Qualitative interpretation of photoelectron angular distributions for linear molecules

CHCLIICAL

Volume 87, number 3

QUALITATIVE

INTERPRETATION

26 bli~rch 1981

PHYSICS LETTERS

OF PHOTOELECTRON ANGULAR DISTRIBUTIONS

FOR LINEAR MOLECULES Walter THIEL

Rcccircd Z lanu~~

1982

A parrluonm; scheme ior pholoclcctron acymmclr)’ pxamcrcrs oi lincx n~olmdcs IS proposed LIIKIzpphcd IO he rcsoCO: C ‘Yl lomwtion as acll~~s IIIC non-rcronxu N2 A ‘il u iomzalion Ryd cncrgy-dcpsndcnt ch.mp, in photoclcclron angular distrlbulions can be rauonalizcd b) such 3 pxllllonmg analysis.

nm~

I Introduction there has been considerable

Recently molecular

The partial waves arc Jsymptotically

photoelectron

ton energies III the

[I-IO]

mentally

VXUUIII

UV

asymmetry

3 qualnarive

[?,lO-20]_

the components

of the polarization

12m211p3) ISa Clcbsch-Gordan moment

dl-

the observed ones

mterpretahon

numbers1

phase shift, and D1,,l,,,

parameters are usu-

ally m reasonable agreement wth

[l-20]

at pho-

region. both elpcrr-

and theoretically

though the calculated

interest in

angular distributions

[71,7_3].

vector. (II/n,, coetiicicnt.

a complc~

S-matrl\

TIC denommator

r~ a

dipole trmsition

which refers ?o a contmuum

ing the appropriate

charactcrixd

and 111whdc MI, dcuotcs

by the quantum

boundary

function

obey-

condttlons

To m eq. (I) IS cas~ly evaI.

uated from eq. (7).

of their energy

dependence stall seems to be lacking. The present paper supSesrs a theoreucal interpretarron

framework

ior such an

and reports some applications

T o

to

= 1 c ID ,,,,,,, ,I” 3 Il~rnr.,

The nurncrator T? in cq. (1) may be written

hnesr molecules.

form by pair-wlsc combmatlon 1ft011, # I’rrr’ru; stmphfications

2. Theory Assummg a dipole transition

and a partial wave cl-

asytnnietry parameter f3 for a freely rotating molecule ts gwen bp [10.11]

T, = (z/75)1/7

pansion for the final stale, the photoelectron

P = TzjTu

.

[(I!f t I)(Y

in cq (2). With sonle further

obvious

we obtain c c (_l)lfJ;+lfl /l,l1,1, I’frr’sI; + I)] ‘Q/o,

1’0120)

(1) (-ly”;+“‘n

[(2+ l)(Z’-+

x (2Jt I)-q10, r’OlJO)(lO, x (I -fQ

x

III real

of the terms with

I rq

X I’ -I

i)] 1/Z

IOIJO)

Re(D, ,,,,,, rDk,+

- sin(qf - r?,p) Tm(Dlrnml. D;,,M~)]

112;--111J1 -?lZ,I’m’lJrn’4l)

x I’-I exddq-v~)l D~,,l,n7D,~,,,~m~ .

[cos(nf - qr)

(2)

,

(4)

where Re(z) and Im(a) are the real and imaginxy part of z, respectively. since the coefficient

The factor ir -I (10,1’0]10)

is +I or -I

IS non-zero only for 2-19

0 009-26 14/82/0000-0000/S

OZ.75 0 1982 North-Holland

CHCMICAL

PHYSICS

I’ = I or I’ = I + 1. Hence WCm3y partitron r, as well 3sfi mto d13gonal .md oif-dt3gon.d terms. with respect to I3nd I’. hl3hing use oi rhc equivalence between the 0fi-dr3g0n31 lernis IL’.1’1nnd specifying the summation linms c\phsi1lv WCobt3m

‘6

LCI-TCRS

,-

I

7 3 4 5 The c\presslons for tl~c d13&0n31terms/311 and tl1c oiGdi3gon31 terms P,,. which idlow drrcctly from eqs. (3) .md (4) contain quJdruplc sums over 111quantum numbers .md 3re rhfficult to an3lyre in 3 qualitative m3rmer. Thcreiorc we restrict our dlscussion to linear 111olcculcs where the quadruple SUII~S 3re rcplaccd by double SUIIIS due to the following relations bctwecn [he 111 quantum numbers ?I1= 111 -I

ior X ionir3tions

IfI = Ml7 + I

for fl ioni73tions.

.

(6)

II

080 057 0 53 0.52

1.40 0.99 0.65 0.49

2 80 1.43 1.20 1.12

051

0.40

1.06

1

-040

2 3

-0.29

-1.10

March

-040

4

-049 -0.17 -0 33 -0.26 -0.25

0.29 140 1.21 0.40 1.46 0 89 0-I-I 1.48 0 70

5

-0.96

0.16

-0.70

1.982

I.-$9

058

Theoff-diagonal terms 4,~ are partirroncd analogous manner. the numerical factors@ collected in table 2.

0.10

0.00 -0.10 -0.15

in an being

(11)

(7)

In the c3sc of Imear molecules, the diagonal terms flIr drc thus composed oi intrachJnnel conrnbutions fl;,” and uiterchmiel contributions @,” where the channels a, /3xc defined In the usualway(a for to = 0.7r ior ItrfI= I. etc.).

The preceding derivation suggests 3n analysis of asymmetry parameters of linear molecules v13 diagonal and off-diagonal terns. see eq. (5). composed of intrachannel and interchannel contributions.

see eqs. (8)-

(12). The examples given in sectIon 3 demonstrate

[hat such an analysis may explain tl1e energy dependence of pl1otoelectron angular distrrbutions. (Eqs. (9) 3nd (I 0) arc 3150 valid for D = p.) According to cqs. (9) and (10) exh contribution is the product of J numerical fxtor .-IpIpsnd 3 hctor Bsd depending on the lransition 1noments. Euphcit formulas ior 4;” arc avadablc from eqs. (Z)-(7) and the snJlytic3l expressions for the Clebsch-Cordan coeilicrents [ 331. For the ske of brevity. we omit these iormulas .md only give tl~e numericti Kdues of .-tiJ up lo I = 5 (see table I) which IS sufficient ior 3pphc.rtions in the low-energy region. Smce degeneracy kctors 3rc contained in A, Oci, the transition niomnts DIo, etc. in eq. (IO) always refer to si’ngle components oi degenerate channels (e g. 71.S), and the summation in the denommator of eq. (10) includes sumnation over these components, if applicable (C Ionizations). 250

3. Results and discussion The present formabsm was applied both to resonant and non-resonant photoionlzation. To illustrate our general arguments, computations in the dipole length approximation were carried out for the C ‘Xl ionization of carbon dioxide and the A ?llu lonization of nitrogen, using numerical procedures described previously [IO]. In both cases, the initial state was represented by ab initio molecular orbitals with an extended basis set, i.e. (1 ls7p2d/54p2d) for N2 and (1 ls6pld/Ss3pld) for CO, [25]. The fmal state was described by multiple-scattering continuum functions [23]. The numerical muffin-tin

Volume

87, number

3

CHEMICAL

Ptl\ SICS

LCI-I-ERS

no

nn

26 U3rd1

1982

Table 2 Coefficients Als)’ for off-diagonal terms I

IIU

OII

P

0

-1.79

1 2 3

-1.57 -1.53 -1.52

-3.10 -2.51 -a 42 -2.35

n

polential

0 I 1 3

0.89 079 077 0.76

for non-overlapping

calculated

[ 261 with

3 (4)

in

1.19 I.43 L.71 1.10

of the

a local chchange approxlparameter Q = 1. The

expansions were truncated

at I =

regions I, and I= 5 (5) in region III for

N2 (CO?)

The contmuum

functions

-0.19 -0.89 -0.93

-I ‘8 -1.40 -1.44

bo

bn

i.b

-1.43 - 1.40 -1.35

031 0.42

0.49 06-l

0.49 0.60

Our calculations

caused by the intcrchanncl

contributions A;”

and _:A?

lecular orbitals.

tnburions

oi

parameter fi around a resonance. In

the case of an extremely

contributions

both to chc

Z/$‘,

on the other hand. remain fairly

which is due lo the fact that the coefficients

ucs (see tsbk

As a first example we discuss the behaviour

m the fl

diagonal and off-d-diagonal terms. The intrachanncl

gonalized with respect to the occupied ab imrio mo-

the asymmetry

yield a deep minimum

curve around 40 eV (see fig. I) which is cvidcntly

constant

were ortho-

nb

2.57 2.80 7.88

touching sphcrcs was

the cxhange

multiple-scattering

1.57 1.77 1.86

1 55 I.28 1.21 1.18

from the 3b imtio wavcfunction

neutral molecule employing Ination

ob

lonizstion

are quite similar ior all rclcvant I WII). The minima in the mtcrchanncl

appear because thcsc contnburions

COII-

happen

to bc positive oufsldc the rcsonancc. and close to zero a1 the resonance due to the dominance

of the

strong resonance in the par-

teal wave I of channel (Y. the transitlon

moment Dla

is much larger rhan all others. Hence, fl is approxunarely piren by @’

smce all other contributrons are

negligble. see eqs (8)-(

12). More specifically. B = ApP smce BjjO z 1 under these conditions. see eq. (lo),

except for a n-type resonance in a Z ionuation

where we have $la

= $ and p =&AT

due to de-

generacy (see above). The hmitingfl value for a strong resonance of a given type IS thus available directly from table 1. Around of the asymmetry

the resonance, rapid variations

parameter are expected

due IO rhe

disappearance and reappearance of the interchannel contributions. To check the vahdrty of these general considerations, fig. 1 shows the results for the C “Xi of carbon dio\lde

in the 30-54

ionization

eV reDon. The u,-

type resonance around 40 eV is well characterized theoretically

[10,13,14,27-291

responsible for the mimmum

[ 10). Tbe effects of vibrational presently)

and is thought motion

(not included

reduce and broaden the calculated

nance features

to be

reso-

[ 13,271 resulting in better agreement

with eaperlment

[lo].

LO

50 h

in the measured fl curve

d2vY)

Ftg. I. Asymmetry paramctcrp. diagonal terms pt , + ~~3+ lemts pt 3 + p35, and intrachattncl contribuuons rq[o for the CO2 C 2X;toniWtion as functions of the photon energ.. Experimental p vslucs [4.10] are included.

pss.ofkh~gonal

CHfhllCAL

\‘olumu 87. nunlbcr 3

moment Dj,.

Ir3nsIIIon

Our nunMc3l

26 March 198,

PHYSICS LLITERS

results I~US

111~gen2ral arguments given above. even

srlpporr

the limiting

though

nancc is certamly

c3se of an extremely

strong r2so-

not reahz2d In carbon dioxide.

WIIII regard IO orher rzson3nccs. our analysis pro-

vklcs 3nalogous c\planaI1ons

for the sm3llcr minim3

m 1112cJlcul31ed fi curv2s for [he N, X’s,’ CO S ‘S+

Our second applic~rion plIoIoiomz31ion. The c3kulaIions

deals ~th

of nitrogen

ior the

in Ihe 17-30

predict .I rapid rise

rry paramr’ter abov2 threshold, c\pcrimcnt

non-resonant

FIN. :! shows IIIC results

ioniz3tIon

A ?II,

oi the

wlrcrc3s the di3gonJ

[ 12.

SIudIes

terms

terms remam approxunately

(~22 fig. 2). Th2 bchaviour

of the ch+onul

terms IS due to [he f3ct Ih3t the r&live

oi the

2V region. 3symmc-

in agr2cm2nI WII~

[6.7] 3nd oth2r lh2oxtIc31

l-11. This rlsc is caus2d by 1112off-diagonal ionst3nt

and

[ I?. I-& 16.7-O]

1onI73tions

rclcv3nI rrmsition

rhdnnel

leads IO 3lmos1 const3nt

3nd mtcrchannel

n3l terms. see eqs. (8)-( 3rr’ dominJIed

15

25

20

30

35

E,.” WI

mOm2nts do not chnge

IIIUCII in the 17-25 2V region. with I Dla 19 1Do,1 >

IDz,l > lD~.,l. which

10

5

magnitudes

contrlbutlons

rrg. 3. Cosines of differences bcr\\cen Coulomb phax shlfis as iuncuons of photoelectron limr~~c energy

to the Ih3go-

IO). The oii-dIa&onal

by rhe contrlbuIIon

mtraterms

$jS smcc the ix-

large because oi the tIansitmn

tor SE! IS partxul3rly momems

vel

mvolvrd.

see eq. (13). Closer inspection

that the second term in the numerator

re-

of eq.

(13) is much smaller than the first one so that S$

I

1Oi

I+

A 2rl,

and p$

3re roughly

proportional

IO the cosine term.

FIN. 3 shows the energy dependrnc2 chff2rences between Coulomb

of the cosme of

phase shifts [3

I]. and

It IS obvious that the curv2 for I = 0 closely resembles the calculated fl curve in fig. 3-. Hence, th2 rapid rise of the asymmrtry

parameter above threshold

by the rapid variation photoelectron

is caused

of Coulob phase shifts for low

energies.

The analysis grvrn above is valid for other nonresonant cases. too. In the framework analogous explanations

pendence of the asymmetry A ?n [6,7],

02 a4n,

[‘_I ionizations

[7] and C,H,

above threshold.

may be generally important

rig. 2. Asymmerry pxamerer p, dlagond turns fizz + oJ4. offdiagonti terms Do? + 024. and intrachanncl contnbutions ‘$,a for the N1 A h ,, ionmtlon as functions of rhe photon cncrgy. Expcrimrntal P values [ 1.6,7,30] are Included.

parameters for the CO

+ A”&

because they may exhibit transition

of our approach.

are found for the energy deX ‘ll,

Phase-shift effrcts

in the low.ensrgy stronger variations

moments for non-resonant

rsgion than the

photoionizarion.

4. Conclusions The energy dependence of photoslectron

asym-

Volume 87, number 3

CHEMICAL

rnetry parameters can be analyzed usmg the partittoning scheme proposed here. In the examples studled, rapid

changes

in the photoelectron

tions are due to rapid contributions moments

which

variations are caused

in the resonant

in the non-resonant

angular

dlstribu-

of the mterchannel

by the transltion

case, and by the phase shifts

PHYSICS LE-t-KRS

[IO]

T-A. Grimm, J.D. Allen Jr.. T.A. Carlson. h1.0. Krausr. D. htchaify, P.R. Keller and J.W. Taylor, J. Cbcm.

[I I ]

J.L. Dchmcr, D. DIU and S. W3U3cc. Phys. Rev Lcttcrr

Phys. 75 (1981)

91.

43 (1979) 1005. [ 111 S. Mdlscc, D. Ddl snd J.L. Dchmer. J. Phyi. 817 (1979) L-117.

[ 131

case.

26 March 1982

[I-I]

J-R Swnson, (1981) L207.

D. Ddl snd J.L. Dehmcr, J. Phys. Ill4

I-.& Grimm, T.A. Carlson, W.B. Dress, P. Apron. J.O. Thomson and J.W. Dwenport,

Acknowledgement

(1980)

[ 151 hf. Rochc, D.R. SUrub This work

was supported

schungsgcmemschaft.

The calculations

out with

the TR 440

Marburg.

Thanks

the intttsl-state

RN.

Holmes

by the Deutsche

computer

were carried

of the Universlt5t

are due to Dr. H. Meyer wavefunctions

for providing

For-

available, experimental

for making

and to Dr. data.

Spcctry. 19 (1980)

[ 161 B. &rchrc [ 171 G. Rxcev. [IS]

Phys. 72

and R.P. hlcssmcr, J. Elcclron

273.

B.R. Tambe. J. Phbs Bt 3 (1980) L221 H. L.c Ruozo and H. Lcfcbvrc-Bnon. J.

3rd

Chem. Phys. 72 (1980) 5701 G. Rsscev. H. Lcicbvrc-Brion.

H. Lc Ruozo

3rd

A.L

Roche, J. Chcm. Phys. 74 (1981) 6686. [ 191 R R. Lucchcse snd B.V. hfcKoy. J. Phys B t-t (t 98 t ) L629. [ZO] W. Iluel. Chcm Phys. 57 (1981) 727.

[ 211

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