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Interpretation of intensities in electron-momentum and photoelectron spectroscopiesag
Journal of Electron Spectroscopy and Related Phenomena, 36 (1985) 37-68 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
INTERPRETATION OF INTENSITIES IN ELECTRON-MOMENTUM AND PHOTOELECTRON SPECTROSCOPIES
I.E. MCCARTHY Institute for Atomic Studies, S.A. 5042 (Australia)
The Flindels
(First received 3 July 1984;in
University
of Sou th Australia,
Bedford
Park,
final form 15 October 1984)
ABSTRACT Relative intensities for the photoelectron reaction on atoms and molecules are not related to structure calculations in the same way as those for the noncoplanar symmetric (e, 2e) reaction. The photoelectron dipole matrix element is dependent on recoil momentum only through the unique relationship of recoil momentum to the photon energy and is much harder to calculate for chemically-interesting momenta. Relative intensities for binary (e, 2e) reactions are independent of total energy at high enough energies and strongly dependent on symmetry and recoil momentum, for which an intensity profile can be measured for values starting at zero. In comparing with structure calculations, binary (e, Se) intensities for low recoil momentum may be compared directly with pole strengths in calculations of the one-electron Green’s function. In the case of states within a single symmetry manifold the relative intensities will be independent of recoil momentum up to some maximum, usually at least a few atomic units.
1. INTRODUCTION
After some years there is still confusion between two types of ionization spectroscopy, electron-momentum and photoelectron spectroscopies, hereafter abbreviated as EMS and PES. EMS, interpreted by calculations of the one-electron Green’s function or by configuration-interaction calculations, has begun to provide a complete quantitative understanding (within experimental error and resolution) of valence-shell electronic structure for singly-ionized small molecules. EMS is the understanding of target and ion structure resulting from the binary (e, 2e) reaction in which two electrons of equal energy are detected at 45” on opposite sides of the incident electron direction. A profile of the recoil momentum of the ion is measured for each energy-resolved ion state by varying the azimuthal angle of one of the detectors. 936%2048/86/$03.30
0 1985 Elsevier Science Publishers B.V.
38
PES is the understanding resulting from the (y, e) (photoelectron) or dipole (e,2e) reaction. In the latter, two electrons are detected with extremely unequal energies. The fast outgoing electron has a momentum very close to the incident electron. The small momentum transfer simulates the negligible photon momentum in (y, e) and the information obtainable from the energy distribution of resolved ion states is analogous to that obtained from (7, e). In both cases the angular distribution of slow electrons contains limited information owing to the dominance of the dipole selection rule. We will not consider the dipole (e, 2e) reaction further, since its structure application is identical to that of (y, e). Intensities in both spectroscopies are characterized more precisely by the differential cross sections for the reactions. In both spectroscopies the energies of the states of the final ion are measured in principle. Experimental limitations will be discussed. In both spectroscopies the reaction dynamics, which depends on the wave functions of the target and the ion, determines the cross section for a reaction that leaves the ion in a particular state. In PES the kinematic limitations of a two-body reaction leave the incident photon energy as the only non-trivial variable for which a profile of such cross sections can be determined. In particular there is a one-to-one correspondence between recoil momentum and photon energy. The extra kinematic freedom of the three-body final state in EMS allows a profile of recoil momentum to be determined at arbitrary incident energy, and in particular at very high energies. The structure information is deduced from profiles of differential cross sections by comparing them with calculations of the reaction dynamics. At low energies (and hence low momentum for PES) the dynamics depends on many degrees of freedom of the many-body problem. At high enough energies (but still at low momentum for EMS) fast electrons can be represented by plane waves and the reaction dynamics is determined by a simple impulsive interpretation in which the structure wave functions directly determine the momentum profile. There are simple experimental criteria for determining the validity of the impulsive interpretation. For high energies both (r, e) and (e, 2e) reactions have an impulsive interpretation. Corrections to the impulsive interpretation introduce high momentum components, and it is an interesting feature of the study of the reaction mechanism to determine the range of validity of the approximation that the momentum profile is determined only by structure. Structure-related high momentum components are very sensitive to the accuracy of the structure calculation and would be an excellent probe for details of the structure calculation. In a detailed review article [l] Cederbaum and Domcke explain the relationship of PES to the one-electron Green’s function. There is no correspondingly-detailed derivation for EMS. In Section 2 we discuss the type of structure phenomena in which we are interested and their representation by the energy-momentum spectral
39
function. In Section 3 we review the physical limitations of the (7, e) and binary (e, 2e) reactions and the computation of approximate cross sections closely related to the spectral function. In Section 4 we discuss the approximate cross sections in the light of more-detailed reaction theories. In Section 5 we supply the argument relating EMS to the Green’s function formalism in the form of a paraphrase of the derivation of Cederbaum and Domcke for PES and review the effects of electron correlations on cross sections. In Section 6 we give two detailed examples of the main points. 2.THEENERGY-MOMENTUMSPECTRALFUNCTION
In ionization spectroscopy our understanding of electronic structure is equivalent to an understanding of the energy-momentum spectral function &.(E, q) for each symmetry manifold r of the molecule. The spectral function is a property only of the ground state $“, of an N-electron molecule and the state vector @-’ of the (N- l)-electron ion produced by separating an electron from the target at an energy cost E. It is defined as
The spherical momentum average is due to the unknown orientation of the molecule in a gas target. Equation (1) is explained further by expanding in the coordinate representation. s,(e,q)
= 1 dtiIAld”r,.
. . d3i-N(27r)-3’2
ewiqsrl$f-‘* (r2, . . , , r, ) J/t@,,
. . . ,rN )I*
c-4)
where A is the antisymmetrization operator. If the ion cannot decay by electron emission the spectral function is S&9 a) = Fe(r) (a) 3 (c - Q,))
(3)
where e, is the energy eigenvalue of the electronic state s of the ion and F,(q) is the momentum profile for the state s. Normally we will consider a particular symmetry manifold and omit the dependence of s on r from the notation. In eqn. (3) we have made the approximation that only electronic degrees of freedom are relevant. We have averaged over the photon-emission, rotation and vibration or dissociation widths of the electronic state s. It is possible to observe single-ionization energies e above the threshold for a second ionization. In this case E is an electronic continuum, characterized at low energies by resonances centered at es which correspond closely to positive-energy states of the ion. The spectral function is now strictly defined over the two-dimensional e, q continuum, although for lower ionization energies it consists mainly of slices close to the energies E,. For
40
larger ionization energies there are many channels for decay of the hole state and the width of the resonance may be quite large [2,3]. The quantity q is the coordinate in the momentum representation of the target-ion overlap in eqn. (1). In the independent orbital picture it is the momentum of the electron in the target. We usually use the approximation of labelling a symmetry manifold r in addition by the principal quantum number n of an orbital, i.e., nr, since the energies E,(,) are usally clustered within a few eV of the orbital (Hartreeinteraction the state s(r) contains the Fock) energy enr. With confiiration one-hole configuration (nr)-’ . In nearly every case the coefficients are negligible for all but one value of n if the target Hartree-Fock representation is used. Some exceptions are known, for example in CO+ [4], but cross sections are usually small for such states. If this is true and the approximation is valid that $“, consists mainly of the Hartree-Fock determinant, then enr is the centroid of the cluster of states [5]. In PES this is often called “satellite” structure, the state with the highest PES intensity being considered as a main state in the quasiparticle sense, with the others as satellites of this state. We instead use the term “components of the orbital nr”, which includes the situation where the correlations are so strong that the quasiparticle picture breaks down. Each component is identified in an EMS experiment by its momentum profile F,(q) [5] .
3. THE PHOTOELECTRON AND BINARY (e, 2e) REACTIONS
In the photoelectron reaction a photon of energy E, ionizes a molecule at the cost of energy E, producing an electron of energy E,. The photon has very small momentum, so the ion recoils with momentum p, where, in atomic units p2 = 2E e
(4)
The differential cross section for the reaction depends on the polar angle of p through the asymmetry parameter /I. For polarized incident radiation it also depends on the azimuthal angle of p. We are interested only in the magnitude of p from the spectroscopic point of view, not in its angular dependence which merely reflects the dipole selection rule. For a given separation energy e (essentially this means for a given ion state s, apart from degeneracies) the recoil momentum p depends only on the photon energy E,, since E, = E,--E
(5)
The e, p continuum can be scanned experimentally only by varying E, and hence E,. For example, for E, = 200 and 2000 eV, p is respectively about 4 and 12. Forp = 1 we have E, = 13.6 eV.
41
In the (e, 2e) reaction an electron of momentum k, ionizes a molecule at the cost of energy Q, producing electrons of momenta k, and k, which are detected in coincidence. Since the mass of the ion is large compared to the electron mass the energy of the ion recoil is essentially zero. The momentum of recoil, p, is given by k,, = k, + k, + p
(6)
and the energies (labelled to correspond with the momenta) are related by E,--E
= EA +E,
= E
(7)
The total energy is E. In the absence of electron spin polarization the cross section is rotationally invariant about kO. This fact, together with the energy constraint of eqn. (7), means that there are four degrees of freedom in the final state. We consider these to be p and the magnitude of the momentum transfer K, defined by K = k,-kk,
(8)
where A labels the faster outgoing electron. From the spectroscopic point of view we are not interested in the angles 0 or in K. The experimental scanning of the e,p continuum affords a major contrast with photoelectron spectroscopy. The extra kinematic degrees of freedom enable us to scan through p for a given E without varying any of the electron energies. In scanning E it is convenient to fix the total energy E and vary E,, . In scanning p for spectroscopic purposes we choose a geometry in which K is fixed at a value that enables p s 0 to be observed. K is maximized by requiring E, = EB, since this simplifies the calculation of the cross secticga. This is noncoplanar symmetric geometry. The polar angles of k, and k, are equal and fixed at a value 8 near 45’ that gives p S 0 when k,, k, and k, are coplanar (the exact value of 8 for p = 0 depends on E and is less than 45O). The azimuthal angle $Iof kB is varied to scanp. In this geometry p = [(2k,
cos 9 - k,)’
+ 4122, sin’ 8 sin2 f $1 1’2
(9)
To estimate the relevant kinematic ranges we choose E,, - 1 keV, for which p -,//zk,
sin $r#~
(10)
Due to experimental limitations the maximum value of sin & is about l/3, so p may be varied from 0 to - kB/2. For example, the maximum observable value of p for EB = 200 eV is about 2. At 2000 eV it is about 6. We have expressed p in terms of k, for direct comparison with photoionixation. By convention, B labels the ionized electron (antisymmetry will be discussed in Section 5), so EB corresponds to E,. It is an extremely interesting and valuable area of atomic and molecular physics to obtain a complete understanding of the photoelectron and (e, 2e)
42
reactions. By this it is meant to be able to calculate the differential cross section in all experimentally-accessible kinematic ranges as a function of all the degrees of freedom. To do this we must know the target and ion state vectors $$’ and $/F-’ . However, in order to understand the spectral function it is sufficient to find a limited kinematic range which is sensitive to the spectral function and for which we can calculate the differential cross section. We shall show that in this respect the binary (e, 2e) reaction has a tremendous advantage over the photoelectron reaction for low Q, since the spectral function can be measured directly from q = 0 to some experimentally-limited maximum value qmax and does not require theoretical interpretation of the experiment. In general, qmax is less than the largest experimentally-attainable momentum. For q “>qmax the interpretation of the photoelectron reaction is equally simple, so that the two reactions have complementary roles to play in spectroscopy. For q ? qmax the spectroscopy is understood in the photoelectron case only through a detailed calculation of the cross section. The spectral function can be calculated directly from the Green’s function formalism, which gives this formalism a special role in ionization spectroscopy. We first give the simple approximations to the cross sections for the reactions in order to see how they are used in the spectroscopic application. To illustrate the procedure we take the binary (e, 2e) reaction. In subsequent sections we discuss the validity of the simple expressions in the context of more-complete theories. The simplest approximation to the photoelectron cross section is do/@
= 4x20(K&)S,(e,
a)
(11)
where (Yis the fine-structure constant and q=-p=k
(12)
e
This expression comes from approximating the N-electron final state by the
product of a plane wave exp(iq r) and \clc-’ in the following matrix element, whose calculation is not so easy, but which is valid at much lower energies l
(13) T(,, e) =
= (2x) 4 $$
0
fe,S,(GP)
(14)
43
where f,, is the half-off-shell Mott-scattering cross section given by
fee =
1 271v exp (2~) - 1 I Ik, - kA I4
1 (~~2
)2
1 -Ik,-kA,2 v = l/lk,,
1 Ik,-kk,li
cos
1 +.
tk, -kd4
vlnIk,_kk,,2 lko - kB’2])
-k,)
(16)
This is the plane-wave impulse approximation. In noncoplanar symmetric geometry Ik, - kA I = IkO - kB I = K is fixed and, for small angles 3 #, v is essentially constant, so that f,, is constant within about 1% over the range of E,P relevant to valence shells. The differential cross section is thus proportional to S, (E,P) over this range. Furthermore, it is unnecessary to measure or calculate absolute cross sections in order to obtain the constant of proportionality. If e is scanned far enough to observe all the structure for the manifold nr, the spectral function may be normalized by the sum rule [ 51 z GE”?P,,(S) = 1
(17)
Here P,,(s) is the pole strength in the language of Cederbaum and Domcke [l] , in terms of which S,(e,p) is usually approximated by
WGP)
= P,,@9jul(Pl~“,,l’
WE-G)
(18)
where (p )qb,,) is the momentum representation of the orbital nr. Note that the sum rule [ eqn. (17)] is strictly valid only when ground-state correlation effects are neglected. The influence of such effects is discussed in relation to eqns. (49) and (50). The approximation (18) is valid for electron energies EB comparable to those for which eqn. (11) is valid for the photoelectron reaction, namely, a few hundred eV or higher. The difference is that at such energies the relevant range e,p can be scanned completely up to a value pmax which is of the order of 5 for E, values of a few keV, whereas for eqn. (11) the range is restricted to p 5 4. Since eqn. (14) is a high-energy approximation we must have a criterion for its validity. The criterion is purely experimental. The incident energy E,, must be so high that the experimentally-determined S,(c,p) is independent of E. over a substantial range of e,p. Note that in approximation (14) the momentum coordinate q is identical to the experimentally-measured momentum p. There is now a large body of experimental evidence [6] that the necessary energy-independence is achieved for E,, 5 400 eV. In order to apply the spectroscopic sum rule of eqn. (17) we must know
44
which ion states s are components of the orbital nr. This again is decided experimentally. The momentum profiles for different nr are easily distinguished in most cases, so that their shape is used to identify nr. An absolute assignment of the representation is obtained by calculating the orbital momentum profile P,,(P)
= z,,il, F 8(P) = J. W(Pb#Jnr)12
(19)
starting with the same orbitals that are used in the Green’s function calculation. The calculation of eqn. (19) is described, for example, by McCarthy and Weigold [ 51. In practice some convolution with the experimental energy-resolution function [ 71 modifies the procedure. The assignment of states s to representations nr is confirmed by comparing the summed momentum profiles [eqn. (19)] with momentum-profile calculations for different nr. If the whole procedure is correct the summed momentum profiles must have the same relative relationship as the calculated orbital momentum profiles for a substantial range of p starting with p = 0. The procedure has been verified over the past few years in a large number of cases [6,7]. Note that both approximations (11) and (14) involve plane waves in a state where the residual system is charged. At the high energies where the approximations are valid there is little difference between plane waves and Coulomb waves over the part of the integration space where the valence wave function gives a significant contribution to the integrand. The asymptotic forms of the continuum functions are irrelevant.
4. REACTION CALCULATIONS
Since we are interested in the use of the photoelectron and (e, 2e) reactions in spectroscopy it is not our purpose to give a detailed account of the computation of their amplitudes. Nevertheless, it is necessary to see what is involved in such calculations and how well they work in different kinematic conditions. We can thus form a judgement of the best conditions for our purpose. It is best to consider first the calculation of the reactions for simple targets whose structure is well known. If the calculation can be performed correctly in these cases, i.e., if we understand the reaction mechanism, we can expect the extension of the same mechanism to spectroscopically more-complicated systems to throw light on the spectral function. The photoelectron amplitude is, in the one-photon approximation T(,,,,(k,)
= @(LA,
lPnl St’)
cw
where IF) is a many-body state vector with one electron obeying outgoing scattering boundary conditions (plane wave and ingoing spherical waves). A, -P, is represented by either the dipole length or velocity operator. These
45
operators are strictly equivalent when acting on the exact state IF), but may give different results for a particular approximation to IF). Methods of calculation all involve an expansion of IF) in channel functions. These are configuration-interaction functions for the ion coupled with the appropriate symmetry to one-electron functions. IF) is essentially the time-reversal of the elastic scattering function for an electron on the observed state $,“-I, coupled to J/,“- ‘. It contains smaller terms representing time-reversed inelastic scattering if coupling of the channel functions cannot be ignored. Channel coupling is less important at higher energies. We will not give details of methods for calculating elastic scattering wave functions (distorted waves), which are relevant to both the photoelectron and (e, 2e) reactions. It is however useful to note that all methods can be related to the solution of a one-body scattering equation in the optical potential [ 81, which includes the potential for the target ground state (static exchange approximation) and a polarization potential representing the effect of all excited states. An example of the present detailed understanding available to the photoelectron reaction on atoms is the R-matrix calculation [9] for low-energy reactions on inert gases. Parameters of resonances at very low E,, which are very sensitive to details of the calculation, are reproduced very well by both length and velocity formalisms with the results apparently converging to each other and to experiment as more configuration interaction is taken into account. For a particular E, the differential cross section is quite well produced as a function of E, up to about 70eV using an average over the discrete pseudostructure introduced at high energies by the use of discrete pseudostates to represent the ion continuum. The importance of configuration interaction is shown for the highly-split 3s-’ orbital of the argon ion, where it is necessary even to obtain a qualitatively-correct cross section at energies E, near 13.6 eV 0, = 1). Configuration interaction in the Rmatrix method is responsible for channel coupling as well as for splitting of the orbital. As an example of a detailed photoelectron calculation for a molecule we consider a calculation of photoexcitation and photoionization of the water molecule by Diercksen et al. [lo]. This involves only the electronic degrees of freedom. Closure is used for rotational and vibrational states. The staticexchange (uncoupled channels) approximation was used and the continuum was approximated from the discrete excitations of pseudostates by the StieltjesTchebycheff moment technique [ll]. For the water molecule the three outer-valence orbitals are essentially unsplit. The curves of differential cross section vs. E, are only qualitatively correct below about 20eV, but have errors less than 20% in comparison with experiment [ 121 for 20 eV ? E e ? 50 eV. The distribution of differential cross section as a function of e, over states which are components of the inner-valence 2ai orbital is quite well reproduced.
46
We may summarize the detailed calculation of the photoelectron amplitude by saying that it is very good below about 50 eV if all effects are taken into account, notably channel coupling and configuration interaction. In the next section we will consider the method suggested by Cederbaum and Domcke [l] for comparing pole strengths for photoelectron reactions at higher energy, and thus using the one-electron Green’s function technique as a method of understanding the splitting of an orbital by configuration interaction. This method rests on the factorization of the many-body wave function ]F) into a time-reversed el~tic-scatte~g distorted wave and the ion wave function in its observed state IF) = lx:-‘(k,))l3/,“-‘> The factorization is correct if inelastic scattering (core excitation) can be ignored in the electron--ion system. The same final state can be arrived at with core excitation in a two-step process, in which the amplitude consists of a factor representing photoionization to an ion state g-l and a factor representing the change of @‘-I to J/,”-i in the electron-scattering process. Such amplitudes are not formally included in eqn. (20) with the factorization (21), but they are included in the full coupled-channels or R-matrix treatment of IF). The cumulation of xg) involves many ion states explicitly at low energies. At higher energies the static-exchange approximation becomes more valid. Here x2-) is obtained from a scattering calculation including exchange on the state J/f-‘. At very high energies a plane wave becomes a reasonable approximation. From the calculation of Diercksen et al. [,lO] for water it appears that the static exchange approximation is good for E, 5 20 eV, i.e., for p 5 1.2. This is also true for a simplified version of the static-exchange approximation used by Hilton et al. [ 131. The range of validity of the planewave approximation is roughly that for the Born approximation to elastic scattering, perhaps E, > 200 eV, i.e., p 5 4. In this range the dipole velocity expression reduces the cross section to the form of eqn. (11) using the definition (1) of S,(e,p) and making the appropria~ sums and averages over degeneracies. The (e, 2e) amplitude may be written formally as
tw Instead of including the many-body complexities of the reaction in the final state IF) we have included them in an operator T which acts on the planewave states and the target and ion eigenstates. The first approximation we must make is the binary-encounter approximation in which T is a two-body operator depending on the coordinates of the incident and struck electrons, but not on those of the ion. This eliminates many exchange terms involving the overlap of a one-electron cont~uum function with a bound orbital, as well as channel coupling in the electron injections with a bound system.
47
The overlap of a plane wave with a valence orbital is negligible for planewave momentum k greater than about 3, i.e., for continuum-electron energies greater than about 100 eV. This condition is necessary for the validity of the binary-encounter approximation. We now have T(e,ze)(kA,k,A
= (k*kz ITl(9,N-‘l$$‘)k,)
(23)
Equation (23) already contains the main spectroscopic point. The (e, 2e) amplitude depends on the spectroscopy only through the overlap amplitude <$r-’ 11): ). It is a three-body transform of the overlap. To conform with the notation of ref. 1 we introduce a basis of oneparticle states (or orbit&) 1$J* ), the best a priori definition for which is the target Hartree-Fock procedure, which eliminates one-particle one-hole configurations from the confiiuration-interaction expansion of $J,”. We define creation and annihilation operators aT,a, for an electron in 1#1) and express eqn. (23) in its second quantized form. 7&&kA,kz) rkl
=
(h@k
= &A&~-’
Ill&,)
hItif)
(24)
(25)
where 7 is a new interaction operator defined so that the plane-wave state (kz > for the “ionized” electron [antisymmetry is of course taken into account in eqn. (22)] is replaced by the target Hartree-Fock continuum orbital.
1x$-)(kB1) = Ih)
(26)
This is the static exchange approximation to I x&j ). In eqn. (24) we usually make the approximation that I$, N-1 > contains the one-hole configuration al I $$ > for only one characteristic orbital I&). The quantity ($r-’ IaII I,!J~) is the spectroscopic amplitude for the orbital I in the ion state s. The quantity 7kl of eqn. (25) is the one-orbital (e, 2e) transform. The investigation of its form under a wide range of different kinematic conditions has been the subject of intensive effort in atomic physics since the pioneering (e, 2e) experiments of Ehrhardt et al. [14]. These experiments all involve small momentum transfer K and consequently small magnitudes of kB. The first-order theory for this situation is the Born approximation in which x$‘N(kB) is given by eqn. (26) and 7 is the electron-electron Coulomb potential u. Much unnecessary worry about the difficulty of understanding the (e, 2e) reaction has been caused by the qualitative inability of first-order theories to reproduce these data. Considerable improvement for helium at E,, = 500 eV has recently been obtained by Byron et al. [15] who use the second Born approximation to 7kl. However, these experiments are done in a kinematic range that offers a challenge to the full power of atomic reaction theory. Experiments that offer less challenge to theory have been carried out by Giardini-Guidoni et al. [ 161 and the Flinders group [17-191 for atoms
48
whose structure is well understood. The main point of these experiments is that the energy condition for the binary-encounter approximation is met for all continuum electrons. The distorted-wave impulse approximation, in which T includes operators that distort all the continuum-electron waves in various approximations to the optical potential for elastic scattering in the appropriate two-body subsystem, and the electron--electron interaction is treated by using the full scattering operator t instead of the potential u, reproduces the data excellently for total energies E above about 400 eV. The matrix element (k’( t(E) Ik) is the exact antisymmetrized amplitude for the scattering of two electrons with initial and final relative momenta k, k’ by the Coulomb potential. In the (e, 2e) application we require halfoff-shell matrix elements [20], in which k’ corresponds to the energy E for which the electron-scattering Schrijdinger equation is solved, but k in general corresponds to a different energy. The spin average of the absolute square of this amplitude is the half-off-shell Mott scattering cross section (15). For reasons stated in the previous section, noncoplanar symmetric geometry is most appropriate for the spectroscopic application. This is wellunderstood theoretically as shown for example for the 2s and 2p orbitals of neon [ 181, where differential cross sections relative to one unknown experimental intensity are correctly described at 600 eV for 0
=
(hb
ItlWo)
(27)
49
This factorizes as follows TkZ
=
(k’(tlk)(pl$+)
(23)
where k’, k and p are given respectively by eqns. (15) and (6). The electronelectron amplitude is spin-dependent. On squaring eqn. (24), using eqn. (28), spherically averaging, averaging over initial-state and summing over finalstate spin degeneracies, we obtain the form (14) for atomic targets, where
&(E,P) =
l~119F)12 SawP14d12w-~,)
w-’
(29)
The definition (29) of &(e,p) is the same as eqn. (18) with the orbital nr now denoted by 1. The pole strength &(s) is given by
4(s)
= Kti-’
laAJ/~)12
(39)
The sum rule (17) for Pi(s) is obtained from the application of the orthonormality and closure relations to eqn. (30). For molecules we have additional rotational and vibrational degeneracy. It has been shown for the cases of H2 and D2 [17] that an explicit vibrational sum is equivalent to calculating the (e, 2e) amplitude at the nuclear equilibrium positions. This approximation is always used. Rotational degeneracies are accounted for by the spherical average.
5. RELATIONSHIP
TO THE ONE-ELECTRON
GREEN’S
FUNCTION
The formalism of the one-electron Green’s function has been related to photoionization by Cederbaum and Domcke [ 11. The total cross section P(w)for photoionization with the energy conditions o
=
E,, a0 = E,
is expressed in terms of target Hartree-Fock photoionization amplitudes rkl
=
(#k
IA-PI@,)
(31) orbitals, &, and one-orbital (32)
where $!$ is some optimum one-particle scattering function, for example the static-exchange continuum orbital of eqn. (21) or the scattering function for the appropriate optical potential
Ix,‘-‘(W) =
I$k)’
(33)
We paraphrase the argument to discuss the total cross section P(o)for the set of (e, 2e) reactions initiated by incident electrons of momentum k0 , with the “scattered” electron having momentum kA . Antisymmetrization of the final state with respect to the two continuum electrons will be performed
50
explicitly. Apart from initial- and final-state degeneracies we have P(0)
= (2a)4 ‘9
w=EA-i-EB,
&(Fl(k,
0 w,,
]r1ko)l\1/;)12
(34)
=Eo
(35)
The operator connecting the N-electron final state IF) and initial state $2 is expressed in the second-quantized form (24), giving
kAkB
= (27r)4 k o
P(0)
(- ll~)%n,
6,~~
Im CiGtimn (- 00
---iv)1
(36)
where rkr is given by eqn. (25) and Gulmnis particle-hole component of the two-body Green’s function, whose calculation is difficult. In order to express P(o) in terms of the one-electron Green’s function we formally factorize IF) into a one-electron distorted wave IXk) (kB)) for the “ionized” electron B and the observed ion state I I#-’ >. IF)
=
Ixg’(kB))I$:-l)
(37)
defining the factorized amplitude rsl (corresponding to rel in the photoelectron case [ 11) by rs1 = (kAXB-)(kB)]r]Q+ko)
(38)
We do not have to make a factorizability approximation if we define the operator r to include the required operator describing many-body effects including core excitation (two-step processes). The conditions under which r can be represented by various approximations have been thoroughly discussed in Sections 3 and 4. The neglect of core excitation corresponds to the binary-encounter approximation. For our spectroscopic application we confine P(w) to noncoplanar symmetric reactions. We choose E. 5 400 eV and use the plane-wave impulse approximation TBl
=
(3%
%&BitI&kO)
We now have P(O)
= (2n)4 ‘+
&,,?$,,~B,
hl
[Gh(a
- Cd0 - i?‘j)]
0
where 2, signifies the appropriate sum and average over final- and initialstate vibrational, rotational and electron spin degeneracies and GI,(o---iq)
=
&
(I# Id I J/~-‘w,“-’ w + Efl -Et
bll4w -iq
(41)
The forms of eqns. (40) and (41) are now identical (apart from the kinematic constant and the omission of the energy constraint eqn. (7) from the
51
formalism) to eqns (2.4) of [ 11. The spectral function &(e,p) is given by consideration of the poles of GI, (w - ir,~).The numerator of the s term of GnI is the product of two spectroscopic amplitudes MS) = QY-’
Ial
(42)
I tit>
We first consider final-state correlations in terms of the target HartreeFock approximation in which we neglect initial-state correlations, i.e., 3/t is a pure Hartree-Fock configuration Qt. J/f’-’ contains the configuration aI 1@r ) (the Z-hole configuration) with an amplitude &(s). Other configurations of J/,“-’ have zero overlap with a2I G$ ) if they involve electrons in orbitals unoccupied in at. G1, is then diagonal and its pole strength 9(s) is given by eqn. (30) and obeys the sum rule (17) for states s in the symmetry manifold characterized by 1. There are two reasons why G,, is not in general diagonal. The first occurs occasionally in molecules but not in atoms where states with the same principal quantum number are well-separated in energy. Occasionally $r-’ has significant contributions from the I- and n-hole states, where orbitals I and n have the same symmetry. An example is the CO+ ion, where in a particular calculation [4] there was a state at 24.7 eV with t5,, = - 0.169, tdo = 0.239 and t so = 0.080.
The second and most important reason for nondiagonal G1, is initial-state N-l belongs to the symmetry manifold 1, i.e., it concorrelations. Suppose tiB tains the Z-hole configuration al (!E$ ). I)$ must contain some configurations with electrons in orbitals n which are unoccupied in @‘. The corresponding configurations in J/r-’ now have finite overlap with a, I $I: ). A specially simple example is helium, where the final states are uncorrelated. It is discussed in Section 6. In general the cross section is related to the eigenvalues Dk (0) of the matrix G(w), whose elements are given by eqn. (41). P(w) bu(w)
4 kdB
= (277) - xk =
T,,(O)
ITB,(o)
=
In2
(2,
tr
[b(w
-
w.
-
iv)]}
0 1’oi(o)
(43) (44)
6,
(45)
IZji7gjMij(O)
where M is the eigenvector matrix of G. The eigenvalues Dk are related to the pole strengths Pk (s) by Im
@k
(0
-
iv)}
=
nz,Pk
(S)
6 (W
-t #-’
-
EON)
(46)
We may now write P(0)
= (27r)4
kAkB --crskITBk(d12Pk(S)
ko
(47)
52
Since the plane-wave impulse approximation to rzk factorizes in the form of eqn. (28), we antisymmetrize the matrix element with respect to the final state continuum electrons by noting that (p I r#~~ ) is symmetric under exchange of A and B and that the two-electron t-matrix element (k’ (t Ik) must be antisymmetrized to describe the two degenerate spin states singlet and triplet. The sum and average Za now givesP(o) in the form P(0)
= (2x) 4 y
where
f,, is given by eqn. (15) and
Sk(%P)
=
fee&kSk
te,p)
Pk(S)IdljI~:I(PI~I)Mk,(e)12
s(E-%,)
(49)
The review article of Cederbaum and Domcke [l] describes the calculation of Pk (s), Mkl(e) and E, in great detail. Note that the energy momentum spectral function eqn. (49) is a direct result of the calculation, involving only the structure of the target molecule and the ion, and not any explicit properties of the reaction mechanism except for the relation q = p connecting the observed recoil momentum and the coordinate in the momentum representation. We now consider the spectral function S, , eqn. (49), under different conditions. If Gkl (and therefore &l) is diagonal then the p-dependence of Sk is the momentum profile for the orbital k. The pole strengths Pk (5) sum to unity according to eqn. (17). If there is a contribution to J/,N-1 from more than one Z-hole configuration then the pdependence of Sk arises from a superposition of p-profiles for each relevant orbital. This is an unusual situation normally confined to states with small pole strengths. If there is appreciable initial-state correlation then Mkl is not diagonal but off-diagonal elements are usually small. It is reasonable to make the practical approximation Sk(%p)
=
&(s)~@h’b#‘k)i2
h(e-%)
(59)
In this case, however, the sum rule (17) for the Pk (s) must be modified to LZP,
(s) =
IW
I !@)I”
(51)
In experiments which measure cross sections relative to one unknown quantity we may normalize the summed pole strengths for all manifolds k to unity, since the sum is independent of the manifold. This normalization is equivalent to considering the pole strength as the square of the coefficient of the k-hole configuration in J/y-’ . The practical approximation eqn. (50) is expected to become invalid at high momentum p. For example the valences orbitals of neon and xenon have nodes atp Z 3 and 1.5, respectively. Here small Mkl may give dominant
53
contributions to eqn. (49) and the corresponding photoionization amplitudes, eqn. (13). Thus high-momentum cross sections are sensitive to confiiuration interaction. This is both a problem and an advantage for highmomentum ionization spectroscopy, where photoionization is easy to compute and has a considerable count-rate advantage over binary (e, Ze). The problem is that the analysis eqn. (50) cannot be used and that a very accurate calculation of configuration interaction is needed for the initial state. The advantage is that such a calculation can be tested very accurately. In Section 6 we give the example of helium, where high-momentum photoionization cross sections have been measured. We now note a major difference between EMS and PES concerning the application of the Green’s function formalism to experiments. For EMS the plane-wave impulse approximation, assisted by the spectroscopic sum rule, eqn. (17), directly relates the experiment to the pole strengths Pk (s) for states s resulting from the splitting of the orbital k. The experimentalist needs only to calculate the spherically-averaged square of the momentumspace orbital in order to identify the orbital k for each ion state s. For the application of PES it is necessary to calculate the corresponding one-orbital amplitude from eqns. (32) and (33), which is a much more difficult task (described in Section 4) requiring considerable theoretical interpretation. The much simpler plane-wave calculation is valid only for very high energies E, and recoil momentap. Cederbaum and Domcke note that the ratio of the pole strength Pk (s) to the pole strength for the orbital k (which is usually normalized to 1) may be more easily obtained from experiment. In terms of two ion states r, s resulting from the splitting of k, the ratio of photoionization cross sections is equal to the ratio of the pole strengths provided the corresponding oneorbital cross sections are equal, i.e., provided (using the notation of [l] ) lQ&)12
= lr,,&,)l”
(52)
In this case, if one can identify the orbital k to which s belongs (an easy task in EMS but not in PES), one can use the same analysis as described for EMS at the end of Section 3. For a given incident photon energy o. the cross section depends rather strongly on E, (= w. -e, ), since this determines p. For small molecules and E, >20 eV, where static-exchange calculations of x$-)(p) are reasonably valid, the one-orbital cross section may change by a factor of roughly 2 over a 10 eV interval (typical of the splitting of inner-valence orbitals) [lo, 121 and therefore the intensity is not proportional to the pole strength. The change is more rapid for molecules such as HBr, which contain large atoms. A calculation at least as detailed as the staticexchange approximation is therefore needed for interpretation. If the experiment is arranged to observe the cross sections for both r and s at the same value of E,, rather than wo, the condition (52) is more-nearly satisfied. At very high energies the validity
54
of eqn. (52) is subject to considerations of initial-state configuration interaction. Note that similar considerations apply to eqn. (50) at high momenta, but not necessarily at high energies since the three-body kinematics enables observation of low momenta at high energies. The Green’s function formalism of this section has been applied to EMS of several molecules [6,7,21] in which pole strengths for split inner-valence states are qualitatively reproduced by the structure theory. The mostdetailed calculation is for water [22] where the convergence of the structure calculations with respect to basis size has been investigated. Here the calculation of the Green’s function poles in the extended Tamm-Dancoff approximation [23] reproduces the inner-valence pole structure quite well. Somewhat better agreement is obtained by direct calculation of I&,“-’ and tig using the multireference double-excitation configuration interaction method [24]. Examples of the application of the formalism to PES are given in ref. 25.
6. CONCLUSIONS WITH EXAMPLES
In this section we give detailed illustrations of the two main points of the discussion. The first is that EMS can be used to identify symmetry manifolds nr in the spectrum of the ion and, if configuration interaction is not very important in the target, it can give pole strengths for states within each manifold that obey the requirements of energy independence and other consistency rules. The 3p and 3s manifolds of the argon ion are easily identified by the shapes of their momentum profiles. The former has a minimum at p = 0, while the latter has a maximum. For the 3p manifold only one state, at 15.76 eV, is significant. The sum rule (17) gives its pole strength Psp(15.76) as unity. CI calculations [26] give about 0.97 for this pole strength, which is within experimental error of the value 1. The 3s manifold has several states, which can all be cleanly resolved experimentally so that reliable intensity measurements are available. Intensities of all states in the 3p and 3s manifolds are measured relative to each other. The strongest state in the 3s manifold is at 29.3 eV. Its pole strength is given by the sum rule (17) to be 0.53 + 0.05 for a 400 eV (e, 2e) experiment [ 271. The condition for applicability of the sum rule is not that we can understand details of the reaction, for example sufficiently to calculate correct momentum profiles, although these are given within experimental error by the distorted-wave impulse approximation [28] as they are for neon and xenon [18]. The sum rule is applicable under the purely-experimentallyverifiable condition that momentum profile shapes for all states within the manifold are the same over a significant range of p from zero up to some
55 TABLE 1 ENERGIES IONa
AND POLE STRENGTHS
EMS e(eV)
29.3 36.7 38.6 41.2 42-43.4 >43.4
FOR THE 3s MANIFOLD
Mitroy P$:) 0.647 0.032 0.176 0.074 0.041 0.122
* 0.019 f 0.008 + 0.011 f 0.007 + 0.006 f 0.008
OF THE ARGON
PES
MUP
E (eV)
PJZ)
E (eV)
Pi:)
e (eV)
Pjf)
28.7 36.8 39.1 41.8 >42
0.649 0.013 0.161 0.083 0.081
29.4 36.5 38.6 41.2
0.61 0.01 0.17 0.20
29.3
0.81
38.6 41.2
0.13 f 0.02 0.06 f 0.02
*Column headings are described in the text. EMS data are from ref. 28. PES data are from ref. 29.
maximum value. An essential condition is that the pole strength ratios must be independent of the total energy E of the (e, 2e) experiment. Both these conditions are satisfied for the argon 3s manifold [5]. The momentum profile shape is given by the plane-wave impulse approximation with the argon Hartree-Fock 3s orbital up to p = 1 at 1200 eV for states at 29.3,38.6, 41 and 44 eV. This is approximately true also for weak continuum states at 48 and 51 eV. Pole strengths are independent of energy from 200 to 1200 eV. Their values are summarized in Table 1. A stricter test of the understanding of the (e, 2e) reaction on argon is giving by comparing the intensities of the 29.3 eV component of the 3s-hole state and the 3p hole state at 15.76 eV. Here we must have a reaction theory that affords a reliable comparison over a range of p. The plane-wave impulse approximation correctly describes momentum profile shapes up to p = 1 [ 51 but obtains a pole strength of 0.28 f 0.03 at 400 eV for the 29.3 eV state. The distorted-wave impulse approximation however describes shapes up to p = 2.8 at 1000 eV [28] and 1.8 at 400 eV essentially within experimental error, and gives a pole strength of 0.55 f 0.05 at 400 eV. Both these values are completely consistent with the sum rule (17) and complete the total understanding available for the noncoplanar symmetric (e, 2e) reaction. It is interesting to consider the pole strengths for the argon 3s manifold given by different CI theories and by the ratio interpretation (52) of the photoelectron reaction [29]. These are summarized in Table 1. Two CI theories are given. The first is an ab initio calculation by Mitroy [26]. The second uses phenomenological matrix elements of the Hamiltonian determined by fitting photon spectra. It describes only bound states of the ion and was calculated by McCarthy et al. [303 (denoted by MUP). Clearly the CI calculations must be improved before they are capable of reproducing
56 TABLE 2 RATIO OF n = 2 TO n = 1 INTENSITIES IN THE SEPARATION-ENERGYINDEPENDENT INTERPRETATION OF PES FOR IONIZATION OF HELIUMa
m P
2
3
4
5
2 3 4 7 10
4.1 6.4 7.3 6.7 5.9
4.3 6.2 6.7 5.6 5.1
4.3 6.1 6.6 5.9 5.4
4.3 6.1 6.6 5.9 5.3
aRatio are given in unit8 of lo-‘. by m and describedin the text.
The quality of the CI calculationfor helium is denoted
the EMS data. The ratio interpretation of PES intensities cannot be applied to present [29] data for argon, since the data are in the form of peaks whose magnitude is unmeasured because of an unknown background. The figures for Psd(e) given in column PES of Table 1 are certainly an overestimate. The second main point of the discussion is the possibility of using PES to distinguish details of theory at large values of p(” lo), where the (7, e) reaction has a distinct intensity advantage over EMS. To illustrate this we use the example of the n = 2 to n = 1 intensity ratio for the helium ion states. This is an extremely simple example, since the wave functions for the ion states are known exactly. We assume that the (7, e) reaction mechanism is independent of the energy difference for the high energy ionization of helium to n = 1 and n = 2 states. The intensity ratio then depends only on the details of the ground-state wave function of helium. For illustration we use progressively-improved CI calculations of helium [ 311, characterized by an integer m which signifies inclusion of natural orbital sets up to na in the CI basis. The number and symmetry of natural orbitals for each m is given in the form (m, number of orbitals, symmetry label) by (1,5,s), (2, 4,p), (3,3,d), (4,2,n, (5,l,g). The sensitivity of the intensity ratio to the quality of the CI wave function is shown by Table 2. Clearly the CI expansion has converged for m = 5. The ratio of intensities at p = 10 depends sensitively on the quality of the CI wave function. The validity of the assumptions we have made about the reaction mechanism is shown by the experimental value of the ratio, which is (5 + 1) x 10m2 at p = 10.37 [32].
57 ACKNOWLEDGMENTS
I would like to acknowledge support from the Australian Research Grants Scheme and ENEA, Frascati, where this work was begun. I would also like to acknowledge help and encouragement from Professors C.E. Brion, R. Cambi, L.S. Cederbaum, A. Sgamellotti and E. Weigold.
REFERENCES 1 2 3 4 5 6
7
8 9 10 11
12 13 14 15 16 17 18 19
L.S. Cederbaum and W. Domcke, Adv. Chem. Phys., 36 (1977) 205. H.S. KohIer, Nucl. Phys., 88 (1966) 529. U. WiIIe and R. Lipperheide, Nucl. Phys. A, 189 (1972) 113. S. Dey, A.J. Dixon, K.R. Lassey, I.E. McCarthy, P.J.O. Teubner, E. Weigold, P.S. Bagus and E.K. Viinikka, Phys. Rev. A, 15 (1977) 102. I.E. McCarthy and E. Weigold, Phys. Rep., 27C (1976) 275; E. Weigold and I.E. McCarthy, Adv. At. Mol. Phys., 14 (1978) 127. E. Weigold, S. Dey, A.J. Dixon, I.E. McCarthy, K.R. Lassey and P.J.O. Teubner, J. Electron Spectrosc. ReIat. Phenom., 10 (1977) 177; A Giardini-Guidoni, R. Tiribelli, D. Vinciguerra, R. Camilloni and G. Stefani, J. Electron Spectrosc. Relat. Phenom., 12 (1972) 405; A. J. Dixon, S.T. Hood, E. Weigold and G.R.J. Williams, J. Electron Spectrosc. Relat. Phenom., 14 (1978) 267; C.E. Brion, I.E. McCarthy, I.H. Suzuki and E. Weigold, Chem. Phys. Lett., 67 (1979) 115; C.E. Brion, S.T. Hood, I.H. Suzuki, E. Weigold and G.R.J. WiIIiams, J. Electron Spectrosc. Relat Phenom., 21 (1980) 71; I.H. Suzuki, C.E. Brion, E. Weigold and G.R.J. WiiBams, Int. J. Quantum Chem., 18 (1980) 275. See also refs. 4,5 and 21. R. Cambi, G. CiuIIo, A. Sgamellotti, F. TaranteIIi, A. Giardini-Guidoni and A. Sergio, Chem. Phys. Lett., 80 (1981) 295; R. Fantoni, A. Giardini-Guidoni, R. ThibeIIi, R. Cambi, G. CiuIIo, A. SgameIIotti and F. TaranteIIi, Mol. Phys., 15 (1982) 839. I.E. McCarthy and A.T. Stelbovics, Phys. Rev. A, 22 (1980) 502; I.E. McCarthy, C.J. Noble, B.A. Phillips and A.D. Turnbull, Phys. Rev. A, 15 (1977) 2173. P.G. Burke and K.T. Taylor, J. Phys. B, 8 (1975) 2620; K.A. Berrington, P.G. Burke, W.C. Fon and K.T. Taylor, J. Phys. B, 15 (1980) L603. G.H.F. Diercksen, W.P. Kramer, T.N. Rescigno, G.F. Bender, B.V. McKay, S.R. Langhoff and P.W. Langhoff, J. Chem. Phys., 76 (1982) 1043. P. W. Langhoff, in B.J. Dalton, S.M. Grimes, J.P. Vary and S.A. Williams (Eds.), Theory and Applications of Moment Methods in Many-Fermion Systems, Plenum, New York, 1980, pp. 191-212. K.H. Tan, C.E. Brion, Ph. E. Van der Leeuw and M.J. Van der Wiel, Chem. Phys., 29 (1978) 299. P.R. Hilton, S. Nordhohn and N.S. Hush, Chem. Phys. Lett., 64 (1979) 515. H. Ehrhardt, K.H. Hesselbacher, K. Jung and K. Wilhnann, Case Stud. At. Phys., 2 (1971) 159. F.W. Byron, Jr., C.J. Joachain and B. Pireaux, J. Phys. B, 13 (1980) L673. A. Giardini-Guidoni, R. Fantoni, R. Tiribeih, R. Marconero, R. Camilloni and G. Stefani, Phys. Lett., 77 (1980) 19. I. Fuss, I.E. McCarthy, C.J. Noble and E. Weigold, Phys. Rev. A, 17 (1978) 604. A.J. Dixon, I.E. McCarthy, C.J. Noble and E. Weigold, Phys. Rev. A, 17 (1978) 597. E. Weigold, C.J. Noble, S.T. Hood and I. Fuss, J. Electron Spectrosc. ReIat. Phenom., 15 (1979) 253.
58 20 21 22 23 24 25
26 27 28 29 30 31 32
W.F. Ford, Phys. Rev. B, 133 (1964) 1616. A. Minchmton, I. Fuss and E. Weigold, J. Electron Spectrosc. ReIat. Phenom., 27 (1982) 1; A. Minchmton, C.E. Brion, J.P.D. Cook and E. Weigold, Chem. Phys., 76 ii983j 86 R. Cambi, G. CiuIlo, A. Sgamellotti, C.E. Brion, J.P.D. Cook, I.E. McCarthy and E. Weigold, Chem. Phys., 91(1984) 373. 0. Walter and J. Schirmer, J. Phys. B, 14 (1981) 3806; J. Schirmer, L.S. Cederbaum and 0. Waiter, Phys. Rev. A, 28 (1983) 1237. R.J. Buenker and S.D. Peyerimhoff, Theor. Chim. Acta, 36 (1974) 33; 39 (1975) 217; R.J. Buenker, S.D. Peyerimhoff and W. Butacher, Mol. Phys., 35 (1978) 771. W. von Niessen and L.S. Cederbaum, Mol. Phys., 43 (1981) 897; W. von Niessen, G. Bieri, J. Schirmer and L.S. Cederbaum, Chem. Phys., 65 (1982) 157; P.W. Langhoff, S.R. Langhoff, T.N. Rescigno, J. Schirmer, L.S. Cederbaum and W. Domcke, Chem. Phys., 58 (1981) 71. J.D. Mitroy, Ph.D. Thesis, University of Melbourne, 1983; J.D. Mitroy, K.A. Amos and I. Morrison, J. Phys. B, 17 (1984) 1659. S.T. Hood, Ph. D. Thesis, The Flinders Univeristy of South Australia, 1974. I.E. McCarthy and E. Weigold, Phys. Rev. A, 31(1985) 160. D.P. Spears, H.J. Fischbeck and T.A. CarIson, Phys. Rev. A, 9 (1974) 1603. I.E. McCarthy, P.E. Uylings and R. Poppe, J. Phys. B, 11 (1978) 3299. I.E. McCarthy, J.D. Mitroy and E. Weigold, unpublished results. T.A. CarIson, M.O. Krause and W.E. Moddeman, J. Phys. (Paris), 10 (1971) C4,76.
INTERPRETATION OF INTENSITIES IN ELECTRON-MOMENTUM AND PHOTOELECTRON SPECTROSCOPIES
I.E. MCCARTHY Institute for Atomic Studies, S.A. 5042 (Australia)
The Flindels
(First received 3 July 1984;in
University
of Sou th Australia,
Bedford
Park,
final form 15 October 1984)
ABSTRACT Relative intensities for the photoelectron reaction on atoms and molecules are not related to structure calculations in the same way as those for the noncoplanar symmetric (e, 2e) reaction. The photoelectron dipole matrix element is dependent on recoil momentum only through the unique relationship of recoil momentum to the photon energy and is much harder to calculate for chemically-interesting momenta. Relative intensities for binary (e, 2e) reactions are independent of total energy at high enough energies and strongly dependent on symmetry and recoil momentum, for which an intensity profile can be measured for values starting at zero. In comparing with structure calculations, binary (e, Se) intensities for low recoil momentum may be compared directly with pole strengths in calculations of the one-electron Green’s function. In the case of states within a single symmetry manifold the relative intensities will be independent of recoil momentum up to some maximum, usually at least a few atomic units.
1. INTRODUCTION
After some years there is still confusion between two types of ionization spectroscopy, electron-momentum and photoelectron spectroscopies, hereafter abbreviated as EMS and PES. EMS, interpreted by calculations of the one-electron Green’s function or by configuration-interaction calculations, has begun to provide a complete quantitative understanding (within experimental error and resolution) of valence-shell electronic structure for singly-ionized small molecules. EMS is the understanding of target and ion structure resulting from the binary (e, 2e) reaction in which two electrons of equal energy are detected at 45” on opposite sides of the incident electron direction. A profile of the recoil momentum of the ion is measured for each energy-resolved ion state by varying the azimuthal angle of one of the detectors. 936%2048/86/$03.30
0 1985 Elsevier Science Publishers B.V.
38
PES is the understanding resulting from the (y, e) (photoelectron) or dipole (e,2e) reaction. In the latter, two electrons are detected with extremely unequal energies. The fast outgoing electron has a momentum very close to the incident electron. The small momentum transfer simulates the negligible photon momentum in (y, e) and the information obtainable from the energy distribution of resolved ion states is analogous to that obtained from (7, e). In both cases the angular distribution of slow electrons contains limited information owing to the dominance of the dipole selection rule. We will not consider the dipole (e, 2e) reaction further, since its structure application is identical to that of (y, e). Intensities in both spectroscopies are characterized more precisely by the differential cross sections for the reactions. In both spectroscopies the energies of the states of the final ion are measured in principle. Experimental limitations will be discussed. In both spectroscopies the reaction dynamics, which depends on the wave functions of the target and the ion, determines the cross section for a reaction that leaves the ion in a particular state. In PES the kinematic limitations of a two-body reaction leave the incident photon energy as the only non-trivial variable for which a profile of such cross sections can be determined. In particular there is a one-to-one correspondence between recoil momentum and photon energy. The extra kinematic freedom of the three-body final state in EMS allows a profile of recoil momentum to be determined at arbitrary incident energy, and in particular at very high energies. The structure information is deduced from profiles of differential cross sections by comparing them with calculations of the reaction dynamics. At low energies (and hence low momentum for PES) the dynamics depends on many degrees of freedom of the many-body problem. At high enough energies (but still at low momentum for EMS) fast electrons can be represented by plane waves and the reaction dynamics is determined by a simple impulsive interpretation in which the structure wave functions directly determine the momentum profile. There are simple experimental criteria for determining the validity of the impulsive interpretation. For high energies both (r, e) and (e, 2e) reactions have an impulsive interpretation. Corrections to the impulsive interpretation introduce high momentum components, and it is an interesting feature of the study of the reaction mechanism to determine the range of validity of the approximation that the momentum profile is determined only by structure. Structure-related high momentum components are very sensitive to the accuracy of the structure calculation and would be an excellent probe for details of the structure calculation. In a detailed review article [l] Cederbaum and Domcke explain the relationship of PES to the one-electron Green’s function. There is no correspondingly-detailed derivation for EMS. In Section 2 we discuss the type of structure phenomena in which we are interested and their representation by the energy-momentum spectral
39
function. In Section 3 we review the physical limitations of the (7, e) and binary (e, 2e) reactions and the computation of approximate cross sections closely related to the spectral function. In Section 4 we discuss the approximate cross sections in the light of more-detailed reaction theories. In Section 5 we supply the argument relating EMS to the Green’s function formalism in the form of a paraphrase of the derivation of Cederbaum and Domcke for PES and review the effects of electron correlations on cross sections. In Section 6 we give two detailed examples of the main points. 2.THEENERGY-MOMENTUMSPECTRALFUNCTION
In ionization spectroscopy our understanding of electronic structure is equivalent to an understanding of the energy-momentum spectral function &.(E, q) for each symmetry manifold r of the molecule. The spectral function is a property only of the ground state $“, of an N-electron molecule and the state vector @-’ of the (N- l)-electron ion produced by separating an electron from the target at an energy cost E. It is defined as
The spherical momentum average is due to the unknown orientation of the molecule in a gas target. Equation (1) is explained further by expanding in the coordinate representation. s,(e,q)
= 1 dtiIAld”r,.
. . d3i-N(27r)-3’2
ewiqsrl$f-‘* (r2, . . , , r, ) J/t@,,
. . . ,rN )I*
c-4)
where A is the antisymmetrization operator. If the ion cannot decay by electron emission the spectral function is S&9 a) = Fe(r) (a) 3 (c - Q,))
(3)
where e, is the energy eigenvalue of the electronic state s of the ion and F,(q) is the momentum profile for the state s. Normally we will consider a particular symmetry manifold and omit the dependence of s on r from the notation. In eqn. (3) we have made the approximation that only electronic degrees of freedom are relevant. We have averaged over the photon-emission, rotation and vibration or dissociation widths of the electronic state s. It is possible to observe single-ionization energies e above the threshold for a second ionization. In this case E is an electronic continuum, characterized at low energies by resonances centered at es which correspond closely to positive-energy states of the ion. The spectral function is now strictly defined over the two-dimensional e, q continuum, although for lower ionization energies it consists mainly of slices close to the energies E,. For
40
larger ionization energies there are many channels for decay of the hole state and the width of the resonance may be quite large [2,3]. The quantity q is the coordinate in the momentum representation of the target-ion overlap in eqn. (1). In the independent orbital picture it is the momentum of the electron in the target. We usually use the approximation of labelling a symmetry manifold r in addition by the principal quantum number n of an orbital, i.e., nr, since the energies E,(,) are usally clustered within a few eV of the orbital (Hartreeinteraction the state s(r) contains the Fock) energy enr. With confiiration one-hole configuration (nr)-’ . In nearly every case the coefficients are negligible for all but one value of n if the target Hartree-Fock representation is used. Some exceptions are known, for example in CO+ [4], but cross sections are usually small for such states. If this is true and the approximation is valid that $“, consists mainly of the Hartree-Fock determinant, then enr is the centroid of the cluster of states [5]. In PES this is often called “satellite” structure, the state with the highest PES intensity being considered as a main state in the quasiparticle sense, with the others as satellites of this state. We instead use the term “components of the orbital nr”, which includes the situation where the correlations are so strong that the quasiparticle picture breaks down. Each component is identified in an EMS experiment by its momentum profile F,(q) [5] .
3. THE PHOTOELECTRON AND BINARY (e, 2e) REACTIONS
In the photoelectron reaction a photon of energy E, ionizes a molecule at the cost of energy E, producing an electron of energy E,. The photon has very small momentum, so the ion recoils with momentum p, where, in atomic units p2 = 2E e
(4)
The differential cross section for the reaction depends on the polar angle of p through the asymmetry parameter /I. For polarized incident radiation it also depends on the azimuthal angle of p. We are interested only in the magnitude of p from the spectroscopic point of view, not in its angular dependence which merely reflects the dipole selection rule. For a given separation energy e (essentially this means for a given ion state s, apart from degeneracies) the recoil momentum p depends only on the photon energy E,, since E, = E,--E
(5)
The e, p continuum can be scanned experimentally only by varying E, and hence E,. For example, for E, = 200 and 2000 eV, p is respectively about 4 and 12. Forp = 1 we have E, = 13.6 eV.
41
In the (e, 2e) reaction an electron of momentum k, ionizes a molecule at the cost of energy Q, producing electrons of momenta k, and k, which are detected in coincidence. Since the mass of the ion is large compared to the electron mass the energy of the ion recoil is essentially zero. The momentum of recoil, p, is given by k,, = k, + k, + p
(6)
and the energies (labelled to correspond with the momenta) are related by E,--E
= EA +E,
= E
(7)
The total energy is E. In the absence of electron spin polarization the cross section is rotationally invariant about kO. This fact, together with the energy constraint of eqn. (7), means that there are four degrees of freedom in the final state. We consider these to be p and the magnitude of the momentum transfer K, defined by K = k,-kk,
(8)
where A labels the faster outgoing electron. From the spectroscopic point of view we are not interested in the angles 0 or in K. The experimental scanning of the e,p continuum affords a major contrast with photoelectron spectroscopy. The extra kinematic degrees of freedom enable us to scan through p for a given E without varying any of the electron energies. In scanning E it is convenient to fix the total energy E and vary E,, . In scanning p for spectroscopic purposes we choose a geometry in which K is fixed at a value that enables p s 0 to be observed. K is maximized by requiring E, = EB, since this simplifies the calculation of the cross secticga. This is noncoplanar symmetric geometry. The polar angles of k, and k, are equal and fixed at a value 8 near 45’ that gives p S 0 when k,, k, and k, are coplanar (the exact value of 8 for p = 0 depends on E and is less than 45O). The azimuthal angle $Iof kB is varied to scanp. In this geometry p = [(2k,
cos 9 - k,)’
+ 4122, sin’ 8 sin2 f $1 1’2
(9)
To estimate the relevant kinematic ranges we choose E,, - 1 keV, for which p -,//zk,
sin $r#~
(10)
Due to experimental limitations the maximum value of sin & is about l/3, so p may be varied from 0 to - kB/2. For example, the maximum observable value of p for EB = 200 eV is about 2. At 2000 eV it is about 6. We have expressed p in terms of k, for direct comparison with photoionixation. By convention, B labels the ionized electron (antisymmetry will be discussed in Section 5), so EB corresponds to E,. It is an extremely interesting and valuable area of atomic and molecular physics to obtain a complete understanding of the photoelectron and (e, 2e)
42
reactions. By this it is meant to be able to calculate the differential cross section in all experimentally-accessible kinematic ranges as a function of all the degrees of freedom. To do this we must know the target and ion state vectors $$’ and $/F-’ . However, in order to understand the spectral function it is sufficient to find a limited kinematic range which is sensitive to the spectral function and for which we can calculate the differential cross section. We shall show that in this respect the binary (e, 2e) reaction has a tremendous advantage over the photoelectron reaction for low Q, since the spectral function can be measured directly from q = 0 to some experimentally-limited maximum value qmax and does not require theoretical interpretation of the experiment. In general, qmax is less than the largest experimentally-attainable momentum. For q “>qmax the interpretation of the photoelectron reaction is equally simple, so that the two reactions have complementary roles to play in spectroscopy. For q ? qmax the spectroscopy is understood in the photoelectron case only through a detailed calculation of the cross section. The spectral function can be calculated directly from the Green’s function formalism, which gives this formalism a special role in ionization spectroscopy. We first give the simple approximations to the cross sections for the reactions in order to see how they are used in the spectroscopic application. To illustrate the procedure we take the binary (e, 2e) reaction. In subsequent sections we discuss the validity of the simple expressions in the context of more-complete theories. The simplest approximation to the photoelectron cross section is do/@
= 4x20(K&)S,(e,
a)
(11)
where (Yis the fine-structure constant and q=-p=k
(12)
e
This expression comes from approximating the N-electron final state by the
product of a plane wave exp(iq r) and \clc-’ in the following matrix element, whose calculation is not so easy, but which is valid at much lower energies l
(13) T(,, e) =
= (2x) 4 $$
0
fe,S,(GP)
(14)
43
where f,, is the half-off-shell Mott-scattering cross section given by
fee =
1 271v exp (2~) - 1 I Ik, - kA I4
1 (~~2
)2
1 -Ik,-kA,2 v = l/lk,,
1 Ik,-kk,li
cos
1 +.
tk, -kd4
vlnIk,_kk,,2 lko - kB’2])
-k,)
(16)
This is the plane-wave impulse approximation. In noncoplanar symmetric geometry Ik, - kA I = IkO - kB I = K is fixed and, for small angles 3 #, v is essentially constant, so that f,, is constant within about 1% over the range of E,P relevant to valence shells. The differential cross section is thus proportional to S, (E,P) over this range. Furthermore, it is unnecessary to measure or calculate absolute cross sections in order to obtain the constant of proportionality. If e is scanned far enough to observe all the structure for the manifold nr, the spectral function may be normalized by the sum rule [ 51 z GE”?P,,(S) = 1
(17)
Here P,,(s) is the pole strength in the language of Cederbaum and Domcke [l] , in terms of which S,(e,p) is usually approximated by
WGP)
= P,,@9jul(Pl~“,,l’
WE-G)
(18)
where (p )qb,,) is the momentum representation of the orbital nr. Note that the sum rule [ eqn. (17)] is strictly valid only when ground-state correlation effects are neglected. The influence of such effects is discussed in relation to eqns. (49) and (50). The approximation (18) is valid for electron energies EB comparable to those for which eqn. (11) is valid for the photoelectron reaction, namely, a few hundred eV or higher. The difference is that at such energies the relevant range e,p can be scanned completely up to a value pmax which is of the order of 5 for E, values of a few keV, whereas for eqn. (11) the range is restricted to p 5 4. Since eqn. (14) is a high-energy approximation we must have a criterion for its validity. The criterion is purely experimental. The incident energy E,, must be so high that the experimentally-determined S,(c,p) is independent of E. over a substantial range of e,p. Note that in approximation (14) the momentum coordinate q is identical to the experimentally-measured momentum p. There is now a large body of experimental evidence [6] that the necessary energy-independence is achieved for E,, 5 400 eV. In order to apply the spectroscopic sum rule of eqn. (17) we must know
44
which ion states s are components of the orbital nr. This again is decided experimentally. The momentum profiles for different nr are easily distinguished in most cases, so that their shape is used to identify nr. An absolute assignment of the representation is obtained by calculating the orbital momentum profile P,,(P)
= z,,il, F 8(P) = J. W(Pb#Jnr)12
(19)
starting with the same orbitals that are used in the Green’s function calculation. The calculation of eqn. (19) is described, for example, by McCarthy and Weigold [ 51. In practice some convolution with the experimental energy-resolution function [ 71 modifies the procedure. The assignment of states s to representations nr is confirmed by comparing the summed momentum profiles [eqn. (19)] with momentum-profile calculations for different nr. If the whole procedure is correct the summed momentum profiles must have the same relative relationship as the calculated orbital momentum profiles for a substantial range of p starting with p = 0. The procedure has been verified over the past few years in a large number of cases [6,7]. Note that both approximations (11) and (14) involve plane waves in a state where the residual system is charged. At the high energies where the approximations are valid there is little difference between plane waves and Coulomb waves over the part of the integration space where the valence wave function gives a significant contribution to the integrand. The asymptotic forms of the continuum functions are irrelevant.
4. REACTION CALCULATIONS
Since we are interested in the use of the photoelectron and (e, 2e) reactions in spectroscopy it is not our purpose to give a detailed account of the computation of their amplitudes. Nevertheless, it is necessary to see what is involved in such calculations and how well they work in different kinematic conditions. We can thus form a judgement of the best conditions for our purpose. It is best to consider first the calculation of the reactions for simple targets whose structure is well known. If the calculation can be performed correctly in these cases, i.e., if we understand the reaction mechanism, we can expect the extension of the same mechanism to spectroscopically more-complicated systems to throw light on the spectral function. The photoelectron amplitude is, in the one-photon approximation T(,,,,(k,)
= @(LA,
lPnl St’)
cw
where IF) is a many-body state vector with one electron obeying outgoing scattering boundary conditions (plane wave and ingoing spherical waves). A, -P, is represented by either the dipole length or velocity operator. These
45
operators are strictly equivalent when acting on the exact state IF), but may give different results for a particular approximation to IF). Methods of calculation all involve an expansion of IF) in channel functions. These are configuration-interaction functions for the ion coupled with the appropriate symmetry to one-electron functions. IF) is essentially the time-reversal of the elastic scattering function for an electron on the observed state $,“-I, coupled to J/,“- ‘. It contains smaller terms representing time-reversed inelastic scattering if coupling of the channel functions cannot be ignored. Channel coupling is less important at higher energies. We will not give details of methods for calculating elastic scattering wave functions (distorted waves), which are relevant to both the photoelectron and (e, 2e) reactions. It is however useful to note that all methods can be related to the solution of a one-body scattering equation in the optical potential [ 81, which includes the potential for the target ground state (static exchange approximation) and a polarization potential representing the effect of all excited states. An example of the present detailed understanding available to the photoelectron reaction on atoms is the R-matrix calculation [9] for low-energy reactions on inert gases. Parameters of resonances at very low E,, which are very sensitive to details of the calculation, are reproduced very well by both length and velocity formalisms with the results apparently converging to each other and to experiment as more configuration interaction is taken into account. For a particular E, the differential cross section is quite well produced as a function of E, up to about 70eV using an average over the discrete pseudostructure introduced at high energies by the use of discrete pseudostates to represent the ion continuum. The importance of configuration interaction is shown for the highly-split 3s-’ orbital of the argon ion, where it is necessary even to obtain a qualitatively-correct cross section at energies E, near 13.6 eV 0, = 1). Configuration interaction in the Rmatrix method is responsible for channel coupling as well as for splitting of the orbital. As an example of a detailed photoelectron calculation for a molecule we consider a calculation of photoexcitation and photoionization of the water molecule by Diercksen et al. [lo]. This involves only the electronic degrees of freedom. Closure is used for rotational and vibrational states. The staticexchange (uncoupled channels) approximation was used and the continuum was approximated from the discrete excitations of pseudostates by the StieltjesTchebycheff moment technique [ll]. For the water molecule the three outer-valence orbitals are essentially unsplit. The curves of differential cross section vs. E, are only qualitatively correct below about 20eV, but have errors less than 20% in comparison with experiment [ 121 for 20 eV ? E e ? 50 eV. The distribution of differential cross section as a function of e, over states which are components of the inner-valence 2ai orbital is quite well reproduced.
46
We may summarize the detailed calculation of the photoelectron amplitude by saying that it is very good below about 50 eV if all effects are taken into account, notably channel coupling and configuration interaction. In the next section we will consider the method suggested by Cederbaum and Domcke [l] for comparing pole strengths for photoelectron reactions at higher energy, and thus using the one-electron Green’s function technique as a method of understanding the splitting of an orbital by configuration interaction. This method rests on the factorization of the many-body wave function ]F) into a time-reversed el~tic-scatte~g distorted wave and the ion wave function in its observed state IF) = lx:-‘(k,))l3/,“-‘> The factorization is correct if inelastic scattering (core excitation) can be ignored in the electron--ion system. The same final state can be arrived at with core excitation in a two-step process, in which the amplitude consists of a factor representing photoionization to an ion state g-l and a factor representing the change of @‘-I to J/,”-i in the electron-scattering process. Such amplitudes are not formally included in eqn. (20) with the factorization (21), but they are included in the full coupled-channels or R-matrix treatment of IF). The cumulation of xg) involves many ion states explicitly at low energies. At higher energies the static-exchange approximation becomes more valid. Here x2-) is obtained from a scattering calculation including exchange on the state J/f-‘. At very high energies a plane wave becomes a reasonable approximation. From the calculation of Diercksen et al. [,lO] for water it appears that the static exchange approximation is good for E, 5 20 eV, i.e., for p 5 1.2. This is also true for a simplified version of the static-exchange approximation used by Hilton et al. [ 131. The range of validity of the planewave approximation is roughly that for the Born approximation to elastic scattering, perhaps E, > 200 eV, i.e., p 5 4. In this range the dipole velocity expression reduces the cross section to the form of eqn. (11) using the definition (1) of S,(e,p) and making the appropria~ sums and averages over degeneracies. The (e, 2e) amplitude may be written formally as
tw Instead of including the many-body complexities of the reaction in the final state IF) we have included them in an operator T which acts on the planewave states and the target and ion eigenstates. The first approximation we must make is the binary-encounter approximation in which T is a two-body operator depending on the coordinates of the incident and struck electrons, but not on those of the ion. This eliminates many exchange terms involving the overlap of a one-electron cont~uum function with a bound orbital, as well as channel coupling in the electron injections with a bound system.
47
The overlap of a plane wave with a valence orbital is negligible for planewave momentum k greater than about 3, i.e., for continuum-electron energies greater than about 100 eV. This condition is necessary for the validity of the binary-encounter approximation. We now have T(e,ze)(kA,k,A
= (k*kz ITl(9,N-‘l$$‘)k,)
(23)
Equation (23) already contains the main spectroscopic point. The (e, 2e) amplitude depends on the spectroscopy only through the overlap amplitude <$r-’ 11): ). It is a three-body transform of the overlap. To conform with the notation of ref. 1 we introduce a basis of oneparticle states (or orbit&) 1$J* ), the best a priori definition for which is the target Hartree-Fock procedure, which eliminates one-particle one-hole configurations from the confiiuration-interaction expansion of $J,”. We define creation and annihilation operators aT,a, for an electron in 1#1) and express eqn. (23) in its second quantized form. 7&&kA,kz) rkl
=
(h@k
= &A&~-’
Ill&,)
hItif)
(24)
(25)
where 7 is a new interaction operator defined so that the plane-wave state (kz > for the “ionized” electron [antisymmetry is of course taken into account in eqn. (22)] is replaced by the target Hartree-Fock continuum orbital.
1x$-)(kB1) = Ih)
(26)
This is the static exchange approximation to I x&j ). In eqn. (24) we usually make the approximation that I$, N-1 > contains the one-hole configuration al I $$ > for only one characteristic orbital I&). The quantity ($r-’ IaII I,!J~) is the spectroscopic amplitude for the orbital I in the ion state s. The quantity 7kl of eqn. (25) is the one-orbital (e, 2e) transform. The investigation of its form under a wide range of different kinematic conditions has been the subject of intensive effort in atomic physics since the pioneering (e, 2e) experiments of Ehrhardt et al. [14]. These experiments all involve small momentum transfer K and consequently small magnitudes of kB. The first-order theory for this situation is the Born approximation in which x$‘N(kB) is given by eqn. (26) and 7 is the electron-electron Coulomb potential u. Much unnecessary worry about the difficulty of understanding the (e, 2e) reaction has been caused by the qualitative inability of first-order theories to reproduce these data. Considerable improvement for helium at E,, = 500 eV has recently been obtained by Byron et al. [15] who use the second Born approximation to 7kl. However, these experiments are done in a kinematic range that offers a challenge to the full power of atomic reaction theory. Experiments that offer less challenge to theory have been carried out by Giardini-Guidoni et al. [ 161 and the Flinders group [17-191 for atoms
48
whose structure is well understood. The main point of these experiments is that the energy condition for the binary-encounter approximation is met for all continuum electrons. The distorted-wave impulse approximation, in which T includes operators that distort all the continuum-electron waves in various approximations to the optical potential for elastic scattering in the appropriate two-body subsystem, and the electron--electron interaction is treated by using the full scattering operator t instead of the potential u, reproduces the data excellently for total energies E above about 400 eV. The matrix element (k’( t(E) Ik) is the exact antisymmetrized amplitude for the scattering of two electrons with initial and final relative momenta k, k’ by the Coulomb potential. In the (e, 2e) application we require halfoff-shell matrix elements [20], in which k’ corresponds to the energy E for which the electron-scattering Schrijdinger equation is solved, but k in general corresponds to a different energy. The spin average of the absolute square of this amplitude is the half-off-shell Mott scattering cross section (15). For reasons stated in the previous section, noncoplanar symmetric geometry is most appropriate for the spectroscopic application. This is wellunderstood theoretically as shown for example for the 2s and 2p orbitals of neon [ 181, where differential cross sections relative to one unknown experimental intensity are correctly described at 600 eV for 0
=
(hb
ItlWo)
(27)
49
This factorizes as follows TkZ
=
(k’(tlk)(pl$+)
(23)
where k’, k and p are given respectively by eqns. (15) and (6). The electronelectron amplitude is spin-dependent. On squaring eqn. (24), using eqn. (28), spherically averaging, averaging over initial-state and summing over finalstate spin degeneracies, we obtain the form (14) for atomic targets, where
&(E,P) =
l~119F)12 SawP14d12w-~,)
w-’
(29)
The definition (29) of &(e,p) is the same as eqn. (18) with the orbital nr now denoted by 1. The pole strength &(s) is given by
4(s)
= Kti-’
laAJ/~)12
(39)
The sum rule (17) for Pi(s) is obtained from the application of the orthonormality and closure relations to eqn. (30). For molecules we have additional rotational and vibrational degeneracy. It has been shown for the cases of H2 and D2 [17] that an explicit vibrational sum is equivalent to calculating the (e, 2e) amplitude at the nuclear equilibrium positions. This approximation is always used. Rotational degeneracies are accounted for by the spherical average.
5. RELATIONSHIP
TO THE ONE-ELECTRON
GREEN’S
FUNCTION
The formalism of the one-electron Green’s function has been related to photoionization by Cederbaum and Domcke [ 11. The total cross section P(w)for photoionization with the energy conditions o
=
E,, a0 = E,
is expressed in terms of target Hartree-Fock photoionization amplitudes rkl
=
(#k
IA-PI@,)
(31) orbitals, &, and one-orbital (32)
where $!$ is some optimum one-particle scattering function, for example the static-exchange continuum orbital of eqn. (21) or the scattering function for the appropriate optical potential
Ix,‘-‘(W) =
I$k)’
(33)
We paraphrase the argument to discuss the total cross section P(o)for the set of (e, 2e) reactions initiated by incident electrons of momentum k0 , with the “scattered” electron having momentum kA . Antisymmetrization of the final state with respect to the two continuum electrons will be performed
50
explicitly. Apart from initial- and final-state degeneracies we have P(0)
= (2a)4 ‘9
w=EA-i-EB,
&(Fl(k,
0 w,,
]r1ko)l\1/;)12
(34)
=Eo
(35)
The operator connecting the N-electron final state IF) and initial state $2 is expressed in the second-quantized form (24), giving
kAkB
= (27r)4 k o
P(0)
(- ll~)%n,
6,~~
Im CiGtimn (- 00
---iv)1
(36)
where rkr is given by eqn. (25) and Gulmnis particle-hole component of the two-body Green’s function, whose calculation is difficult. In order to express P(o) in terms of the one-electron Green’s function we formally factorize IF) into a one-electron distorted wave IXk) (kB)) for the “ionized” electron B and the observed ion state I I#-’ >. IF)
=
Ixg’(kB))I$:-l)
(37)
defining the factorized amplitude rsl (corresponding to rel in the photoelectron case [ 11) by rs1 = (kAXB-)(kB)]r]Q+ko)
(38)
We do not have to make a factorizability approximation if we define the operator r to include the required operator describing many-body effects including core excitation (two-step processes). The conditions under which r can be represented by various approximations have been thoroughly discussed in Sections 3 and 4. The neglect of core excitation corresponds to the binary-encounter approximation. For our spectroscopic application we confine P(w) to noncoplanar symmetric reactions. We choose E. 5 400 eV and use the plane-wave impulse approximation TBl
=
(3%
%&BitI&kO)
We now have P(O)
= (2n)4 ‘+
&,,?$,,~B,
hl
[Gh(a
- Cd0 - i?‘j)]
0
where 2, signifies the appropriate sum and average over final- and initialstate vibrational, rotational and electron spin degeneracies and GI,(o---iq)
=
&
(I# Id I J/~-‘w,“-’ w + Efl -Et
bll4w -iq
(41)
The forms of eqns. (40) and (41) are now identical (apart from the kinematic constant and the omission of the energy constraint eqn. (7) from the
51
formalism) to eqns (2.4) of [ 11. The spectral function &(e,p) is given by consideration of the poles of GI, (w - ir,~).The numerator of the s term of GnI is the product of two spectroscopic amplitudes MS) = QY-’
Ial
(42)
I tit>
We first consider final-state correlations in terms of the target HartreeFock approximation in which we neglect initial-state correlations, i.e., 3/t is a pure Hartree-Fock configuration Qt. J/f’-’ contains the configuration aI 1@r ) (the Z-hole configuration) with an amplitude &(s). Other configurations of J/,“-’ have zero overlap with a2I G$ ) if they involve electrons in orbitals unoccupied in at. G1, is then diagonal and its pole strength 9(s) is given by eqn. (30) and obeys the sum rule (17) for states s in the symmetry manifold characterized by 1. There are two reasons why G,, is not in general diagonal. The first occurs occasionally in molecules but not in atoms where states with the same principal quantum number are well-separated in energy. Occasionally $r-’ has significant contributions from the I- and n-hole states, where orbitals I and n have the same symmetry. An example is the CO+ ion, where in a particular calculation [4] there was a state at 24.7 eV with t5,, = - 0.169, tdo = 0.239 and t so = 0.080.
The second and most important reason for nondiagonal G1, is initial-state N-l belongs to the symmetry manifold 1, i.e., it concorrelations. Suppose tiB tains the Z-hole configuration al (!E$ ). I)$ must contain some configurations with electrons in orbitals n which are unoccupied in @‘. The corresponding configurations in J/r-’ now have finite overlap with a, I $I: ). A specially simple example is helium, where the final states are uncorrelated. It is discussed in Section 6. In general the cross section is related to the eigenvalues Dk (0) of the matrix G(w), whose elements are given by eqn. (41). P(w) bu(w)
4 kdB
= (277) - xk =
T,,(O)
ITB,(o)
=
In2
(2,
tr
[b(w
-
w.
-
iv)]}
0 1’oi(o)
(43) (44)
6,
(45)
IZji7gjMij(O)
where M is the eigenvector matrix of G. The eigenvalues Dk are related to the pole strengths Pk (s) by Im
@k
(0
-
iv)}
=
nz,Pk
(S)
6 (W
-t #-’
-
EON)
(46)
We may now write P(0)
= (27r)4
kAkB --crskITBk(d12Pk(S)
ko
(47)
52
Since the plane-wave impulse approximation to rzk factorizes in the form of eqn. (28), we antisymmetrize the matrix element with respect to the final state continuum electrons by noting that (p I r#~~ ) is symmetric under exchange of A and B and that the two-electron t-matrix element (k’ (t Ik) must be antisymmetrized to describe the two degenerate spin states singlet and triplet. The sum and average Za now givesP(o) in the form P(0)
= (2x) 4 y
where
f,, is given by eqn. (15) and
Sk(%P)
=
fee&kSk
te,p)
Pk(S)IdljI~:I(PI~I)Mk,(e)12
s(E-%,)
(49)
The review article of Cederbaum and Domcke [l] describes the calculation of Pk (s), Mkl(e) and E, in great detail. Note that the energy momentum spectral function eqn. (49) is a direct result of the calculation, involving only the structure of the target molecule and the ion, and not any explicit properties of the reaction mechanism except for the relation q = p connecting the observed recoil momentum and the coordinate in the momentum representation. We now consider the spectral function S, , eqn. (49), under different conditions. If Gkl (and therefore &l) is diagonal then the p-dependence of Sk is the momentum profile for the orbital k. The pole strengths Pk (5) sum to unity according to eqn. (17). If there is a contribution to J/,N-1 from more than one Z-hole configuration then the pdependence of Sk arises from a superposition of p-profiles for each relevant orbital. This is an unusual situation normally confined to states with small pole strengths. If there is appreciable initial-state correlation then Mkl is not diagonal but off-diagonal elements are usually small. It is reasonable to make the practical approximation Sk(%p)
=
&(s)~@h’b#‘k)i2
h(e-%)
(59)
In this case, however, the sum rule (17) for the Pk (s) must be modified to LZP,
(s) =
IW
I !@)I”
(51)
In experiments which measure cross sections relative to one unknown quantity we may normalize the summed pole strengths for all manifolds k to unity, since the sum is independent of the manifold. This normalization is equivalent to considering the pole strength as the square of the coefficient of the k-hole configuration in J/y-’ . The practical approximation eqn. (50) is expected to become invalid at high momentum p. For example the valences orbitals of neon and xenon have nodes atp Z 3 and 1.5, respectively. Here small Mkl may give dominant
53
contributions to eqn. (49) and the corresponding photoionization amplitudes, eqn. (13). Thus high-momentum cross sections are sensitive to confiiuration interaction. This is both a problem and an advantage for highmomentum ionization spectroscopy, where photoionization is easy to compute and has a considerable count-rate advantage over binary (e, Ze). The problem is that the analysis eqn. (50) cannot be used and that a very accurate calculation of configuration interaction is needed for the initial state. The advantage is that such a calculation can be tested very accurately. In Section 6 we give the example of helium, where high-momentum photoionization cross sections have been measured. We now note a major difference between EMS and PES concerning the application of the Green’s function formalism to experiments. For EMS the plane-wave impulse approximation, assisted by the spectroscopic sum rule, eqn. (17), directly relates the experiment to the pole strengths Pk (s) for states s resulting from the splitting of the orbital k. The experimentalist needs only to calculate the spherically-averaged square of the momentumspace orbital in order to identify the orbital k for each ion state s. For the application of PES it is necessary to calculate the corresponding one-orbital amplitude from eqns. (32) and (33), which is a much more difficult task (described in Section 4) requiring considerable theoretical interpretation. The much simpler plane-wave calculation is valid only for very high energies E, and recoil momentap. Cederbaum and Domcke note that the ratio of the pole strength Pk (s) to the pole strength for the orbital k (which is usually normalized to 1) may be more easily obtained from experiment. In terms of two ion states r, s resulting from the splitting of k, the ratio of photoionization cross sections is equal to the ratio of the pole strengths provided the corresponding oneorbital cross sections are equal, i.e., provided (using the notation of [l] ) lQ&)12
= lr,,&,)l”
(52)
In this case, if one can identify the orbital k to which s belongs (an easy task in EMS but not in PES), one can use the same analysis as described for EMS at the end of Section 3. For a given incident photon energy o. the cross section depends rather strongly on E, (= w. -e, ), since this determines p. For small molecules and E, >20 eV, where static-exchange calculations of x$-)(p) are reasonably valid, the one-orbital cross section may change by a factor of roughly 2 over a 10 eV interval (typical of the splitting of inner-valence orbitals) [lo, 121 and therefore the intensity is not proportional to the pole strength. The change is more rapid for molecules such as HBr, which contain large atoms. A calculation at least as detailed as the staticexchange approximation is therefore needed for interpretation. If the experiment is arranged to observe the cross sections for both r and s at the same value of E,, rather than wo, the condition (52) is more-nearly satisfied. At very high energies the validity
54
of eqn. (52) is subject to considerations of initial-state configuration interaction. Note that similar considerations apply to eqn. (50) at high momenta, but not necessarily at high energies since the three-body kinematics enables observation of low momenta at high energies. The Green’s function formalism of this section has been applied to EMS of several molecules [6,7,21] in which pole strengths for split inner-valence states are qualitatively reproduced by the structure theory. The mostdetailed calculation is for water [22] where the convergence of the structure calculations with respect to basis size has been investigated. Here the calculation of the Green’s function poles in the extended Tamm-Dancoff approximation [23] reproduces the inner-valence pole structure quite well. Somewhat better agreement is obtained by direct calculation of I&,“-’ and tig using the multireference double-excitation configuration interaction method [24]. Examples of the application of the formalism to PES are given in ref. 25.
6. CONCLUSIONS WITH EXAMPLES
In this section we give detailed illustrations of the two main points of the discussion. The first is that EMS can be used to identify symmetry manifolds nr in the spectrum of the ion and, if configuration interaction is not very important in the target, it can give pole strengths for states within each manifold that obey the requirements of energy independence and other consistency rules. The 3p and 3s manifolds of the argon ion are easily identified by the shapes of their momentum profiles. The former has a minimum at p = 0, while the latter has a maximum. For the 3p manifold only one state, at 15.76 eV, is significant. The sum rule (17) gives its pole strength Psp(15.76) as unity. CI calculations [26] give about 0.97 for this pole strength, which is within experimental error of the value 1. The 3s manifold has several states, which can all be cleanly resolved experimentally so that reliable intensity measurements are available. Intensities of all states in the 3p and 3s manifolds are measured relative to each other. The strongest state in the 3s manifold is at 29.3 eV. Its pole strength is given by the sum rule (17) to be 0.53 + 0.05 for a 400 eV (e, 2e) experiment [ 271. The condition for applicability of the sum rule is not that we can understand details of the reaction, for example sufficiently to calculate correct momentum profiles, although these are given within experimental error by the distorted-wave impulse approximation [28] as they are for neon and xenon [18]. The sum rule is applicable under the purely-experimentallyverifiable condition that momentum profile shapes for all states within the manifold are the same over a significant range of p from zero up to some
55 TABLE 1 ENERGIES IONa
AND POLE STRENGTHS
EMS e(eV)
29.3 36.7 38.6 41.2 42-43.4 >43.4
FOR THE 3s MANIFOLD
Mitroy P$:) 0.647 0.032 0.176 0.074 0.041 0.122
* 0.019 f 0.008 + 0.011 f 0.007 + 0.006 f 0.008
OF THE ARGON
PES
MUP
E (eV)
PJZ)
E (eV)
Pi:)
e (eV)
Pjf)
28.7 36.8 39.1 41.8 >42
0.649 0.013 0.161 0.083 0.081
29.4 36.5 38.6 41.2
0.61 0.01 0.17 0.20
29.3
0.81
38.6 41.2
0.13 f 0.02 0.06 f 0.02
*Column headings are described in the text. EMS data are from ref. 28. PES data are from ref. 29.
maximum value. An essential condition is that the pole strength ratios must be independent of the total energy E of the (e, 2e) experiment. Both these conditions are satisfied for the argon 3s manifold [5]. The momentum profile shape is given by the plane-wave impulse approximation with the argon Hartree-Fock 3s orbital up to p = 1 at 1200 eV for states at 29.3,38.6, 41 and 44 eV. This is approximately true also for weak continuum states at 48 and 51 eV. Pole strengths are independent of energy from 200 to 1200 eV. Their values are summarized in Table 1. A stricter test of the understanding of the (e, 2e) reaction on argon is giving by comparing the intensities of the 29.3 eV component of the 3s-hole state and the 3p hole state at 15.76 eV. Here we must have a reaction theory that affords a reliable comparison over a range of p. The plane-wave impulse approximation correctly describes momentum profile shapes up to p = 1 [ 51 but obtains a pole strength of 0.28 f 0.03 at 400 eV for the 29.3 eV state. The distorted-wave impulse approximation however describes shapes up to p = 2.8 at 1000 eV [28] and 1.8 at 400 eV essentially within experimental error, and gives a pole strength of 0.55 f 0.05 at 400 eV. Both these values are completely consistent with the sum rule (17) and complete the total understanding available for the noncoplanar symmetric (e, 2e) reaction. It is interesting to consider the pole strengths for the argon 3s manifold given by different CI theories and by the ratio interpretation (52) of the photoelectron reaction [29]. These are summarized in Table 1. Two CI theories are given. The first is an ab initio calculation by Mitroy [26]. The second uses phenomenological matrix elements of the Hamiltonian determined by fitting photon spectra. It describes only bound states of the ion and was calculated by McCarthy et al. [303 (denoted by MUP). Clearly the CI calculations must be improved before they are capable of reproducing
56 TABLE 2 RATIO OF n = 2 TO n = 1 INTENSITIES IN THE SEPARATION-ENERGYINDEPENDENT INTERPRETATION OF PES FOR IONIZATION OF HELIUMa
m P
2
3
4
5
2 3 4 7 10
4.1 6.4 7.3 6.7 5.9
4.3 6.2 6.7 5.6 5.1
4.3 6.1 6.6 5.9 5.4
4.3 6.1 6.6 5.9 5.3
aRatio are given in unit8 of lo-‘. by m and describedin the text.
The quality of the CI calculationfor helium is denoted
the EMS data. The ratio interpretation of PES intensities cannot be applied to present [29] data for argon, since the data are in the form of peaks whose magnitude is unmeasured because of an unknown background. The figures for Psd(e) given in column PES of Table 1 are certainly an overestimate. The second main point of the discussion is the possibility of using PES to distinguish details of theory at large values of p(” lo), where the (7, e) reaction has a distinct intensity advantage over EMS. To illustrate this we use the example of the n = 2 to n = 1 intensity ratio for the helium ion states. This is an extremely simple example, since the wave functions for the ion states are known exactly. We assume that the (7, e) reaction mechanism is independent of the energy difference for the high energy ionization of helium to n = 1 and n = 2 states. The intensity ratio then depends only on the details of the ground-state wave function of helium. For illustration we use progressively-improved CI calculations of helium [ 311, characterized by an integer m which signifies inclusion of natural orbital sets up to na in the CI basis. The number and symmetry of natural orbitals for each m is given in the form (m, number of orbitals, symmetry label) by (1,5,s), (2, 4,p), (3,3,d), (4,2,n, (5,l,g). The sensitivity of the intensity ratio to the quality of the CI wave function is shown by Table 2. Clearly the CI expansion has converged for m = 5. The ratio of intensities at p = 10 depends sensitively on the quality of the CI wave function. The validity of the assumptions we have made about the reaction mechanism is shown by the experimental value of the ratio, which is (5 + 1) x 10m2 at p = 10.37 [32].
57 ACKNOWLEDGMENTS
I would like to acknowledge support from the Australian Research Grants Scheme and ENEA, Frascati, where this work was begun. I would also like to acknowledge help and encouragement from Professors C.E. Brion, R. Cambi, L.S. Cederbaum, A. Sgamellotti and E. Weigold.
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