Correlation effects and electron momentum distributions in the valence orbitals of ethylene

Journal of Electron Spectroscopy and Related Phenomena, 14 (1978) 267-275 o Elsevler Sclentlflc Pubhshmg Company, Amsterdam - Prmted m The Netherlands

CORRELATION EFFECTS AND ELECTRON MOMENTUM DISTRIBUTIONS IN THE VALENCE ORBITALS OF ETHYLENE

A J DIXON,

S T

HOOD

Institute for Atomzc Studres, S A 5042, (Australra)

and E WEIGOLD

The Flrnders

Unwerszty

of South

Austraha,

Bedford

Park,

G R J WILLIAMS

Department (Australza)

of Theoretzcal

(First received 27 February

Chemzstry,

1978,

Unwerslty

of Sydney,

Sydney,

IV S

W 2006,

m fmal form 15 March 1978)

ABSTRACT The non-coplanar symmetric (e, 2e) reaction has been apphed to CzH4 at 1000-eV incident electron energy An ion state at 27 4 eV separation energy IS strongly excited and IS ldentlfled by its angular correlation as a confi uratlon mteractlon state of symmetry 2Ag associated with the single hole state 2ai Y The spectroscopic strength of the state compares reasonably well with theoretical predlctlons Angular correlations have also been measured at separation energies correspondmg m the molecular orbital picture to removal of an electron from the lbs,, lb3,, 3ag, lbm, 2b lu and Zap orbltals

INTRODUCTION

(e, 2e) spectroscopy 1s a powerful tool for identifying and asslgnmg ion states produced by the sudden removal of a bound electron from a target atom or molecule’ The technique comprises the measurement of the relative differential ionization cross-sectlon either as a function of energyloss m the lomzmg colhslon or as a function of the recoil momentum of the ion where this 1s varied by changing the geometry of detection of the outgomg electrons The first measurement gives the separation energy of the varrous final ion states, m a way analogous to photoelectron spectroHowever, it 1s the second measurement, the angular correlation SCOPY measured at a fur&l separation energy, which can be decisive m determmmg the assignment of a particular final ion state Thrs IS closely related to the square of the momentum space wave functions of the eJected electron’, since it measures the cross-section as a function of this ion recoil momentum

268

q, which m the plane wave limit 1s equal and opposite to the momentum of the target electron (m the mdependent particle model) In the molecular orbital picture the ground state of the ethylene molecule 1s

lazlb$,2ai

2b$,lbz,3az

lb&lb:,(‘A,)

The four outermost states have been observed m UV photoelectron spectroscopy2 The final ion states produced by removal of an electron from either the 2a, or 2b,, orbltal (with which this paper 1s particularly concerned) have been observed m photoelectron spectroscopy with Al Ko! (1486-eV) X-rays3 and Mg KCY (1254-eV) and Y -Mr (132-eV) X-rays4 These three X-ray photoelectron spectra have a similar appearance weak structure at energies correspondmg to ejection of an electron from the four outer orbltals, two strong peaks at energies of 19 3 eV and 23 7 eV corresponding to removal of an electron from the 2b3, and 2a, orbltals, and a thud strong peak at ca 27 4 eV not identifiable as a molecular orbltal quasi-particle hole state On the basis of an XCYcalculation of the energy of the lowest excited state of the 2biA and 2ai1 ions, Berndtsson et al 3 identified the third peak as a conflguratlon interaction state of the ion of symmetry 2BSu and therefore assigned it to the 2bi,j hole state In this paper It IS shown that the momentum dlstrlbutlon at 27 4 eV 1s mconslstent with this interpretation but m agreement with the assignment 2Ag It 1s concluded that the state at 27 4 eV 1s a conflguratlon interaction state of the ion associated with the 2a;’ smgle hole state EXPERIMENTAL

AND

THEORETICAL

BACKGROUND

The colhslon geometry of the experiment was non-coplanar symmetric with the two outgoing electrons havmg equal energies and detected at 8 = 42 3” This IS the same geometry as used by us m previous experiments on molecules’ ’ The mcldent electron energy was 1000 eV The energy resolution was 2 5 eV, adequate for the study of the mam states of mterest The outermost states would be better studied with a higher energy resolution but m our experiment this could only be achieved at the expense of a lower srgnal-to-background ratio and longer data acqulsltlon time The electron gun cathodes available to us were not so partial to ethylene that we could afford very long data runs The experiments were carried out under computer control For the separation energy spectra this entailed measurmg the comcldence signal as a function of the energy of the incident electrons The separation energy e 1s then given by E = E, -E,-EE, where E, 1sthe mcldent outgoing electrons

electron

energy and E, , E2 the energies of the

269

For the angular correlations E, (= E, ) was kept fixed and E, was chosen to give the required separation energy e The comcldence signal was then measured as a function of the azlmuthal angle @ at which one of the outgoing electrons IS detected (#1 = #, & = 7~) This probes the dependence of

the signal of the recoil momentum q = k, -k?, where k,,

-k2

kl,

respectively q =

[(2k,

4 gwen by

k2 are the momenta

of the incident

and outgomg

electrons

In our geometry the magnitude of q IS given by

cos 8 - k,)2

+ 4kT sm2 8 sin2 3 @]I/2

The theory of the (e, 2e) reactlon on molecules has been investigated m some detail ‘, 6 An incident electron energy IS chosen such that the free electron wave functions can be approximated as plane waves and the reaction cross-section by the expression (~0112’1~z

(0

I di’Z(2n)-3

I 1

d3rexp

(lq

lr)(FIG > I2 IO)

(1)

This 1s an Impulse approxlmatlon T 1s the half off-shell Coulomb t-matnx for electron--electron scattering The integral indicated by Dlrac brackets 1s an average over the ground-state vlbratlons and the integral over 5-2 IS the rotational mtegral The quantum number p,. describes degeneracles only in electromc states F and G are the wave functions descrlbmg the final ion state and target ground state respectively For the case where it IS reasonable to neglect correlations m the target, the overlap of F and G has the simple form

(2) where tiL, 1s the characterlstlc orbltal of the ion elgenstate 1F) In the language of the independent particle model $, 1s the orbital from which the electron was removed to form the Ion state IF) Equation (2) contams the assumption that there 1s only one characterlstlc orbital for each ion state This entails the neglect of mteractlon between different single hole states of the same symmetry t$ 1s the expansion coefficient of the final ion state for that conflguratlon conslstmg of a hole m orbital c of the target groundstate wave function With the assumptions we have made where the sum 1s taken over all ion states which derive mtenslty from the characterlstlc orbital I&~ n, 1s the number of electrons m the target orbital or twice the dlmenslon of the u-reducible representation r of the molecular pomt group From Eqns (1) and (2), it follows that the angular correlations to all Ion states “belongmg” to the same characterlstlc orbital will have the same shape, independent of the magnitude of the strength I t,‘:) 12, the most Intense

270

or mam line being usually rndentlfled with the hole m the particular orbital It IS this fact that 1s explolted to answer the question 1s the state at 27 4 eV associated with the mam 2a,-’ hole state at 23 7 eV or the 2bgA single hole state at 19 3 eV9 Other states can be ruled out for reasons to be given m the next section

RESULTS

AND

Assignment

DISCUSSION

of the configuration

mteractlon

state at 27 4 eV

In Fig 1 1s shown a separation energy spectrum taken at a total energy (E, + E, ) of 1000 eV and 6 = 0” (coplanar geometry) The energy scale is calibrated using He as a reference Structure due to lomzatlon from the valence orbltals 1s evident In the simple MO picture the peaks at 19 3 eV and 23 7 eV are usually identified with the 2b,, and 2a, orbltals The peak at 27 4 eV has previously been observed m X-ray photoelectron spectroscopy Figure 1 also shows the Gaussian deconvolutlon of the two states allowmg for the experrmental energy resolution (2 5 eV FWHM) and the natural widths of the states obtamed from the PE spectra There 1s also evidence of contrnuous structure going to higher energies Berndtsson et al found mconcluslve evidence of a discrete peak at 31 2eV Although our data are slmllarly mconcluslve we have made momentum dlstrlbutlon measurements at this energy as well as at 27 4 eV In Fig 2, plotted as a function of the momentum Q, are shown the angular correlations measured at the four separation eneraes 19 3 eV, 23 7 eV,

E, = E,

c6 0 i

m L

Figure 1 IOOOeV molecular the text

8,=02=0 A

al=

o”

= +E =42

= 500eV 3” I

The separation energy spectrum m the symmetric (e, 2e) reactlon on Cs HQ at The arrows mdlcate the separatzon energies for the mam transltlons and the orbltals with which they are ldentlfled The state at 27 4 eV 1s dlscussed m

271

Figure 2 Non-coplanar symmetric (e, 2e) cross-sectlons for CzH4 at 1000 eV for fixed separation energies (e) plotted as a function of the momentum (I The dashed curves show the momentum dlstrlbutlons calculated from the generalized overlap amplitudes for the 2b3, (19 3 eV) and 2a, (23 7-eV) transltlons

27 4 eV and 31 2 eV The measurements were made m the same data run to enable the relative mtenslty of the angular correlations to be measured This IS mdxated on the vertical scale The relative rntensltles measured m thx way are consistent with the relative mtensltles at I$J= 0” taken from the separation energy spectrum (Fig 1) The angular correlations at, f = 19 3 eV and 23 7 eV, identified with the 2bgJ and 2a;l single hole states, are compared with the momentum dlstnbutlons given by the direct computation of the overlap function (F( G) (eqns 1 and 2) These generalized overlap amplitudes for “hole” states are calculated with the one-particle Green’s function formalism’ They appear m the spectral resolution of the one-particle Green’s function, the residue at the poles being the spectroscopic strength S2f) of the transltlons (for an HF target Sd” 1s JUSt (@ I*, eqn 2) The advantage of the Green’s function formahsm 1s that, although only an SCF calculation for the target ground state 1s needed, part of the ground-state correlation as well as additional lomc-state correlation effects are automatically taken mto account The atomic basis used to generate the particle hole basis m the present calculations was a contracted (5s, 3p/3s) Gaussian basis set The resulting strengths and momentum dlstrlbutlons for the dominantly smgle-hole states are given respectively m Table 1 and Figs 2 and 3 The orbital separation eneraes (Koopmans’ energies) obtamed from the basis are also included m Table 1 The lomzmg process for the dominant transltlons should be well described by the removal of a Hartree-Fock particle (eqn 2), since the Green’s

272

functions are well approximated by quasi-particle propagators m the nelghbourhood of poles with large pole strengths Fourier transforms of the orbital wave functions of the above basis are indeed nearly identical to the generalized overlap momentum dlstrlbutlons m shape The orbital wave functions of Snyder and Basch 8 also give very slmllar shapes for the momentum dlstrlbutlons As explained earlier the assignment of the state at 27 4 eV proceeds by ldentlfymg that ion state of large spectroscopic strength with which it shares a slmllar angular correlation It may happen of course that more than one such Ion state can be found In ethylene the quasiparticle hole states 3a;’ and 2a,-I both have angular correlations very slmllar to the angular correlation at 27 4 eV The single hole state 3a;’ can however be ruled out as the dominant parent of the 27 4 eV state for two reasons First the 3ai1 state 1s much weaker than the 27 4 eV state m the X-ray photoelectron spectrum, but not m the (e, 2e) spectrum Second the 3a;’ 1s much further away m energy from the 27 4 eV state than 1s the state usually identified with the 2%-’ hole (15 eV versus 3 7 eV) We, therefore, feel confident m assoclatmg the state at 27 4 eV with the 2a, characterlstlc orbital A x2 test of the slmllarlty m shape gives a statlstlcal SlgdlCanCe Of 30% By contrast the same test applied between the 27 4 eV and 19 3 eV angular correlations gives a slgmflcance
1

GENERALIZED OVERLAP THE MAIN TRANSITIONS

Orbital

AMPLITUDE SPECTROSCOPIC IN Cz H4 (e, 2e) C2 Hi

Orbital

energy

STRENGTHS

S,(‘)

FOR

s (0 c

VIP

(ev)

lblu lblg

3%

lb, 2bu

2ag lb,

1%

-10 -13 -16 -17 -21 -28 -305 -305

3 8 0 6 6 3

10 12 14 15 19 23 8 9

5 9 7 9 3 7

0 0 0 0 0 0 0 0

93 93 91 86 83 70 58 58

273

cross-sectlon for CZ H4 at Figure 3 The non-coplanar symmetric (e, 2e) dlfferentlal 1000 eV plotted as a function of the momentum 4 for, in order of lncreasmg E, the main Zba, and 2ap transltlons The curves show the shapes of the lbl,, IbIg, 3ag, lbzu, momentum dlstrlbutlons derived from the generahzed overlap amphtudes of the states at 23 7 eV and 27 4 eV to be 0 66 and 0 20 The next strongest state 1s predicted to be at 31 1 eV and also strength 0 03 The spectroscopic of symmetry ‘Ap, with spectroscopic -I hole state at 23 7 eV has also been calculated strength of the prmclpal 2a, by us using the generahzed overlap amplitude method The predicted value 1s 0 70 (Table l), m good agreement with the CI calculation of Martm and Davidson From our experimental results we cannot deduce reliably the spectroscoplc strength of the 2a;’ single hole state However, an estimate of the strength of the state at 27 4 eV relative to the strength of the 2ai1 state aves 0 61 ? 0 06 compared wth the value 0 3 of Martm and Davidson’

scoplc

strengths

respectively

Anguh

correlattons measured at selected energies Fig 3 are shown angular correlations measured at separation energies of 10 OeV, 12 6 eV, 14 6 eV, 16 6 eV, 19 3 eV and 23 7 eV The angular correlations at 19 3 eV and 23 7 eV are the same as those shown m Fig 2 The other four angular correlations are taken at energies correspondmg roughly to lomzatlon from the orbltals lb,, (10 5eV), lb,, (12 9eV), 3a, In

274

(14 7eV) and lbZ, (15 9eV), where the energes m parentheses are the vertical lornzatlon potentials measured by Rabelals et al * The eneraes 10 0 eV and 16 6 eV were chosen respectively below and above the energy of the correspondmg state to mmlmlze interference from the nearest nelghbourmg state The energy resolution of our experiment was not sufficient to eliminate interference between close states Additionally, the photoelectron spectrum of the 3a, and lbau orbltals shows mtrmslc overlap attnbutable to their vibrational structure The six expemmental angular correlations have been plotted on an lsometrlc drawing The results are plotted m this way to emphasize the fact that the angular correlation represents a ‘section’ taken at constant separatlon energy of a three-dimensional surface Even with poor energy resolution the angular correlation can be very informative, particularly regarding the ordermg of the states The relative vertical scale m Fig 3 1s taken from Fig 1 This Dves the relative strengths of the angular correlations only very roughly since some are measured on the sides of very steep peaks so that any slight uncertamty m energy will cause a large uncertainty m relative mtensity The theoretical curves plotted m Fig 3 are angular correlations of the mam transitions m the valence region calculated by the generalized overlap method The curves have been separately normalized to give the best visual fit to the experimental data The experimental angular correlations of Fig 3 are m general consistent mth the theoretical expectations, although the angular correlation at 16 6 eV shows an excess of low momentum components, presumably owing to interference from the nelghbourmg state at 14 6 eV The angular correlation at 10 0 eV IS the pure ground-state transition uncontammated by other states and is excellently desclnbed by the calculated lbl, momentum dlstnbutlon

CONCLUSION

The peak at ca 27 4 eV m the X-ray photoelectron spectrum of ethylene has been shown to be associated with the 2ai1 smgle hole state from a compmson of the pertinent angular correlations For this purpose the (e, 2e) technique has a flexlblhty and declslveness not matched by photoelectron spectroscopy Further, because it probes the low momentum part of the wave function it eJects those electrons which, on average, are far from the nucleus It then becomes reasonable to approximate the free electron wave functions by plane waves In this case It becomes possible to measure spectroscopic strengths although m the present experiment hmltatlons of energy resolution allowed only a measurement of the relative spectroscopic strength of the states at 23 7 eV and 27 4 eV

275 ACKNOWLEDGEMENTS

We are grateful to the Australian Research Grant’s Committee for fmancial support and one of us (S T H ) IS grateful for a Queen Elizabeth Fellowship

REFERENCES I E McCarthy and E Welgold, Phys Rep, 27C (1976) 275 J W Rabalals, T P Debles, J L Bertosky, J J Huang and F 0 Elhson, J Ckem Pkys, 61 (1974) 516 A Berndtsson, E Baslher, U Gehus, J Hedman, M Klasson, R Nllsonn, C Nordlmg and S Svensson, Phys Scrlpta, 12 (1975) 235 M S Banna and D A Shn-ley, J Electron Spectrosc Relat Phenom 8 (1976) 225 S Dey, A J Dixon, I E McCarthy and E Welgold, J Electron Spectrosc Relat Phenom 9 (1976) 397 S Dey, A J Dixon, K R Lassey, I E McCarthy, P J 0 Teubner, E Welgold, P S Bagus and E K Vnmkka, Phys Reu A, 15 (1977) 102 G R J Wllhams, I E McCarthy and E Welgold, Ckem Pkys, 22 (1977) 281 L C Snyder and H Basch, Molecular Wave Fun&Ions and Propertzes, Wdey, New York, 1972 R L Martm and E R Davldson, Chem Phys Lett , 51 (1977) 237