Journal of Electron Spectroscopy and Related Phenomena, 31 (1983) 151-160 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

CHARACTERIZATION OF RESONANCES IN PHOTOIONIZATION

WALTER THIEL*

Fachbereich Physikalische Chemie der Universitiit Marburg, D-3550 Marburg (W. Germany)

(Received 21 September 1982)

ABSTRACT

Multiple-scattering calculations are used to characterize the au resonances in the Nz X”’ I;: and COz c”? Xi ionizations by means of radial density plots and to identify the resonant components in the corresponding continuum wavefunctions and cross-sections.

INTRODUCTION

Final-state resonances cause rapid variations of experimental photo- ionization cross-sections and asymmetry parameters [ 1, 21, particularly in vibrationally resolved spectra [3-51. These resonance effects are repro- duced reasonably well by recent approximate calculations [ 5-151. From a theoretical point of view, resonances have also been characterized by a rapid rise in the corresponding eigenphase sum [9, 16, 171, by eigenchannel contour maps [ 181, and by Stieltjes orbitals [ 191.

The present paper makes use of well-known general concepts to illustrate certain resonance features in a qualitative manner. Resonances are visualized by means of radial density plots obtained from S-matrix-normalized multiple- scattering functions, which are found to be more suitable than the corre- sponding eigenchannel wavefunctions [ 181. Previous discrete-basis-set calculations have established a connection between resonances and unoc- cupied virtual orbitals [8,13,19] ; this paper shows how this connection can be derivea by an analysis of the resonant multiple-scattering wavefunction. Finally, the origin of the resonant oscillator strength is traced within the framework of the multiple-scattering approach.

RADIAL DENSITY PLOTS

The radial density pr can be obtained from the expansion of the continuum function \k, around the center of gravity of a molecule using polar coordi- nates r, 8, r#~ and spherical harmonics YJ

* Present address: FB 9-Theoretische Chemie, Universitat Wuppertal, D-5600 Wuppertal 1, W. Germany.

036%2048/83/$03.00 0 1983 Elsevier Science Publishers B.V.

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Three types of continuum functions ‘k, are available for treating photo- ionization, i.e., K-matrix-normalized functions \kE, S-matrix-normalized functions \kE, and eigenchannel functions \kf, which are related by the linear transformations [ 18,201

(3)

(4)

where L = (2, m) labels the quantum numbers characteristic of the asymptotic behaviour. I and K are the unit and K matrix, respectively, and U, denotes the coefficients of the eigenvectors which diagonalize the K matrix. Making use of the asymptotic forms of the functions \kE, \kE and \k,E as given in refs. 18 and 20, the following integrals are easily evaluated

(‘P: 1 ‘I+) = 6(E’ - E)[6,,! + 1 KLLf~KLttL~] (5) L”

(‘I’; I’I’$) = 6(E’ - E)G,,+os2 (T/J,) (7)

where c(,, is an eigenphase, 6,l the Kronecker delta, and 6(E’ -E) the Dirac h-function referring to the Rydberg energy scale.

When comparing radial density plots for different continuum functions and different energies, it is obviously essential that all functions share the same normalization. According to eqns. (5)-(7), this requirement is fulfilled only by the S-matrix-normalized functions \ki and by the renormalized eigen- channel functions

*:,,norm = !P,E cos (RC(,) (8)

whereas the functions \kE and !I!: are not acceptable with regard to this criterion.

Figures 1 and 2 show plots of the radial densities p; (r) for the u, reson- ances in the N2X2 Z: and CO2 c2 2:: ionizations [l-16], with L = (I, 0) and odd 1 owing to uu symmetry. The densities pi(r) were derived from con- tinuum functions $S, obtained via the multiple-scattering (MS) formalism [ 201 using numerical techniques described elsewhere [ 151. The MS potentials were calculated from the ab initio 4-31G MO [21] wavefunctions for the

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2 4 6 8 rlau 1

Fig. 1. Radial densities pi(r) for N 2, with L = (I, 0), as functions of distance r from the center of gravity.

neutral molecules, employing a local exchange approximation [ 221 with the exchange parameter (Y = 1. The resulting potentials were numerical in regions Ii and III, and constant in region II. The MS expansions were truncated at 1 max = 3(5) in regions Ii(II1). The MS continuum functions were orthogon- alized with respect to the occupied 4-31G molecular orbit& [ 151.

Figures 1 and 2 show clear resonance features. At the resonance energy, the radial densities of the resonant channels are drastically enhanced in the molecular region, both with respect to other channels and other energies (lower or higher). The resonance is associated with the 1 = 3 channel in N, [ 16,181, but with the 1= 5 and 1= 1 channels in CO*. In the latter case, the radial densities at resonance show strong maxima close to the atoms and a deep minimum between them (see Fig. 2) which indicates a node in the resonant continuum wavefunction. The radial density plots are thus suf- ficiently detailed to exhibit the dominant I-character and the nodal pattern of resonances. Being simpler than wavefunction contour maps [ 181, they may therefore serve as a useful tool for visualizing resonances.

According to the preceding discussion, it should be possible to construct alternative radial density plots from the renormalised eigenchannel functions

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:I, ’ 1 1 I

; A 1,

- I,

P=l 1.3

3 ,))‘; - -.-

--- 1.5

2 1 6 8

Fig. 2. Radial densities p:(r) for CO2 , as in Fig. 1.

Fig. 3. Eigenphases p for a, eigenchannels in Nz as functions of photon energy &AL The dominant I-components of the eigenchannel functions A and B are indicated at low and high energies.

155

\k&,rm (see eqn. (8)). This approach, however, leads to certain problems which can be understood from the eigenphase plots for Nz [9,16] shown in Fig. 3. Close to the resonance energy, i.e., within the width of the reson- ance, there is an avoided crossing [9,23] between eigenchannels A and B. Consequently the resonance is associated with eigenchannel B below the avoided crossing, and with eigenchannel A above, which is reflected in the corresponding radial density plots. Hence, the avoided crossing obscures the resonance features by “distributing” them over two eigenchannels. When using the functions Jr:, on the other hand, the resonance in Nz is confined to a single channel (see Fig. 1). Hence, the S-matrix-normalized continuum functions \Eg seem to be the most suitable choice for visualizing resonances by means of radial density plots.

RELATION TO BOUND-STATE ORBITALS

Inspection of the radial density plots suggests a decomposition of the continuum function \kf into two orthogonal components according to

*, = *,c?, + *xl, (9)

The component \k,, is confined to the molecular region and is assumed to be of dominant importance at the resonance, whereas \k,, is a scattering function whose amplitude varies only slowly with energy. The function \k,,, is presently defined by an expansion in the basis of the unoccupied bound- state virtual orbit& (VO’s) \kj of the neutral molecule

with

Cfi = (\kf I\ki) = (\k,s,I\ki) (11)

This definition is justified by the resulting behaviour of \k,, at resonance (see below), although it implies that \k,, does not vanish completely off- resonance. The coefficients cfi are obtained by inserting eqn. (1) and ex- pansion [ 151

into eqn. (ll), giving

cfi = C (4fNIhW) J

(13)

Using the MS continuum functions \kt (see preceding Section ) and 4-31G VO’S !I!,, all 0, -type coefficients Cfi for NZ and COZ have been calculated

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10 Ekln ido Ll

Fig. 4. Absolute coefficient values I Cfi I for Nz , with \k, = \kt and L = (I, 0), as functions of photoelectron kinetic energy Eb.

as functions of photon energy. The coefficients involving the 30, 4-31G VO of N2 and the 50, 4-31G VO of CO2 turn out to be by far the most import ant ones at resonance. Figures 4 and 5 illustrate the energy dependences of the corresponding absolute values I Cfi I. It is obvious that the contributions of the specified VO’s to the resonant continuum functions (I = 3 in Nz, 1= 5 and 1= 1 in CO2 ) reach a pronounced maximum at the resonance energy. Hence, to a good approximation, the component \k,, can be rep- resented by a single bound-state VO, namely 3a, in N2 and 50, in CO*. These 4-31G VO’s correspond to compact, valence-like, antibonding u* orbitals, being the lowest a,-type VO in N, and the second-lowest one in CO?. By analogy to discrete-basis-set studies [ 8,13,19], the present analysis thus establishes a direct connection between multiple-scattering continuum functions at resonance and unoccupied antibonding orbitals of the neutral molecules considered.

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E kin (eV)

Fig. 5. Absolute coefficient values 1 Cfi 1 for CO2, as in Fig. 4.

ORIGIN OF RESONANT OSCILLATOR STRENGTH

The cross-section for the one-electron photoionization process \kMo + \kt is proportional to the absolute square IDi I 2 of t4e corresponding dipole transition moment. According to eqn. (9), it is possible to write

Df=D +D res (14)

I Dt I 2 = ID,.;’ + D,,,D:r + D&D,, + ID,, I 2 (15)

D res = CVODVO (16)

where Dvo is the dipole transition moment involving the ionized orbital \kMo and the VO Ova which contributes most heavily to \k,, (see pre- ceding Section), via the coefficient c v. = cfi. The energy dependence of the resonant component D,,, is given by cvo , since Dvo is constant.

Figures 6 and 7 show the results obtained for N2 and CO2 using MS functions \kE (see above) and 4-31G orbitals \kMo and \k,,. The total cross-sections refer to the production of all o,-type photoelectrons and thus involve summations over three transitions \kMo + \kt (1 = 1,3,5). On the other hand, the resonant contributions shown include only the

Ekln (eV)

Fig. 6. Photoionization cross-sections for Nz ( -) together with their resonant contri- butions (---), as functions of photoelectron kinetic energy I&,, (see text for further details). The MS options chosen exaggerate the strength of the resonance [ 1,8-11).

Ek,, (eV)

Fig. 7. Photoionization cross-sections for CO2, as in Fig. 6. The curves imply that the total contribution from the cross-terms in eqn. (16) is negative beyond Eb = 19 eV. The MS options chosen exaggerate the strength of the resonance [ 141.

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component ) II,,, I 2 for the resonant channels (I = 3 in N2, 1= 5 and 1 = 1 in C02) (see eqn. (15)).

It is evident from Figs. 6 and 7 that the prominent peak in the cross- sections is due mainly to the resonant contributions defined above, which provide 50-75s of the total cross-section. For comparison, the non-resonant contributions arising from the term ID,, I2 in eqn. (15) amount to less than 10% of the cross-section at resonance. Hence, the resonant component

cvo \k v. of the continuum wavefunction carries the bulk of the available oscillator strength. Therefore it seems justified to describe the u, resonances in the N2X2 Zi and CO2 c”’ Zs ionizations as two-step, one-electron processes: a valence-like u + u* transition is followed by the emission of a photoelectron.

ACKNOWLEDGEMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (through a Heisenberg-Stipendium) and the Fonds der Chemischen Industrie. The calculations were carried out using the TR 440 computer of Universitit Marburg.

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