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STIELTJES IMAGING TECHNIQUE APPLIED TO INNER SHELL IONIZATION PHENOMENA IN
MOLECULES
V. Carravetta, H. Agren
To cite this version:
V. Carravetta, H. Agren. STIELTJES IMAGING TECHNIQUE APPLIED TO INNER SHELL ION-
IZATION PHENOMENA IN MOLECULES. Journal de Physique Colloques, 1987, 48 (C9), pp.C2-
769-C2-772. �10.1051/jphyscol:19879134�. �jpa-00227244�
Colloque C9, suppl6ment au n012, Tome 48, dkcembre 1987
STIELTJES IMAGING TECHNIQUE APPLIED TO INNER SHELL IONIZATION PHENOMENA IN MOLECULES
V. CARRAVETTA and H. AGREN"
Istituto di Chimica Quantistica e Energetica Molecolare del CNR, Via Risorgimento 35, 1-56100 Pisa, Italy
* ~ n s t i f x t e of Quantum Chemistry, University of Uppsala, PO Box 518. 5-751 20 Uppsala, Sweden
ABSTRACT
We devise Stiel t j e ' s imaging techniques for high-energy electronic continua in molecules and demonstrate some appl icati ons for Auger resonances, shake-of f spectra, lifetime broadening and discrete-conti
nuum
interaction energy shifts.1. INTRODUCTION
Theoretical descriptions of bound-conti nuum electronic transitions in molecules are complicated by the anisotropy of molecular ionic fie1 d. ~2 methods, especial 1 y the Stiel t j e s imaging method, have been quite promising in overcoming these problems, and have late1 y been appl ied to shape resonances and cross sections in molecular photoelectron spectra C1,21. However, except for photoionization a t low photon energies, the use of these methods has been rather sparse.
In the present work we show how Stieltjes imaging can be generalized to include high energy electronic continua and also other types of inner she1 1 ionization phenomena in molecules. This means that both shape resonances (open channels) and
"del ta-"or Auger resonances (closed channels) can be addressed with the same, Stiel t j e s imaging, approach.
The technique i s applied to shake-off transitions, Auger resonances, 1 i fetime broadening and t o di screte-conti nuum interaction energy shift and results are a1 so used to analyse fine structure in the emission spectra emerging from 1 i fetime-vi brational interference and variation of lifetime with nuclear conformation. Sample calculations on these phenomena for the water molecule are presented.
2. THEORY AND APPLICATIONS
Stieltjes imaging i s a moment theory method, which utilizes spectral moments S(-k) obtained by integrable basis set calculations of a theoretical spectrum in the discrete and continuous energy regions. Spectral 'distributions describing the transition from a discrete initial state to a final state in the electronic continuum are obtained by the Stiel t j e s derivative (SD)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879134
C9-770 JOURNAL DE PHYSIQUE
of a function F ( ~ ) ( x ) = t ~ ~ x I(" deriving from t h e principal spectral
representation of order
'n
C X ! ~ ) ; 1 t h a t reproduces t h e f i r s t 2n spectral momentsS(-k) = I i = l , n I!")(x{"))-~ k = 0 , 1 , 2 ,...( 211-1).
In t h e usual photoionization cross section calculations C1,21, the discretized theoretical spectrum, from which t h e S(-k) a r e computed, c o n s i s t s of e x c i t a t i o n energies and osci 1 la t o r strengths
.
In t h e present contest, i t i s defined by t h e Ferrni Go1 den Rule f o r t h e Auger decay; 2nl
< P ; - ' I S ~ ~ ( G - E ) ( p N z l
71 and by t h e Sudden Approxi mati on f o r the shake-of f processC ($+ Ef ) ;
1 <$y-2 1 iEv i,, 1
5> I
2 Iwhere $: i s the molecule ground s t a t e , $-'is a core hole s t a t e , :'$; s t h e appropriate double ionized final s t a t e , E, i s t h e energy of t h e secondary emitted electron and Ef i s t h e shake-off threshold C3,41.
The mny-el ectron continuum i s basi c a l l y approximated by an anti symetri zed product of a fixed t a r g e t function f o r the molecular ion and a continuum o r b i t a l f o r t h e outgoing electron obtained by dtagonal izing a static-exchange hami 1 tonian C2-41, projected on a ~2 basis s e t containing special o s c i l l a t i n g functions C21, t h a t , f o r t h e water molecule, has t h e fol 1 owing expressions
r\ A h fi n A , - . A A
hS = T + V + t i (=XI y) (ZJi-Ki) + CJx + Jy -(1/2)Kx -(1/2)Kyl(1-6xy)
where S(sing1et) and T ( t r i p l e t ) , s p e c i f ~ t h e ,spin of t h e ion final s t a t e ( double hole i n t h e o r b i t a l s x and y ), T
,
V, J and K a r e the k i n e t i c , nuclear a t t r a c t i o n , Coulomb and exchange operators respectively.In contrast t o photoionization o r shake-off t r a n s i t i o n s C1,2,41, where t h e cross sections a r e obtained from the SO on a conti nous range of energies, t h e Auger process, r e f e r s t o only one energy value E, corresponding t o t h e resonance energy of the considered decay channel C31.
Fig. 1
electron momentum) corresponding to the d-wave contribution to the channels 3i:, 3 ~ i l 6 2 ( ~ ) , 3iil1.5; (S), 16; in the Auger decay of the Hz0 core hole state, are reported for example; the resonance condition i s mrked by crosses. By these functions, the energy shift of the core hole state due to the interaction with the Auger continua can be. computed
the total value obtained amounts to
=
0.06 eV.The described approach has been appl ied at several approximati on 1 eve1 s starting with the SCF approximation for the initial and final states and the independent wave approximation for the emitted electron. The effect of the electronic relaxation for the ionic states and of the different channel interaction has been subsequently i nvestigated; this 1 ast effect was found rather important especi a1 1 y towards the 1 ow kinetic ener v mergv part-of the Auger spectrum.
!Ig.2 CORE ELECTRON SHAKE-OFF SPECTRUM O F MATER INDEPENDENT CHANNEL CALCULATION RELAXED ORBITAL SHAKE-OFF EDGES
SHAKE-OFF ENERGY ( e V >
Fig. 3 AUGER SPECTRUM OF WATER: HOLE-MIXING CALCULATIONS
I
K I N E T I C ENERGY CeV)
C9-772 JOURNAL DE PHYSIQUE
The 4 flgures are a short review of different applications of the Stieltjes method for inner shell ionization processes in the water molecule. Fig. 2 reports the total core electron shake-off spectrum (broken line) together with partial contributions from several channels identified by the label of the valence orbital involved in the shake-off and the spin label of the final ionic state E41. In Fig.3 the theoretical Auger spectrum including the interaction of the 16 different decay channels i s compared with the experimental spectrum C51. Since the fluorescence yield i s very small for core hole state o f water as for inner vacancies of a1 1 small-Z elements, the lifetime (and lifetime broadening) i s directly given by the total Auger rate.
Sumning a1 1 the theoretical independent channel rates we obtain a 1 i fetime broadening of 0.15 eV for the core hole state of water.
Fig. 4 TOTAL AUGER DEACY RATE (LIFETIME WIDTH)
Angl e(HOH) (degree)
1.6 1.8 2.0
R(0-H) (a. u. ) I n Fig.4 we display the dependence of the lifetime on the nuclear conformation C63.
I t i s seen that most of the variatfon (from united atom (Ne) t o the free atom (0) 1 imi t s ) i s a1 located to internuclear distances short of the equi 1 i bri um geometry.
The 'constant resonance approximation1 seems thus to hold for core ionization of water. Although the lifetime variation only marginally influences the band shape in the core emission spectra of water, (viz. Auger and X-ray emission spectra) the total lifetime gives rise to non-negligible distortions of the bands due to vi brational-lifetime interference effects. A formal ism for band shape analysis including 1 ifetime inteference and lifetime geometry variation has recent1 y been derived and applications to water core emi ssion spectra are in progress.
REFERENCES
C13 P. W. Langhoff, in : Methods in Computational Molecular Physicsleds G. H. F. Diercksen and S. W 1 son (Rei dell Dordrecht, 1983) pp. 299-333 i C21 I.Cacelli, R.Moccfa e V.Carravetta ,Chem. Phys. 90 (1984) 313;
J. Phys. B 18 (1985) 1375; Molec. Phys. 59 (1986) 385.
E33 V. Carravetta and H.Agren, Phys. Rev.A 35 (1987) 1022.
C41 H. Agren and V. Carravetta, J. Chem. Phys. 87 (1987) 370.
C51 H. Siegbahn, L. Asp1 und and P. Kel fve, Chem. Phys. Lett. 35 (9175) 330.
C61 H.Agren, A. Cesar and V. Carravetta, Chem. Phys. Lett.
