the present knowledge on the energy-dependent photoionization probabilities of a)Electronic address: [email protected]

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alkali-metal clusters is incomplete. In addition to few theoretical works on Li2,21–23 most of the available experimental and theoretical studies are devoted to small Nan and Kn clusters.24–26 To our knowledge, almost nothing is known about the photoionization of small Lin clusters. One of the main goals of the present work is, therefore, to close this gap. Medium and large size metal clusters are often studied using different modifications of jellium model approximation27,28 with the so-called super-atom description of the wave functions. On the contrary, sophisticated ab-initio methodologies can be applied to study small clusters. For instance, photoionization of Na2−8 and K2−8 was treated in Ref. 26 within the multiple scattering approach. Photoionization of relatively small clusters composed mainly of surface atoms is known to be dominated by one-electron process in which one electron absorbs the whole energy of a photon and leaves the cluster.29 The influence of plasmon excitations and their subsequent decay either by emitting a photoelectron or by heating the cluster starts to play a role for larger clusters having atoms on the bulk sites.29 Therefore, theoretical investigation of the photoionization of small alkali-metal clusters within the single (independent) particle approximation for a photoelectron in the continuous spectrum is well justified. In the present work, we systematically investigate the angle-resolved photoionization spectra of all electron shells of Li2−8 clusters. The main aim of our study is to reveal general trends in the differential photoionization cross sections with respect to the size of a cluster. In order to accurately describe partial waves of a photoelectron in the continuum spectrum, we use the Single Center (SC) method.30,31 It was recently formulated and successfully applied to investigate angularresolved ionization and decay spectra of diatomic and linear molecules.32–37 Here, we extend the SC method to the angularresolved studies of polyatomic molecules and apply it to Lin clusters within well justified approximations and at reasonable computational costs. The paper is organized as follows. In

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Sec. II, we briefly discuss our theoretical approach and relevant computational details. The present theoretical results are analyzed and discussed in Sec. III. We conclude in Sec. IV with a short summary. II. COMPUTATIONAL APPROACH

The angular distribution of photoelectrons emitted from randomly oriented molecules ionized by linearly polarized light is given by the well known formula for the differential photoionization cross section38 dσ(ω) σ(ω) [1 + β(ω)P2(cos θ)]. (1) = dΩ 4π In order to compute the photoionization cross section σ(ω) and photoelectron angular distribution parameter β(ω), we use equations derived in our previous work for polyatomic molecules37 (see Eqs. (8) and (9) there). The calculation of σ(ω) and β(ω) requires molecular orbitals (MOs) of the occupied shells of a cluster and of the photoelectron in the continuum spectrum. The former core MOs were obtained by using the PC GAMESS (General Atomic and Molecular Electronic Structure System) version Alex A. Granovsky (www http://classic.chem.msu.su/gran/gamess/index.html) of the GAMESS (US) QC package.39 The calculations were performed within the Hartree-Fock (HF) approximation at the equilibrium internuclear geometries for the ground state (energetically most stable) configuration of nuclei in neutral Lin clusters, which were optimized in Ref. 7 within the density functional theory (DFT). We used a triplet zeta valence (TZV) basis set.40 The photoelectron MOs were computed by the SC method and code.30,31 Briefly, one-particle MOs are represented in the SC method via expansions over spherical harmonics with respect to a single center  Pεℓm (r) Ψε (⃗r ) = Yℓm (θ,ϕ). (2) r ℓm Here, Pεℓm (r) stands for the radial part of the partial harmonic with a given angular momentum ℓ and its projection m on to the chosen quantization axis in the molecular frame. In the independent particle approximation, these radial parts satisfy the system of coupled HF equations (see Eq. (2) in Ref. 31). The system includes potentials for the nuclear-electron interaction V ne , local direct J ee , and nonlocal exchange K ee electrostatic Coulomb interactions of the photoelectron with the ionic core. Those potentials were calculated within the frozen core approximation using molecular orbitals of the occupied electron shells of a cluster obtained as the linear combination of atomic orbitals (MO LCAO) as described above and represented subsequently via SC expansion (2). Because of the exchange Coulomb interaction, the original system of HF equations for the photoelectron contains non local terms. It can be solved, e.g., iteratively.30 An alternative way is to use a noniterative procedure suggested in Refs. 41 and 42, which is stable and robust. It consists in the formulation of a united system of coupled differential equations relative to both, partial harmonics and generalized potentials describing

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exchange interaction of the photoelectron with the ionic core. For diatomic (linear) molecules, the noniterative procedure is described in details in the Appendix C of Ref. 31. There, the magnetic quantum number m is well defined and summation over index m in SC expansion (2) can be omitted. As a consequence, in the multipole expansion of the exchange potential K ee over the spherical potentials with multiplicities k and q (see Eq. (C4) in Ref. 31), the value of index q is strictly fixed and corresponding summation can be omitted as well. This allows one to introduce the generalized potential Yck (r) (see Eq. (C5) in Ref. 31), which uniquely represents the harmonic of multiplicity k of the exchange interaction of the photoelectron with a given core electron (the later is enumerated c and represented by the molecular orbital P c  by index = Pℓ c m c ). Below, we briefly discuss how the noniterative procedure and the basic equations discussed in Ref. 31 need to be modified for polyatomics. In the case of a general polyatomic molecule, summations over both indices ℓm in SC expansion (2) are infinite, and the radial harmonics Pεℓm (r) are complex functions. Therefore, using indices ck only is not sufficient for a unique identification of the exchange potentials in the multipole expansion. Here, one has to introduce the generalized spherical potential Yck q (r), which represents harmonic of multiplicities kq of the exchange interaction of the photoelectron with the core electron P c . Let us now combine partial harmonics Pεℓm (r) of the photoelectron and all generalized potentials Yck q (r) into the single united vector solution P(r) {Pεℓm }+ . (3) P=*  , Yck q As is explicitly demonstrated in the Appendix C of Ref. 31, this vector satisfies the following united homogeneous system of coupled differential equations:  ′ ′  ck q  F Fℓ m d2 P ˆ * ˆ =  ℓm′ ′  cℓm ′k ′q ′ + . = F (4) P, with F ℓm 2 dr F F ck q , ck q For a polyatomic molecule, the matrix elements of Fˆ are given in atomic units as ( ) ℓ(ℓ + 1) ne ℓ ′m ′ Fℓm = − 2ε δ ℓℓ′δ mm′ + 2Vℓmℓ ′m ′(r) r2 ee + 2Jℓmℓ (5a) ′m ′(r),   2b ′ c ck q Fℓm = (−1)m c (2ℓ c′ + 1)(2ℓ + 1) r ′ ′ ℓc mc

ℓm Fck q ′



k ℓ + ℓ ′ k ℓ + * ℓ c′ Pℓ′c m′c , (5b) ×* c ′ , 0 0 0 - ,−mc q m  (2k + 1)  =− (−1)m c (2ℓ c + 1)(2ℓ ′ + 1) r ℓ m c

c

k ℓ′ + ∗ ℓc k ℓ +* ℓc ×* ′ Pℓ c m c , (5c) , 0 0 0 - ,−mc q m k(k + 1) c ′k ′q ′ Fck = δ cc ′δ k k ′δ qq ′. (5d) q r2 ℓ ′m ′ Matrix elements Fℓm (5a) consist of the centrifugal kinetic energy and of two potentials for the nuclear-electron ′

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interaction, V ne , and the local Coulomb interaction of the photoelectron with the ionic core, J ee . The latter potentials are non-diagonal and couple partial harmonics with ℓm , ℓ ′m ′. All these potentials and the one-electron energy ε are given in c ′k ′q ′ atomic units. Matrix elements Fck q (5d) describe generalized spherical potentials Yck q (r) of multiplicities kq for the exchange interaction of the photoelectron with the core electron c. These unknown exchange potentials are incorporated in the equations for the partial harmonics via coupling matrix ck q elements Fℓm (5b). Simultaneously, as described by matrix ℓ ′m ′ elements Fck (5c), the exchange potentials depend on the unq known partial harmonics. Note also that both matrix elements (5b) and (5c) depend explicitly on the multiplicity q. System (4) can thus be solved at once (i.e., noniteratively) for whole vector solution (3).41,42 The method for numerically solving the system of HF equations (4) for the photoelectron in the continuous spectrum, resulting in observable incoming partial photoelectron waves normalized on the energy scale and satisfying the condition of mutual orthogonality, is described in detail in Ref. 31. Let us now discuss some relevant computational details. The SC expansions of the occupied MOs of a cluster were restricted to ℓ c mc ≤ 29. This allows us to describe the 1s inner MOs within the accuracy of about 99.5% and 2s outer MOs better than 99.9%. In order to converge (see Table I and also discussion below), also the partial wave expansion of the photoelectron MO has to include harmonics with ℓm ≤ 29. Thereby, the photoelectron is represented by (ℓ + 1)2 = 900 continuum partial waves with different values of ℓm. The maximal possible multiplicity k in the expansion of the exchange potential over multipole harmonics is, therefore, k = ℓ c + ℓ = 29 + 29 = 58. As a result, exchange interaction of the photoelectron with one occupied electron shell is fully represented by (k + 1)2 = 3481 harmonics of the generalized exchange potentials with different values of kq. For instance in Li4, where ionic core consists of 6 occupied MOs, one has to TABLE I. Convergence of the photoionization cross sections σ and photoelectron angular distribution parameters β, computed in the length (L) and velocity (V ) gauges for the 1a 2g core orbital of Li4 cluster of D 2h symmetry and the photoelectron kinetic energy of ε = 0.5 a.u., with respect to the number of harmonics ℓm of the partial photoelectron waves and to the number of harmonics k q of the generalized exchange potentials included in the calculation. The data are shown relative to the final result obtained with ℓm ≤ 29 and k q ≤ 9, for which absolute values are also presented. ℓm/k q

σL

σV

βL

βV

Final result, absolute data 29/9

3.163 Mb

2.542 Mb 1.665 Convergence over ℓm, relative data

1.660

14/9 19/9 24/9 29/9

1.071 1.018 1.007 1

0.998 0.914 1.002 0.975 1.001 0.990 1 1 Convergence over k q, relative data

0.982 0.992 0.996 1

29/3 29/5 29/7 29/9

1.101 1.037 1.010 1

0.979 0.985 0.995 1

0.937 0.976 0.993 1

0.993 0.997 0.999 1

solve the united system of 900+6·3481 = 21786 equations (4), which requires enormous computational efforts. In order to carry out calculations for Lin clusters at reasonable computational costs, we have restricted multiplicities of the harmonics included in the multipole expansion of the exchange potential to an acceptable value. As will be demonstrated below, including harmonics with kq ≤ 9 guarantees convergence of the present calculations. In this case, the number of generalized exchange potentials with different values of kq reduces to (k + 1)2 = 100 for one occupied electron shell. For the example of Li4, one has to solve now the united system of 900 + 6 · 100 = 1500 equations (4), which is feasible. Such a single-energy-point calculation of the photoelectron MO in Li4, represented on 500 grid points covering radial interval of 0–40 a.u., requires about 30 Gb memory and can be solved by one core of a contemporary processor within about 10 h. Similar calculation in Li8 for a system of 900+12 · 100 = 2100 equations requires about 48 Gb memory and lasts about 20 h. In both cases, the memory requirements can be considerably reduced by storing part of the working data on a disk without considerable loss of efficiency. As a final point, we discuss convenance of the present computational results with respect to the number of harmonics ℓm of the partial photoelectron waves and to the number of harmonics kq of the generalized exchange potentials incorporated in the calculation. In what follows, we refer to Table I, which summarizes the relevant data obtained for the photoionization of the 1ag2 orbital of Li4(D2h ) at the photoelectron energy of ε = 0.5 a.u. The absolute photoionization cross sections and photoelectron angular distribution parameters computed in the length and velocity gauges in the most accurate approximation (including ℓm ≤ 29 and kq ≤ 9) are listed in the first line. One can see that the final values of σ L and σV agree within about 20%. This difference can be attributed to restrictions of the presently used single particle HF approximation and it is due to many-electron correlations43 neglected in the calculations. For the chosen example, the final values of β L and βV agree very well. However, both σ and β values computed in the present work in the length and velocity gauges for different cluster sizes, orbitals, and photoelectron energies differ on average by about 25%, which can be considered as the accuracy of the present calculations. The middle and lower parts of Table I illustrate convergence of the computed data with respect to ℓm and kq, respectively. These data are shown on the relative scale with respect to the final result (first line). As one can see from Table I, the σV and βV values converge very fast to the final result. Already at rather moderate ℓm values and almost immediately with respect to kq, the relative error does not exceed 2%. On the contrary, σ L and β L results illustrate slower convergence. In the most inaccurate approximations (i.e., at ℓm/kq ≤ 14/9 and ℓm/kq ≤ 29/3), the relative error is close to 10% and it decreases slowly when more ℓm/kq harmonics are included. This observation is found to be a general trend for different cluster sizes, orbitals, and photoelectron energies considered in the present work. We, thus, conclude that the differential cross sections obtained in the present work in the velocity gauge are more reliable. A similar result was also obtained in the previous theoretical study of Li2.22 Therefore, only σV

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and βV values are reported in Sec. III. One can also see from Table I that the final result computed with ℓm/kq ≤ 29/9 (first line) has converged with sufficient accuracy. In both gauges, the relative error of the final result does not exceed 1% (cf. line “24/9” with line “29/9” and also line “29/7” with line “29/9”).

III. RESULTS AND DISCUSSION A. The dimer

Results of the present calculation of the differential photoionization cross sections for the lithium dimer are collected in Fig. 1. Shown are the photoionization cross section σ (panels (a) and (c)) and the photoelectron angular asymmetry param2 eter β (panels (b) and (d)) computed for the inner 1σg,u and outer 2σg2 shells of the dimer. Where possible, the present results are compared with the previous theoretical data.21–23 As one can see from Figs. 1(c) and 1(d), the σ(ε) and β(ε) computed in the present work (solid curves) for the outer shell of Li2 are in overall good quantitative agreement with the

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results from Refs. 22 and 23 (dashed curves). Being compared to the latter, the presently computed cross section (Fig. 1(c)) is slightly lower in its maximum by about 5%-10%, and the maximum is slightly shifted toward the low electron energy side by about −1.5 eV. The present cross section and the results from Refs. 22 and 23 differ qualitatively from the results of Ref. 21, which exhibit a double maximum structure just above ionization threshold. This difference can be attributed to the restrictions of the computational approach of Ref. 21 (see below in this section). Although it is difficult to resolve experimentally the individual contributions of nearly degenerate inner shells of Li2, the calculated differential cross sections for 1σg2 and 1σu2 shells are illustrated separately in Figs. 1(a) and 1(b). Here, only the cross sections from Ref. 21 are available for comparison (short-dashed and dashed-dotted-dotted curves). One can see that σ(ε) computed in the present work for 1σg2 and 1σu2 (solid and dashed curves) differ from each other, especially for photoelectron energies below 50 eV. The 1σg2 cross section is rather smooth, whereas the 1σu2 one possesses broad oscillations as a function of energy. On the contrary, the 1σg2 cross section computed in Ref. 21 exhibits a sharp maximum and the 1σu2 cross section is smooth. To our opinion, the reason of these disagreements is due to the following restrictions of the computational approach in Ref. 21. Those calculations were performed by the multiple scattering method implying sphere and volume averaged muffin-tin potential and a local exchange potential. Importantly, only the partial harmonics up to ℓ = 2 were included for bound states of the dimer, and continuum states were represented by partial waves with up to ℓ = 7, which, according to our experience, is not sufficient to describe photoionization of Li2. B. The Li3−8 clusters

FIG. 1. The differential photoionization cross sections computed in the present work for inner shells (panels (a) and (b)) and outer shell (panels (c) and (d)) of Li2 in the velocity gauge (solid curves) in comparison with the available theoretical data from Refs. 21–23 (broken curves).

The differential photoionization cross sections computed in the present work for Li3−8 clusters are depicted in Figs. 2–7, respectively. Shown in these figures are the cross section σ (panels (a) and (c)) and photoelectron angular distribution parameter β (panels (b) and (d)) for all inner and outer shells of a cluster. For the open-shell clusters with odd number of atoms, the total differential cross sections including contribution from both singlet and triplet residual ionic states are shown in the case of ionization of any doubly occupied orbital. Unfortunately, no theoretical or experimental data, which can be compared with the present results, are available in the literature. The following most stable structures of clusters were used in the calculations: for Li3, an isosceles triangle of C2v symmetry; for Li4, a planar rhombus of D2h symmetry; for Li5, a triangular bipyramid of C2v symmetry; for Li6, a square bipyramid of D4h symmetry; for Li7, a pentagonal bipyramid of D5h symmetry; and, finally, for Li8, a square antiprism of D4d symmetry. These structures of the clusters are illustrated in each of Figs. 2–7 in the insets. The equilibrium internuclear distances for the structures were taken from Ref. 7. The ground state electron configurations of each cluster are indicated in the captions to Figs. 2–7. Below, we briefly discuss individual results for each cluster, whereas a comparative analysis of all results is given in the Sec. III C.

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FIG. 2. The differential photoionization cross sections computed in the present work for inner shells (panels (a) and (b)) and outer shells (panels (c) and (d)) of Li3(C 2v ) in the velocity gauge (see legends). Geometry of the cluster used in the calculations is illustrated in the inset. The ground state electron configuration of the neutral cluster is 1a 121b 222a 123a 124a 11.

As one can see from Fig. 2 for Li3(C2v ), the inner-shell photoionization cross sections (panel (a)) and angular distribution parameters (panel (b)) do not differ much between MOs of different symmetry. Exception is β(ε) for the 2a1-orbital which possesses shallow dip around the threshold. The energy dependencies are all rather smooth possessing only weak and broad oscillation. Outer-shell cross sections (Fig. 2(c)) and asymmetry parameters (Fig. 2(d)) differ noticeably from each other. Indeed, at the threshold σ for the highest occupied molecular orbital (HOMO) is larger than that for HOMO-1, in spite of the fact that the former is populated by only one electron. Simultaneously, variations of β(ε) for HOMO are more pronounced. Similar trends can be seen in Fig. 3 for inner and outer shells of Li4(D2h ). In addition to that, prominent maxima in σ(ε) of the 1ag and 2ag inner shells start to develop just above ionization thresholds (Fig. 2(a)). The cross section for now doubly occupied HOMO of Li4 is almost twice larger than that for Li3, and the maximum in σ(ε) for HOMO-1 of Li4 moves toward larger photoelectron energies, being now

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FIG. 3. The differential ionization cross sections computed for Li4(D2h ) cluster (see notations in Fig. 2). The ground state electron configuration of 2 2a 2 1b 2 3a 2 2b 2 . the neutral cluster is 1a 2g 1b 1u g g 2u 2u

more pronounced than that for HOMO-1 of Li3 (cf. solid and separately dashed lines in Figs. 2(c) and 3(c)). The differential cross sections of all inner shells of Li5(C2v ) are again rather similar to each other (Figs. 4(a) and 4(b)), with the following two exceptions. First, σ(ε) of the 1a1-orbital exhibits a very sharp maximum, and, second, β(ε) of the 2a1orbital possesses no dip around the threshold. Photoionization probabilities of the two outermost HOMO and HOMO-1 of Li5(C2v ) are very unlike to that of the innermost HOMO-2 (Fig. 4(c)): they are several times larger around threshold (note that HOMO is occupied by only one electron) and fall much faster as a function of the photoelectron kinetic energy. It is also seen from Fig. 4(d) that β(ε) of HOMO does not fall significantly at the threshold. The following applies to Li6(D4h ). The innershell cross sections (Fig. 5(a)) start to differ noticeably, i.e., σ(ε) for all MOs, except 1a2u -orbital, exhibit maxima of different sharpness at different photoelectron energies. Similar to the Li5(C2v ) case, σ(ε) of HOMO and HOMO-1 of Li6(D4h ) are significantly larger around threshold than that for HOMO-2 (Fig. 5(c)). In addition, the former two decrease rapidly, such that the latter starts to prevail at larger energies. Consequently,

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FIG. 4. The differential ionization cross sections computed for Li5(C 2v ) cluster (see notations in Fig. 2). The ground state electron configuration of the neutral cluster is 1a 121b 122a 123a 121b 224a 122b 225a 11.

oscillations in β(ε) of HOMO and HOMO-1 are significantly more pronounced than for HOMO-2 (Fig. 5(d)). The individual inner-shell photoionization cross section of Li7(D5h ) shown in Fig. 6(a) differ noticeably from each other. They all exhibit a feature above ionization thresholds. For 1a1′ , 1e1′ , 2e1′ , and 2a1′ orbitals, it is a sharp maximum, whereas for 1e2′ , 2e2′ , and 1a2′′ orbitals—a relatively low hump at larger kinetic energy. It can also be seen from Fig. 6(b) that β(ε) of 1e1′ and 2e1′ orbitals fall around the threshold significantly deeper (down to about 0.3) than that for other inner shells. The differential cross sections for outer shells of Li7(D5h ) (Figs. 6(c) and 6(d)) vary significantly, e.g., σ(ε) of all MOs except HOMO possess pronounced maxima at different energies, and pronounced oscillations in β(ε) for HOMO and HOMO-3 are in the counter phase. Very similar trends can be seen from Fig. 7 for Li8(D4d ) cluster. The maxima in σ(ε) for the inner shells (Fig. 7(a)) are somewhat broader and shifted to slightly larger photoelectron energy. For 1a1-orbital, σ(ε) exhibits an additional high-energy hump and oscillations in β(ε) are very prominent at the threshold. All outer-shell photoionization probabilities of Li8(D4d ) exhibit

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FIG. 5. The differential ionization cross sections computed for Li6(D4h ) cluster (see notations in Fig. 2). The ground state electron configuration of the 2 1e 2 2e 2 1b 2 2a 2 1a 2 3a 2 3e 2 4e 2 . Note that shown neutral cluster is 1a 1g u u u u 1g 1g 2u 1g data are indistinguishable for each pair of the degenerate MOs, i.e., for 1e u and 2e u , and, separately, for 3e u and 4e u .

now a maximum, which, unlike to Li7(D5h ), is present even for HOMO (cf. Figs. 6(c) and 7(c)). Among all outer shells, β(ε) for HOMO-3 possesses the strongest oscillation (Fig. 7(d)). Finally, we would like to briefly comment on the following main effects which were not included in this study, but may affect the presently computed differential cross sections. These are, of course, electron correlations in the initial and final photoionization states, including also coupling between different continuum channels (inter-channel correlations). Correlation may result in sizable changes of σ(ε) and β(ε) close above the ionization threshold and are expected to be larger for the outer electron shells.43 In addition to the direct ionization studied here, photoionization via the excitation and subsequent autoionization of Rydberg or doubly excited electronic states may produce sharp resonant structures in the cross sections.44 The nuclear motion, like, e.g., vibration, rotation, bending, torsion, or dissociation, could also play an important role. It results in the redistribution of the computed photoionization probability according to the Franck-Condon principle and is,

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FIG. 6. The differential ionization cross sections computed for Li7(D5h ) cluster (see notations in Fig. 2). The ground state electron configuration of the ′2 ′2 ′2 ′2 ′2 ′′2 ′2 ′2 ′2 ′′1 neutral cluster is 1a ′2 1 1e 1 2e 1 1e 2 2e 2 2a 1 1a 2 3a 1 3e 1 4e 1 2a 2 . The data are indistinguishable for each pair of the degenerate MOs, i.e., for 1e ′1 and 2e ′1, for 1e ′2 and 2e ′2, for 3e ′1 and 4e ′1.

FIG. 7. The differential ionization cross sections computed for Li8(D4d ) cluster (see notations in Fig. 2). The ground state electron configuration of the neutral cluster is 1a 121b 221e 122e 121e 322e 321e 222e 222a 123e 124e 122b 22. The data are indistinguishable for each pair of the degenerate MOs, i.e., for 1e 1 and 2e 1, for 1e 3 and 2e 3, for 1e 2 and 2e 2, for 3e 1 and 4e 1.

usually, an efficient mechanism for a broadening of features in σ(ε) and β(ε) of polyatomic molecules.45 Incorporating nuclear dynamics in the calculations is, however, beyond the scope of the present work.

here only the total differential cross sections of both multiplets. In order to account for this multiplet structure of the ion, a simple statistical consideration can be used as the first approximation. On the contrary, ionization of the inner shells of different symmetry (or of the 1s-shells stemming from differently located atoms) results in the formation of the group of several close-by photoelectron lines. The estimated energy splitting of less than 0.3 eV within each group of lines is due to small chemical shifts of 1s-electrons. It is, therefore, difficult to resolve experimentally individual photoelectron line in each group of the doublet, or singlet, or triplet states in the 1s-photoelectron spectra of Lin . In addition, the individual differential cross section for different inner shells of all clusters differ not significantly (see panels (a) and (b) of Figs. 2–7). It is, thus, meaningful to introduce the total differential cross section for all inner shells of a cluster as follows:

C. Comparative analysis

We first analyze the inner-shell photoionization of Li2−8. For clusters with even number of atoms, the inner-shell ionization results in the formation of the doublet electronic states of the residual ion, whereas for open-shell clusters with odd number of atoms, it creates two well-separated singlet and triplet electronic states. The energy splittings between the singlet and triplet states of the ion are quite substantial (e.g., ionization potentials of the 1s12s1 3 S and 1s12s1 1 S states of Li are equal to 64.41 and 66.31 eV,46 respectively). The respective two groups of lines can thus be resolved experimentally. As was mentioned in the preceding section, this splitting of the ionic states of the open-shell clusters into singlet and triplet states was neglected in the present calculation, and we analyze

σt ot =

n  i=1

n

σi and βt ot =

1  σi βi . σt ot i=1

(6)

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Since index i in Eq. (6) runs over all n inner shells of a cluster, σt ot grows with the cluster size. In order to facilitate a direct comparison of the data for different clusters, we introduce the total cross section per one electron. We thus divide σt ot by the total number of inner-shell electrons. Parameter βt ot in Eq. (6) is a relative quantity and need not to be normalized to the cluster size (number of electrons). The total cross section per one electron and the total angular distribution parameter for inner-shell photoionization of Li2−8 clusters are depicted in Figs. 8(a) and 8(b), respectively. Similar data set computed for the Li atom is also shown in this figure for comparison. One can see from Fig. 8(a) that the total photoionization cross sections per one innershell electron are very similar for the considered Lin clusters and they all resemble that for Li atom. For larger clusters, a slight deviation from the atomic-like cross section is, however, evident. Indeed, a small hump just above the ionization threshold shows up for Li4 and Li5. For Li6−8, it develops already to a sharp maximum at photoelectron energy of 5–7 eV. Nevertheless, Fig. 8(a) illustrates that the approximation of the independent atomic centers can largely be applied to model the total 1s-photoelectron spectra of Lin . In the nonrelativistic HF approximation, the inner-shell asymmetry parameter for lithium atom is equal to 2, being independent of the photoelectron energy (solid line in

FIG. 8. Comparison of the present results averaged over all inner shells of each Lin cluster. Panel (a): the total photoionization cross section per one electron. Panel (b): the total photoelectron angular distribution parameter.

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Fig. 8(b)). This is the result of the spherical symmetry of the 1s-electron, which favors emission of photoelectrons along the direction of the polarization of the electric field. For Li2−8 clusters, βt ot (ε) deviate from the atomic case. They all fall significantly at the threshold (down to about 1), and only asymptotically at large photoelectron energies approach the value of β = 2. This is owing to the loss of a spherical symmetry in Lin clusters which is, of course, more relevant for slow photoelectrons. As the size of a cluster grows, the deviation of βt ot (ε) from 2 grows as well. We now turn to the outer shells of Li2−8. Here, ionization potentials of different non-degenerate MOs may differ significantly by a few eV. Therefore, except of the states which are degenerate by symmetry, photoelectron lines of outer shells do not overlap and can be experimentally resolved. In addition, the differential photoionization cross sections of outer shells change significantly for different MOs of a cluster (see panels (c) and (d) of Figs. 2–7). We thus analyze below the individual differential photoionization cross sections for all clusters. In order to facilitate a direct comparison of data for different clusters, we compare the cross sections per one electron. As before, for the open-shell clusters, we report the total data for the whole multiplet structure of the ion. The individual cross

FIG. 9. Comparison of the present results for the individual outer shells of Li n clusters. Panel (a): photoionization cross section per one electron. Panel (b): photoelectron angular distribution parameter. Note that for each pair of the degenerate MOs, the shown data are indistinguishable, as indicated in the legend.

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section per one electron and the angular distribution parameter for outer-shell photoionization of Li2−8 clusters and also for lithium atom are collected in Figs. 9(a) and 9(b), respectively. As one can see from Fig. 9, the differential cross section of outer shells of various Lin clusters differ significantly from each other, and these individual data cannot be unified for all clusters, as it was possible for inner-shell photoionization. Nevertheless, some general trends can still be formulated. For Li3−6, photoionization probabilities computed for the outermost valence shells are very different from that for deeper valence shells. In particular, the threshold values of the cross sections per one electron for the outermost MOs are almost twice larger than those for the deeper ones, and, simultaneously, photoionization probabilities of the former MOs decrease much faster as a function of the photoelectron kinetic energy than those for the latter ones. For Li7 and Li8, photoionization probabilities per one electron stabilize becoming closer to each other. One should note that in spite of the very different behavior of σ(ε), the integral photoionization probability per  one electron, i.e., σ = σ(ε)dε, is very similar for all outer shells of all clusters. They differ from the average value of about 21 Mb×eV by at maximum +16%/−25%. Finally, loss of the spherical symmetry in Lin clusters is more relevant for the outer shells than for the inner ones. Therefore, β(ε) of outer shells deviate even stronger from the non-relativistic atomic value of β2s = 2 (cf. Figs. 8(b) and 9(b)). As the size of a cluster grows, β(ε) parameters possess larger variations and deviations from 2. IV. SUMMARY

The differential photoionization cross sections of Li2−8 clusters were investigated theoretically by the single center method. The calculations were performed in the Hartree-Fock approximation at the equilibrium internuclear distances for the most stable geometry of each cluster. We report the individual cross sections and asymmetry parameters for all electron shells of the considered clusters in a broad photoelectron energy range. For the lithium dimer, our results agree well with the previous theoretical results. For Li3−8 clusters, the presently reported results are new. It was obtained that the individual differential cross sections of all inner shells Li(1s) change only moderately between different molecular orbitals and with respect to the size of a cluster. Taking into account a near-degeneracy of the inner electron shells, it is possible to introduce the total cross section for all inner shells of a cluster divided by the number of 1selectrons. With the help of this inner-shell ionization cross section per one electron, the 1s-photoelectron spectra of Lin can easily be modeled in the independent atoms approximation. In contrast to Li(1s), the differential photoionization cross sections of outer Li(2s) shells change significantly between different cluster sizes and molecular orbitals. The individual cross sections and photoelectron asymmetry parameters exhibit different energy dependencies, and photoionization probabilities per one electron computed for the outermost valence shells are much larger than that for deeper valence shells. Taking also into account the non-degeneracy of the outer shells, these differential cross sections cannot be unified and individual data

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are required for the interpretation of photoelectron spectra of valence shells of Lin clusters. In conclusion, as far as we know there are no experimental results for either σ or β of Li2−8 in the literature. These data are, however, extremely desired for the interpretation of, e.g., experimental ion-mass-spectra of Lin . We hope that the present theoretical study will motivate photoionization experiments on small lithium clusters.

ACKNOWLEDGMENTS

This work was supported by the State Hessen Initiative for the Development of Scientific and Economic Excellence (LOEWE) within the focus-project Electron Dynamics of Chiral Systems (ELCH). Financial support by the Deutsche Forschungsgemeinschaft (DFG Project No. DE 2366/1-1), and by the Southern Federal University within the inner Project No. 213.01.-07.2014/11PPIG is gratefully acknowledged. K.J. acknowledges the Research Council for Natural Sciences and Engineering of the Academy of Finland for support. B.M.L. would like to thank the Institute of Physics, University of Kassel, and Ph.V.D. acknowledges Research Institute of Physics, Southern Federal University for the hospitality during their research stays there. 1V.

Bonaˇci´c-Koutecký, P. Fantucci, and J. Koutecký, Chem. Rev. 91, 1035 (1991). 2W. A. de Heer, Rev. Mod. Phys. 65, 611 (1993). 3M. B. Knickelbein, Annu. Rev. Phys. Chem. 50, 79 (1999). 4F. Calvayrac, P. G. Reinhard, E. Suraud, and C. A. Ullrich, Phys. Rep. 337, 493 (2000). 5I. Boustani, W. Pewestorf, P. Fantucci, V. Bonaˇ ci´c-Koutecký, and J. Koutecký, Phys. Rev. B 35, 9437 (1987). 6F. Wang, N. Andriopouios, N. Wright, and E. I. von Nagy-Felsobuki, J. Cluster Sci. 2, 203 (1991). 7X. H. Hong and F. Wang, Phys. Lett. A 375, 1883 (2011). 8D. Yepes, S. R. Kirk, S. Jenkins, and A. Restrepo, J. Mol. Model. 18, 4171 (2012). 9N. Goel, S. Gautam, and K. Dharamvir, Int. J. Quantum Chem. 112, 575 (2012). 10K. B. Rao and P. Jena, Phys. Rev. B 32, 2058 (1985). 11O. Sugino and H. Kamimura, Phys. Rev. Lett. 65, 2696 (1990). 12J. Jellinek, V. Bonaˇ ci´c-Koutecký, P. Fantucci, and M. Wiechert, J. Chem. Phys. 101, 10092 (1994). 13M. W. Sung, R. Kawai, and J. H. Weare, Phys. Rev. Lett. 73, 3552 (1994). 14F. Himo and L. A. Eriksson, J. Chem. Soc., Faraday Trans. 91, 4343 (1995). 15P. Fantucci, V. Bonaˇ ci´c-Koutecký, J. Jellinek, M. Wiechert, R. J. Harrison, and M. F. Guest, Chem. Phys. Lett. 250, 47 (1996). 16A. Rubio, J. A. Alonso, X. Blase, L. C. Balbs, and S. G. Louie, Phys. Rev. Lett. 77, 247 (1996). 17R. O. Jones, A. I. Lichtenstein, and J. Hutter, J. Chem. Phys. 106, 4566 (1997). 18R. Rousseau and D. Marx, Phys. Rev. A 56, 617 (1997). 19J. Blanc, M. Broyer, J. Chevaleyre, P. Dugourd, H. Küling, P. Labastie, M. Ulbricht, J. P. Wolf, and L. Wöste, Z. Phys. D: At., Mol. Clusters 19, 7 (1991). 20J. Blanc, V. Bonaˇ ci´c-Koutecký, M. Broyer, J. Chevaleyre, P. Dugourd, J. Koutecký, C. Scheuch, J. P. Wolf, and L. Wöste, J. Chem. Phys. 96, 1793 (1992). 21J. W. Davenport, G. J. Cosgrove, and A. Zangwill, J. Chem. Phys. 78, 1095 (1983). 22I. Cacelli, R. Moccia, and A. Rizzo, J. Chem. Phys. 102, 7131 (1995). 23R. Moccia and R. Montuoro, Chem. Phys. Lett. 368, 430 (2003). 24M. M. Kappes, M. Schfir, U. Röthlisberger, C. Yeretzian, and E. Schumacher, Chem. Phys. Lett. 143, 251 (1988). 25W. A. Saunders, K. Ctemenger, W. A. de Heer, and W. D. Knight, Phys. Rev. B 32, 1366 (1985).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.39.62.90 On: Mon, 23 Feb 2015 04:13:32

034306-10 26B.

Galitskiy et al.

Wäistberg and A. Rosén, Z. Phys. D: At., Mol. Clusters 18, 267 (1991). Brack, Rev. Mod. Phys. 65, 677 (1993). 28K. Jänkälä, M. Tchaplyguine, M.-H. Mikkelä, O. Björneholm, and M. Huttula, Phys. Rev. Lett. 107, 183401 (2011). 29M. Koskinen and M. Manninen, Phys. Rev. B 54, 14796 (1996). 30Ph. V. Demekhin, D. V. Omelyanenko, B. M. Lagutin, V. L. Sukhorukov, L. Werner, A. Ehresmann, K.-H. Schartner, and H. Schmoranzer, Opt. Spectrosc. 102, 318 (2007). 31Ph. V. Demekhin, A. Ehresmann, and V. L. Sukhorukov, J. Chem. Phys. 134, 024113 (2010). 32Ph. V. Demekhin, I. D. Petrov, V. L. Sukhorukov, W. Kielich, P. Reiß, R. Hentges, I. Haar, H. Schmoranzer, and A. Ehresmann, Phys. Rev. A 80, 063425 (2009); 81, 069902(E) (2010). 33Ph. V. Demekhin, I. D. Petrov, T. Tanaka, M. Hoshino, H. Tanaka, K. Ueda, W. Kielich, and A. Ehresmann, J. Phys. B 43, 065102 (2010). 34Ph. V. Demekhin, I. D. Petrov, V. L. Sukhorukov, W. Kielich, A. Knie, H. Schmoranzer, and A. Ehresmann, Phys. Rev. Lett. 104, 243001 (2010). 35Ph. V. Demekhin, I. D. Petrov, V. L. Sukhorukov, W. Kielich, A. Knie, H. Schmoranzer, and A. Ehresmann, J. Phys. B 43, 165103 (2010). 27M.

J. Chem. Phys. 142, 034306 (2015) 36Ph.

V. Demekhin, B. M. Lagutin, and I. D. Petrov, Phys. Rev. A 85, 023416 (2012). 37A. Knie, M. Ilchen, Ph. Schmidt, Ph. Reiß, C. Ozga, B. Kambs, A. Hans, N. Mueglich, S. A. Galitskiy, L. Glaser, P. Walter, J. Viefhaus, A. Ehresmann, and Ph. V. Demekhin, Phys. Rev. A 90, 013416 (2014). 38B. Ritchie, Phys. Rev. A 13, 1411 (1976). 39M. W. Schmidt et al., J. Comput. Chem. 14, 1347 (1993). 40T. H. Dunning, J. Chem. Phys. 55, 716 (1971). 41K. Smith, R. J. W. Henry, and P. G. Burke, Phys. Rev. 147, 21 (1966). 42P. G. Burke and M. J. Seaton, “Numerical solutions of the integro-differential equations of electron–atom collision theory,” in Atomic and Molecular Scattering. Methods in Computational Physics (Academic Press, London, 1971), Vol. 10, pp. 1–80. 43M. Y. Amusia, Atomic Photoeffect (Plenum Press, New York, 1990). 44U. Fano, Phys. Rev. 124, 1866 (1961). 45G. Herzberg, Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand Reinhold, New York, 1966). 46NIST Atomic Spectra Database, National Institute of Standards and Technology, Gaithersburg, MD, 2014, http://www.nist.gov/pml/data/asd.cfm.

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Hartree-Fock calculation of the differential photoionization cross sections of small Li clusters.

Cross sections and angular distribution parameters for the single-photon ionization of all electron orbitals of Li2-8 are systematically computed in a...
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