Random phase approximation: from Giant to Intra-doublet resonances

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Radiation Physics and Chemistry 70 (2004) 237–251

Random phase approximation: from Giant to Intra-doublet resonances M.Ya. Amusiaa,b,* a

The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel b Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia

Abstract We discuss here the history and current achievements of one of the most powerful approaches of 20th century physics—the random phase approximation (RPA) that permits us to study collective or multiparticle effects in atoms, nuclei, molecules and clusters, as well as in quantum liquids. We concentrate on RPA application to studies of isolated atoms where it permits one to disclose the collective multielectron nature of so-called Giant resonances and predict a number of others, like Interference and Intra-doublet resonances. We present general theory as well as results of concrete calculations for a number of atoms. r 2003 Elsevier Ltd. All rights reserved.

1. Introduction—historic remarks

Here Oo is the plasma oscillation frequency, given by the relation:

We celebrate an important event in the history of the theoretical physics of the 20th century: almost 50 years ago, Pines and Bohm (1952) developed the quantummechanical approach to an infinite in space electron gas, which due to some inessential reasons was called the random phase approximation (RPA). They found that this system has collective coherent oscillations of all electrons—plasmons, and have determined their dispersion law O ¼ OðqÞ; namely the dependence of the frequency O upon the linear momentum q of the oscillation. The oscillation’s wavelength l is connected to q1

O2o ¼ 4pr;

l ¼ 2p=q:

ð1Þ

For small q the dispersion function is: OðqÞDOo þ aq2 :

ð2Þ

*The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel. E-mail address: [email protected] (M.Ya. Amusia). 1 Atomic system of units is used throughout this paper: the electron mass m; electron charge e and Planck’s constant _ are equal to 1: m ¼ e ¼ _ ¼ 1:

ð3Þ 1

where r is the electron gas density . Just as for the Hartree–Fock (HF) equations, the RPA equations can be easily applied to systems with other than pure Coulomb inter-electron interaction. Thus, RPA was used (Brown, 1967) and is used till now in the nuclear many-body problem as well. In fact, a simplified version of RPA was applied to the elementary particle physics under the name of the Tamm–Dankoff equation long before RPA was suggested for the electron gas. Landau (1956) created the theory of Fermi-liquids, being an infinite in space homogeneous system of strongly interacting particles, the forces between which are of short range. An example of such a system is liquid helium, consisting of He3 atoms. Landau discovered that in Fermi-liquids collective oscillations, so-called zerosound, exists, the frequency of which is proportional to the momentum q; so that the dispersion law became the following: O ¼ v0 q:

ð4Þ

Here v0 is the velocity of the zero sound. This collective frequency is essentially different from that for the electron gas, i.e., from (3).

0969-806X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2003.12.014

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Collective oscillations were also found in nuclei. It was discovered, that in the cross-section of photon absorption a powerful maximum exists at the frequency O; connected to the total number of nucleons A in a nuclei by the following relation: O ¼ bA1=3 :

ð5Þ

The maximum at the frequency O given by (5) was called Giant resonance. Migdal (1944) has described it as a collective oscillation, or to say more precisely, a coherent motion of all protons relative to all neutrons of a given nuclei. Quite a natural question could be asked already at that time: whether in atoms similar collective oscillations or atomic Giant resonances exist, which represent coherent motion of all electrons relative to the nuclei. In fact however, this question was asked and answered to some extent much later. It is essential to note that the equations describing the collective motion of an electron gas in RPA, in the theory of the Fermi-liquid and in the microscopic theory of nuclei, which led to expressions (3)–(5) are similar. In fact, they are the same equation, applied to various physical objects with different interparticle interaction. The deep connection between different collective oscillations, namely giant resonances, plasmons and zero sound were recently reviewed (Amusia and Connerade, 2000). At the end of the 5th decade of the 20th century it became clear that this equation and its different modifications could be derived relatively simply and elegantly. It was done using the technique of the socalled Feynman diagrams. Feynman (1961) invented this technique in 1947–48 in order to formulate quantum electrodynamics (QED).

2. RPA and RPA with exchange in atoms RPA, along with HF, is a very powerful method that is widely applied to studies of different many-particle objects, namely solids, liquids, molecules, atoms and atomic nuclei. The essential and specific feature of this approximation in its application to atoms comes from two facts: (a) the interparticle interaction is well known—it is a simple Coulomb interaction potential, and (b) the total number of interacting particles is not too big, permitting accurate numerical solution of corresponding equations. This feature demanded a refinement of RPA by making it self-consistent with the approximation that is used for description of the one-electron motion, which is treated in atoms in the frame of the HF approximation. This refinement required direct inclusion of the exchange into the RPA frame that has led to formulation of the RPAE—random phase approximation with

exchange (Amusia et al., 1969, 1971; Amusia, 1990). In studies of heavy atoms or tiny effects in medium heavy atoms relativistic effects are of importance. Inclusion of relativism into RPAE led to the relativistic random phase approximation (RRPA) (Johnson and Cheng, 1979; Johnson et al., 1980) that combines relativistic one-electron description in the frame of Dirac–Hartree– Fock (DHF) method with account of pure Coulomb interelectron interaction, i.e., neglecting the relativistic correction to it, such as, e.g. Breit potential (Berestetskii et al., 1980). Using RPAE and RRPA, photoionization calculations for very many atoms were performed. Partial and total cross-sections were considered, as well as dipole (see, e.g. Amusia, 1990) and non-dipole (Amusia and Cherepkov, 1975; Amusia et al., 1999; Johnson et al., 1999) photoelectrons’ angular distribution and their spin polarization (Cherepkov, 1972). In calculations, atoms with closed and semi-closed electron subshells were the primary objects. RPAE was successfully applied also to theoretical description of fast electron inelastic scattering (Amusia et al., 2001a, b), Compton scattering and of a number of other processes including Auger and radiative decays (Amusia and Lee, 1993). The most important achievement of RPAE is the prediction of a number of specific resonances that manifests profoundly in different characteristics of the photoionization processes. In the current paper our main attention will be given to these resonances. The most well known among them is the Giant resonance that manifests itself as a powerful maximum in photoabsorbtion cross-section of such different scale objects as nuclei, atoms, molecules and fullerenes. The Giant resonance is a manifestation in finite systems of the same excitation as plasmon in electron gas, or electronic liquids in metals, or zero-sound excitation in extended homogeneous objects, like Fermi-liquids, e.g., He3. Other resonances, that will be considered, are more specific for atoms.

3. RPAE equations There exist a number of derivations of the RPAE equations. It is important to emphasize that these equations are connected to the so-called time-dependent Hartree–Fock (TDHF) approximation (Thouless, 1961). In its frame it is assumed that the total wave function of a multi-electron system in the presence of a strong timedependent field is an anti-symmetrized product of oneelectron wave functions. When the external timedependent field is weak, one can leave in TDHF equations only the leading terms in the external field, which gives the RPAE equations (see e.g. Thouless, 1961; or Amusia, 1990).

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Instead of accurate derivation that can be found elsewhere (see, e.g. Amusia, 1990) we present here a symbolic derivation. Let us concentrate on the case of photon absorption (photoionization). The amplitude of photon absorption with excitation or ionization of a single electron D# can be presented as a sum: # # DðoÞ ¼ d þ DDðoÞ;

ð6Þ

where d is the respective amplitude in the one-electron HF approximation. Let us assume that DDðoÞ comes from the following process. Let the initial photon be absorbed by any other atomic electron, which is determined by the amplitude # DðoÞ: Then the created electron–vacancy pair exists or, as it is very often said, propagates, during a time period w# ðoÞ ¼ w# 1 ðoÞ þ w# 2 ðoÞ (that is connected by the uncertainty principle to the photon frequency o), where # # w# 1 ðoÞ ¼ 1=ðo  o0 Þ and w# 2 ¼ 1=ðo þ o0 Þ with o0 being the excitation energy of the virtual electron–vacancy state. Since w# 1 ðoÞ has a singularity at o ¼ o0 ; a ‘‘prescription’’ is added how to handle this singularity: by adding ig; g- þ 0 to the denominator so that one has # w# 1 ðoÞ ¼ 1=ðo  o0 þ igÞ: The propagator w# 1 ðoÞ represents the lifetime of an electron–vacancy created as a result of photon absorption, while w# 2 ðoÞ represents the lifetime of a pare existing in parallel to the incoming photon. Due to interelectron interaction U; which is a combination of the direct Vd and exchange Ve Coulomb potentials, U# ¼ V# d  V# e ; the intermediate electron– vacancy state is transformed into the final one. In the # frame of the chosen approximation DDðoÞ can be # # # thus leading to the presented as DDðoÞ ¼ DðoÞ# wðoÞU; RPAE equation: # # # DðoÞ ¼ d þ DðoÞ# wðoÞU:

ð7Þ

Note that the matrix elements of d; dif (with i; f being the initial and final one-electron HF states, respectively) are complex because of the presence of the imaginary parts exp id originating from the one-electron wave function. Here d is the photoelectron’s elastic scattering phase. The matrix elements of Dif ðoÞ are, of course, complex, and can be presented as Dif ðoÞ ¼ jDif ðoÞjexp iðDif þ dÞ

ð8Þ

thus including additional phase Dif : This is a result of the fact that w# 1 ðoÞ is complex, having ig in the denominator. The photoionization cross-section sðoÞ is connected to the square modulus of the photoionization amplitude, in HF with dif and in PRAE with Dif ðoÞ: X sHF ðoÞB jdif j2 dðo  ef þ ei Þ; i;f

sRPAE ðoÞB

X i;f

jDif ðoÞj2 dðo  ef þ ei Þ;

ð9Þ

239

where ef ;i are the one-electron initial and final state oneelectron HF energies. In non-symbolic form Eq. (7) is a rather complex integral equation. Therefore it is much more convenient for us to analyze instead (7). Indeed (7) allows a rather simple solution: # # DðoÞ ¼ d=½1  w# ðoÞU : ð10Þ In reality (7) for atomic cases can be solved only numerically, using the system of computing codes presented and described in detail by Amusia and Chernysheva (1997). The solutions of RPAE equations have some specific features that are typical for exact solutions. Namely, in the dipole approximation, when the incoming photon momentum is neglected, the results of calculations in socalled length and velocity forms (with operators, ~; describing photon–electron interaction as or and ir respectively) coincide and the dipole sum rule is fulfilled (Amusia, 1990).

4. Resonances as possible solutions of RPAE equations 4.1. Giant and interference resonances Note that the photoionization amplitude and hence the cross-section are enhanced when the denominator in (10) is small. If the denominator has a solution O determined by the equation: 1  w# ðOÞU# ¼ 0; ð11Þ at O > I; where I is the atomic ionization potential, then the cross-section has a powerful maximum called Giant resonance and O is its energy. This resonance corresponds to plasma oscillations in an infinite electron gas, described above by Eq. (2). To estimate O; one can use Eqs. (2) and (3), substituting there the electron density by that of a considered multi-electron subshell and q by qE1=rs ; where rs is the radius of the considered multielectron subshell. The Giant resonance O is of collective nature in the sense that it appears due to coherent virtual excitation of all electrons of at least one considered multi-electron subshell, like 4d in Xe. To clarify this point, let us note that (11) includes summation over virtual excitations of all participating electron–vacancy pairs: X 1 1 ð12Þ hkjjV jjk  kj i ¼ 0: 2  o2 þ ig o kj k>F ; jpF Here okj ¼ ek  ej ; ek;j are the one-electron HF energies of electrons on unoccupied ðk > FÞ levels and vacancies (or holes)—jpF ; with kðjÞ denoting the energies or principal quantum numbers ek ; nk;j ; angular momenta lk;j ; their projections mk;j and spin projections sk;j : Since the interelectron interaction is relatively weak

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and rapidly decreases with the growth of k; (12) can have a solution only because many terms of the sum there contribute coherently. The most famous and the first example of atomic Giant resonance was the photoionization cross-section of 4d in Xe that was discovered in Lukirskii et al. (1964) and described as a multi-electron Giant resonance in Amusia et al. (1971). Most recent example is the whole family of 4d subshells in iodine and its ions, both negative I and positive—Iþ and I2þ ; for which RPAE (Amusia et al., 2000c) and GRPAE (Amusia et al., 2002b) calculations were performed. Fig. 1 presents the results for 4d10 electrons of iodine ion I2þ calculations in RPAE and GRPAE along with the results of remeasurement of the cross-sections for iodine and its ions (Kjeldsen et al., 2000) that led to increase of previous incorrect experimental data by a factor of three and permitted to achieve good agreement with the theory. RPAE helped to predict and discover another sort of multi-electron resonances, that we call now Interference resonances. To describe them, let us consider a situation, in which the direct HF amplitude ds is small, while there are other electrons with big photoionization amplitude Db ; D# b ðoÞbds : Then from (7) one has D# s ðoÞE ds þ D# b ðoÞ#wðoÞU# bs E D# b ðoÞ#wðoÞU# bs bds ;

section a rather complicated structure that was named Interference or Correlation resonance. The partial cross-sections are obtained from (10) if one limits summation over i by only those electrons that belong to the ionized subshell. The first example of interference resonance was predicted in the partial photoionization cross-section of 3s and 5s electrons in Ar and Xe (Amusia, 1990). In the latter case the action of 5p6 and 4d10 electrons completely modifies the HF cross-section leading to two extra maxima—at the 5p6 threshold and in the region of the 4d10 Giant resonance. The first minimum is just above the first maximum, while the second is at rather high energy, well above the Giant resonance. The same is the situation for when one subshell interacts with a powerful resonance far from the threshold of the first one, where its cross-section is already small, e.g. when 4d Giant resonance acts upon 5p transition far from the latters threshold. The interference resonances are seen in very many cases, both in atoms and in ions. A recent example is the case of photo-ionization cross-section of 5p6 and 5s2 electrons in Xe+ in the vicinity of 4d-threshold (Amusia et al., 2000c, d), which are detected by measuring the Xe2+ yield. The results are presented in Fig. 2 and good agreement with experimental data (Koizumi et al., 1996) is seen. The prominent maximum is the result of the Giant resonance action.

ð13Þ 4.2. Giant autoionization

if the inter-transition interaction Ubs is not too small. The enhancement of the photoionization amplitude described by (13) manifests itself as a resonance in the partial cross-section of s electrons photoionization. Very often the term D# b ðoÞ#wðoÞU# bs and d are of opposite sign, so that the total amplitude acquires along with an extra maximum two minima, thus forming in the partial cross-

Of special interest is autoionization – photoionization in such a case, when a strong discrete excitation of an electron from an intermediate (or even inner) subshell is superimposed with the continuous spectrum of outer electrons. The interaction between them leads to photoionization cross-section of a rather specific form

Fig. 1. Photoionization cross-section of 4d10 electrons in I2þ : Calculations from Amusia et al. (2000a, 2002b), experiment from Kjeldsen et al. (2000).

Fig. 2. Photoionization cross-section of 5p6 and 5s2 electrons in Xe+ Calculations from Amusia et al. (2000b), experiment from Koizumi et al. (1996).

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that was named Fano profile (Fano and Cooper, 1968). If the discrete and continuous spectrum excitations are both of ‘‘one electron–one vacancy’’ type, the Fano cross-section can be calculated using RPAE. To do this, one has to isolate the discrete excitation in w# ðoÞ presenting it as a sum of a pure pole at the discrete excitation energy od and the rest, w# 0 ðoÞ: w# ðoÞ ¼ 1=ðo  od Þ þ w#0 ðoÞ;

ð14Þ

where w#0 ðoÞ is a non-singular term that include contributions from remote discrete excitation and that of continuous spectrum. Particularly interesting is the autoionization of atoms with big oscillator strengths, which is typical for atoms with semi-closed subshells, such as Mn, Cr, Eu, etc. For instance, in Mn with its semi-filled 3d3 subshell the discrete transition from 3p into semi-empty level 3d, 3p-3d; has a very big oscillator strength f ; f E1:98; and interacts strongly with the continuous spectrum transitions of the 3d electrons, 3d-ef ; p: However, to treat atoms with semi-filled subshells, the HF and RPAE approaches had to be properly modified. HF and RPAE are most simple and straightforward when applied to closed shell objects. The presence of open shells leads to a lot of complications that have nothing to do with the many-body nature of the considered objects. It appeared, however, that half-filled shell atoms can be treated almost as simple and consistent as closed subshell objects. This is a consequence of the Hund rule, according to which all electrons in a half-filled subshell must have the same spin projection. As a result, one can consider all the subshells being split into two levels totally filled with two kind of electrons, that correspond to their different spin projection, namely up; m and down; k electrons. The ‘‘up’’ and ‘‘down’’ electrons cannot exchange and therefore can be treated as different particles. Therefore, HF and RPAE were generalized for systems with two kinds of electrons (Amusia et al., 1983). This is easy to achieve presenting (7) in the matrix form: ðD# m ðoÞD# k ðoÞÞ ¼ ðd#m ðoÞd#k ðoÞÞ þ ðD# m ðoÞD# k ðoÞÞ

U# mm V# km

V# mk U# kk

! :

w# mm

0

0

w# kk

!

ð15Þ

Here the arrow mðkÞ denote the function related to ‘‘up’’(‘‘down’’) electrons, respectively. Note the presence of pure direct (without exchange) interaction between electron–vacancy pairs created by exciting either ‘‘up’’ or ‘‘down’’ electrons. As an example of Giant autoionization, Fig. 3 presents the results of Eu and its ions (Amusia et al., 2000).

Fig. 3. Giant Autoionization resonance in Eu atom. Calculations, from Amusia et al. (2000), experiment from Koizumi et al. (1996) and Kojima et al. (1998).

Fig. 4. Continuous spectrum autoionization in Si. Calculations from Gribakin et al. (1992), experiment from Balling et al. (1993).

4.3. Continuous spectrum autoionization RPAE is well suited to describe not only atoms and positive ions, but also so-called simple negative ions A that can be formed already in the HF approximation. In negative ions the spectrum, contrary to the case of a neutral atom, usually has no discrete excitation. Therefore it would be only natural that the photoionization cross-section of A would not have such structures as Fano profiles. However, it appeared that Fano profile for its existence does not necessarily require the presence of a strong discrete level. Instead, interference between two continua can form a Fano profile leading to continuous spectrum autoionization that was predicted in RPAE calculations (Amusia et al., 1990a, b; Gribakin et al., 1992). This is exemplified in Fig. 4 for the case of

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Si, with calculations from Gribakin et al. (1992) and experiment from Balling et al. (1993).

spherically symmetric objects like close shell atoms. So, beyond RPAE we limit ourselves to its approximate generalizations (Amusia, 1996).

4.4. Quadrupole resonances 5.1. Generalized RPAE Above we concentrated on dipole excitations only, since usually the wavelength of radiation that interacts with atoms having cross-sections of essential size is much bigger than the radius of the latter. However, atomic excitations can be of any multipolarity, just as the solutions of Eq. (7). Quadrupole Giant resonances are determined by the equation similar to (7) # # # QðoÞ ¼ q# þ QðoÞ# wðoÞU; ð16Þ where q# is the quadrupole amplitude in HF approximation. Quadrupole Giant resonances are well known in nuclei, where their energy can be estimated using an expression similar to (5) with an additional factor 2 in front of it, Oq E2bA1=3 : They were observed experimentally. Quadrupole Giant resonances can exist in finite multielectron systems also. Their energy Oq can be roughly estimated using again Eq. (2), substituting qq by qq E2=rs ; where rs is the radius of considered multielectron subshell. Recently, it was demonstrated that for 4p6 electrons in Xe there exists a Giant Quadrupole resonance (Johnson and Cheng, 2001). Its direct observation in photoabsorption is almost impossible, since the corresponding cross-section is at least four orders of magnitude smaller, because it includes an extra factor ðors =cÞ2 Ba2 ¼ 1=c2 (c being the speed of light) as compared to the dipole cross-section. However, the quadrupole amplitude leads to noticeable corrections to the angular distributions of photoelectrons where their relative contribution is considerably bigger, since it includes the above mentioned quadrupole factor only in the first power, as ors =cBa ¼ 1=c: Apart from that, some numerical factors enhance the quadrupole contributions to the angular anisotropy parameters of photoelectrons, as it will be illustrated below in this paper.

5. Important corrections beyond RPAE RPAE includes very essential corrections to the oneelectron picture of the atomic structure and processes. But it is an approximation and there exist known prominent effects that are neglected by this approximation. They cannot be treated as accurately as RPAE corrections since it includes as virtual states those with two and more electron–vacancy excitations. The account of their interaction requires accurate solution of at least a three-body problem and cannot be reduced in essence to one-body equations, as is done in RPAE for

The most important among corrections to RPAE are those connected to the relaxation or rearrangement of electron shells during the process of photoionization. Indeed, when an inner subshell is ionized close to the threshold of its formation, the slow photoelectron leaves the ion in a relaxed field. Namely, when the vacancy is created, all other atomic electrons feel its presence, and are attracted by the vacancy. This leads to additional screening of the vacancy that decreases the field acting upon the outgoing electron and thus decreases the photoionization cross-section. This effect was noticed for the first time in connection with ionization of 2p6 electrons in Ar, where RPAE gave at threshold a crosssection that is by a factor 2 too big. That was improved by accounting for the effect of a 2p vacancy creation upon all other atomic electrons (Amusia, 1990). The inclusion of relaxation along with the RPAE correction led to development of the Generalized RPAE or GRPAE, through use of which a number of successful calculations of photoionization cross-sections were performed (Amusia, 1990; Amusia and Chernysheva, 1997). The symbolic form of GRPAE equation is given, just as for RPAE, by (7), but the matrix elements are calculated with HF wave functions that take into account the presence of the vacancy self-consistently, i.e. for all vacancy and photoelectron states, calculating all of them in the field of an ion with vacancy j: 5.2. Post-collision interaction If the inner vacancy is deep enough, it can decay via Auger effect by emitting of at least one fast Auger electron. As a result of this decay the slow photoelectron will be essentially affected by instant alteration of the field acting upon it, since it will change from the field of an ion with a single vacancy to that of an ion with at least two vacancies. The simplest way to take this into account is to calculate the photoelectron’s wave function in the real final state field (Amusia and Chernysheva, 1997). However the Auger decay of an inner vacancy accompanied by emission of a slow photoelectron leads to a rather complex phenomenon called post collision interaction (PCI) (see e.g., Kuchiev and Sheinerman, 1989 and references therein). RPAE effects must be taken into account in describing the formation of the inner vacancy, while the next step, the account of Auger decay, requires consideration outside the RPAE frame (Amusia and Chernysheva, 1997). Due to Auger decay and instant increase of the field acting upon the

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photoelectron, the latter can be recaptured back by the final state ion, into its highly excited state, leading to a single instead of a doubly charged ion. As a result, just above the inner subshell ionization threshold the single charged ion’s yield increases profoundly (Amusia et al., 1977). This interesting effect was observed for the first time in Ar ionization just above the 2p threshold (Van der Wiel et al., 1976). 5.3. Direct knockout Another effect that is outside the RPAE frame is direct knockout—a process, in which the initially created photoelectron on its way out of the atom collides with other atomic electrons and eliminates at least one of them. The importance of this process in removal of a photoelectron from Xe and Ba above their 4d threshold was recognized in Amusia et al. (1990b). Excellent agreement with experiment (Becker et al., 1989) was achieved. It appeared that while the first part of this process, photon absorption, is described in the RPAE (GRPAE) frame, the absorption of the photoelectron wave is accounted for by the factor exp ½Im dðeÞ ; where the imaginary part of the scattering phase Im dðeÞ is responsible for the inelastic scattering of the photoelectron wave with energy e: It appeared that with good accuracy the cross-section for elimination of a single electron sþ ðoÞ could be presented as sþ ðoÞ ¼ sRPAE ðoÞexp ½2 Im dðeÞ :

ð17Þ

Direct knockout modifies not only the sþ ðoÞ crosssection, but also the angular distribution of photoelectrons and their spin polarization. However, these problems have not yet been discussed in the literature, but seem to be interesting subjects for future studies.

6. Intra-doublet resonances Recently a new type of resonances was discovered. It was observed (Kivim.aki et al., 2000) that the partial cross-section of 3d5=2 level in Xe has two maxima while that of 3d3=2 has only one. This immediately led to an assumption confirmed by comparison with the results of calculation (Amusia et al., 2002a, b): s5=2 ðoÞ is strongly affected by the 3d3=2 electrons near their ionization threshold, which leads to appearance of the second maximum. The influence of 3d5=2 electrons upon 3d3=2 is much weaker. It was demonstrated in Amusia et al. (2002c) that 3d Xe is not unique and at least the same kind of phenomena, but even stronger, exists in 3d in Cs and Ba. Fig. 5 presents the results for 3d Xe, both experimental and calculational. It is seen that an additional maximum appears, when the interaction between 3d5=2 and 3d3=2 electrons is taken into account. This maximum was called Intra-doublet resonance. To

Fig. 5. Intra-doublet resonance in 3d10 Xe. Calculations from Amusia et al. (2002c), experiment from Kivim.aki et al. (2000).

consider this case, 3/2 and 5/2 electrons were treated as different filled levels with the same and equal number of electrons, N ¼ 5: This permitted one to apply the GRPAE technique for semi-filled shell atoms and correspondingly the Eq. (15) straightforwardly. However, while solving (15) wmm and wkk were multiplied by correcting factors, 6/5 and 4/5, respectively. The same corrections were introduced into the cross-sections. This non-relativistic calculation was compared to results of the relativistic treatment using the RPA method (Radojevich et al., 2003). The agreement proved to be good enough.

7. Angular distributions of photoelectrons The angular distribution of photoelectrons from a given subshell j ¼ nl with the principal quantum number n and angular momentum l; that takes into account the first-order terms in its expansion in powers of ors =c51 can be presented for incoming non-polarized light in the following form (Amusia et al., 1974): dsnl ðoÞ snl ðoÞ ¼ 1  bnl ðoÞP2 ðcos yÞ dO 4p  þkgnl ðoÞP1 ðcos yÞ þ kZnl ðoÞP3 ðcos yÞ ; ð18Þ where k ¼ o=c; P1;2;3 ðcos yÞ are the Legendre polynomials, y is the angle between photon and photoelectron momenta, bnl ðoÞ is the dipole, while gnl ðoÞ and Znl ðoÞ are so-called non-dipole angular anisotropy parameters. In RPAE they are expressed via the matrix elements of # # dipole DðoÞ and quadrupole QðoÞ photoionization amplitudes that are solutions of Eqs. (7) and (15), respectively. There are two possible dipole transitions from subshell l; namely l-l71; that correspond to two # matrix elements of DðoÞ: We denote them as D7 nl : There

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are three possible quadrupole transitions l-l72; l and, respectively, three matrix element that are denoted as Q7;0 nl : The dipole parameter is given by the formula bnl ðoÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ  2  2 ðl þ 2ÞjDþ nl j þ ðl  1ÞjDnl j þ 6 lðl þ 1ÞjDnl Dnl j 2  2 ð2l þ 1ÞðjDþ nl j þ jDnl j Þ

:

ð19Þ The HF expression for bnl ðoÞ is obtained when 7 substituting in Eq. (19) D7 nl by dnl : Note that for selectrons in the non-relativistic approximation bn0 ðoÞ ¼ 2; as it follows from (19). However, it was demonstrated for 5s in Xe (Cherepkov, 1978) by taking into account two outgoing interfering photoelectron waves with total momentum j ¼ 1=2 and 3=2; and including the action of the 4d Giant resonance that b5s ðoÞa2 and has a very deep minima close to the location of the minimum in the s5s : Using RRPA, the results rather close to now available experimental data were obtained by Johnson et al. (1980). It is seen from (19) that bnl ðoÞ contains the dipole matrix elements in different combinations than snl ðoÞ  and, since usually Dþ nl is noticeably bigger than Dnl ; the parameter bnl ðoÞ that includes an interference term    Dþ nl Dnl is more sensitive to Dnl behavior as a function of o than the partial nl cross-section. The presence of the above discussed Giant, Interference, and Giant autoionization resonances affects bnl ðoÞ impressively. A first example is b5p ðoÞ in Xe where the action of Giant resonance led to an additional broad and powerful maximum (Amusia and Ivanov, 1976; Amusia, 1990) that was observed later experimentally. In Amusia (1990) one can find examples of the influence of other above-mentioned resonances bnl ðoÞ: The influence of intra-doublet resonances is clearly seen in the corresponding bnl ðoÞ parameters. While in 3d of Xe this effect is quite small (Amusia et al., 2002a, b; Radojevich et al., 2003), the modifications of b3d5=2 ðoÞ in Cs and Ba are really impressive (Amusia et al., 2003a, 2004), which is illustrated by Figs. 6 and 7. In Cs a rather specific maximum is created, while in Ba a minimum has appeared instead. Of special interest are the two non-dipole parameters, gnl ðoÞ and Znl ðoÞ: In general, they are described by much more complex formulas than (19), that include both 7;0 dipole D7 matrix elements. nl and quadrupole Qnl Corresponding expressions can be found in Amusia et al. (2001a, b), both in HF and RPAE approximations. The account of the electron correlations in general and the Giant resonances in particular affects the nondipole parameters impressively, adding to them, as functions of o; extra oscillations (Amusia et al., 2001a, b; Johnson and Derevianko, 1999). It is important to note that these corrections are particularly strong

Fig. 6. Dipole angular anisotropy parameter b3d5=2 ðoÞ in Cs. Calculations from Amusia et al. (2003b).

Fig. 7. Dipole angular anisotropy parameter b3d5=2 ðoÞ in Ba. Calculations from Amusia et al. (2003b).

near thresholds and in the vicinity of the abovediscussed resonances. The general expressions for non-dipole parameters are simplified drastically when s-subshells are considered. Indeed, in this case one has gns ðoÞ ¼ Zns ðoÞ and the non-dipole parameter contribution is given by the following expression: kgns ðoÞ ¼ 6k

jQþ ns j cosðd2 þ D2  d1  D2 Þ: jDþ ns j

ð20Þ

A rough estimate for the non-dipole parameters contribution is og=cBors =cBaD1=137: However already quite early, in Amusia et al. (1974), it was recognized that the smallness of the factor a could be compensated even for low frequencies by the large magnitude of the ratio Qns =Dns : For 3s in Ar this ratio is about 50, so the non-dipole parameter contribution

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reaches a value of as big as about one (Amusia et al., 1974, 1999, 2001a). Note, that close to ionization thresholds the ratio Qns =Dns changes with energy rather slowly, while the phase shifts are varying very fast. As a result, cosðd2 þ D2  d1  D2 Þ can have a narrow maximum leading to a maximum or rapid variation in gns ðoÞ that could be called phase resonance. As we saw, the non-dipole corrections even close to outer shell thresholds can be as big as one. Generally speaking, gnl ðoÞ and Znl ðoÞ has no such limitation as bnl ðoÞ; for which the following boundaries exist: 1obnl o2: Indeed, gnl ðoÞ and Znl ðoÞ are enhanced at quadrupole autoionization resonances, as it was illustrated by a concrete transition in the Ar atom (Amusia and Dolmatov, 1980). However, due to a numerical mistake the magnitude of Qns at the resonance was taken there too small. It was recently corrected by Dolmatov and Manson (1999) demonstrating great role for quadrupole resonances in non-dipole parameters. Obviously, they are enhanced at zeroes of dipole matrix elements whether they are connected to zeroes in the one-electron approximation (Cooper minima), or at the ‘‘wings’’ of Interference and Giant autoionization resonances, where, as it was mentioned in connection to consideration of these resonances, the dipole amplitudes can reach zero value or be very close to it. The role of dipole and quadrupole Giant resonances as well as electron correlations at photon frequencies starting from the threshold of such an outer subshell as 5s in Xe is demonstrated by recent experiments (Ricz et al., 2003; Hemmers et al., 2003). Fig. 8 presents the results from Hemmers et al. (2003) demonstrating the overall reasonable agreement between theory and experiment and shows profound traces of the Giant dipole resonance as well as an additional maximum at the high-energy slope, coming from the quadrupole resonance. It is interesting at the same time to note that the agreement between theory and experiment is not perfect leaving room for refinement of the theory.

Fig. 8. Non-dipole anisotropy parameter g5s ðoÞ of 5s2 electrons in Xe (Hemmers et al., 2003).

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Fig. 9. Non-dipole anisotropy parameter g3d5=3;3=2 ðoÞ of 3d10 electrons in Cs (Amusia et al., 2003a).

Fig. 10. Phase resonanses in Cs 3d5=2 transitions. Calculation, from Amusia and Chernusheva (2002a).

An interesting example of the role of the interelectron interaction is the modification of the non-dipole parameters under the action of intra-doublet resonances (Amusia et al., 2003a). Our calculations were performed for Cs 3d5/2 and 3d3/2 using the same approach as for the calculations of the partial cross-sections in Amusia and Chernysheva (2002c). The results for the g3d5=2;3=2 ðoÞparameters are presented in Fig. 9. It is seen that a completely new maximum appears when the intradoublet interaction is taken into account. Apart from the additional maximum, it is remarkable how fast the non-dipole parameters are changing in the near threshold region in both HF and RPAE approximations. As it was found (Amusia and Chernysheva, 2002), this rapid variation is connected to a resonance structure—Phase resonance—in the cosine of the phase shifts that is exemplified in Fig. 10. It is seen there a maximum, being remarkably narrow and well pronounced. The resonances mentioned above manifest themselves not only in total and partial cross-sections and angular distributions, particularly in corresponding non-dipole

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parameters, but also in spin polarization of the photoelectrons. Even in the dipole approximation the latter are determined (Cherepkov, 1994) by combinations of the matrix elements D7 nl different from that in snl ðoÞ or bnl ðoÞ and are therefore of special interest. The spin polarization of photoelectrons is strongly affected by the above-mentioned resonances (Cherepkov, 1994).

8. Correlations at high x The considered above RPAE and GRPAE corrections are limited to energies that are relatively close to the respective subshell ionization thresholds. For RPA this is correct in general, since at obO; with O given by (2) and qE1=rs the RPA corrections are decreasing as O=o or even as ðo=OÞ2 51: The situation in RPAE is different due to the presence of exchange: there are, as we will see, some corrections that are not vanishing with o increase. Direct calculations performed using RRPA code demonstrated that at quite high o; up to 1 keV region, s2p ðoÞ in Ne acquires an almost non-decreasing growth correction with o due to the action of 2s electrons upon 2p (Dias et al., 1997). It appeared that with account of the 2s electron influence s2p ðoÞ proved to be in good agreement with experiment, thus eliminating a difference of about 25%. While the action in the opposite direction, namely 2p6 upon 2s2 in the near threshold region was well known since 1972 (see in Amusia, 1990), the result obtained in Dias et al. (1997, 1999) called for a general explanation. To achieve this, a case of photoionization of two interacting subshells, nl and nl 0 at high but non-relativistic o was considered. At such o one can neglect the difference between the energies of subshells nl and nl 0 ; enl and enl 0 ; considering them as degenerate. Without inter-electron interaction taken into account, the photoionization cross-sections at high but nonrelativistic o decreases as s0nl ðoÞjo-N B1=o7=2þl :

ð21Þ

It was demonstrated in Amusia et al. (2000a) that the electrons with smaller l 0 always affect the cross-section snl ðoÞ at any o; including the limit o-N: It appeared that as a result of the interaction the asymptotic behavior is altered qualitatively, leading instead of (21) to: snl ðoÞjo-N B1=o9=2

ð22Þ

and thus becoming l-independent. So, already for d electrons the RPAE corrections due to the presence of s-electrons start to dominate. Relativistic corrections to the cross-section DR snl ðoÞ are of the order of snl ðoÞo=c2 ; while the electron correlation correction DC snl ðoÞ that led from (21) to (22) can be estimated as snl ðoÞo=Vns;nd : Since Vns;nd is of

the order of the nl-electron binding energy, DC snl ðoÞbDR snl ðoÞ if o5c2 ; i.e. up to about 50 keV. Now we will show that there are correlation effects that lead to the cross-section with even a slower decrease with the frequency growth, namely to that which coincides with the s-electron cross-section asymptotic sðoÞB1=o7=2 : We will present here two examples of such a behavior. To clarify the main features of the considered effect, let us assume that in an atom two initial states exist, a simple one (s), with the wave function js ðrÞ (e.g. one vacancy state) and a complex one (c), with its wave function jc ðrÞ: This state, in addition to a vacancy, includes at least one electron–vacancy pair. The interaction between s and c states is determined by the matrix element Vsc : It is quite natural to assume, that while sð0Þ s ðoÞ is nonzero, the direct photoionization of the state c is impossible, so that sð0Þ c ðoÞ ¼ 0: By accounting for the interaction between s and c states new states with wave functions cs ðrÞ and cc ðrÞ are formed: cs ðrÞ ¼ Fs1=2 js ðrÞ þ ð1  Fs Þ1=2 jc ðrÞ;

ð23aÞ

cc ðrÞ ¼ ð1  Fs Þ1=2 js ðrÞ þ Fs1=2 jc ðrÞ:

ð23bÞ

Here Fs characterize the admixture of c to s states and is called spectroscopic factor of the s-state. If the interaction Vsc is small compared to the energy difference Desc  jes  ec j; then Fs1=2 EVsc =Desc :

ð24Þ

At high enough photon energies o the s and c photoionization cross-section are, respectively ð0Þ ss ðoÞ ¼ Fs sð0Þ s ðoÞ; sc ðoÞ ¼ ð1  Fs Þss ðoÞ:

ð25Þ

This leads to a prediction that at high enough o the ratio sc ðoÞ=ss ðoÞ is o independent sc ðoÞ=ss ðoÞ ¼ ð1  Fs Þ=Fs :

ð26Þ

The first example of this effect is the behavior of the ionization cross-section of a complex, ‘‘two vacancy–one excited electron’’ level, which is connected via Coulomb interaction to an s-vacancy. As such level we can choose 3p2 3d excitation of Ar connected to 3s1 vacancy. Only the two-vacancy state 3p2 with total spin S ¼ 1 can contribute to this admixture. If the ratio ðV =DeÞ is not small, expression (26) evolves into the following relation: s3p2 3d ðoÞ ¼ s3s1 ðoÞð1  F3s Þ=F3s ;

ð27Þ

where F3s is the spectroscopic factor of the 3s vacancy. Due to the same mechanism the observable crosssection of the 3s1 photoionization cross-section decreases by the same factor F3s1 : This behavior is illustrated by the results of concrete calculations of s3s1 ðoÞ in Ar (Hansen et al., 1999). Indeed the experimental cross-section s3s1 ðoÞ at high o is notice-

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ably less than the RPAE value, which includes interactions between 3s2, 3p6, 2s2 and 2p6 electrons. We have calculated F3s and found that F3s E0:79; which lead to reasonably good agreement of s3s-1 ðoÞ ¼ sRPAE 3s-1 ðoÞF3s with the experimental cross-section s3s1 ðoÞ (see Hansen et al., 1999). As a result of the discussed above admixture, the cross-section of the 3p2 3d excitation behaves at high o as sns ðoÞB1=o7=2 : As to the 3p1 ionization crosssection, it decreases as 1=o9=2 (Amusia et al., 2000a). Thus, we see, that the ratio s3p2 3d ðoÞ=o3p1 ðoÞBo at o-N: Expression (27) is valid in any case, if a 3p2 3d or some other states are connected to ns1 : The factor o can easily compensate the ratio ðV =DeÞ2 : So, in the high o region even a small admixture of s1 greatly amplifies the np2 3d cross-section at o-N: Other states, like two vacancies nd 2 with a discrete excited electron can also be mixed to a s1 state. If one neglects the influence of the s-vacancy upon d-vacancy the latter’s excitation cross-section decreases with o growth as 1=o11=2 ; while the nd 2 n0 cð1 SÞ states ionization cross-section decreases as 1=o7=2 : If we take into account the interaction between nd 1 and ns1 ; the asymptotic behavior of nd 1 photoionization crosssection is 1=o9=2 : Thus, the nd 2 n0 lð1 SÞ; ionization cross-section is bigger than that of nd 1 by a factor of o: The described above correction affects the twoelectron photoionization cross-section dramatically. Indeed, it is a common view that at o-N the ratio of double-to-single photoionization cross-sections, s2þ ðoÞ=sþ ðoÞ; at o-N is o-independent. This is correct however, when there are only s-subshell electrons in the ionized target. In more complicated cases the situation is qualitatively different. Indeed, let us consider the simultaneous elimination, for example, of two atomic electrons from the d-subshell in Xe. Due to the binding of 4d2 es0 ; g and 4s1 states by the Coulomb interaction, the cross-section s2þ ðoÞ acquires a term 7=2 proportional to sþ : The single-electron 4s ðoÞBo ionization cross-section sþ ðoÞ is decreasing (Amusia 4d et al., 2000a) as o9=2 ; so the ratio þ R4d ðoÞ  s2þ 4d ðoÞ=s4d ðoÞBo

ð28Þ

rapidly increases with o growth. Thus we conclude, that an admixture of an s-electron state to the more complicated ones can alter qualitatively the photoionization cross-section at high o: For isolated atoms the single vacancy states have defined angular momenta. However, in e.g. molecules the total one-electron field is not spherically symmetric. Therefore in these objects atomic d-electrons, for example, acquire an admixture of an s-state. As a result, at high o their cross-sections behaves as o7=2 instead of o9=2 : Such prediction deserves experimental verification.

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9. Specifics of near-threshold regions of negative ions The field that acts in a negative ion A upon a photoelectron removed from an outer level is of short range. Therefore the photoionization cross-section of A do not ‘‘jump’’ at threshold from a zero to a nonzero value as it does in atoms, but increases instead at threshold according to the Wigner law, as s ðoÞ ¼ aðo  I Þ1=2 : In fact, the Wigner law holds only very close to thresholds and then the cross-section starts to increase very rapidly although the photoelectron moves in a short-range field of the neutral residual ion A: This is a consequence of the fact that the additional electron that forms the A is located well outside the neutral atom’s core. Large distances correspond to small photoelectron energies. For higher o the field acting upon the photoelectron is almost the same in A and A : As for the inner subshells, the situation is quite different. Indeed, due to inner vacancy Auger decay, the photoelectron moves in the field of a single charged ion. If slow enough, it will be recaptured to one of the high levels of the residual ion, thus leading to increase in the yield of neutral atoms close to the inner shell detachment threshold, similar to what happens in inner subshell ionization of neutral atoms (see Amusia et al., 1977). At higher o the photoelectron is not recaptured any more. Thus, the Wigner behavior for the inner shell must be limited to a relatively narrow region, with the width of the order of the PCI width of the inner vacancy. The process of recapture for negative ion H was recently discussed in detail (Sanz-Vicario et al., 2002). For Li the inner shell photoionization cross-section is measured by Berrah et al. (2001) and calculated using the R-matrix approach by Gorczyca et al. (2003). The role of recapture proved to be decisively important. Within the relatively narrow o-region, of the order of the PCI vacancy width, the photoabsorption crosssection is increasing rather fast, similar to the situation for neutral atoms. However, one must have in mind that the relative role of recapture is different. Indeed, while for neutrals the cross-section would ‘‘jump’’ at threshold even without recapture, the situation for A is different. There it is the recapture that makes the cross-section non-zero. With subsequent increase of o the situation simplifies and the photoelectron moves in the field of a normally non-excited positive ion. For instance, in 4d ionization of I and I the photoelectrons move in the fields of I2þ and Iþ ; respectively.

10. Resonances in endohedral systems The fullerene C60 and similar objects’ shells affect all the above-discussed resonances of the so-called endohedral atoms, or atoms ‘‘staffed’’ inside the fullerene shell. The effect depends upon the typical energy of the

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outgoing electron. Thus, for instance, one can expect that since in Xe@C60 the Xe Giant resonance decays emitting slow electrons, with eE27 eV; the reflections of these electrons from the C60 walls and corresponding modulation of the 4d Xe Giant resonance is highly probable. As to, e.g. Eu@C60 ; where the energies of electrons that are emitted in the process of the Giant resonance decay are much higher, eE111 eV; their reflections by C60 shell must be considerably weaker than in Xe@C60 : To develop an RPAE approach for such a multiatomic object as A@C60 that works at a level of accuracy achieved for isolated atoms is almost impossible. Therefore simplifications become inevitable. The most simple is to substitute the C60 potential acting upon a photoelectron, that goes from A, by an ‘‘orangeskin’’ potential UðrÞ ¼ U0 dðR  rÞ; where U0 o0 is determined by the experimentally known affinity energy C 60 (see e.g., Amusia et al., 1998). This potential is so simple that permits one to express the cross-section and other photoionization parameters of A@C60 via that for an isolated atom A (Baltenkov, 1999). The effect of C60 is clearly seen for the case of 5s electrons in Xe, where the presence of the C60 skin adds additional and prominent structure (Amusia et al., 2000b). Recently, the same approach was applied to studies of the dipole and non-dipole angular anisotropy parameters in Ne@C60 ; for Ne 1s, 2s and 2p electrons. Strong oscillations were found that do not exist on smooth parameters that correspond to the isolated Ne (Amusia et al., 2003a,b, 2004), which is illustrated in Fig. 11. These oscillations were called in (Connerade et al., 2000) confinement resonances.

Fig. 11. Non-dipole anisotropy parameter g1s ðoÞ of Ne1s2 electrons in Ne@C60. Calculations, from Amusia et al. (2004).

Of course, the discussed above d-type potential is a crude simplification. And it is not applicable to clusters and atoms imbedded into clusters. There completely different potentials are required. An important issue is, of course, to take somehow into account the atomic structure of the fullerene or cluster. There are a number of attempts in this direction, of which we will mention the jellium model for a metallic cluster (Guet and Johnson, 1995) and for an atom inside a metallic cluster (Wendin and W.astberg, 1993), an attempt to take into account the atomic structure of C60 (Decleva et al., 1999) and a more realistic potential than the d-type one for RPAE calculations of A@C60 (Connerade et al., 2000).

11. Landau theory for atoms The generalization or improvement of RPAE is dictated by the fact that in a number of cases this approximation is not accurate enough to describe the real structure and behavior of multi-electron atoms. There are several more sophisticated methods, e.g. GRPAE, that are describing some cases of atomic photoionization more accurate than RPAE. However, all these methods suffer from a lack of self-consistency that is a feature of RPAE based on HF. It is possible to compliment RPAE by accounting for some of corrections to it perturbatively. This is also quite difficult to perform self-consistently, but, what is most important, very often not accurate enough. The reason is that in some aspects the interelectron interaction in atoms, and even more in fullerenes and clusters, is not weak even after non-perturbative HF and RPAE contributions are taken into account. Landau in his Fermi-liquid theory (Landau, 1956) suggested an approach and derived corresponding equations that permit one to treat systems with strong enough interaction. For our case the equation looks similar to (7): # # # # GðoÞ; # þ DðoÞ DðoÞ ¼ dðoÞ XðoÞ ð29Þ where dðoÞ is the amplitude that goes beyond HF by % taking into account the ‘‘two electron–two vacancy’’ and # more complex excitations, GðoÞ is not pure Coulomb, but instead an effective interaction that accounts as well # for complex virtual excitations, and XðoÞ describes propagation of ‘‘quasi-electron—quasi-vacancy’’ pairs. The interparticle interaction in Fermi-liquids (Landau, 1956, 1957) was of short range, while Coulomb longrange potentials make the situation much more complex. Perhaps, a combination of RPAE that rigorously includes Coulomb interaction with corrections consistently treating the addition to U# in (7), DU# non-Coul ; as a short-range interaction could be useful and effective. It is possible that to calculate DU# non-Coul one should apply that version of the local density approximation that has

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no arbitrariness connected to the choice of correlation energy as a function of electron density (Amusia and Shaginyan, 1992).

12. Conclusions and perspectives Unexpectedly, RPAE and GRPAE, in spite of their age, are still effective and successfully applied to new effects and properties of atomic photoionization and some other atomic processes. Even in the field of isolated atoms there is room for further development of RPAE, for example the description of open shell or exited atoms. It is reasonable to expect that soon these same approximations will be applied to diatomic molecules with the numerical accuracy that is already achieved for isolated atoms. First steps in this longstanding problem are already made (Semenov and Cherepkov, 1998; Semenov et al., 2000) and they are rather encouraging. Similar method can be applied to an atom in a strong external electric field. To study multi-electron atoms in strong time-dependent fields the TDHF approximation (Amusia and Chernysheva, 1997) has to be developed further, particularly the methods of numerical solutions of the corresponding equation. Two-atomic molecules or atoms in external electric field are axially symmetric and therefore the RPAE equations for them are two-dimensional. The envisaged progress in computing power will make it possible to apply ‘‘brutal force’’ HF and RPAE approaches to three-dimensional problems, like three-atomic molecules and atoms in a combinations of fields. The other problem is to go beyond RPAE and to do it self-consistently, which was briefly discussed in the previous section. No doubt that RPAE is fully applicable to studies of multi-atomic objects, such as atomic clusters and fullerenes. We discussed above some really first steps in these directions. The problem is of great interest and perspective but the difficulties are tremendous in achieving the same level of consistency that is obtained for isolated closed subshell atoms.

References Amusia, M., 1990. Atomic Photoeffect. Plenum Press, New York, London, pp. 1–317. Amusia, M., 1996. Theory of photoionization: VUV and soft X-ray frequency region. In: Becker, U., Shirley, D. (Eds.), Photoionization in VUV and Soft X-ray Energy Region. Plenum Press, New York, London, pp. 1–45. Amusia, M., Cherepkov, N., 1975. Many-electron correlations in the scattering processes. Case Stud. At. Phys. 5 (2), 47–179.

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