On the angular distributions of molecular photoelectrons: dipole cross-sections for fixed-in-space and randomly oriented molecules

5 October 2001

Chemical Physics Letters 346 (2001) 341±346

www.elsevier.com/locate/cplett

On the angular distributions of molecular photoelectrons: dipole cross-sections for ®xed-in-space and randomly oriented molecules J.C. Arce a, J.A. Sheehy b, P.W. Langho€ b,c,d,*, O. Hemmers e, H. Wang e,f, P. Focke g, I.A. Sellin g, D.W. Lindle e a Departamento de Quimica, Universidad del Valle, A.A. 25360 Cali, Colombia Air Force Research Laboratory, AFRL/PRS, Edwards AFB, CA 93524-7680, USA c Department of Chemistry, Indiana University, Bloomington, IN 47405-4001, USA San Diego Supercomputer Center, University of California, 9500 Gilman Drive, La Jolla, CA 92093-0505, USA e Department of Chemistry, University of Nevada, Las Vegas, NV 89154-4003, USA f MAX-Lab, Lund University, P.O. Box 118, 22100 Lund, Sweden g Department of Physics, University of Tennessee, Knoxville, TN 37996-1600, USA b

d

Received 11 June 2001

Abstract New theoretical expressions are devised employing a dynamical perspective for photoionization cross-sections differential in electron ejection angles for both ®xed-in-space and randomly oriented molecules, and comparisons made with K-shell ionization measurements in molecular nitrogen. Closed-form cross-sectional expressions are obtained in the dipole limit in terms of molecular body-frame transition moments and related normalized angular-distribution amplitudes which can be calculated employing interaction-prepared states without reference to speci®c scattering boundary conditions, and which reduce to more familiar atomic expressions in appropriate limits. Ó 2001 Published by Elsevier Science B.V.

Synchrotron-radiation-based experiments have been reported recently on new aspects of the angular distributions of electrons photoejected from atoms [1±5] and spatially-®xed molecules [6±9]. The former reveal signi®cant deviations from dipole distributions due to retardation terms in the interaction between radiation and matter [10], whereas the latter provide the angular distributions of photoejected electrons relative to an ap-

*

Corresponding author. Fax: +1-858-534-5113. E-mail address: langho€@drifter.sdsc.edu (P.W. Langho€).

propriate internal molecular body frame. Although standard dipole cross-sections [10,11] and their non-dipole extensions [12±14] are useful in interpreting the atomic data, the corresponding theory for randomly oriented [15,16] and ®xedin-space [17,18] molecules provides largely computational prescriptions [19,20], rather than conveniently employed and potentially insightful analytical expressions. Accordingly, new di€erential photoionization cross-sectional expressions are needed which are better suited for timely interpretations of the recently reported molecular data, and for corresponding ab initio calculations.

0009-2614/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 9 2 0 - 4

342

J.C. Arce et al. / Chemical Physics Letters 346 (2001) 341±346

In the present Letter, new theoretical expressions are devised for molecular photoionization cross-sections for both spatially-®xed and randomly oriented molecules on a common basis, and comparisons are made with concomitant measurements of dipole anisotropy factors that are free from contamination due to (non-dipole) retardation e€ects. Following an early development of Bethe [21±23], the photoejection process is treated as a dynamic or initial value problem employing the time-dependent Schr odinger equation, avoiding at the outset particular choices of scattering boundary conditions to describe molecular photoionization states, and providing considerably more ¯exibility in representations of ejection amplitudes than can be conveniently had in the customary static theory. The di€erential cross-section is obtained in a simple closed form from the spatially and temporally asymptotic ejected ¯ux or probability amplitude, which is seen to contain (complex) outgoing waves modulated in their angular variations by the molecular ®eld experienced by the departing electrons. All dependences of the di€erential cross-section upon the orientation of the molecule in the laboratory frame are obtained in analytical forms, aiding interpretations of ®xedin-space photoelectron measurements in the absence of detailed calculations. Interaction-prepared states [24], which can be calculated without explicit reference to continuum eigenfunctions [25], provide the required bodyframe transition moments and corresponding angular variations of the outgoing waves. The resulting expression for the leading (E1) dipole anisotropy factor for randomly oriented molecules is similar in form to, but distinct in detail from, the familiar Bethe±Cooper±Zare [10,26] expression for atomic electron photoemission anisotropies, to which it reduces in appropriate limits. The cross-section di€erential in the direction of ^e ˆ ^r) for a molecule having a outgoing electrons (k particular orientation speci®ed by Euler angles ^ ˆ …a; b; c† in the laboratory frame is obtained R from the expression [21±23] ^ ^ dr…R† …k hx† e;  2 ˆ …8ph2 =mA20 †r2 jW…1† …r; t†j ; dXk^e

…1†

where the ®rst-order time-dependent wave function W…1† …r; t†, evaluated in the asymptotic (r ! 1, t ! 1) limit, is obtained from the Schr odinger equation to ®rst-order [W…r; t† ˆ W…0† …r; t† ‡ W…1† …r; t† ‡


] in the radiation ®eld o ih †W…1† …r; t† ‡ H^ …1† …t†W…0† …r; t† ˆ 0: ot Here, H^ …0† is the molecular Hamiltonian,

…H^ …0†

…2†

H^ …1† …t† ˆ …e=mc†^p
A0 eikp
r cos…xt†

…3†

is the familiar semi-classical interaction Hamiltonian for linearly polarized incident radiation [11], W…0† …r; t† ˆ Ug …rb †e iEg t=h refers to the non-degenerate many-electron vibronic ground state of the molecule, with Ug …rb † expressed in center-of-mass body-frame electron coordinates (rb ), a sum over electrons is implicit when required, and hkp , A0 and hx are the incident photon momentum, vector potential and energy, respectively. Solution of Eq. (2) is obtained for ®xed orien^ of the molecule in the laboratory frame tation (R) by transforming to the body-frame in the usual manner [27]. Retaining the ®rst term in a Rayleigh expansion of the exponential factor in the vector potential, W…1† …r; t† is obtained in the form W…1† …r; t† ˆ …eA0 =mc†

‡1 X kˆ 1

…1† ^ …k† Dk;0 …R†W …rb ; t†;

…4†

^ are rotation matrices [27], and where the Dk;0 …R† …k† the W …rb ; t† are solutions of the inhomogeneous body-frame equations   o …0† ^ H W…k† …rb ; t† ih ot …1†

ˆ Fg…k† …rb † cos…xt†e

iEg t= h

:

…5†

1=2 ^ …k† Y1 …^pb †Ug …rb †

…6†

Here, Fg…k† …rb † ˆ

j0 …kp rb †…4p=3†

are dipole `test' functions generated by the indicated harmonic-polynomial body-frame momen…k† tum operators (Y^1 …^pb †; k ˆ 0; 1) acting on Ug …rb †, and j0 …kp rb † is the zeroth-order spherical Bessel function [27]. Solutions of Eq. (5) for a particular ionic channel are obtained employing the so-called kprepared dipole eigenstates of H^ …0† [24,25]. These

J.C. Arce et al. / Chemical Physics Letters 346 (2001) 341±346

can be written for purposes of analysis in the forms P …c;a† …c;a† …k† c;a UE …rb †hUE jFg i …k† UE …rb † ˆ n ; …7† o 1=2 P …c;a† …k† 2 c;a jhUE jFg ij …c;a†

where the UE …rb † can be chosen to be any of the energy-normalized channel eigenstates of H^ …0† [28], and the sums over c and a correspond to forming a projector over the total continuum state degener…c;a† acy at molecular energy E. The particular UE …rb † employed here for purposes of analysis are standing-wave eigenchannel functions of H^ …0† , where c labels the molecular irreducible symmetry representations and a enumerates the eigensolutions within a given irreducible representation [28±30]. In this case, the energy normalized kprepared dipole eigenstates of Eq. (7) have the asymptotic …rb ! 1† forms 1 X …m† …k† UE …rb † ! …k† Yl …^rb † lE l;m E 2m 1=2 X …c;a† D …c;a† Ul;m …E† UE jFg…k†  ph2 kb c;a   o n 1 …c†  sin kb rb ‡ dl …kb ; rb † ‡ d…a† c …E† ; rb …8† where …k† lE

P …c;a† …k† 2 D E c;a jhUE jFg ij …k† …k† ˆ UE jFg ˆn o1=2 P …c;a† …k† 2 jhU jF ij g E c;a

…9†

is the (real, non-negative) body-frame dipole transition moment of k symmetry, rb and ^rb refer to the body-frame radial and angular coordinates, respectively, of the ejected electron,  hkb is the outgoing (body-frame) electron asymptotic linear momentum, the labels (l; m) are asymptotic angu…c† lar momentum values, dl …kb ; rb † is the l-wave Coulomb phase shift (including a factor lp=2), U…E† is the unitary matrix which diagonalizes the real symmetric standing-wave K…E† matrix constructed in the (l; m) representation, and d…a† c …E† are the eigenchannel phase shifts [28±30]. In the case of linear molecules with 1 R ground states, the irreducible label c ˆ k ˆ m ˆ 0; 1 in Eqs. (7)±(9),

343

with the in®nite sums there consequently performed over l and a only. Employing the prepared states of Eqs. (7)±(9), the solutions of Eq. (5) are found to have the asymptotic (rb ! 1, t ! 1† forms [21±23]  1=2   mp 1 …k† ^ …k† …^ r † W…k† …rb ; t† ! lEx U Ex b rb 2h2 kb  ei…kb rb

Ex t= h†

;

…10†

for rb  …hkb =m†t, but to vanish for rb > …hkb =m†t. Here, Ex ˆ Eg ‡ hx is the resonance energy, and ( X …c† 1 …k† …m† ^ E …^rb † ˆ Yl …^rb †eidl …kb ;rb † U x …k† lEx l;m ) E …a† X D …c;a† …k† idc …Ex † …c;a† UEx jFg e Ul;m …Ex †  c;a

…11† is the unity-normalized angular emission amplitude of k symmetry. It is seen that the integration in time required to solve Eq. (5) has transformed the standing waves of Eq. (8) into a `Bethe front' comprised of waves having net outgoing electron momentum [21±23]. Employing the time-dependent wave function of Eqs. (4) and (10) in Eq. (1), the partial-channel photoionization cross-section di€erential in the ^e for molecular direction of outgoing electrons k ^ is obtained in the form orientation (R) 2 ^ ^ ‡1 dr…R† …k hx† 4p2 e2 X e;  …1† ^ …k† ^ …k† ^ ˆ 2 Dk;0 …R†lEx UEx …kb † ; dXk^e m cx kˆ 1 …12† …k†

where the body-frame moments lEx are given by Eq. (9) and the corresponding angular amplitudes ^b † by Eq. (11). It should be noted that the ^ E…k† …k U x expressions of Eqs. (7)±(11) are employed for analytical purposes only, with the transition moments and angular amplitudes required in Eq. (12) obtained from solution of Eq. (5) without explicit reference to scattering states in computational applications [25]. Although the general case is not any more complicated, it is convenient for later purposes to write out the predictions of Eq. (12) for the 1 R

344

J.C. Arce et al. / Chemical Physics Letters 346 (2001) 341±346

states of linear molecules. In this case, the k-prepared angular amplitudes [Eq. (11)] can be written ^ …k† ^ E…kˆ0† …^rb † ˆ H U Ex …hb †…2p† x ^ …kˆ1† …^rb † U Ex

ˆ

1=2

;

1=2 i/b ^ …?† …1†H e ; Ex …hb †…2p†

…13a† …13b†

^ E …hb † are (complex) functions of the where the H x body-frame polar angle hb of the ejected electron, and /b is the corresponding body-frame azimuthal angle. Employing Eqs. (13a) and (13b), Eq. (12) ^ ˆ /R ; hR ; 0) takes the form (R  ^ ^ 2 dr…R† …k hx† 2pe2  …k† 2 ^ …k† e;  lEx cos2 hR H ˆ 2 …h † b Ex m cx dXk^e 2  2 ^ …?† …?† ‡ 2 lEx sin2 hR cos2 /b H Ex …hb † p …k† …?† 2lEx lEx sin hR cos hR cos /b n o …?†  ^ …k† ^  2Re H …h † H …h † ; b b Ex Ex …k;?†

is the corresponding anisotropy factor, and …?† …k† q…hx† ˆ lEx =lEx is the indicated ratio of bodyframe perpendicular and parallel transition moments. Eq. (16) recovers the standard partialchannel cross-section [11], whereas Eq. (17) gives a new closed-form expression for the corresponding anisotropy factor in terms of the parameter q…hx† and the angle averages performed in the bodyframe (hb ) over the indicated amplitudes of Eqs. (13a) and (13b). As an illustration, the development is applied to ionization of the degenerate K-shells (1r2g ; 1r2u ) of molecular nitrogen. In this case, the anisotropy factor can be written in the form  2 ^ …k† b… hx† ˆ hH Ex j cos hb

^E i 1=3jH x …k†

^ E j cos2 hb 2q… hx†2 hH x …?†

^E i 2=3jH x …?†

^ E j sin hb cos hb jH ^E i …1 ‡ 2q… hx†2 †hH x x …45†

.

…14†

…45†

…1=3†…1 ‡ 2q… hx†2 †;



…18†

where Ref


g refers to the real value of the generally complex quantity in brackets. The foregoing results are of interest in connection with the expression obtained from Eq. (12) for the photoionization cross-section of an ensemble of randomly oriented linear molecules. Performing the required molecular orientational average over Eq. (12) while holding the laboratory-frame di^e ˆ …he ; / † of the ejected electrons conrection k e stant gives the familiar result

employing Eq. (14) for /R ˆ 90°, hR ˆ 45° with /b ˆ 0° to replace the last term in the numerator of Eq. (17). The three (measurable) unity-nor2 ^ …k† malized angular distributions jH Ex …hb †j , …?† …45† 2 2 ^ E …hb †j and jH ^ E …hb †j in Eq. (18) are crossjH x x section-weighted sums of appropriate individual channel (1rg=u ! kru=g , kpu=g ) contributions to the 2 measured distributions, and the parameter q…hx† is now given by

^e ;  dr…k hx†=dXk^e ˆ fr… hx†=4pg

q…hx† ˆ

 f1 ‡ b… hx†P2 …cos he †g; where r… hx† ˆ

   2  2 4p2 e2 …k† …?† …1=3† l ‡ 2 l Ex Ex m2 cx

…15†

…16†

is the angle-integrated partial-channel photoionization cross-section,  2 ^ …k† ^ …k† b…hx† ˆ hH 1=3jH Ex j cos hb Ex i 2 ^ …?† ^ …?† 2=3jH 2q… hx†2 hH Ex j cos hb Ex i  p …?† ^ …k† ^ ‡ 2q…hx†2 RehH j sin h cos h j H i b b Ex Ex . 2 …17† …1=3†…1 ‡ 2q… hx† †

…?†

2

ˆ

2

…?†

…l1rg !kpu † ‡ …l1ru !kpg † …k†

2

…k†

…l1rg !kru †2 ‡ …l1ru !krg †2 2 babs …hx† ; 2…1 ‡ babs …hx††

…19†

where babs …hx† is the molecular photoabsorption anisotropy [31±33]. The predictions of Eq. (18) shown in Fig. 1(a) are seen to be in precise accord with anisotropy measurements obtained in molecular nitrogen employing a spectrometer designed to isolate the dipole terms of Eq. (15) [5], and in general accord with earlier experimental values obtained in the absence of this correction [34,35]. Fig. 1(b) shows the photon energy variations of the three angular averages required in Eq. (18), and of the parameter q…hx† constructed employing Eq. (19) and the

J.C. Arce et al. / Chemical Physics Letters 346 (2001) 341±346

345

Fig. 1. (a) Photoionization anisotropy factor b… hx† for the K-shell (1r2g , 1r2u ) of molecular nitrogen: () present experimental values; previous measurements as indicated; (- - -) Eq. (18). Error bars for the present measurements are smaller than the data symbol shown. (b) Photon energy variations of q…hx† and of the body-frame angle-averages employed in Eq. (18). The former is inferred from measured photoabsorption anisotropy data [31±33] and the latter from measured body-frame photoelectron angular distributions [6], employing the results of static exchange calculations and high-photon energy limits (arrows) to provide continuous curves, as discussed in the text.

measured photoabsorption anisotropy values [31±33]. The former have been estimated from single-channel calculations adjusted to agree with values obtained from integrals over measured body-frame angular distributions [6], and to correctly approach their asymptotic ( hx ! 1) limits (3/5, 1/5, )4/15) obtained from outgoing atomiccentered 1s ! kp waves. The q… hx† values inferred from absorption anisotropy measurements are seen to approach unity with increasing photon energy, in which limit b…hx† ! 2, in accordance with the measured photoemission anisotropy values of Fig. 1. Conclusion The present development for molecular photoionization cross sections provides a common basis for interpretations of measurements performed on ®xed-in-space and randomly oriented molecules, and for corresponding ab initio calculations of di€erential cross sections. Comparisons between theory and experiment in molecular nitrogen illustrate the general theoretical development and provide dipole anisotropy values in good accord with K-shell measurements. Additional applications of the new formalism to both ®xed-in-

space and randomly oriented molecules are in progress and will be reported subsequently. Acknowledgements Support provided in part by grants from the National Research Council, the US Air Force Oce of Scienti®c Research, the National Science Foundation and the Department of Energy (DOE) is gratefully acknowledged. We thank Dr. Peter R. Taylor for his hospitality to PWL at the San Diego Supercomputer Center during the course of the investigation. The authors thank the sta€ of ALS for their support. The ALS is funded by the DOE, Materials Sciences Division, Basic Energy Sciences, under Contract No. DEAC03-76SF00098. References [1] B. Krassig, M. Jung, D.S. Gemmell, E.P. Kanter, T. LeBrun, S.H. Southworth, L. Young, Phys. Rev. Lett. 75 (1995) 4736. [2] M. Jung, B. Krassig, D.S. Gemmell, E.P. Kanter, T. LeBrun, S.H. Southworth, L. Young, Phys. Rev. A 54 (1996) 2127. [3] O. Hemmers, G. Fisher, P. Glans, D.L. Hansen, H. Wang, S.B. Whit®eld, D.W. Lindle, R. Wehlitz, J.C. Levin, I.A.

346

[4] [5] [6] [7] [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

J.C. Arce et al. / Chemical Physics Letters 346 (2001) 341±346 Sellin, R.C.C. Perera, E.W.B. Dias, H.S. Chakraborty, P.C. Deshmukh, S.T. Manson, J. Phys. B 30 (1997) L727. A. Derevianko, O. Hemmers, S. Oblad, H. Wang, S.B. Whit®eld, R. Wehlitz, I.A. Sellin, W.R. Johnson, D.W. Lindle, Phys. Rev. Lett. 84 (2000) 2116. D.W. Lindle, O. Hemmers, J. Electron Spectrosc. Relat. Phenom. 100 (1999) 297. E. Shigemasa, J. Adachi, M. Oura, A. Yagishita, Phys. Rev. Lett. 74 (1995) 359. F. Heiser, O. Gessner, J. Viefhaus, K. Wieliczek, R. Hentges, U. Becker, Phys. Rev. Lett. 79 (1997) 2435. E. Shigemasa, J. Adachi, K. Soejima, N. Watanabe, A. Yagishita, N.A. Cherepkov, Phys. Rev. Lett. 80 (1998) 1622; E. Shigemasa, J. Electron Spectrosc. Relat. Phenom. 88±91 (1998) 9. O. Gessner, F. Heiser, N.A. Cherepkov, B. Zimmermann, U. Becker, J. Electron Spectrosc. Relat. Phenom. 101±103 (1999) 113. H.A. Bethe, E.E. Salpeter, Quantum Mechanics of Oneand Two-Electron Atoms, Springer, Berlin, 1957. J. Berkowitz, Photoabsorption, Photoionization, and Photoelectron Spectroscopy, Academic, New York, 1979. A. Bechler, R.H. Pratt, Phys. Rev. A 39 (1989) 1774; A. Bechler, R.H. Pratt, Phys. Rev. A 42 (1990) 6400. J.W. Cooper, Phys. Rev. A 42 (1990) 6942; J.W. Cooper, Phys. Rev. A 47 (1993) 1841. A. Derevianko, W.R. Johnson, K.T. Chen, At. Data Nucl. Data Tables 73 (1999) 153. J.C. Tully, R.S. Berry, B.J. Dalton, Phys. Rev. 176 (1968) 95. D. Dill, J.L. Dehmer, J. Chem. Phys. 61 (1974) 692. D. Dill, J. Chem. Phys. 65 (1976) 1130. J.W. Davenport, Phys. Rev. Lett. 36 (1976) 945.

[19] D. Dill, J. Siegel, J.L. Dehmer, J. Chem. Phys. 65 (1976) 3158. [20] N.A. Cherepkov, S.K. Semenov, K. Ito, S. Motoki, Y. Yagishita, Phys. Rev. Lett. 84 (2000) 250. [21] H. Bethe, Ann. Phys. 4 (1930) 433; G. Breit, H. Bethe, Phys. Rev. 93 (1954) 888. [22] L.A.D. Siebbeles, J.M. Schins, W.J. van der Zande, J.A. Beswich, Chem. Phys. Lett. 187 (1991) 633. [23] J.C. Arce, Quantum Spectra and Dynamics, Ph.D dissertation, Indiana University, Bloomington, 1992. [24] U. Fano, J.W. Cooper, Phys. Rev. 137 (1965) A1364. [25] T.J. Gil, C.L. Winstead, P.W. Langho€, Comput. Phys. Commun. 53 (1989) 123. [26] J. Cooper, R.N. Zare, J. Chem. Phys. 48 (1968) 942. [27] A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton UP, Princeton, NJ, 1957. [28] R.G. Newton, Scattering Theory of Waves and Particles, Springer, Berlin, 1982. [29] R.F. Barrett, L.C. Biedenharn, M. Danos, P.P. Delsanto, W. Greiner, H.G. Washweiler, Rev. Mod. Phys. 45 (1973) 44. [30] D. Loomba, S. Wallace, D. Dill, J.L. Dehmer, J. Chem. Phys. 75 (1981) 4546. [31] N. Satio, I.H. Suzubi, Phys. Rev. Lett. 61 (1988) 2760. [32] K. Lee, D.Y. Kim, C.I. Ma, D.A. Lapiano-Smith, D.M. Hanson, J. Chem. Phys. 93 (1990) 7936. [33] E. Shigemasa, K. Ueda, Y. Sato, T. Sasaki, A. Yagishita, Phys. Rev. A 45 (1992) 2915. [34] D.W. Lindle, C.M. Truesdale, P.H. Kobrin, T.A. Ferrett, P.A. Heimann, U. Becker, H.G. Kerkho€, D.A. Shirley, J. Chem. Phys. 81 (1984) 5375. [35] B. Kempgens, A. Kivimaki, M. Neeb, H.M. K oppe, A.M. Bradshaw, J. Feldhaus, J. Phys. B 29 (1996) 5389.