photoionization of oriented molecules

Z. Phys. D - Atoms, Molecules and Clusters 7, 271-280 (1987) Atoms, Molecules and Clusters ~r PhysikD

© Springer-Vertag 1987

Photoionization of oriented molecules

N.A. Cherepkov* FakuWit fiir Physik, Universit/it Bielefeld, D-4800 Bietefeld, Federal Republic of Germany

V.V. Kuznetsov Institute of Aviation Instrument Making, 190000 Leningrad, USSR

Received 25 March 1987; final version 20 May 1987

A general expression for the angular distribution of photoelectrons with defined spin polarization ejected from oriented molecules is derived in the electric-dipole approximation in the limit of a weak radiation field. An analysis of its geometrical part permits to draw definite conclusions without calculating matrix elements. For linear molecules it is shown that in Hund's cases (a) and (b) photoelectrons may be polarized only parallel to the molecular axis, while in Hund's case (c) they may be polarized in any direction. Appearance of a circular dichroism for nonchiral oriented molecules of relatively low symmetry is predicted. Dependence of a circular dichroism in the angular distribution of photoelectrons on the symmetry of molecules is demonstrated. The results may serve as the framework for studying molecules oriented on surfaces, in liquid crystals, or by molecular beam techniques.

PACS: 33.80.E

Introduction

It is now welt known that in the case of atomic photoionization an angular distribution of photoelectrons with defined spin polarization (ADSP) is de- scribed by five independent parameters, namely, by partial photoionization cross section, angular asymmetry parameter, and by three polarization parameters, describing projections of a photoelectron spin on three mutually perpendicular directions [1, 2]. Since in dipole approximation the number of theoretical parameters describing the photoionization process is also equal to five (three dipole matrix elements and two phase-shift differences), there is a possibility to perform complete, or "perfect", experiment, that is to deduce all five theoretical quantities from measured values.

*Permanent address: A.F. Ioffe Physical-Technical Institute, 194021 Leningrad, USSR

ADSP for unoriented molecules has the same structure as for atoms, but now the number of theoretical parameters is, in principle, infinite, since in molecules the orbital angular momentum of electron is not a good quantum number, while we still have to use the partial wave expansion for continuous spec- trum wave functions. Therefore the partial wave summation extends tilt infinity, giving the infinite number of dipole matrix elements, and the problem of extract- ing theoretical quantities from the measured parameters becomes much more complicated.

The only way to increase the number of measur- able parameters for molecules is to exclude averaging over orientations, that is to fix molecules in space. In our theoretical treatment of oriented molecules we will simply accept that the rotational degree of free- dom is absent. There are at least two cases, when molecules are oriented in space, namely, molecules adsorbed at a surface [3], and molecules in liquid

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crystals. There is also a possibility to produce a beam of oriented molecules [4, 5], though the degree of orientation of molecules in this case is lower. Therefore, it is quite feasible now to study oriented molecules experimentally.

Up to now, there were only few theoretical investi- gations on photoionization of oriented molecules. In electric-dipole approximation, which solely will be considered in this paper, the first study of the fixed- molecule angular distribution was made by Kaplan and Markin [6]. They pointed out that due to the interference of outgoing waves from two nuclei of a diatomic molecule the angular distribution of photoelectrons will have an oscillatory structure. Dill [7] for the first time obtained the general expression for the fixed-molecule angular distribution of photoelectrons, while Dill et al. [-8] and Davenport [-9] per- formed calculations for several particular cases of oriented diatomic molecules. ADSP for oriented molecules has been discussed already in [10] for nonchiral molecules and in [11, 12] for chiral molecules. One of predictions made in [-11], namely, an appearance of a So called circular dichroism in the angular distribution (which means a difference in photoelectron fluxes ejected at a definite angle by left and right circularly polarized light) in the case of oriented diatomic molecules, was further supported by numerical calculations for CO molecules in [13].

Here we present a general theory, which enables one to investigate any phenomenon appearing in ADSP for oriented molecules. The main difference of the present consideration from the previous one consists in another choice of a laboratory frame. In previous papers [-10-12] the laboratory frame was defined by a photon beam while here the laboratory frame is taken to be coincident with the molecular one, which is defined by the molecular axis. This choice enables one to simplify greatly the expression for ADSP at the expense of losing a simple possibility to make a transition to the case of unoriented molecules. But since the case of unoriented molecules has been already studied in detail [-1, 12, 14], it is justified to restrict our attention to the case of oriented molecules only.

General assumptions

Our consideration of a molecular photoionization will be restricted to non-relativistic photon energies where the electric-dipole approximation is applicable. One- photon and one-electron processes only are discussed (that is processes in a weak radiation field with one particle and one hole in the final state). Vibrational

X

÷-, I k s

J hu(m=+ 1)

Fig. 1. Molecular coordinate system, which is used also as a laboratory one in this paper

excitations of molecules are not considered explicitly though all equations are written in a form which enables one to include them easily. A procedure used here to derive ADSP has much in common with corresponding procedure for atoms, reviewed in [1].

In earlier papers [15, 16, 10] the laboratory frame has been usually defined by the photon polarization vector for linearly polarized light or by the photon spin (or photon momentum) for circularly polarized light. This choice enables one to make the transition to the case of unoriented molecules very simply, but instead in the case of oriented molecules ADSP contains all degrees of the photon polarization vector (or the vector of a photon spin) up to inifinity [-1, 12], while it should not contain degrees higher than the second one. Indeed, ADSP is proportional to the second degree of dipole matrix elements, and each of them is linear in the photon polarization vector. The only way to exclude higher degrees of the photon polarization vector from ADSP is to take the molecular frame as the laboratory one.

The particular choice of the molecular frame axes depends on the structure of a molecule under consideration and will be discussed later. In the simplest case of a linear molecule the molecular axis will always be chosen as the Z-axis of the molecular frame.

The complete quantum mechanical description of the photoionization process consists of the following. Light of defined polarization is absorbed by oriented molecules with molecular axes directed along n (see Fig. 1). It is necessary to find the probability of ejection of electrons in a given direction ~¢ with their spins oriented along some other direction s (n, ~ and s are the unit vectors).

As in atoms, the spin polarization of molecular photoelectrons appears through a spin-orbit (or spin- axis) interaction which manifests itself in the multiplet

splitting of molecular levels and in a difference between wavefunctions corresponding to different projections of the total angular momentum of electrons on the molecular axis. Both effects are included in our formalism.

The initial state wavefunction of an N-electron molecule is Tff(r 1, r2 , . . . rN, R), where r i is the posi- tion vector of the i-th electron in the molecular frame, and R represents internuclear distances. Usually ~0 N corresponds to the zero values of all quantum numbers (though not necessarily), and for general consideration we do not need any specification of properties of To N or approximation to which it is calculated.

After absorption of a photon there are the molecular ion in a state defined by some set of appropriate quantum numbers k, and a photoelectron moving in the direction r with spin quantized along the molecular axis. A corresponding wave function is

% = ~u ' (rl, r2 . . . . ru-1, R) ~pp,(rs), (1)

where p is the Z-projection of the photoelectron spin in the molecular frame.

The photoelectron wavefunction ~b~-,(r) contains in the asymptotic region the superposition of a plane wave propagating in the direction of the electron momentum p(t¢ = P/I P l), and a converging spherical wave. The most general expression for it is

~bffu (r) = E Yz*(~) ~l,. .(r), (2) lm

where Fe/mu(r ) already does not depend on the direction of photoelectron propagation and may be inde- pendently expanded in spherical waves [17, 18], e is the photoelectron energy, ~=p2/2 (atomic units h = m = e = 1 are used in this paper). Again, we do not need any further specification of the wavefunctions ~N-1 and ~mu(r) except for the statement that the latter is a complex function, normalized to an energy f-function.

The dipole operator in the molecular frame is

= ~ ~ r~ Ylm,(Oi, ¢~)D~,,~(O), (3) m' i = 1

where Dlm, m(~?) is the Wigner rotation matrix [19] with the Euler angles £2 = {~, fl, 7}, which define the orientation of the polarization vector e for linearly polarized light (m = 0) or the orientation of the photon momentum q for circularly polarized light (m = 4-I, q is the unit vector) in the molecular frame. It is convenient to present a matrix element of the operator (3) in the form

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YYl ¢ ~ l m l

. ( k t 1 1 ~ 1 ~ 1 [ £ , t 0 5 , (4)

where the matrix element

( k reml in I~m' [05 = (%"-x" ~,~lU~l

• - - r~ Y ~ , (0. ¢~) I ~'o ~5 (5) i = 1

is completely defined in the molecular frame. It in- cludes transitions between both electronic and vibrational states•

Equation for ADSP

ADSP is the differential photoionization cross section for the ejection of photoelectrons in some direction 1¢ with the spin oriented along another direction s. The general expression for ADSP is

-6~-. ~-~ N , ~ N - i tk(r ,s ,e) =~°)p -- ~ (T6 Id*l "%:1) ~ " g d.s • . , / 1 1 / ~ 2

- (l+sa)~i.~ ( ~ ' - 1. ~,L~I d~ 1 ~g'), (6)

where ½(1 +sa ) is the spin projection operator with being the Pauli matrix vector, ~ = 1/137 is the fine-

structure constant, o is the photon energy, and ng is the total statistical weight of the initial state. The outer summation extends over all degenerate states of both molecule and ion, and the vector e in the cases of circularly polarized and unpolarized light should be replaced by q.

Expanding the spin projection operator in terms of state multipotes [20], one can reduce it to the form

}(i +s . ) . , .2 = E ( - 1 ) ' ~ - " ~ l / ~ Ys-M.(~) SMs

#1 --#2 - -Ms " (7)

Substituting (4) and (7) into (6) and using the coupling rules for the product of two spherical functions, or two rotation matrices with the same arguments [19], one arrives finally at the following expression for ADSP

I~ (to, S, e)= ] / / ~ ak(o~)(- 1) 1 - m

L M J M j

aJMj SM~(S), (8)

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where Im,~ is the highest orbital momentum 1 retained in the expansion (2) (in principle, /ma,= O0), ak(~O) is the photoionization cross section

~cop ° 'k(c°):-~ng E E E [(O)J*'tkllmllXl)[ 2,

d.s. l l r t l l ~/lnl ~

(9)

parameters A J M J ~'LMSMs are expressed through the dipole matrix elements

jM ~ /g- ALM~SMs=~- ~. ~ Z [l~,12,L, Jl 1/2

d,s. l l m l rn 'm" [2m2 # 1 ~1~2

.(__l)m'+mz-u2-1/2(l~ 12 ool

.( 11 t2 L ~(1/2 1/2 Ss) -m l m2 --M]\#I -1~2 M

(1 1 --m' m" Mj

• (01 3", Iktlm~lA)(kl2m2#:l ~,,-t0), (10)

[/1 -=(2/+ 1), and

B = Z ~ Z I(Ola*'lkllmlPl)l 2. (11) d.s, l lml mt~l

The parameters A.rMj ~-'LMSMs are normalized by the condition

0 0 A00o0 = 1. (12)

Equation (8) is presented in the very convenient form of a product of the geometrical (three spherical functions) and the dynamical (the parameters A JMJ de- L M S M s '

pending on the photon energy) factors. The main ad- vantage of the present (8) consists in the fact that there is only one infinite summation over L (including summation over M), while in the case when the laboratory frame has been defined by the photon beam ((2) of [101 or (83) of [11), there were two infinite summations (over L and T, including corresponding projections). Of course, all results of [1, 10] remain valid, but every term in (8) here corresponds to a sum of many terms of (83) in [1].

If direction of a photoelectron spin is not detected, one should sum (8) over spin directions

I r (~, e) = I~" (r, s, e) + I r (r, - s, e) = ]/~ a k (co) Z/max 2 / ' 1 J )

"( - 1 ) l - m ~ jM~i - - - m m

I I J M j " ~-.LMOO YLM(~) YJMj(e)- (13)

This is the angular distribution of photoelectrons for oriented molecules, obtained previously by Dill ((15) of [71).

Integrating (8) over photoelectron ejection angles, one obtains an expression for the fixed molecule total photoionization cross section with defined spin polarization of photoelectrons:

I~" (s, e) = ~ I~" (r, s, e) df2~

= 2~z]/~at(c°)(" 1) 1 - " Z 1 J M j S M s l~ l~

J M j ^ ^ • AoosM~ YjMj(e) YSMs(S). (1.4)

It is important to note that all summations here are restricted. In (8), (t3), (14), as well as in all analogous equations below, the vector e in the cases of circularly polarized and unpolarized light should be replaced by q.

Transition to the vector form of A D S P

For general analysis of ADSP it is more convenient, instead of using the expansion in spherical waves (8), to deal with an expression presented in the form of products of characteristic vectors for the process, which is independent of a particular choice of coordinate system. From the vector form of ADSP it is possible also to make a transition to the case of unoriented molecules. The process under investigation is characterized by the vectors n, r, s, e (or q) introduced above. For nonlinear molecules the vector n alone could not describe the space orientation of a molecule, and we shall use also the unit vectors in the directions of the X and Y axes of the molecular frame, nx and ny, respectively• Thus, the memory on a molecular orientation will be contained both in the dipole matrix elements calculated in a particular molecular frame, and in the vector structure of ADSP.

It is convenient to present the dependence on the photon polarization state in (8) in the following form:

1, J = 0 ;

m J = l ;

( 2 - 3 m 2) J = 2 .

(15)

From here it is seen that for linearly polarized as well as for unpolarized light the terms with J = 1 do not contribute• Therefore, the vector e for linearly polarized light and the vector q for unpolarized light

can enter ADSP either in the second degree or in zero degree, while in the case of circularly polarized light the vector q can appear also in the first degree.

From the limits of summations in (8) it is seen that the vector s could not enter in a degree higher than the first one, and that the vector ~, as well as n, can enter in any degree up to 2t~,~ for r and (2/re, x+ 3 ) for n.

If a molecular symmetry group contains a centre of symmetry, then L in (8) had to be even. Indeed, due to inversion symmetry both initial and final state wave functions should have a definite parity, therefore in (10) both l~ and 12 should be either even or odd, and from the first 3j-symbol it follows immediately that Lis always even. As a consequence, for molecules having a centre of inversion the vector r can appear in ADSP only in even degrees.

If a molecular symmetry group contains a plane of symmetry with the molecular axis in it (for definite- ness we will always suppose that if there are planes of symmetry, one of them coincides with the YZ-plane of the molecular frame), then from the invariance of the dipole matrix element under reflection in this plane it follows that

(Old~lktlm,#l)=fOIc~mtkt~-rna -#~). (16)

This equality leads also to some relation between the parameters asM~ To derive it, one should change I L M S M S .

the signs of all projections of momenta, over which there are summations in (10), and simultaneously also change the signs of all projections in all 3j-symbols in (10) according to the standard rule [19, 21]. The result is

A J - M j _ I ] J + S A J M . r r-Ms-~t.~- ( - (17) ~ I E X L M S M s "

As a consequence, in the particular case when all projections are equal to zero, M = M s = Ms = 0, we have

J 0 AL0SO=0, if J+S isodd. (18)

A D S P for l inear m o l e c u l e s

Hund's cases (a) and (b)

Consider now general conclusions, which can be drawn from the expression (8) for ADSP, starting from the simplest case of linear molecules. In the Hund's cases (a) and (b), which in the absence of rotation coincide, projections of the angular momentum and spin on the molecular axis are conserved separately. Denoting by Ao, 2,'0 and Ak, Sk the corre-

275

sponding quantum numbers for the initial and final (ionic) states respectively, one obtains

m ' + A o = A k + m l , Z~O = Z~k-~ f l l (19)

m'+Ao=Ak+m2, ZO=Zk+g2.

Subtracting one line from another and combining with the conditions for 3j-symbols in (10) to be different from zero, we find

M + Ms = 0, Ms = 0. (20)

From the condition Ms=O it immediately follows that the vector of photoelectron spin s can appear in ADSP only through the scalar product (sn), and therefore photoelectrons in the Hund's cases (a) and (b) are polarized always in the direction of the molecular axis irrespective of the direction of a photon beam, its polarization, or the direction of photoelectron ejection r. This conclusion is valid also for the total photoionization cross section, derived from (14):

,, ak(O))(1 (2-3m2) 20 ~'- Ik (S, e) = - - ~ - - _ ~ 2 A°°°°'P2(ne)

3m lO ~ A001 o . (n s) (n e)) , (21)

where Pz (n%) is the Legendre polynomial. The parameter 2o Aoooo here plays the same role as the angular asymmetry parameter fl does in the angular distribution of atomic photoelectrons. It is expressed through only square moduli of dipole matrix elements

1 2O A ° ° ° ° - B]//2a.s ~. t~m'~ (2--3m'2)

• 1(0[ d*, Iklaml#l)] 2. (22)

The degree of polarization of the total photoelectron flux

I~' (s, q ) - - 1~ ( - - s, q) P" (s, q) =

I~' (s, q) + I~" ( - - s, q)

3m 10 - - A o o i o" (ns) (n q) ~/2 _ _ (23)

1 + ~ 20 A Aoooo "P2(nq)

is proportional to the same parameter A Az introduced in [14], which defines the degree of polarization of the total photoelectron flux in the case of unoriented

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molecules (see (20) of [14])

1 A ° ° l ° - B I~ml

• m'l(OId*,lkltm~#~)le----1/~A A~. (24)

It is worthwhile to note that in the case of unoriented molecules belonging to the Hund's case (a) all three components of the photoelectron polarization vector are different from zero [14], while for oriented molecules only one component survives which is directed along the molecular axis. Physically this property is quite evident since the dipole operator does not change the direction of the photoelectron spin, which is quantized always along the molecular axis. The rotation of molecules corresponds to the averaging over all orientations of the spin quantization axis in the laboratory frame connected with the photon beam, thus enabling for spin to have in this frame all three components different from zero.

In the case s[lq (23) above coincides with (20) of [10].

As an example, consider another situation when the number of terms of different types in ADSP appears to be finite, namely, when r ± n and e L n (or q 2_n). Then ADSP depends on the angle 0~ between s and n and on the angle ~ = ~b~-~b~ between K and e (or q), where ~b~ and ~b~ are azimuthal angles of the vectors r and e, respectively, in the molecular frame. Neglecting for simplicity all parameters AJM, with L>2, and using relations (17) one ob- LMSMs tains

lr(Os, c0= -ak, (~m) ( T - ( 2 - 3m z) 3V5~ A2a~ zoo.COS 2c~ ~n \ 4V 2

3m 111 +3mV~All l locosO.cosc~ - i /~AI ,o-sine

m2 ) 3 ~/lg sin 2 e] (25) +i(2--3 4~/~ AZ2-2ao.cos0~ • l

where

T = I ~ oo -- A2oooq ( t1~2--3m2,| 20 V ~- 20 \ Aoooo- A ooo)

(26)

Inclusion of terms with L > 2 will bring new adden- dums to T and to all other coefficients in (25), but the angular dependence will remain unchanged.

All terms of ADSP with L=<2 for Hund's cases (a) and (b) are presented in Appendix.

If we neglect by the influence of the spin-orbit interaction on the wave functions of both initial and final states in closed shell molecules, then it is easy to prove that

A J M J ( 1 / 2 A ~ _ aJMa" ( 1 / 2 A L M 0 0 k J*A+l/2, t--"CXLMOOt "~XA-1/2) (27)

A J M J (1/2 A ~ _ A J M j t ' l /2 A LM1Ms~ ~*A+~/2J----'~LM1Ms~ ~'A-1/2)" (28)

Therefore in this approximation, like in atoms [1], for different fine-structure sublevels the polarization parameters (with S = 1) are inversely proportional to the statistical weights of these states and have the opposite signs, while the parameters describing the angular distribution of photoelectrons (with S = 0) are equal. As a result, if the fine-structure splitting of the ionic state is not resolved, all polarization effects in this approximation disappear, and all effects independent of spin remain unchanged. Due to the spin-orbit interaction relations (27), (28) become only approxi- mate, and especially in the regions of autoionization resonances deviations from these relations can be strong. Experimental measurements for unoriented molecules [22, 23] in many cases support these infer- ences, though for more definite conclusions available data are insufficient.

Hund's case (c)

In Hund's case (c) only O, the projection of the total angular momentum J = L + S on the molecular axis, is defined. Denoting by O o and O k the corresponding projections for the initial and final molecular states respectively, one obtains from the conservation law for the projections

m' +Oo=Ok+ml +t~1,

m"+ Oo=f2k+m2 +#2. (29)

Subtracting again one line from another and combining with the conditions for 3j-symbols in (10) to be different from zero, we find

M + M ~ + Ms=O (30)

instead of more strong conditions (20) in Hund's cases (a) and (b). All new terms in Hund's case (c) corre- spond to Ms 4= 0 and describe the polarization of photoelectrons. In particular, the photoelectron spin can be directed perpendicularly to the molecular axis since it is not quantized now along the molecular axis. For the total photoelectron flux instead of (21) we will now have

im, , ak(CO){1 (2 -3m2) 20 k ts, e ) = - - ~ 7~ AooooP2(~)

3m I1 -(es) + ~ A o o l - 1

3m [ A I O j _ a i 1 ........ ~ o o l o ~ o o i 1)(ne)(ns)

(2--3m 2) -1. (ne)(s [ne])}. (3t) 3iA2oll

/2 As compared to the Hund's cases (a) and (b) this equation contains two new terms, one of which is proportional to the degree of polarization of photoelectrons when a beam of circularly polarized radiation is directed perpendicularly to the molecular axis

3 11 ,/;;. Aoo1-1 "(sq) V~ P ± 1 (S)[q±n: + - 1 9 - 3/2A20 ' (32)

-- ~ .c~O000

and the second term gives the polarization of photoelectrons ejected by linearly polarized or unpolarized light, with the degree of polarization defined by equation

p0(s, e) = 3 ] / 2 i A ~ 1_ 1 • (ne)(s [ne]) (33) 1 -- ]//2 A~°oo- P2 (ff'e)

for the particular case of linearly polarized light. With the help of equality (16) expressions for two new parameters in (31) can be written in a rather simple way

2 A ~ o l l - 1 = ~ E E

d•s • . l l m l

• Re(<01 3* I k l l m l l ) ( k l l m 1 - ½ [ d_ 110)), (34)

2i 2 A001-1=~ 2 .% /lml

• Ira((01 ~* I k l lml½)@l lm I -~-1 d-1 [0)). (35)

There are no diagonal terms here, and from M s ~ 0 it follows that #i 4= #a. The latter condition could not be fulfilled in the Hund's cases (a) and (b).

New terms as compared to those presented in Ap- pendix, appear in the Hund's case (c) also in the general expression for ADSP, and all of them describe the polarization of photoelectrons• We will not present them explicitly here.

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Nonlinear molecules

If molecular axes of nonlinear molecules are oriented along some direction n but molecules are randomly oriented around this direction, the condition (30) still remains valid. If, in addition, the molecular symmetry group contains a plane of symmetry and therefore the relation (16) is also fulfilled, then there are no new terms in ADSP as compared to the Hund's case (c).

If all molecules are identically oriented in space, the condition (30) is not fulfilled, and new terms appear in ADSP which should be added to the terms contributing in the Hund's case (c). These new terms are connected with a particular orientation of a molecule relative to the molecular frame axes X and Y, and contain the corresponding unit vectors n~ and ny.

If molecules do not have plane of symmetry, and in particular, if they are optically active (chiral), that is they have no So axis of symmetry, the condition (16) is not valid. Therefore, in all terms presented above the parameters AJL~}Ms with at least one of M, Ms, Ms projections different from zero, have to be replaced by the expression

AJM,r _+ I~AJM.~ a_t_ l~J+S~S-M~ ~ (36) L M S M s ~ \ ~ : t L M S M s ~ \ ~ 1 ~ * L -- M S - - M s ] "

For nonlinear molecules the number of different terms in the expansion (8) is too high, therefore, instead of writing them explicitly, consider new phenomena, which can be predicted from this expansion.

Circular Dichroism

As is well known [24], circular dichroism, i.e. the difference between photoionization cross sections for absorption of left and right circularly polarized light, in the case of unoriented chiral molecules is the result of the interference between electric-dipole and mag- netic-dipole terms. For oriented molecules the circular dichroism appears already in the electric-dipole approximation. The corresponding expression is

a£ ~1 (q) -- O-k -1 (q) = ] /4 a k (CO)( -- A o ~ o o o" (n q)

+~F2 ReA~oo . (qnx ) - [ f2 ImA~oo.(qny)), (37)

where a~(q) is the result of integration of (13) over electron ejection angles,

1o -- [/3 d* 1kll--m 1 - # l ) l 2 Aoooo X (1<oi t l l m ,

,ul

-1<01 d* lkllml~l> t2), (38)

278

A~OO-B~ /2V Z Z Z (<Ola?~lkllmll~l> d.s . l l m l

#1

• < k t l m t # 1 [ d o [0> + <01 d* Ikll -rnl-/~i> • <kll-rnl-#i ]~-i ]0>). (39)

If the molecular symmetry group contains a plane of symmetry, then from relations (16), (17) it follows that

1 0 1 1 ImAo0oo Ao o o o = 0, = 0. (40)

If the molecular symmetry group contains two mutually perpdendicular planes of symmetry (YZ and XZ planes of the molecular frame), then from invariance of ADSP under reflection in the XZ plane the addi- tional restriction follows

AJMa --a if M+Ms+Mj is odd, (41) L M S M s - - "J~

and all terms in (37) turn to be equal to zero. There- fore, circular dichroism in the electric-dipole approximation manifests itself only in molecules with rather low symmetry.

Equation (37) has been obtained previously in [11], but under simplifying assumptions made there all terms have cancelled.

Circular dichroism in the angular distribution (CDAD)

CDAD appears already in the simplest case of oriented linear molecules [1 I, 25] :

ICDAD (K, q) = I~ 1 (K, q)-- Ik- 1 (K, q)

= ~ u 2 cr k (co)(r [-n q])

• Im{AIl_loo+~/5A~[ioo(rn)

+ ~ A~ 1_ 100(5(~n)2-- 1)+ ...}. (42)

From this equation it immediately follows that three vectors n, q, and ~ should not be coplanar, so that they may be considered as a basis for a right or left coordinate system. The photoelectron flux will depend on the fact whether the kind of this system coincides with the kind of photon circular polarization or not. Calculations of CDAD for CO molecule per- formed recently in [,13] support the earlier prediction [,11] that CDAD is of the same order of magnitude as the differential cross section itself for the same angle.

The lower is the symmetry of a molecule, the higher is the number of terms contributing to CDAD. So, for linear molecules having a centre of symmetry

only terms with even L contribute in (42). For nonlinear molecules having two mutually perpendicular planes of symmetry, one should add to (42) the following terms

3 2 ~U~U2ak (CO) ((q n~) (to ny) + (q ny) (r n~))

• f 1 1 1 1 Im/A1 lOO + ]//5A2 lOO • (~n)+ ...}. (42a)

If a molecule has a plane (or planes) of symmetry but not two mutually perpendicular planes of symmetry, one should add also

3 2re ak(co){(qn)(rn') Im(Al°°° + ]~ A~°°°(rn)+ "'')

+(~n)(qnr)Im(A~oo+~A~2~oo(rn)+...)}. (42b)

And at last, for molecules having no planes of symmetry, one should add (only terms with the lowest possible L are retained)

3•k(co) 4 ~V~ { - 2(nq)(nr)" A l °°°° + ((rq)--(nq)(nr))

1 1 1 - - 1 - (A 1 - 1 o o + A 11 o o) - - ((q nx) (re nx) - - (q ny) (~ n,))

• (AI ~oo + A I _- ~ o o ) - / 2 ( q n ) ( r n ~ )

• l 0 A ~ (Al~oo+ ~ o o ) - l f 2 ( r n ) ( q n~) 1 1 1 - 1 • (A100o+A100o)}. (42c)

It is seen that for nonlinear molecules CDAD appears already when the vectors n, q and r are coplanar. For molecules which do not have a plane of symmetry (and for all chiral molecules) there is contribution which does not depend on the vector n, and therefore remains after averaging over orientations of molecules, It gives CDAD for unoriented molecules, discussed previously in [1, I2]. There is a simple connec- tion between the parameter D k, introduced there, and the parameters SM~ ALMSM~ of this paper

Dk L ( A l l + A ] - I a lo ~ (43) - 1 0 0 1 0 0 - - ~ 1 1 0 0 0 . / "

1/2

C o n c l u s i o n

We have shown that though the total number of terms in ADSP for oriented molecules is infinite, it is quite easy to separate some infinite sequences of terms which are responsible for definite physical phenomena, and the number of these sequences is finite. The complete quantum mechanical experiment for mole-

cules, in contras t with the a tomic case, is impossible. F r o m ano the r side, in fo rmat ion obta inable f rom measurements of A D S P for oriented molecules can be used to invest igate the details of molecu la r structure, or the direct ion and degree of molecules or ientat ion. On the example of C D A D it was demons t r a t ed tha t the n u m b e r of terms and the complexi ty of the phe- n o m e n o n itself depend on the symmet ry of molecules. Therefore, exper imenta l measu remen t s of C D A D can reveal, for example, the influence of a surface on the symmet ry proper t ies of adsorbed molecules.

F r o m the measured angula r dependence of a par - ticular p h e n o m e n o n (say, C D A D for or iented linear molecules, (42)) on the angle between t~ and n it is possible to make a definite conclusion on the conver- gence of the expansion over L in (8). This in fo rmat ion is very impor t an t bo th for pract ical calculat ions of A D S P and for the in te rpre ta t ion of exper imenta l results. With the precis ion to which it would be possible to restrict s u m m a t i o n over L in (8) we m a y speak abou t the comple te q u a n t u m mechanica l exper iment for molecules.

In m a n y cases p h e n o m e n a considered in this pape r are expected to be of order 1. In part icular , it is so for the degree of po lar iza t ion of the total pho- toelect ron current f rom oriented molecules [10], for C D A D from oriented l inear molecules 1-11,13]. At present it is not clear wha t is the order of magn i tude of terms appear ing in chiral molecules due to the absence of a plane or centre of symmetry . Therefore, numerical calculat ions are desired.

The first theoretical results for po lar iza t ion parameters in the case of unor ien ted linear molecules are a l ready avai lable [,18, 26], and these calculat ions can easily be extended to the case of or iented molecules. This, together with the rapid deve lopmen t of exper imenta l techniques in surface physics [-27, 28] m a y lead in a nearest future to a substant ia l progress in unders tanding of the p h e n o m e n a men t ioned in this paper.

One of the authors (NAC) is greatly indebted to Prof. U. Heinz- mann, Dr. G. Sch6nhense, and Dr. G. Raseev for many stimulating discussions. He acknowledges the hospitality of the Bielefeld Univer- sity extended to him during his visit, and financial support of the Deutsche Forschungsgemeinschaft, SFB 216 ,,Polarisation und Kor- relation in atomaren Stogkomplexen".

A p p e n d i x

To illustrate ra ther compl ica ted s t ructure of A D S P for or iented molecules, we present here explicity all terms with L < 2 for Hund ' s cases (a ) and (b) , which do not conta in the p roduc t (ten) (except for terms with J = S = 0)

279

I~(te, s, e ) = ~ ) ( 1 + ~ A ° ° o o - ( t e n )

+] /~AOOoo.P2(Rn ) ( 2 - 3 m 2 ) A2Ooo.P2(ffe ) /2

+ (2-- 3 m2) ~22 Aa21-1 oo (n el ((te e) -- (ten)(ne))

2 3 ] /~ 22 -- (2 -- 3 m ) 4 ~ A2 - 2 o o {((tee) - (ten)(n e)) 2

2 3ira 1 - - (te [ n e ] ) } - - ~ All_ lOO(K[-ne])

3m - - - - A ~ ° l o ( n s ) ( n e )

+ 3 m~A~ 1_11 o (n s)((te e ) - (te n) (n e))

+ ( 2 - - 3 m 2) 3i]/~A2~_ ~ l o (n s) (n e) (k [,ne])

- ( 2 - 3 m 2) 3 i ] f i ~ A 2 z 2 1 o(nS)(te[ne])

• ((te e ) - (te n ) ( n c ) ) +...]. /

(A.1)

As usual, for m = + 1 the vector e here should be re- p laced by q.

Since s u m m a t i o n s over J and S in (8) are restricted by values 2 and 1, respectively, A D S P is a lways qua- drat ic in e (or q), and l inear in s, while the vectors te and n m a y enter in any degree. Therefore the vector s t ructure of te rms with L > 2 will differ f rom the vector s t ructure of terms presented in (A.1) by mul t ip l ica t ion of t hem on the p rope r degree of the scalar p roduc t (ten), including zero degree, as it is seen f rom (42). It is possible to say, tha t apa r t f rom the degrees of this p roduc t the n u m b e r of different te rms in A D S P is finite.

R e f e r e n c e s

1. Cherepkov, N.A.: Adv. At. Mol. Phys. 19, 395 (1983) 2. Kessler, J.: Polarized electrons, 2nd edn. Berlin, Heidelberg, New

York: Springer 1985 3. Plummer, E.W., Eberhardt, W.: Adv. Chem. Phys. 49, 533 (1982) 4. Beuhler, R.J., Bernstein, R.B.: J. Chem. Phys. 51, 5305 (1969) 5. Kaesdorf, S., Sch6nhense, G., Heinzmann, U.: Phys. Rev. Lett.

54, 885 (1985) 6. Kaplan, I.G., Markin, A.P.: Dokl. Akad. Nauk SSSR 184, 66

(1969) (Sov. Phys.-Dokl. 14, 36 (1969)) 7. Dill, D.: J. Chem. Phys. 65, 1130 (1976) 8. Dill, D., Siegel, J., Dehmer, J.L.: J. Chem. Phys. 65, 3158 (1976) 9. Davenport, J.W.: Phys. Rev. Lett. 36, 945 (1976)

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10. Cherepkov, N.A.: J. Phys. B14, L623 (1981) 11. Cherepkov, N.A.: Chem. Phys. Lett. 87, 344 (1982) 12. Cherepkov, N.A.: J. Phys. B16, 1543 (1983) t3. Dubs, R.L., Dixit, S.N., McKoy, V.: Phys. Rev. Lett. 54, 1249

(1985) 14. Cherepkov, N.A.: J. Phys. B14, 2165 (1981) I5. Tully, J.C., Berry, R.S., Dalton, B.J.: Phys. Rev. 176, 95 (1968) 16. Buckingham, A.D., Orr, B.J., Sichel, J.M.: Phil. Trans. R. Soc.

London, Ser. A268, 147 (1970) 17. Dill, D , Dehmer, J.L.: J. Chem. Phys. 61,692 (1974) 18. Raseev, G-, Keller, F., Lefebvre-Brion, H.: Phys. Rev. A (in press) 19, Rose, M.E.: Elementary theory of angular momentum. New

York: Wiley 1957 20. Blum, K.: Density matrix theory and applications. New York:

PIenum 1981

21. Sobel'man, I.I.: An introduction to the theory of atomic spectra. Oxford: Pergamon 1972

22. Sch6nhense, G., Dzidzonou, V., Kaesdorf, S., Heinzmann, U.: Phys. Rev. Lett. 52, 811 (1984)

23. B6wering, N., Miiller, M., Heinzmann, U.: Abstracts of XV ICPEAC, p. 43. Brighton 1987

24. Caldwell, D.J., Eyring, H.: The theory of optical activity. New York: Wiley 1971

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83, 1557 (1985) 27. Heinzmann, U., Sch6nhense, G.: In: Polarized electrons in sur-

face physics. Feder, R. (ed.), Chap. 11, p. 467. Singapure: World Scientific 1985

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photoionization of oriented molecules

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