3188
J. Phys. Chem. 1984,88, 3188-3196
We shall now consider p ( t ) :
p(t) = -iLU(t,t?
p(t3
(B6) Again multiplying both sides by P ( t ) and using eq B1 we get P(t) p ( t ) = -iP(t)L U(t,t? P(t? p(t?
(B7)
Multiplying both sides by A,+ from the left and taking a trace,
where we have used eq 7. Equations B5 and B8 form the basis for our REM. We now solve eq B5 for a(t3 and substitute it in eq B8 resulting in bm(t) =
-ic( (AmILU(t,t?ldt?An))(( (AIU(tJ?ldt?A)
)-‘)nlUl(t)
n.1
FEATURE ARTICLE Photoelectron Dynamics of Moleculest V. McKoy,* Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena. California 91 125
T. A. Carlson, Oak Ridge National Laboratory, Chemistry Division, Oak Ridge, Tennessee 37830
and R. R. Lucchese Department of Chemistry, Texas A & M University, College Station, Texas 77843 (Received: February 17, 1984)
Unlike earlier studies of photoelectron spectroscopy which were carried out with traditional light sources, the availability of intense tunable radiation provided by synchrotron sources has made possible remarkable progress in the characterization of the important dynamical aspects of molecular photoionization. In this article we review both the current experimental thrust in molecular photoionization and the theoretical models which are being developed to adequately describe the important new features which emerge from these experiments. Progress in this area continues to reveal the rich dynamical content of molecular photoionization.
Introduction Molecular photoelectron spectroscopy came into prominence with the studies of Turner and his co-workers’ in the early 1960s. The main purpose of these studies of photoelectron spectroscopy was to characterize a molecule in terms of a simple energy level scheme based on molecular orbitals. Although these earlier studies contributed significantly to our understanding of both molecular electronic structure and of photoelectron dynamics,2 they were carried out with traditional light sources and hence did not provide photoelectron spectra over a continuous range of photon energy. Measurements of these photoelectron spectra over a continuous and wide range of photon energies are clearly needed to characterize the dynamical aspects of the molecular photoionization process. Synchrotron radiation provides the intense tunable source of photons needed to study the continuous variation of atomic and molecular photoionization cross sections with photon energy. The increasing availability of synchrotron radiation, coupled with the advent of high-resolution, angle-resolving electron spectrometers, is making it possible to study the structure and dynamics of the electronic continua of atoms and molecules at a highly differentiated level.3 Such experiments, along with related theoretical developments, have led to remarkable progress in our under+Contribution No. 6974.
0022-3654/84/2088-3188$01.50/0
standing of the dynamics of molecular photoionization processes. The main purpose of this article is to review the highlights of recent progress in the experimental and theoretical studies of molecular photoionization. This article is not intended to present a detailed account and critical assessment of developments in this field but is written in the form of an informative survey of both the current experimental thrust in molecular photoionization and the theoretical models which are being developed to adequately describe the important new features which are emerging from these experiments. We shall see that shape resonances or quasi-bound states in which an electron is temporarily trapped by a potential barrier play an important role in molecular photoionization and lead to very pronounced dynamical features in the photoelectrom ~ p e c t r u m . ~Although similar barriers are known in atomic systems, the nonspherical force fields of molecules can lead to very (1) D. W. Turner, A. D. Baker, C. Baker, and C. R. Brundle, “Molecular Photoelectron Spectroscopy: A Handbook of He 584 A Spectra”, Wiley-Interscience, New York, 1970. (2) See, for example, J. Berkowitz and W. A. Chupka, J . Chem. Phys. 51, 2341 (1969). (3) M. 0. Krause in “Synchrotron Radiation Research”, H. Winick and S. Doniach, Eds., Plenum Press, New York, 1980, p 101. (4) J. L. Dehmer, D. Dill, and A. C. Parr in “Photophysics and Photochemistry in the Vacuum Ultraviolet”, S. McGlynn, G. Findley, and R. Huebner, Eds., Reidel Publishing Co., Holland, in press.
0 1984 American Chemical Society
Feature Article
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3189
high angular momentum characeter in molecular barriers. It is not surprising that these shape resonances, along with autoionizing resonances, are the focal p i n t of many of the current experimental and theoretical studies of molecular photoionization. Photoelectron Spectroscopy and the Historical Background of the Experimental Basis of Photoelectron Dynamics Photoelectron dynamics as an experimental measurement is based on photoelectron spectroscopy. Utilizing a line source of radiation, earlier studies measured the energy spectra of photoelectrons ejected from the valence shell of gaseous molecules.’ From each spectra one is able to extract the ionization potentials, IP, from the simple relationship between the photoelectron kinetic energy, E K , and the photon energy, hv; Le., IP = hv - E K . Compilation of such results may be found in the 1iterature.l~~ Within the molecular orbital framwork each ionization potential represents the ejection of an electron from a different orbital. One actually observes a band made up of vibrational and rotational levels. In photoelectron spectroscopy the rotational levels are rarely resolved, but the vibrational structure is frequency observed and may be used for characterizing the photoionization process. In most instances the vibrational structure represents the vibrational spacings in the final state, because with the exception of a limited number of cases, a molecule at room temperature is usually in its ground vibrational level. The observed envelope is generally the result of a Franck-Condon transition since photoionization takes place in a time short compared to the time scale for nuclear motion, and the transition probability for the vibrational states depends solely on the overlap of the initial and final wave functions. Ionization potentials are generally reported either as adiabatic, which is equivalent to a transition to the ground state vibrational level for a given electronic transition, or vertical, which is the measured maximum of the vibrational band. A number of monographs6 have been written on the field of photoelectron spectroscopy, and it will not be our purpose here to further review this rather extensive field. However, it should be noted that, experimentally, photoelectron spectroscopy does more than provide information on the energies for the photoionization processes; it also separates the various processes, allowing one to study the relative cross sections for each individual process by means of the observed intensity. It is this measurement of the relative cross sections (partial cross sections) and cross section as a function of angle between the polarization vector and direction of the ejected photoelectron that makes up the experimental basis for studying photoelectron dynamics. Early results on partial cross sections and angular distribution parameters were limited to a few available line sources such as the resonance lines of N e I(16.67, 16.85 eV), He I(21.21 eV), and He I1 (40.8 eV) and the X-rays of Z r M{ (151.4 eV), Mg Ka,,, (1254 eV), and A1 Ka,,, (1487 eV). By comparing partial cross sections of valence orbitals measured with H e I to those with X-rays, Gelius et ale7were able to correlate the extent of atomic s and p character making up different molecular orbitals. In addition to partial cross sections, experiments were carried out on the intensities of ejected photoelectrons measured as a function of the angle of ejection. In the gaseous state, where the molecules are randomly oriented, this distribution takes the form of du/dO = ( u / 4 a ) [ l + @P,(cos e)] (1) where 8 is the angle between the polarization vector and the direction of the ejected photoelectron and P2is a Legendre polynomial. @ can be determined with unpolarized radiation, but ( 5 ) K. Kimura, S. Katsumata, Y. Achiba, T. Yamazaki and S. Iwata, “Handbook of He1 Photoelectron Spectra of Fundamental Organic Molecules”, Halsted Press, New York, 1981. (6) T. A. Carlson, “Photoelectron and Auger Spectroscopy”,Plenum Press, New York, 1975; J. H. D. Eland, “Photoelectron Spectroscopy”, WileyHalsted, New York, 1974; J. W. Rabalais, “Principles of Ultraviolet Photoelectron Spectroscopy”, Wiley, New York, 1977; J. Berkowitz, “Photoabsorption, Photoionization, and Photoelectron Spectroscopy”, Academic Press, New York, 1979. (7) U. Gelius in “Electron Spectroscopy”, D. A. Shirley, Ed., NorthHolland Publishing Co., Amsterdam, 1972, p 311.
use of polarization enhances the sensitivity of the measurement. The angular distribution parameter reflects the nature of the orbital from which the electron is ejected and is dependent on the photon energy because of the change in the interaction of the continuum channels into which the photoelectron is ejected. A study of @ as a function of photon energy is an ideal complement to data on the partial cross sections. A number of interesting generalizations were discovered in the study of @ using line sources, but interpretation was restricted. Despite the usefulness and historical importance that line sources have given the field of photoelectron dynamics, the measurements of cross sections require studies over a continuous wide range of photon energies in order to make realistic comparisons with theory, to make generalizations as to orbital behavior, or to investigate resonances. Such a source of photons is synchrotron radiation, and it will be the main task of the experimental portion of this paper to discuss recent results obtained with this source of radiation. Recent Experiments with Synchrotron Radiation To effectively carry out an experimental program on the partial cross sections and angular distribution parameters, one requires a source of photons whose energies are continuous over a wide range, and to be practical, one must also be able to pass the photons through a monochromator and thus vary the wavelength at will. Such a photon source must be also of sufficient intensity that one can extract a beam of reasonably well-resolved wavelength yet carry out angle-resolved photoelectron spectroscopy. Such a source, thus, must be quite intense. If the source of photons is polarized, this will facilitate measurement of the angular distribution parameters. The photon source that will accomplish the above goals is synchrotron radiation. Synchrotron radiation arises from acceleration of high-energy electrons under vacuum. Facilities dedicated to the use of synchrotron radiation are usually storage rings in which electrons are circulated at energies in the neighborhood of 1 GeV to produce usable radiation up to energies of about 1 keV. In addition, wigglers and undulators are being increasing used to greatly enhance the intensities and energies of the photon source. For experimental studies on photoelectron dynamics, it is often desirable to make measurements at energies considerably above the ionization threshold for the valence shells (the order of 100 eV or more). In addition, one wishes to study effects above the threshold of core-shell binding energy; in the case of the elements in the second row of the periodic table (carbon, nitrogen, oxygen, and fluorine), this implies a need for photons up to nearly 1 keV. Special studies on the deeper core shells of heavier elements may require still higher energy photons, but most of the experimental needs of photoelectron dynamics will be satisfied with a range of energies from 10 to 1000 eV. It is necessary to have not only a storage ring that will deliver photons with the appropriate energy range but also a monochromator. For energies below 1 keV, some sort of movable grating is normally required. Monochromators are designed for efficiency, resolution, or a particular energy range. The choice of parameters depends on the experimental needs and response. Further information on synchrotron radiation and monochromators can be found in ref 8. The basic nature of the experimental procedures used in angle-resolved photoelectron spectroscopy is outlined below. In these experiments a beam of monochromatic photons passes through a source volume. The target gas is allowed to leak into the source volume and maintained at pressures approximately 10-3-104 torr. Differential pumping keeps the pressure in the first chamber before the monochromator in the region of torr so that windowless experiments with gases are possible without contamination of the monochromator or storage ring. Most commonly, the ejected photoelectrons are allowed to emerge from a field-free source volume, which fixes the angle of emergence. The electrons then (8) H. Winick and S. Doniach, Eds., “Synchrotron Radiation Research”, Plenum Press, New York, 1980.
3190
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 CCI, (200
I
I
I
approximation, p can be extracted from such data as shown in Figure 1 by the relationship 4(R - 1 ) (2) = 3P(R + 1) - ( R - 1)
(hv=30eV)
I
I
l
e = 90"
9 = 0 " 200sec
l
1
600 sec
io00
where R is Z(Oo)/Z(900) and Z(Oo) and Z(90°) are the intensities of photoelectrons moving in the direction parallel and perpendicular to the polarization vector. P is the polarization of the photon source. It depends on both the nature of the synchrotron radiation and how it is influenced by the monochromator. Integrated angular cross sections as a function of photoelectron energy are extracted from the relationship
800
c u)
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McKoy et al.
600
400
200
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12
i4
i6
18
20
42
t4
16
18
20
uI P ( e V ) 18
16
14
12
10
48
16
14
l2
(0
Figure 1. Typical data taken in angle-resolved photoelectron spectroscopy using synchrotron radiation. Three variables are under consideration: (1) the photon energy which is set at 30 eV, (2) the angular distributions measured at 0 = 0 and 90°, and (3) the energy of the photoelectrons, whose spectra reveal the first four bands associated with the first four orbitals of CC1,.
either pass directly into an electron spectrometer or pass first through a transfer lens to the spectrometer. The electrons are frequently accelerated or deaccelerated before energy analysis. A variety of electrostatic analyzers9-" have been successfully employed in angle-resolved photoelectron spectroscopy, since once the solid angle has been defined, there is little preference for which dispersion system is employed. For angular studies with synchrotron radiation, it is desirable to place the analyzers on a rotatable platform perpendicular to the synchrotron radiation beam. By having two spectrometers so mounted, one can simultaneously record intensities at two angles, obviating problems of changes in pressure and light intensity. For randomly oriented molecules, measurements at two angles are sufficient for determining the angular distribution parameter, p. If only angleintegrated spectra are required, spectrometers with a large acceptance angle are desirable, such as the cylindrical mirror analyzer.I2 Use of position-sensitive detectors for those analyzers having a substantial focal plane can allow for simultaneous energy analysis, thereby also increasing the efficiency enormo~sly.'~Time-of-flight analyzers" can be particularly valuable, if the synchrotron source puts out short pulses of radiation suitable for efficient triggering. Such analyzers can be highly efficient since the entire energy spectrum is susceptible to analysis for each pulse. Typical photoelectron spectra are shown in Figure 1. A pair of spectra are taken at two different angles, 0, where 0 is the angle between the polarization vector and the direction of the ejeced photoelectron. These spectra are taken as a function of photon energy by adjusting the wavelength of light allowed to pass through the monochromator. Thus, three variables are studied: (1) the energy of the photoelectron, (2) the energy of the photon, and ( 3 ) the intensity as a function of angle 6. From these three experimental variables, two vital pieces of information were obtained as a function of photon energy: the partial cross sections and the angular distribution parameter, p. Within the dipole (9) M. 0. Krause, T. A. Carlson, and P. R. Woodruff, Phys. Rev. A , 24, 1374 (1981). (10) A. C. Parr, R. Stockbauer, B. E. Cole, D. L. Ederer, J. L. Dehmer, and J. B. West, Nucl. Instrum. Methods, 37, 172 (1980). (11) M. G. White, R. A. Rosenberg, G. Gabor, E. D. Poliakoff, G. Thornton, S. H. Southworth, and D. A. Shirley, Rev. Sci. Instrum., 50, 1268 (1979). (12) H. 2.Zar-el, Rev. Sci. Instrum., 38, 1210 (1967). (13) C. D. Moak, S. Datz, F. Garcia-Santibaiiez, and T.A. Carlson, J . Electron Spectrosc. Relat. Phenom., 6, 151 (1975).
where the intensity Z(Oo) is the area of the photoelectron peak taken at Oo, monitored for change in the photon beam and pressure, f ( S ) is a correction for the transfer lens and the spectrometer transmission, and f ( N / N , ) is a correction for the light intensity as a function of wavelength. The second term is a correction for the angular distribution. When comparing angle-integrated intensities, it is frequently convenient to choose the magic angle, which for a given polarization is equal to cor1 ( - 1 / ( 3 P ) ) . Measurements made at the magic angle are independent of the value of 8. To convert relative cross sections as a function of photon energy to a set of absolute cross sections, an absolute cross section at only one energy is required. Absolute cross sections can be measured with an absolute pressure measuring device and a standard gas whose partial cross sections are well-known. As can be seen from Figure 1 , the different photoelectron bands corresponding to ejection of electrons from the various orbitals can be individually studied for their partial cross sections and p values. In addition, vibrational bands usually arising from a distribution of vibrational levels in the final state of the ion can be studied individually, if resolution permits. Other final-state effects that result in the energy separation of different bands in the photoelectron spectrum, such as multiplet splitting due to spin-orbit coupling, the Jahn-Teller effect, and multielectron excitation, can also be studied separately. Survey of Literature. The literature of angle-resolved photoelectron spectroscopy using synchrotron radiation is still limited, but it is growing very rapidly. No attempt will be made to cover atomic systems except to refer to some reviews.I4 In the following sections we shall give a partial listing of experimental studies made on various molecules. This list hass been restricted to angle-resolved photoelectron spectroscopy. Additional references may be found in a recent review by Dehmer et aL4 Small molecules such as diatomic and triatomic molecules have received most of the initial attention. Extension to more complex polyatomic systems is, however, already under way. Interlacing studies on a variety of molecular systems are investigations on a number of various phenomena that characterize the behavior of partial cross sections and angular distribution parameters. We shall now proceed to discuss some of the more important ones. Overall Photon Energy Dependence. One of the first characteristics to evaluate in the photoelectron dynamics of molecules is the overall photon energy dependence of the partial cross sections and angular distribution parameters for the individual orbitals. These energy profiles become a characteristic description for a given molecular orbital. In particular, it is important to make direct comparisons between experiment and theory. This has been done in the case of diatomic molecules for CO,I5N2,15H2,16Cl,," (14) F. J. Wuilleumier, At. Phys., 7, 491 (1981); B. Sonntag and F. Wuilleumier, Nucl. Instrum. Methods, 208, 735 (1983). (15) G. V. Marr, J. M. Morton, R. M. Holmes, and D. G. McCoy, J . Phys. B, 12, 43 (1979). (16) S. Southworth, W. D. Brewer, C. M. Truesdale, P H. Kobrin, D. W. Lindle, and D.A. Shirley, J . Electron Spectrosc., Relat. Phenom., 26, 43 (1982). (17) T. A. Carlson, M. 0. Krause, F. A. Grimm, and T. A. Whitley, J . Chem. Phys., 78, 638 (1983).
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3191
Feature Article 2,
I
I
I
I
2.0
1
r
I
0
(0
I
I
1
I
I
I
60
70
1.6
L
0,
-
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1.2
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0.8
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2
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1’
0.4
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I 30
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40
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50
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Photon Energy ( e V )
Figure 2. Comparison of the multiple-scattering results from ref 63 with accurate Hartree-Fock results from ref 62 for the photoelectron asymmetry parameter for photoionization leading to the C*Z8+state of C02’: (-), asymmetry parameter averaged over the symmetric stretch vibrational mode; (---), equilibrium fixed-nuclei asymmetry parameter; ( O ) , experimental data of ref 31; (H), experimental data of Katsumata and co-workers ( J . Electron Spectrosc. Relat. Phenom., 17, 229 (1979)).
HCl,’* and NO,19for triatomics COz,20CS2,21N20,22,23 and H20,24 and for the simple organic molecules acetylene:5 ethylene,26 and ben~ene.~’In making these comparisons, one should remember that calculations based on one-electron theories do not include the effects of autoionization. One-electron theories include the effects of shape resonances and the Cooper minimum, and these effects will be discussed later. In comparing theory and experiment, it is more important to look for the overall energy dependence of these cross sections than for point by point quantitative agreement. For the studies mentioned a b o ~ e , I ~ -reasonable ~’ success has been achieved with approximate methods in most cases even for the larger molecules. The energy dependence of these cross sections can often be classified according to the type of orbital involved. In particular, the photon energy dependence of /3 has been measured for a orbitals in a wide variety of unsaturated hydrocarbon^.^^-^* In the range of photoelectron energies from 2 to 10 eV, it was found that the /3 values for P orbitals characteristically rose quickly with increasing energy and then leveled off. This behavior contrasted with that of the u orbitals in the same molecules. For example, in a study of azabenzenesZ9it was possible to use the contrasting behavior of the a orbitals and nitrogen nonbonding orbitals to assign the ordering of ionization potentials from analysis of an (18) T. A. Carlson, M. 0. Krause, A. Fahlman, P. R. Keller, J. W. Taylor, T. Whitley, and F. A. Grimm, J . Chem. Phys., 79, 2157 (1983). (19) S.Southworth, C. M. Truesdale, P. A. Kobrin, D. W. Lindle, W. D. Brewer, and D. A. Shirley, J. Chem. Phys., 76, 143 (1982). (20) F. A. Grimm, J. D. Allen, Jr., T. A. Carlson, M. 0. Krause, D. Mehaffy, P. R. Keller, and J. W. Taylor, J. Chem. Phys., 75, 92 (1981). (21) T. A. Carlson, M. 0. Krause, and F. A. Grimm, J . Chem. Phys., 77, 1701 (1982). (22) T. A. Carlson, P. R. Keller, J. W. Taylor, T. Whitley, and F. A. Grimm, J . Chem. Phys., 79, 97 (1983). (23) C. M. Truesdale,S.Southworth, P. H. Kobrin, D. W. Lindle, and D. A. Shirley, J . Chem. Phys., 78, 7117 (1983). (24) C. M. Truesdale, S. Southworth, P. H. Kobrin, D. W. Lindle, G. Thornton, and D. A. Shirley, J. Chem. Phys., 76, 860 (1982). (25) P. R. Keller, D. Mehaffy, J. W. Taylor, F. A. Grimm, and T. A. Carlson, J . Electron Spectrosc. Relat. Phenom., 27, 223 (1982). (26) D. Mehaffy, P. R. Keller, J. W. Taylor, T. A. Carlson, H. 0. Krause, F. A. Grimm, and J. D. Allen, Jr., J. Electron Spectrosc. Relat. Phenom., 26, 213 (1982). (27) D. Mehaffy, P. R. Keller, J. W. Taylor, T. A. Carlson, and F. A. Grimm, J. Electron Spectrosc. Relat. Phenom., 28, 239 (1983). (28) P. R. Keller, J. W. Taylor, T. A. Carlson, and F. A. Grimm, J . Electron Spectrosc. Relat. Phenom., in press. (29) M. N. Piancastelli, P. R. Keller, J. W. Taylor, F. A. Grimm, and T. A. Carlson, J . Am. Chem. Soc., 105, 4235 (1983).
0
20 30 40 50 PHOTOELECTRON ENERGY ( e V )
Figure 3. Plot of p as a function of photoelectron energy for the first four
orbitals of CCI4 and, for comparison, the 3p subshell of argon. See ref 42. Note the dip at about 30-40 eV which is due to a Cooper minimum.
imcompletely resolved photoelectron band in pyridine. Shape Resonances. The effects o f shape resonances may be examined experimentally by several different means. There is, of course, the resonance itself an increase in cross section, viewed as a broad peak spread over as much as several volts. This is usually accompanied by a change in /3 values as a function of photon energy over roughly (but not exactly) the same energy region as the resonance. There can also be profound effects on the vibrational structure, with regard to both changes in the branching ratios and changes in the /3 values for the individual vibrational bands. A nice example for study is the behavior of /3 for the fourth level of COz (Figure 2). A deep dip in the fl-dependence curve was initially predicted3’ in a fixed-nuclei calculation using the continuum multiple-scattering model. Experiment20,31shows clearly the effect of a shape resonance (Figure 2), but the dip in the /3-dependence curve is more shallow and much broader than theory. We will return to a discussion of this resonance in a later section where we will see that experimental studies of effects of such resonances offer a constructive challenge to theory. Other striking examples of the effects of shape resonances have been studied for N2OZ2and N2.32 Autoionization. In autoionization, photoionization takes place via absorption into a discrete excited state followed by decay into an adjacent continuum. The electrons ejected by autoionization carry away the energy difference between the initial ground- and final-state ion. This energy difference can be exactly that corresponding to direct photoionization. Thus, the two processes are often indistuinguishable in terms of the kinetic energy of the ejected electrons. However, the branching ratios for the different final vibrational states, the cross sections, and the /3 values are usually quite different, being dependent on entirely difference processes. Using synchrotron radiation, we have studied several different C0,3402,35 and Nz022)for their cross section molecules (e.g., N2,33 and /3 values for the varioius vibrational levels passing through autoionizing resonances. Dramatic, sharp changes as a function (30) F. A. Grimm, T. A. Carlson, W. B. Dress, P. Agron, J. 0. Thomson, and J. W. Davenport, J . Chem. Phys., 72, 3041 (1980). (31) T. A. Carlson, M. 0. Krause, F. A. Grimm, J. D. Allen, Jr., D. Mehaffy, P. R. Keller, and J. W. Taylor, Phys. Reu. A , 23, 3316 (1981). (32) J. B. West, A. C. Parr, B. E. Cole, D. L. Ederer, R. Stockbauer, and J. L. Dehmer, J . Phys. B, 105, L13 (1980); T. A. Carlson, M. 0. Krause, D. Mehaffy, J. W. Taylor, F. A. Grimm, and J. D. Allen, Jr., J . Chem. Phys., 73, 6058 (1980). (33) A. C. Parr, D. L. Ederer, B. E. Cole, J. B. West, R. Stockbauer, K. Codling, and D. L. Dehmer, Phys. Reu. Lett., 46, 22 (1981). (341 D. L. Ederer, A. C. Parr, B. E. Cole, R. Stockbauer, J. L. Dehmer, J. B. West, and K. Codling, Proc. Roy. Soc. London, Ser. A , 378,423 (1981). (35) K. Codling, A. C. Parr, D. L. Ederer, R. Stockbauer. J. B. West, B. E. Cole, and J. L. Dehmer, J . Phys. B, 14, 657 (1981).
3192 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
of photon energy are frequently seen. Broader features due to autoionizing resonances have also been observed in and other unsaturated In the case of acetylene, this feature is almost certainly due to autoionization of the "20, lrgn valencelike state. The nature of this resonance feature has been discussed by several a ~ t h o r s . ~ ~ ~ ~ ~ - ~ ~ Cooper Minimum. The phenomenon of a Cooper minimum has been known for some time in atomic photoionization.4 The Cooper minimum occurs at a photoelectron energy for which the matrix elements describing a transition into a given continuum channel change sign. If the channel undergoing a Cooper maximum makes up the dominant contribution to the cross sections, the net cross section will show a distinct minimum. Moreover, the /3 value is strongly affected near the Cooper minimum; in fact, the behavior in is often experimentally a more sensitive test of the Cooper minimum than is the cross section. First-row atoms such as nitrogen and neon do not show Cooper minima in photoionization out of the 2p shell since a change in sign of the matrix element requires a node in the orbital.41 However, the 3p, 4p, and 5p orbitals do have nodes and elements such as sulfur, chlorine, argon, bromine, krypton, iodine, and xenon do exhibit Cooper minima for their outermost shell.41 Furthermore, of the two continuum channels available for photoionization of a p orbital, the d channel undergoes a Cooper minimum and the s channel does not. At the Cooper minimum only the s continuum channel is available, and hence the angular distributions will be isotropic, Le., /3 = 0, in the absence of relativistic and electron correlation effects. It seems plausible that lone-pair orbitals of molecules might behavior like their atomic counterparts and show Cooper minima. The molecular Cooper minimum has been studied for a variety of molecules containing lone-pair orbitals derived from sulfur?I c h l ~ r i n e , ~bromine,"2 ~ * l ~ ~ ~and ~ iodine.43 Typical results are seen in Figure 3. It can be seen that the lone-pair orbitals derived from chlorine behave quite similarly to the 3p orbital of argon. More surprisingly, orbitals that do not have lone-pair characters but are bonding also show effects near the Cooper minimum. Calculations based on the multiple-scattering X a method have done well in predicting the qualitative behavior of both the cross sections and /3 values associated with these Cooper minima,17-18,21,42,43 including the effects on the bonding orbitals. Closer examination of these calculations show that in some orbitals, e.g., the 2~ of HC1,I8 2tl of CC14,42and 2 ~ of, CS2?l certain molecular channels play roles closely akin to the atomic channels. In contrast, some lone-pair orbitals have nonnegligible contributions to the cross section that cannot be explained within the atomic framework. It should be remembered that 1 is not a good quantum number for molecular orbitals and that both the initial ground-state and final continuum wave functions for molecules can be represented by a sum of partial waves. In practice, only a few of these make sizable contributions to the photoelectron cross sections. Further studies are needed to clarify the nature of the Cooper minimum in molecular systems. In this regard the behavior of the Cooper minimum is being investigated through a series of homologous compounds such as the hydrogen and the CCl,Fe+, f r e o n ~ . ~ ~ Satellite Structure and Configuration Interaction. The main or principal bands in a photoelectron spectrum correspond to single -+
(36) R. Unwin, I. Khan, N. V. Richardson, A. M. Bradshaw, L. S. Cederbaum, and W. Domcke, Chem. Phys. Lett., 77, 242 (1981). (37) A. C. Parr, D. W. Ederer, J. B. West, D. M. P. Holland, and J. L. Dehmer, J . Chem. Phys., 76,4349 (1982). (38) Z. H. Levine and P. Soven, Phys. Rev. Lett., 50, 2074 (1983). (39) D. Lynch, M.-T. Lee, R. R. Lucchese, and V. McKoy, J . Chem. Phys., in prebs. (40) J. W. Cooper, Phys. Rev., 128,681 (1962); S. T. Manson and J. W. Cooper, ibid., 165, 126 (1968). (41) S. T. Manson, J . Electron Spectrosc. Relat. Phenom., 1, 413 (1973). (42) T. A. Carlson, M. 0 Krause. F. A. Grimm, P. Keller, and J. W. Taylor, J . Chem. Phys., 77, 5340 (1982). (43) T. A. Carlson, A. Fahlman, M. 0. Krause, P. R. Keller, J. W. Taylor, T. Whitley, and F. A. Grimm, J . Chem. Phys., in press. (44) A. Svensson, T. A. Carlson, T. Whitley, and F. A. Grimm, to be submitted for publication.
McKoy et al. electrons ejected from the different bound orbitals in a molecule. In addition, there exists the satellite structure which arises primarily from multiple-electron excitations and is hence shifted from the main band. Satellite structure is effectively studied by synchrotron radiation. Studies in the past with line sources have been useful but limited? Soft X-rays have been used on both the valence and core shells, but the line width of the source is rather broad and cross sections at high energy are rather poor. The He I source (21.22 eV) is not of sufficient energy to reach much of the satellite structure, and the H e I1 source (40.8 eV) has to contend with H e I contamination. In addition, it is important to follow the relative intensities of the satellite structure in order to characterize the origins of the satellite structure. For example, the fifth band in CSz was studied as a function of photon energy2' and the partial cross sections as a function of photon energy were seen to be in reasonable agreement with the assumption that the band arose from electron shake-up as the result of photoionization of the 2 ~ , 0 r b i t a 1 . ~ ' ~Synchrotron ~~ radiation has also been applied to the study of satellite structure in the photoelectron spectra of N26 and It is important to note that the total contribution of the satellite structure is not negligible. For some inner valence orbitals, electron correlation effects are so large that single-electron ejection occurs in a minority of events. Multielectron excitations clearly play a very important role in photoelectron dynamics. Threshold Effects on Core Shells. One might initially conceive of the behavior of core-shell electrons of molecules to be principally like that of atomic orbitals. At higher photoelectron energies this is indeed essentially true. Near ionization thresholds, however, the emerging photoelectrons can be profoundly affected by the molecular potential. In addition, threshold effects are greatly affected by the presence of shape resonances. Calculations have shown that there can be profound differences both in the cross section and values dependent on the molecule and from which atomic core shell the electron is eje~ted.~-~O An important feature of core-shell studies is that one can study the molecular potential of a given molecule from different vantage points in the molecule. For example, C O and COz can be examined first by ejecting a 1s electron from carbon and then from oxygen.51,52 Additional studies have been carried out on the 3d shell of bromine comp o u n d ~ the , ~ ~2p shell of silicon compounds,54and the 2p shell of sulfur.52
Theoretical Study of Photoelectron Dynamics Physical Picture. The theoretical study of molecular photoionization requires the consideration of both electronic and nuclear motion. As with bound-sate molecular systems, the Born-Oppenheimer approximation is generally valid, and hence the total molecular wave function can be factored into separate electronic and nuclear parts. In this section we shall discuss the current capabilities for the theoretical computation of the electronic part of the molecular photoionization problem and shall only give a brief description of some important vibrational effects which have been examined. Single-photon photoabsorption, and in particular photoionization, is usually studied by assuming that the photon field can 45) J. Schirmer, W. Domcke, L. S. Cerderbaum, W. von Niessen, and L. isbrink, Chem. Phys. Lett., 61, 30 (1979). (46) S. Krummacher, V. Schmidt, and F. Wuilleumier, J . Phys. B, 13, 3993 (1980); F. A. Grimm and T. A. Carlson, Chem. Phys., 80, 389 (1983). (47) S. Krummacher, V. Schmidt, F. Wuilleumier, J. M. Bizan, and D. Ederer, J . Phys. B, 16, 1733 (1983). (48) J. L. Dehmer and D. Dill, J . Chem. Phys., 65, 5327 (1976). (49) F. A. Grimm, Chem. Phys., 53, 71 (1980). (50) J. R. Swanson, D. Dill, and J. L. Dehmer, J. Chem. Phys., 75, 619 (1981). (51) C. M. Truesdale, S. H. Southworth, P. H. Kobrin, U. Becker, D. W. Lindle, H. G. Kerkhoff, and D. A. Shirley, Phys. Rev. Lett., 50, 1265 (1983). (52) C. M. Truesdale, D. W. Lindle, P. H. Kobrin, U. E. Bedser, H. G. Kerkhoff, P. A. Heimann, T. A. Ferrett, and D. A. Shirley, submitted for publication in J . Chem. Phys. (53) T. A. Carlson, M. 0. Krause, F. A. Grimm, P. R. Keller, and J. W. Taylor, Chem. Phys. Lett., 87, 552 (1982). (54) P. R. Keller, J. W. Taylor, F. A. Grimm, P. Senn, T. A. Carlson, and M. 0. Krause, Chem. Phys., 74, 247 (1983).
Feature Article be treated classically. The photoabsorption cross section is then obtained from the first-order transition rate between the initial state and the final excited state. The initial target state in most cases is the ground electronic state of the neutral molecule, and the final ionized state has a photoelectron in the molecular electronic continuum. The initial target state can be obtained by using the usual methods of quantum chemistry, and hence the initial state is generally represented by a Hartree-Fock or configuration interaction (CI) wave function. The continuum orbitals of the final state are more difficult to obtain, and methods for computing such continuum states are of much current interest.55 These difficulties arise primarily from the nonspherical potentials of molecular ions and the necessity to include exchange effects in these potentials. To understand the nature of the problem of determining photoelectron continuum functions, one must first realize that standard quantum chemistry procedures cannot be used directly to solve the continuum problem. First of all, bound-state wave functions are square integrable (L2)and can thus be expanded in L2 basis functions, e.g., Slater functions or Cartesian Gaussian functions. In contrast to this, the continuum functions are not L2 and are in fact 6 function normalized. Thus, such continuum functions cannot be directly expanded in L2 basis sets. Secondly, for continuum states, Rayleigh-Ritz variational expressions for the energy cannot be used, and instead one must use variational expressions of the Kohn type based upon the differential equation form of the Schrodinger equation or variational expressions of the Schwinger type based upon the Lippmann-Schwinger integral equation form of the Schrodinger equation.55 However, the continuum electronic problem is amenable to the same physical approximations as is the bound-state problem. Thus, we can consider the independent-particle approximation, or HartreeFock approximation, as the simplest form of the many electron continuum wave functions. This approximation can be improved upon by including electron correlation effects. This very natural scheme of approximation is reflected in the physics of the photoionization process. In particular, the classification of the resonant processes which occur in photoionization directly reflects this approximation scheme. There are two main resonant processes in photoionization which are classified as oneand two-electron resonances. A one-electron resonance is due to only the average potential which the photoelectron feels due to the other electrons in the molecule. When this independentparticle or Hartree-Fock potential for the photoelectron contains resonant states which are above the ionization threshold in energy, these manifest themselves as “shape” resonances. The name shape resonance comes from the fact that the resonance is produced by the shape of a single-particle potential. The origin of this name is particularly clear when the single-particle potential is a local potential. Two-electron resonances in photoionization are due to discretelike states which can be excited directly. These autoionizing states lie above the threshold for ionization and thus decay into continuum target states via two-electron processes. As is clear from the nature of these processes, a Hartree-Fock level description of the photoionization is suitable for the one-electron resonances, but electron correlation must be included in the study of the two-electron resonances. Model Potentials. Most studies of molecular photoionization cross sections have been performed using the independent-particle model. Among these, the Hartree-Fock model plays a very important role. The Hartree-Fock potential, known also as the static-exchange potential, contains both a static or local potential and an exchange or nonlocal potential. In many methods currently used, the inclusion of the exchange terms in the Hartree-Fock potential greatly increases the computational effort needed to obtain a photoionization cross section. To reduce the computational effort, there have been several methods developed to approximate the exchange part of the potential by a local potential. (55) See, for example, “Electron-Molecule and Photon-Molecule Collisions”, T. N. Rescigno, B. V. McKoy, and B. Schneider, Eds., Plenum Press, New York, 1979.
l5
t
Ob---
20
30
40
50
Photon Energy ( e V )
Figure 4. Comparison of MSM results from ref 61 with the HartreeFock results of ref 6 2 for the photoionization cross section of CO,leading to the CZZgt state of C02+: (-), cross section averaged over the symmetric stretch vibrational mode; (---), equilibrium fixed-nuclei cross section; (e),experimental results of Brion and Tan (Chem. Phys., 34, 141 (1978)).
The most effective local approximation to the exchange potential in electron-molecule-scattering problems is the Hara free-electron gas potential.56 This potential has been widely used in electron neutral scattering but has not yet been tested in photoionization calculations. The Slater exchange potential has been used in a study of photoionization, but detailed comparisons with exact static-exchange calculations have not yet been made.57 Another type of model potential is that of the multiple-scattering method (MSM) which has been used extensively to obtain photoionization cross sections.58 The MSM approach is based on a very simplified model potential which can then be solved to yield readily photoionization cross sections. The MSM model potential is of the “muffin tin” type where the interaction potential is constructed from spherical regions inside of which the potential is either spherically symmetric or constant. The MSM also usually assumes a local approximation for the exchange potential. The MSM approach has the obvious advantage that the method can be applied to fairly large systems, and in general the MSM calculations are qualitatively very useful. In particular, the MSM method was used to obtain the first theoretical description of shape resonances in molecular photoioni~ation,~~ as well as the effect such resonances can have on the final vibrational-state distributions60 An example of a large molecular system studied by the MSM approach would be the study of benzene.27 In that study, the experimental asymmetry parameters were compared to those obtained by using the MSM approach. The MSM results for the first two channels, and (3e2J1, were in good agreement with experiment, but serious disagreement was found for the third (la& channel. On the basis of this disagreement and experimental asymmetry measurements of the photoionization of hexafluorobenzene, Mehaffy et al.27concluded that vibronic mixing was responsible for this disagreement between theory and experiment. In some cases, however, the quantitative predictions of this method can be misleading. The MSM can do poorly in predicting the absolute magnitude of cross sections, especially in the region of shape resonances. An example of this is the cross section predicted for the photoionization of C 0 2 leading to the C2Z,+ state of the ion.61 As shown in Figure 4, this shape resonance was found (56) (57) (58) (59) (60) (1979). (61) (1980).
M. A. Morrison and L. A. Collins, Phys. Reu. A , 23, 127 (1981). C. Duzy and R. S . Berry, J . Chem. Phys., 64, 2421 (1976). D. Dill and J. L. Dehmer, J . Chem. Phys., 61, 692 (1974). J. W. Davenport, Phys. Reu. Le??.,36, 945 (1976). J. L. Dehmer, D. Dill, and S . Wallace, Phys. Reu. L e f f . ,43, 1005
J. R. Swanson, D. Dill, and J. L. Dehmer, J . Phys. B, 13, L231
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The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
to be unphysically narrow in the fixed-nuclei approximation with the MSM approach, and only when vibrational averaging is included do the MSM results seem to be in acceptable agreement with experimental results. This is in contrast to the reasonable agreement between experiment and the accurate fixed-nuclei Hartree-Fock calculations.62 On the other hand, Figure 2 shows that the photoelectron asymmetry parameters predicted for this system by the MSM agree much better with accurate HartreeFock calculations and experimental measurements than do the corresponding absolute computed cross section^.^^,^^ Hartree-Fock Potentials. In recent years exact continuum Hartree-Fock equations for small molecules have been solved numerically. There are three commonly used approaches to solving the exact static-exchange molecular scattering problem. Firstly, there are single-center expansion methods where the scattering equations for the continuum final states are expanded in spherical harmonics, leading to a set of coupled integrodifferential equations for the radial wave functions.” Secondly, there are the Schwinger variational methods which can employ basis set expansions in variational expressions.6s And finally, there are the moment theory approaches,66where the moments of the oscillator strength distributions are computed by using standard quantum chemistry methods. The underlying cross section is then obtained by inverting the resulting moment problem. Direct Integration of Single-Center Expansions. Several methods have been used to solve directly the single-center static-exchange equations of the form (V2
+ k2 - @T(7))\Ei(?) = s f l x ( 7 , 7 ? Qg(7’)
d33’ (4)
In a single-center expansion method the continuum wave function qi(?)is first expanded in products of wave functions and spherical harmonics whose angylar argument is the direction of the asymptotic momentum k:
By making this expansion, one can analytically treat the angular dependence of the photoionization cross section. The wave function qkI, is in turn expanded in a product of spherical harmonics and radial functions
* k d 3 = C*/,IO) YI,,(Qd I
(5b)
where we have assumed that the angular momentum about the z axis is zero. Expansion in spherical harmonics reduces the Hartree-Fock equations to a set of coupled radial equations of the form
and where +p and +q are radial functions of occupied orbitals. All of these direct-solution methods integrate the homogeneous and inhomogeneous differential equations using either an integral equations approach67or a differential equations approach!* The (62) R. R. Lucchese and V. McKoy, Phys. Reu. A , 26, 1992 (1982). (63) J. R. Swanson, D. Dill, and J. L. Dehmer, J. Phys. E, 14, L207 (1981). (64) For a recent review, see N. F. Lane, Rev. Mod. Phys., 52, 29 (1980). ( 6 5 ) R.R.Lucchese, D. K. Watson, and V. McKoy, Phys. Rev. A , 22,421 (1980). ( 6 6 ) See P. W. Langhoff on pp 183-224 of ref 5 5 . (67) M. A. Morrison, N. F. Lane, and L. A. Collins, Phys. Rev. A , IS, 2186 (1977).
McKoy et al. numerical single-center methods differ in the way the exchange potential Pxis treated. TOsolve these equations, Raseev and c o - w o r k e r ~represent ~~ the exchange terms by additional coupled differential equations. The iterative method of Robb and Collins70starts from the static solution and then iteratively solves the static-exchange equations with the exchange contribution computed approximately as a local potential using the wave function from the previous iteration. In a study of electron-molecular ion collisions for the system H2+, CH’, and N2+,Robb and Collins found that usually five iterations were needed for c o n ~ e r g e n c e . ~ ~ A third method for directly integrating the static-exchange equations is to approximate the exchange potential by a separable approximation of the form
P X ( 7 y ) = @P(7,79
=
up) g,(q i
(8)
Substituting this approximate potential into eq 4 reduces the static-exchange equations into a set of homogeneous and inhomogeneous equations: (V2
+ k2 - VST(7))@(3 = 0
(9)
and (V2
+ k2 - V T ( q ) Q k ( r )= fi(3
(10)
A particular solution to the static-exchange equation can then be constructed from the solutions to eq 9 and 10. The separable approximation methods have been developed for electron-neutral molecule and are currently being applied to molecular photoi~nization.~~ Another method for solving eq 6 is to convert these equations to a set of coupled integral equations, introduce a discrete quadrature on the integrals, and solve the resulting set of linear algebraic equations.72 This approach, coupled with the use of a separable exchange potential, has been applied by Schneider and Collins to electron-neutral molecule collisions72and is currently being used to study molecular photoi~nization.~~ The various single-center expansion methods solve the identical set of radial equations and hence should generally lead to the same photoionization cross sections. The main difference between these methods is their computational efficiency. In practice, the published results of the single-center methods do differ for the same system.75 These differences are often due to the different cutoffs used in the required single-center expansions or to the use of slightly different Hartree-Fock potentials. Single-center expansions at present are only practical for relatively small molecules. Photoionization cross sections have been computed for a number of linear molecules including C 0 2 and HCCH.76,39 Single-center expansions have also been applied to nonlinear electron-molecule collisions.77 Thus, it seems possible that such methods could be applied to nonlinear systems with up to two heavy atoms. Some possible systems amenable to the single-centerexpansion approach would de H2CCH2,CH,, HjC0, H,O, - . and NH?. Iteratiue Schwinger Method. Another approach to solving the static-exchange scattering problem is the use of basis set expansions in variational principles. The most successful method of this type as applied to molecular photoionization has been the iterative Schwinger method.65 In general, one could compute these pho(68) P. G. Burke and M. J. Seaton, Methods Comput. Phys., 10, 1 (1971). (69) G. Raseev, H. Le Rouzo, and H. Lefebvre-Brion,J. Chem. Phys., 72, 5701 (1980). (70) W. D. Robb and L. A. Collins, Phys. Reu. A , 22, 2474 (1980). (71) T. N. Rescigno and A. Orel, Phys. Reu. A , 24, 1267 (1982). (72) B. I. Schneider and L. A. Collins, Phys. Reu. A , 24, 1264 (1981). (73) M. E. Smith, V. McKoy, and R. R. Lucchese, Phys. Rev. A , in press. (74) L. A. Collins and B. I. Schneider, Phys. Reu. A , 29, 1695 (1984). (75) R.R. Lucchese, G. Raseev, and V. McKoy, Phys. Reu. A , 25, 2572 (1982). (76) R. R. Lucchese and V. McKoy, Phys. Rev. A , 26, 1406 (1982). (77) F. A. Gianturco and D. G. Thompson, J. Phys. B, 13, 613 (1980).
The Journal of Physical Chemistry, Vol. 88, No. 15, 1984 3195
Feature Article toionization cross sections without resorting to the use of single-center expansions, but the integrals involved in these expressions for molecular systems have as yet no known analytic form. Presently, the matrix elements are computed from single-center expansions. Thus, the current Schwinger method can be viewed as a hybrid approach which uses both basis sets and numerical single-center expansion techniques. The Schwinger method begins by choosing an initial basis set lai: i = 1, 2, ..., N), in which the scattering wave function is expanded. The Schwinger variational expression for the K matrix is then given by
where [D-'Iij is the matrix inverse of the matrix D with elements Dij = ( a i l U - U G q f l j )
(12)
and where Qk,/ are the Coulomb waves and G is the Coulomb Green's function. A coordinate representation of an accurate wave function can be obtained by putting the basis set expansion with the expansion coefficients implicitly defined by eq 11 into the Lippmann-Schwinger equation in an iterative fashion, yielding the following expression: N
These approximate continuum fractions can also be viewed as new basis functions. With these accurate functions added to the initial basis set, the Schwinger expression will yield even more accurate solutions. Thus, by using successively improved approximate continuum wave functions in the Schwinger variational expression, we can obtain the exact solution in an iterative fashion.6s In a study of the photoionization of N2,it was found that if the initial basis was adequate, the iterative Schwinger method converged in one or two iteration^.'^ The converged wave functions can then be used to compute photoionization cross sections. An improved method which is based on the direct use of Schwinger type variational expressions for the required dipole matrix elements has also been de~eloped.'~These more general variational expressions can also be improved in a systematic manner using an iterative method or by using a Pad&approximant scheme. A more careful analysis of the iterative Schwinger method showed that the dipole matrix elements obtained in the original method were variationally stable. However, the original iterative method was not necessarily the optimal choice for improving on the initial basis set calculation. The new iterative method based on Pad&approximants has generally better convergence properties.'* Such methods based on variational principles which rely on the use of single-center expansions will yield solutions which are identical with those obtained from the direct single-center expansion integration procedures provided that the same expansion parameters are used. Moment Theory Methods. An alternative approach to obtaining the photoionization cross sections is to use a moment theory approach.66 In principle, one can obtain the moments of the oscillator strength distribution using only a square integrable (L2) basis set. Thus, with standard quantum chemistry techniques one can obtain these moments of the oscillator strength distributions. This is done by diagonalizing the appropriate electronic Hamiltonian and obtaining what are known as pseudostates. In the bound region of the spectrum the eigenfunctions approximate the true electronic states, whereas in the continuum part of the spectrum, the L2 eigenfunctions of the electronic Hamiltonian do not correctly represent any physical state of the system and are thus called pseudostates. The moments of the oscillator strengths computed by using these pseudostates in the limit of complete oneand N-particle basis sets do converge to the exact moments. However, in practice the basis sets used for molecular systems (78) R. R. Lucchese and V. McKoy, Phys. Reu. A , 28, 1382 (1983).
do not satisfy this completeness condition, and thus one is a t best using only approximate moments of the oscillator strength distributions. Once the moments of the oscillator strengths have been obtained from the computed pseudostates, the moment problem can be inverted to yield an approximation to the continuum oscillator strengths. There are two commonly used procedures for inverting the moment problem. The first approach is the Stieltjes density method, used extensively by Langhoff and co-workers,66and the second approach is the Tchebycheff density method which has as well as by Delaney also been used by Langhoff and co-w~rkers'~ et a1.80 The moment inversion problem cannot be solved reliably when only a few of the moments are known accurately. If one has all exact moments, the moment inversion methods mentioned above converge to the exact oscillator distributions as higher order moments are included. In the case of the Stieltjes density approach, one would obtain, in the limit of infinite order, a histogram representation of the discrete spectrum and a continuous representation of the continuum spectrum. With the Tchebycheff procedure one would obtain, in high order, a &function representation of the discrete spectrum and a continuous representation of the continuous spectrum. When the cross sections are being obtained in high order from a finite-order pseudospectrum, the Stieltjes density method provides a histogram representation of the pseudospectrum and the Tchebycheff density method provides a &function representation of the pseudospectrum. One must be careful to avoid using an inversion procedure of too high an order lest the imaged spectrum begins to show the underlying unphysical pseudospectrum. In this respect the Tchebycheff density method is more unstable when going to too high an order than is the Stieltjes density procedure. For example, the Tchebycheff results of Delaney and co-workersS0show spurious resonance-like structure which is not seen in the Hartree-Fock results of Smith and coworkers.81 Such unphysical structure in the photoionization cross sections obtained by using the Tchebycheff method can, in principle, be removed by using an extrapolation procedure on the recurrence coefficients obtained in the method.79 At present, the moment theory approaches are very suitable for systems which are intermediate between small systems for which single-center expansions are tractable and much larger systems for which model potential methods such as the MSM method are applicable. Thus, on the basis of moment theory methods, systems as large as O3and H 2 C 0 have already been s t ~ d i e d . ~One ~ , drawback ~~ of the moment theory method is that it does not seem possible to obtain photoelectron angular distribution from this approach. Methods which directly solve the scattering equations (e.g., single-center expansion methods and the MSM method) can readily yield the asymmetry parameters. Vibrational Effects. For nonresonant photoionization the final-state vibrational distributions are in large part governed by the Franck-Condon principle, and hence the intensity of a given vibrational line is proportional to the square of the overlap matrix element between the neutral ground-state vibrational wave function and the vibrational wave function of the ionic state. This approximation assumes that the electronic transition matrix element does not vary rapidly with internuclear geometry. However, when the photoionization cross section is enhanced by either a shape or autoionizing resonance, the electronic transition matrix element can depend strongly on the internuclear coordinate. This rapid change in the intensity of the photoionization with geometry is essentially due to the sensitivity of the energy of a resonance to internuclear distance. (79) P. W. Langhoff, C. T. Corcoran, J. S. Sims, F. Weinhold, and R. M. Glover, Phys. Reu. A , 14, 1042 (1976). (80) J. J. Delaney, I. H. Hillier, and V. R. Saunders, J . Phys. E, 15, 1477 ( 1982). (81) M. E. Smith, R. R. Lucchese, and V. McKoy, J . Chem. Phys., 79, 1360 (1983). (82) N. Padial, G. Csanak, B. V. McKoy, and P. W. Langhoff, J. Chem. Phys., 74, 4581 (1981). (83) P. W. Langhoff, S. R. langhoff, and C. T. Corcoran, J . Chem. Phys., 67, 1722 (1977).
3196 The Journal of Physical Chemistry, Vol. 88, No. 15, 1984
35
-
30
-
molecular scattering potential led to quite different results than those using modified Coulomb wave functions to represent the continuum orbitaLS7 The interchannel problem is well understood from atomic problems. One possible approach to solving this problem is to take the Hartree-Fock level solutions obtained by one of the static-exchange methods discussed above and to use them in a random-phase approximation (RPA) formalisms8 or in a manybody perturbation theory (MBPT) type c a l ~ u l a t i o n . ~Both ~ of these approaches have worked well in atomic systems and should be extended to molecular systems. A second approach to the interchannel coupling problem is to use a close-coupling app r o x i m a t i ~ n . ~Formally, ~ the many-body electronic continuum can be reduced to an infinite set of coupled equations to include only those which are most strongly coupled to a particular channel of interest. Finally, the moment theory method is a very attractive approach to solving the interchannel coupling problem since this approach uses only bound-state quantum chemistry methods which can accurately treat the electron correlation effects. Moment theory has been used to obtain molecular photoionization cross sections using both a CI approach for treating ~ o r r e l a t i o nand ~~*~~ the RPA method.92
--
$ 250
z
CII
20-
c u
e
15
-
IO
-
5 -
01 IO
McKoy et al.
I
I
I5
20
25
30
35
40
Photon Energy ( e V 1
Figure 5. Branching ratios for the production of the v’ = 1/v’ = 0 levels of the XzZ,+ state of N2+ by photoionization of N2: (-), accurate Hartree-Fock results of ref 8 4 using the dipole length approximation; (---), results of ref 84 using the dipole velocity approximation; MSM results of ref 60; (0).experimental results of ref 32. (-a-),
An example of non-Franck-Condon effects in vibrational-state distributions occurs in photoionization out of the 3a, level of N2. This cross section is shape resonance enhanced in the a, channel. Using the MSM method, Dehmer and co-workers predicted that this resonance would lead to very non-Franck-Condon branching ratios.60 Subsequent experiments3’ verified these non-FranckCondon distributions, but the deviations predicted by Dehmer and co-workersm were substantially larger than those observed.32 The accurate Hartree-Fock level calculations of Lucchese and M C K O ~shown , ~ ~ in Figure 5, are in very good agreement with experiment showing that, for shape resonances, these nonFranck-Condon effects can be well described within the Hartree-Fock and Born-Oppenheimer approximations. Electron Correlation. Beyond the independent-particle model one must include electron correlation effects. There are two general classes of such many-electron effects which can be described in terms of a coupling of these Hartree-Fock states. This first type of effect is the two-electron resonance which can be thought of as occurring when a virtual bound state interacts with a continuum through a two-electron interaction. The second many-electron effect is interchannel coupling where two distinct continuum channels are coupled by a two-electron interaction. For molecules larger than H2 few studies have been published which have included the explicit treatment of such effects. The autoionization or two-electron resonance has been discussed in detail by Fano as a C I problem.s5 For molecular systems a simpler approach based on the golden rule approximation has been used to obtain the width of such resonances in small molecular systems. This approach has been applied to the dissocitive recombination of CH+ by Raseev and co-workers.s6 In that study it was found that using static-exchange level continuum wave functions which accurately include the nonspherical nature of the (84) R. R. Lucchese and V. McKoy, J. Phys. B, 14, L629 (1981). (85) U. Fano, Phys. Reu., 124, 1866 (1961). (86) G . Raseev, A. Guisti-Suzor, and H. Lefebvre-Brion, J . Phys. E , 11, 2735 (1978).
Summary and Future Goals The availability of the intense tunable radiation provided by synchrotron sources has made possible remarkable progress in our characterization of the important dynamical aspects of molecular photoionization. In this article we have tried to summarize some of the recent progress both in the experimental studies of molecular photoionization and in the development of theoretical methods by which we can gain some additional quantitative and physical insight into the underlying dynamics of these processes. Continued progress can certainly be anticipated in those areas of study which we have discussed in this article. Moreover, important and new information on molecular photoionization dynamics will continue to emerge from experiments in related areas such as (1) studies of oriented molecules adsorbed on surfaces, (2) coincidence measurements between ion fragments, photoelectrons, and Auger electrons, (3) studies of the polarization of photoion fluorescence and the extraction of photoion alignment parameters and cross sections for degenerate photoionization pathways in molecules from these studies:3 (4)studies of the dynamics of resonant multiphoton ionization of molecules and the photoionization of excited ( 5 ) rotationally resolved state distributions of molecular photoions,g5 and (6) studies of the spin polarization of molecular
photoelectron^.^^^^^ Acknowledgment. V.McK. and R.R.L. thank the National Science Foundataion for support of this research under Grant No. CHE-8218 166. T.A.C. acknowledges support of the Division of Chemical Sciences, Office of Basic Energy Science, US.Department of Energy, under Contract W-7405-eng-26 with the Union Carbide Corp. (87) For an example of a study where the Coulomb wave approximation is used, see W. H. Miller, C. A. Slocomb, and H. F. Schaefer, J. Chem. Phys., 56, 1347 (1972). (88) M. Ya. Amus’ya, N. A. Cherepkov, and L. V. Chernysheva, Z h . Eksp. Teor. Fiz., 60, 160 (1971) (Sou. Phys.-JETP (Engl. Transl.), 33, 90 (1971)). (89) H. P . Kelly, Phys. Rev. [Sect.]B, 136, 896 (1964). (90) P. G . Burke, Adu. Phys., 14, 521 (1965). (91) S. V. ONeil and W. P. Reinhardt, J . Chem. Phys., 69,2126 (1978). (92) G . R. Williams and P. W. Langhoff, Chem. Phys. Lett., 78, 21 (1981). (93) See, for example, J. A. Guest, K. H. Jackson, and R. N. Zare, Phys. Reu. A, 28, 2217 (1983). (94) S. T. Pratt, E. D. Poliakoff, P. M. Dehmer, and J . L. Dehmer, J . Chem. Phys., 78, 65 (1983). (95) See, for example, J . E. Pollard, D. J . Trevor, J . E. Reutt, Y . T. Lee, and D. A. Shirley, Chem. Phys. Lett., 88, 434 (1982). (96) U.Heinzmann, F. Schafers, and B. A. Bess, Chem. Phys. Lett., 69, 284 (1980). (97) N. A. Cherepkov, Adu. A t . Mol. Phys., 19, 395 (1983).
