The theory of electron scattering from polyatomic molecules

THE THEORY OF ELECTRON SCATTERING FROM POLYATOMIC MOLECULES

F.A. GIANTURCO

Department of Chemistry, University of Rome, Cittd Universitaria, 00185 Roma, Italy and

A. JAIN

Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309-0440, U.S.A.

NORTH-HOLLAND

-

AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 143, No. 6 (1986) 347-425. North-Holland, Amsterdam

THE THEORY OF ELECTRON SCATTERING FROM POLYATOMIC MOLECULES F.A. GIANTURCO Department of Chemistry, University of Rome, Cittd Universitaria, 00185Roma, Italy

and

A. JAIN * Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309-0440, U.S.A. Received May 1986

Contents: 1. Introduction 2. Summary of experimental electron-impact excitation 2.1. Rotational excitation 2.2. Vibrational excitation 2.3. Electronic excitation 2.4. Total cross sections 2.5. Emitted radiation after excitation 2.6. Electron beam and discharge processes 3. The theoretical ingredients 3.1. The target wave functions 3.2. Interaction forces 3.3. Wave functions for the bound nuclei 3.4. Electron motion versus nuclear motion 4. The scattering equations 4.1. Frames of reference 4.2. Coupled expansions 4.3. R-matrix method 4.4. Dividing angular space 4.5. Numerical methods to solve the scattering equations

349 351 352 353 355 356 356 357 357 359 361 370 374 376 376 377 381 381 383

5. The computational models 5.1. The CMS approach 5.2. SF scattering via model potentials 6. Computed dynamical observables 6.1. Elastic scattering 6.2. Rotational excitation 6.3. The vibrational excitation 6.4. Electronic excitation 6.5. The transport of electrons in gases 6.6, Resonant scattering 6.7. Dissociative attachment processes 7. Specific examples 7.1. CH 4 7.2. SF6 7.3. H20 7.4. CO 2 7.5. H2S 7.6. NH 3 7.7. Sill 4

384 384 385 388 388 389 393 395 395 396 397 398 399 403 404 406 406 408 409

* Present address: Physics Department, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506, U.S.A.

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F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules 7.8. Other studies on polyatomics 8. Conclusions Appendix A: Real spherical harmonics (SLMe(O,40) Appendix B: First Born K-matrix elements for C3v and C2vpoint groups using asymptotic part of the interaction

410 412 414

Appendix C: The first Born approximation for the rotational excitation of asymmetric, symmetric and spherical top molecules References

349

417 419

416

Abstract: A step-by-step, fairly detailed picture is presented of the most recent attempts to find theoretical and computational techniques that can describe the complicated nature of the forces which come into play during collisions of slow electrons with polyatomic, gaseous targets. The principal ingredients that are needed in any of the current methods to get results at an acceptable level of reliability are here described and the recent model approaches used to overcome the most difficult hurdles are also reviewed, The quality of computed quantities with respect to the corresponding dynamical observables is analyzed in detail for the rather limited number of polyatomics that has as yet been studied, and future directions of development are indicated while reporting current methods.

1. Introduction

From the theoretical point of view, the detailed study of electron-molecule (atom) scattering processes began with the development of quantum mechanics and was motivated by the existing data from the early electron scattering experiments involving rare gases that had been conducted in the 1930's. What is now known as the Bethe-Born approximation [1], for instance, was first applied to treat energetic electrons scattered from atomic and molecular gases to better understand the nature of their interaction as well as the structural properties of the molecular medium [2]. In the intervening 50 years since these early applications, a wide variety of methods have been tried to treat electron-atom scattering processes and the processes involving molecular targets. This has been in response to increasing demand for better knowledge of the various kinds of cross sections in the applied sciences where molecules and charged particles are involved (e.g., plasma physics, laser physics, astrophysics, atmospheric and interstellar sciences, isotope separation in fission generators, MHD power generation, electrical discharges, radiation chemistry and radiolysis). Thus, the corresponding theoretical techniques have evolved from the various modifications of plane-wave and semiclassical models to the recent, very elaborate, close-coupling and variational approaches. This intense research activity, both experimental and theoretical, can be seen in the number of review articles, monographs, and books from meetings, workshops, symposia, etc. that have been published in the past few years (some eight review articles and six topical books have appeared between 1980 and 1985). The most recent volumes include comprehensive and exhaustive reviews of almost every aspect of the physics of electron-molecule scattering edited by Christophorou [3], the volume edited by Shimamura and Takayanagi [4], entirely dedicated to e--molecule collisions, and the extensive proceedings of a workshop that dealt with electron-molecule collisions both in the low-energy and in the impulsive regimes, two different but related fields of study that rarely appear together [5]. Although all of the above volumes give abundant references to previous reviews and fundamental papers on the subject, various recent additions that complement their coverage of the subject are also worth mentioning [6-13]. They all discuss both theoretical and experimental aspects of the problem while concentrating mainly on the diatomic molecules as gaseous targets, and all complement and augment the earlier classic review by Massey [14]. Because the electron-molecule force field is more difficult to treat computationally than that arising in electron-atom scattering, and because the experimental data on gaseous molecules were not

350

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

available until more recently, early applications of theoretical models to the electron impact excitation of molecular targets received very little attention, except for the limited work on molecular hydrogen that was carried out in the 1940's [15]. In the early and mid 1960's there was renewed interest in some + of the processes involving H : , H 2 and N 2, both because more accurate measurements were becoming available and because new theoretical tools could begin to rely on better computing facilities. However, the range and breadth of molecular systems examined experimentally and the testing of some of the more sophisticated computational models, for many years remained essentially limited to the above three targets, with a few forays into processes involving O2, NO and CO. Electron collision processes with molecules play a central role in a wide variety of naturally occurring and laboratory-produced phenomena. The importance of excitation processes involving ground-state molecules, and their excited or metastable states, has been acknowledged for many years in the study of ionospheric processes in planetary atmospheres [16], just to cite one important example. Electron collisions with polyatomic molecules can lead to a wide variety of processes: elastic scattering, rotational, vibrational and electronic excitations, ionization, dissociation, electron capture, and combinations of these processes. What we intend to discuss here is how the basic elements of the force field, which determines the above processes are described in various theoretical frameworks and how realistically they are treated by the few computational methods. It is worth mentioning at this point that, unlike the cylindrically symmetrical diatomic systems, most polyatomic molecules are nonlinear (linear polyatomics are similar to diatomics except in a few cases) and belong to a wide variety of point groups. In addition, their rotational and vibrational structure is totally different from that of diatomic targets and requires a knowledge of standard group theory. The purpose of the present review is to introduce the reader to the basic concepts involved in e--polyatomic molecule scattering events (theoretical), concepts which are not present in the diatomic cases already discussed at length in the references given above. Our main concern will be with those molecules which have been recently studied in the laboratory and which are important in various applied sciences such as space, plasma, laser and atmospheric physics: e.g., H20, NH3, CO2, NO2, N20 , CH4, SF6, Sill4, 502, H2S, C6H 6. On a qualitative basis we can divide e--polyatomic molecule scattering into three regions: a. Low-energy region (E <- 10 eV). This region is full of events and is a favorite of both theoreticians and experimentalists. The interplay between the incoming electron and the molecule is most intimate, with maximum overlapping of the respective wave functions. The outer electron has full opportunity to fall in one of the many unoccupied molecular orbitals (MO) (shape resonance) or to excite any of the occupied MO's as it falls into another (Feshbach-type resonance), thus forming a negative molecular ion for a finite time before decaying into one of the many energetically open final channels. This kind of interaction between electron and molecule is manifested by a sharp rise in the cross-section value, well known to be a resonance in the scattering process. Almost all molecular systems studied so far have shown this behavior. In this situation the molecular ion may be formed in a vibrationally and/or rotationally excited state and might even dissociate into several possible neutral or ionic fragments. Interestingly, the lifetime of these resonances varies from very long ( r - 10-4-10 -6 sec) to very shor~ (10-1°-10 -16 sec), depending on the system and the collision energy. It was in fact this essential unity of many different processes happening at a resonance that made such events among the most interesting discoveries in modern atomic physics [17]. These sharp peaks are also observed at very low energies near rotational-vibrational thresholds and their interpretation is still a challenging job for theoreticians and experimentalists. At very small electron velocities where the vibrational channels are closed, the only energy loss mechanism is through rotational excitations. The rotational excitation process turns out

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

351

to be crucial for polar molecules, where this partial cross section becomes a thousand times larger than other cross sections. Finally, there may be a few electronic channels that are also open in this energy range and which might have different vibrational and rotational substructures, thus making this area of analysis so complex and vast that no single polyatomic molecule (or diatomic system except the hydrogen molecule) can be fully explored theoretically or experimentally. Many dissociation channels also fall in this energy region. b. Intermediate and high energy region (E <-1000 eV). In this energy regime, almost all possible e--molecule processes are accessible (except some inner shell excitation or ionization channels in molecules with large-Z atoms; here Z is the number of electrons), therefore it becomes a very difficult region to study theoretically. The elastic scattering dominates, but processes such as ionization, threshold excitation, and dissociation, are also significant and contribute effectively to the total cross section. Because of the complex structure of the target electronic wave function, a realistic description of its response function in the presence of the incoming fast electron rapidly becomes an impossible task and several drastic approximations need to be introduced at various levels of the treatment. Several recent articles have discussed high energy e -molecule scattering [12, 18]. Due to the inevitable limitations of space, the present review is necessarily rather narrow in scope; thus, of the many possible exit channels, we will be concerned mainly with the elastic, rotational, and vibrational channels, while electronic excitation will be touched upon only very briefly. One reason for this choice is the fact that a great deal of experimental and theoretical attention has been paid to rotational and vibrational excitation processes. Although the present review gives preference to theoretical discussion of the various techniques employed to date to calculate elastic, rotational, vibrational (and electronic) excitation cross sections, section 2 attempts to give a very brief summary of the experimental work on electron-polyatomic molecule interactions. It is by no means exhaustive but is intended to introduce the reader to the great wealth of data that are becoming available in this field. In section 3 we depict the basic theoretical ingredients that are indispensable for any electronmolecule calculation of dynamical observables. Section 4 sets up the general form of the scattering equations and the various types of reference frames that are computationally convenient and physically significant in treating the many-channel scattering problem, while section 5 reports the rather limited number of computational models that have been employed thus far to treat polyatomic targets. In section 6 we describe the various physical quantities that can be extracted from the previous computational models, and the final section reports the specific molecular cases that have been dealt with recently. Since electron scattering is closely related to photo-ionization and photo-excitation, we mention them only in relation to their specific importance in the various fields discussed in this review. Atomic units are used throughout unless otherwise noted.

2. Summary of experimental electron-impact excitation There have been many excellent reviews on the experimental aspects of e--molecule excitation processes since the field was started in the early 1930's. Work up to the late 1960's has been summarized by Massey et al. [14] and up to 1982 by Trajmar et al. [13]. Both works have compiled experimental data on e--molecule collisions (elastic, rotational, vibrational and electronic channels only). Other standard texts on the subject were also published in the 1970's [19-21]. Moreover, in just the past 2-3

352

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

years, several updated articles have appeared on almost all experimental aspects of e -molecule interactions [3, 4]. It is not our purpose here to give even a short summary of experimental techniques and applications, but just to sketch recent developments and work done on some polyatomic molecules, specifically their elastic, rotational, vibrational and electronic transitions. As these excitations may involve emitted radiations, this is also discussed briefly. These processes play important roles in electron beam and discharge processes, and we talk a bit about such techniques without attempting a comprehensive or exhaustive discussion of the subject. 2.1. Rotational excitation

At very low energies where rotational excitation is the only energy loss mechanism, the e--molecule cross sections play a crucial role in many applied sciences. Rotational level spacings are of the order of 1 0 - 3 ~ 1 0 - 5 eW (except for H2), and are almost impossible to resolve by present day spectrometers. In one vibrational transition about a hundred (or more) rotational lines may be present within an energy range of approximately 80meV. To date the H 2 molecule is the only system where rotational substructures can be observed [22-25] due to its large rotational spacings. There are two basic techniques used to study electron-impact excitation on molecules: the beam and the swarm methods. The swarm technique is more accurate at very low energies, but becomes doubtful beyond about 0.5 eV. On the other hand, beam methods yield reliable cross sections beyond about I eV and provide a direct way to obtain cross sections. Phelps [26] has reviewed the swarm data on the rotational excitation of molecules in the very low-energy region. There have been several recent attempts to improve beam techniques for higher resolution, but it is not possible to separate the different lines of the rotational levels in any of the polyatomic gases studied so far: therefore, in an energy-loss spectrum, only broadened peaks without any substructure can be obtained, To obtain specific values for individual rotational excitation cross sections one has to employ a line shape analysis with the application of theoretical models [27]. In general, at the working gas temperature, various higher rotational levels are significantly populated and therefore, a high-J theoretical approximation [28] can be adapted to analyze the experimental broad maxima. For instance, Jung et al. [27] reported the first rotational excitation cross sections for a polyatomic molecule (H20) at 2.14 and 6eV in the angular range 15 to 105°. They fit their energy-loss spectra with only three rotational branches (AJ = 0, -+1) and found a good fit at almost all angles (see fig. 1). It can be seen from this figure that there is discrepancy between the sum of the branches and the experimental points for energy loss of about 75 meV, indicating that some other rotational transitions (AJ = +-2 or greater) are occurring with small probability. In this way, Jung et al. [27] obtained the absolute differential cross sections (DCS) for the elastic (AJ = 0) and the sum for rotational excitation (AJ = + 1) and deexcitation (AJ = - 1 ) cross sections at 2.14 and 6 eV: they found that rotational inelastic cross sections (AJ = +-1) were much larger than the elastic (AJ = 0) ones. The overall energy resolution in their measurement was about 10-18 meV (FWHM). Some preliminary observations were reported earlier by Tanaka [29] on the rotational excitation of methane with a resolution of 17 meV in a beam experiment. The energy spread of the incident beam and the energy resolution of the scattered beam were about 11 meV. To further improve the electron beam techniques for studying rotational-vibrational transitions, Field et al. [30] have constructed an electron-beam source with high resolution and high current. This method uses the principle of threshold photo-ionization of Ar employing synchrotron radiation [31]. The advantage of this new source over the conventional source of monochromatic electron beams (using

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

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a filament and electrostatic magnetic analyzers) is that it provides sufficient current needed for high resolution. Field et al. tested their new source on He, Ar, N2, O 2 and CF 4 down to energies of 50 meV. Although they do not present any data on the rotational excitation, their technique might turn out to be very efficient in future measurements at low energies. More work is clearly needed in this area: improving the quality of experimental data on pure rotational excitation processes will not only lead to more stringent testing of the various theoretical models but also to the development of highly sophisticated instrumentation that can help to uncover other aspects of the underlying physics that are still only theoretical speculations (e.g. threshold phenomena or rotational resonances in polar systems with a supercritical dipole moment). 2.2. Vibrational excitation Vibrational excitation levels are much easier to resolve by present day experiments due to their 102-103 times larger energy spacings. Again the same two types of techniques, i.e., the swarm and the beam, are used to obtain vibrational excitation cross sections. Compared to diatomics, the vibrational excitation in polyatomic molecules is much more complicated because of the many vibrational modes (3N - 6 for nonlinear and 3 N - 5 for linear polyatomic molecules, where N is the number of atoms in the molecule) that are present; however, a rather large amount of work has been done on the measurement of vibrational excitation for both linear and nonlinear polyatomic molecules. Trajmar et al. [13] and Trajmar and Cartwright [32] have compiled and reviewed earlier experimental work up to 1983. Interesting examples and several comments on the instrumentation needed are given in a volume edited by Brown [33]. We summarize here only some of the recent measurements on the vibrational excitation of polyatomic species not included in refs. [13] and [32].

354

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

Azria et al. [34] have reported vibrational excitation of N20 (electronically ground state) under high resolution (-20meV) at 40 ° angle between 1.2 and 3.6eV. Their energy loss spectra revealed progressions in various modes, (n00), (nl0) and (n01) with n up to 7, the intensity of which diminished quickly with the energy loss. Absolute DCS have been measured for excitation of the first few vibrational levels of H2S, N20 and BF 3 molecules by Tronc et al. [35] in the range 0-20 eV: they found that, except for the low-energy shape resonances, intermediate energy shape resonances excite specific vibrational modes. Recently, Andric et al. [36] and Andric and Hall [37] have observed vibrational excitation in the SO 2 and N20 molecules respectively. In both studies, low-energy vibrational resonances are detected at 3.4 eV (SO2) and 2.3 eV (N20). In fig. 2, we show a typical energy-loss spectrum in SO 2 at 3.4 eV and 90 ° scattering angle (under a resolution of less than 30 meV). The main vibrational progressions belong to the predominantly symmetric stretch mode (n, 0, 0). The general envelope of the peaks decreases slowly but adjacent peaks have somewhat more rapid variations due to statistics and oscillations in their individual cross sections. Smaller peaks shown in fig. 2 are also due to the symmetric stretch excitation but with one quantum of bending (n, 1, 0). In fig. 3 (for N20 ) this second series (n, 0, 1) is clearly visible along with the (n, 0, 0) series; however, the bending mode is quite weak. The energy-loss spectrum of N20 (fig. 3) was taken at 2.4 eV and 20° angle with about 30 meV resolution. For CH4, most recent data on the vibrational excitation measurements are due to Rohr [38] (v2, 4 modes at 60° from threshold to 4 eV), Rohr [39] (v~, 3 and v2, 4 modes from 20° to 120° and up to 4 eV), Okada et al. [40] (vl, 3 and v2, 4 modes between 2 and 20 eV with a resolution of 30 meV), Kubo et al. [41] (v~, 3 and v2, 4 modes between 3 and 20 eV at 30°-140 ° scattering angles), and Sohn et al. [42] (vl, 3 and v2, 4 modes from threshold to 1 eV). Most of the above measurements were performed by cross-beam techniques, while vibrational excitation cross sections for CH 4 from the swarm data have been extracted recently by Hayashi [43] from earlier swarm experiments. Field et al. [30] have observed I

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vibrational structure in CF 4 by using a new source for the electron beam with high resolution (-25 meV) and Curry et al. [44] have reported measurements on the vibrational excitation of CH 4 and C2H 6 at 30°-130 ° angles between 7.5 and 20 eV. Apart from the rich structure which appears in most of the vibrational excitation cross sections, another interesting region that is worth investigating is near threshold, where almost all vibrational cross sections exhibit marked peaks for polar diatomic molecules but not for non-polar or weakly polar molecules. The situation is quite different in non-polar polyatomic molecules, where, for example, in CH4 and SF6, very narrow threshold peaks are still observed in the vibrational excitation function [45]. Rohr [39] argues that these peaks may be positioned very close to threshold at very low incident energies and that they may be very narrow. This kind of experimental study will require very high resolution and sophistication. We shall talk more about such resonance phenomena in subsequent sections. It is also to be noted that the rotational substructure imbedded in any vibrational energy loss is usually unresolved and therefore the corresponding vibrational excitation cross sections really represent summation over final rotational states, and averaging over initial rotational states, of the molecule under consideration. 2.3. Electronic excitation

Electronic transitions in molecules usually require a few electron volts and, therefore, can be easily resolved in the laboratory. Transitions may occur either to valence or to Rydberg-like orbitals. In general, an electron energy loss spectrum is obtained for the system under investigation; this spectrum carries only relative information on the DCS for various excitation channels. Fortunately, the shape of each DCS is quite representative of the nature of the transition involved. For example, optically

356

F.A. Gianturco and A. .lain, The theory of electron scatteringfrom polyatomic molecules

allowed (dipole) transitions contribute the most to the cross section, and the DCS show a sharp forward peak that decreases by orders of magnitude at intermediate angles; spin-forbidden transitions, where the spin quantum number changes by unity, have nearly isotropic DCS (within a factor of 2 or 3), and scattering may result in metastable molecules; spin-allowed (but symmetry-forbidden) transitions behave somewhat in between, i.e., the DCS is forward peaked but not strongly so. Most of the measurements have been made from the ground electronic and vibrational states of the molecule. A general discussion about the point-group elements and electronic structure nomenclature of polyatomic molecules is given in standard texts [46, 47]. In recent reviews the necessary theory, the experimental work, and the available data on the electronic excitation of polyatomic molecules have been compiled up to 1983 [13, 32]. Trajmar and Cartwright [32] have provided tabular data on some of the molecules that have been studied in electronic excitations, while Trajmar et al. [13] have covered a few more molecules (N20, CO 2, HCN, H20 ) in their recent review. Experimental studies on the excited electronic states and on the electron-impact electronic excitation have been recently made on CH 4 [48], CD 4 [48], HCN [49], CH3CN [49[, C2HsCN [49], C3HTCN [49], NO 2 [50], CO 2 [51-53[, H 2 0 [54], D20 [54], HzS [54], UF 6 [55], C6H 6 [56, 57]. Near-threshold electron-impact excitation of ammonia and methylamine molecules has recently been explored in a trapped electron spectrometer by Abbain et al. [58]. Drift-tube techniques that have been used to investigate electronic excited states of diatomic molecules [59] are still not being applied to polyatomic species. To obtain cross sections for a particular excitation channel the energy loss spectrum is fitted to some theoretical model. Thus, except for a few diatomic molecules (H 2 and N2), the experimental data on the electronic excitation are fragmentary and not very satisfactory. Most such e--impact electronic excitation cross section data have been generated from optical excitation functions, although this procedure faces serious problems due to cascade contributions, radiation trapping, branching ratios, etc. 2.4. Total cross sections

A general review of the measurement techniques for total cross sections has already been given by Trajmar et al. [13]. To avoid repetition, we will summarize here only recent work not included in that review. The total cross section is a relatively easier quantity to measure, compared to partial and/or angular distribution functions; therefore, most of the accurate and fairly comprehensive e--molecule cross section data are available in terms of total cross sections [13]. Commonly used methods to observe the total cross sections are: transmission, recoil, and transmission with time of flight. At very low energies, swarm experiments can yield very accurate data [43]. At thermal energies a total cross section may include pure rotationally elastic and inelastic cross sections, while at low, intermediate, and high energies an increasingly larger number of exit channels with competitive probabilities contribute to this parameter. There have been rather few new measurements of the total cross sections after the publication of refs. [13] and [3]: N20 has been one of the systems measured by Kwan et al. [60], who used a beam transmission technique to obtain its total cross section in the range 1-500 eV. Szmytkowski et al. [61] measured absolute total cross sections for N20 and OCS molecules in the range of 8-40 eV by using a linear transmission technique. Very recently total cross sections for the CH 4 [62, 63[ and Sill 4 [64] molecules have also been measured. 2.5. Emitted radiation after excitation

When molecules are excited by electron impact, the decay of the excited state may give rise to

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

357

photoemission at various wavelengths depending upon the nature of excited states. The spectrum will therefore be broad as it arises from electronic excitations (few hundred A), from vibrational (a few txm) and from rotational excitations (few hundred p~m). Cross sections for each specific excitation process can thus, at least in principle, be evaluated by measuring optical excitation functions, which will in turn require determination of absolute photoemission rates, branching ratios of the decaying channels for the excited states and the collision geometry. Most of the experimental work has been done in the visible region and only recently there has been sizeable activity in the vacuum UV (VUV) region and far infrared regions. The basic techniques for measuring cross sections and detecting emitted radiation are more or less the same as those for atomic targets [65-66]. A review on measuring optical functions was recently given by Zipf [67] for the spectral region of 500 A - 1.3 txm and with respect to the main dissociative channel of e--molecule scattering. Drift tube techniques have been used for such measurements in the visible and near-ultraviolet region from states with short radiative lifetimes [68, 69]. Recently Bulos and Phelps [70] have used drift tube methods for the direct measurement of rate coefficients for excitation of the 4.3 txm bands of the CO 2 molecule (emitted due to vibrational excitation from the 001 level of CO2). Ajello [71] measured the extreme UV spectrum of 0 2, N 2, CO, NO and CO 2 from 40-130 nm at electron impact energies of 0-350 eV and presented absolute emission cross sections. These emitted radiations have also led to the development of electron-photon coincidence experiments, in which emitted radiations are detected in delayed coincidence with the inelastic electrons. Observation of emitted radiation from fragment atoms produced by dissociation of molecules is of fundamental importance in the understanding of highly excited molecular states [72].

2.6. Electron beam and discharge processes Gas discharges have been studied for at least 100 years. The establishment of a discharge in an electrical breakdown process is largely controlled by the relative importance of various e--molecule collision processes. It is also necessary to know the transport properties of electron beams in gases, their drift velocities, diffusion and energy distributions. The phenomenon of electrical breakdown and discharges in gases has been the subject of a recent NATO Advanced Research Institute [73] to acknowledge that in the past 20 years there have indeed been interesting discoveries and developments in this field. Among them, the activities with various magnetic-confinement fusion programs and electron-beam controlled discharge techniques have acquired increasing relevance in the past ten years [74], while the technique of electron-beam discharge has proven successful in producing uniform glow discharges in large volumes over a continuous range of pressure from a few tenths to several atmospheres [75]. These techniques have also been employed to measure electron attachment to molecules such as NF3, HgBr2, CO2, CCI~, SF6, etc. [76]. Due to space limitations, it is not possible to give here even the briefest account of the latest discoveries in this field, but we can recommend refs. [13, 73-80] for a comprehensive discussion of the theoretical and experimental problems pertinent to this area. 3. The theoretical ingredients

In any quantum mechanical treatment of atomic particles, the basic Schr6dinger equation involves an interaction Hamiltonian and the corresponding wave function of the total system, E)

R) = 0

(1)

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

358

where the e--molecule Hamiltonian is given by ~ = ~e(r) + ~ . , ( r , R ) +

Y(m(R).

(2)

Here ~(e(r) is the kinetic energy term for the projectile, 9(m is the Hamiltonian of the isolated molecule (R represents both electronic and nuclear coordinates), V~m(r,R) is the interaction (Couiombic) operator between the electron and the molecule, E is the total energy of the system and 0T is the total eigenfunction of the e--molecule system. The time-independent formulation of eq. (1) is valid within the steady state situation of the incident beam and experimental conditions [81]. The total wave function qJT describes both the scattered electron and the target states and satisfies the following asymptotic boundary condition,

qJx(r,

R)r'~ 0i(r, R) + ~0f(r,R)

(3)

where the initial wave function 0i represents the incident electron as a plane wave and the initial target state, q~i(R), i.e., ~O~(r,R) = exp(ik~ • r) q~(R).

(4)

The final term ~Ofis a superposition of scattered waves associated with all energetically accessible target states ~Of(r,R) = r1 ~ exp(ik,r) f , ( k , , k~) ~ ( R ) .

(5)

Here k i and k, are the initial and the final electron wavevectors respectively, and n denotes a particular molecular state with all its necessary quantum numbers. The scattering amplitude fn is the important quantity relative to cross sections. We thus see that in order to understand any basic scattering event we need to determine specific quantum mechanical quantities that appear in the evaluation of f,(k n • k,), e.g. the electronic wave function of the bound electrons, the wave function governing the motion of the nuclei, and the nature of the interaction forces between target particles and the incident charged particle. Moreover, when trying to determine the unknown form of the scattered electron wave function within the region of action of the above forces one must use the physical nature of the dynamics in selecting the most effective representation for the relevant constants of motion. In other words, different coupling schemes between several angular momenta (which describe all the particles present) have to be chosen according to the nature of the dominant force field under each scattering condition. This technical aspect of the problem is controlled by our physical intuition and will be discussed below in detail. For polyatomic molecules each of the above quantities is naturally more difficult to determine, due to the many-centered nature of the target and the complicated interplay between the motion of molecular electrons and that of the binding nuclei. Within the interaction region, therefore, the incoming particle must take on a "molecular" nature, thereby making its correct description within eq. (5) more difficult. In the following we will briefly discuss the difficulties and requirements for each of the theoretical ingredients mentioned above.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

359

3.1. The target wave functions Although much progress on calculating equilibrium geometries and electronic states of diatomic and polyatomic molecules [82-84] has been made in the past 20 years, determining actual electronic densities around the bound nuclei still requires a large computational effort when high precision is required. Because one is dealing most of the time with the lower-lying vibrational states, the standard Born-Oppenheimer approximation (BOA) is often assumed to hold for the isolated targets. One is therefore left with the problem of accurately describing the target total electronic distribution at some fixed nuclear geometry with the proviso, however, that it should be most effective when depicting the system's response to the perturbing effects of the incoming electron. This requirement makes the usual total electronic energy optimization criteria be, in principle, inadequate as indicators for the goodness of the final target wave function in performing scattering calculations, although practically no model has attempted to obtain quantitative results using any other criterion. In the scattering problems which are of interest to us, we will be concerned mostly with the nonrelativistic nature of the target bound electrons and nuclei interacting through electrostatic forces only. As mentioned before, the final molecular electronic wave functions will be obtained in a fixed-nuclear (FN) geometry. To understand the nature of target wave functions for polyatomic electrons, it is necessary to briefly examine their symmetry properties. Although numerous texts deal with this topic, it seems appropriate to repeat here some of the basic concepts that stem from the multicenter nature of the nuclear geometries of a polyatomic system. Every polyatomic molecule possesses certain kinds of symmetry operations (rotations, reflection, inversion, etc.) under which its Hamiltonian is invariant. If the Hamiltonian Y( of the molecule with eigenfunction ~bis totally symmetric with respect to a group of symmetry operations (Ri, i = 1, 2 , . . . ) , the R i ~ are also eigenfunctions of Y( with the same eigenvalue. Let us assume that E n is an/n-fold degenerate eigenvalue and let the corresponding set of 1n orthonormal eigenfunctions be ~0,k (k = 1, 2 , . . . , l,); by the above result the operation with any symmetry operator g on any one of the l, functions produces another function (with the same eigenvalue), which can be expressed as a linear combination of this orthonormal set of degenerate functions, i.e., in

{¢
k = 1, 2 , . . . , l n .

(6)

k'=l

Here R is the (l n × ln) matrix representation of the operator/~, and a collection of these matrices, one for each operation, is an /n-dimensional representation of the point group for the chosen fixed molecular geometry. This representation can be shown to be irreducible [85, 86]. We conclude that if a Hamiltonian is invariant with respect to a group of symmetry operations, then the eigenfunctions belonging to any eigenvalue form a basis for an l,-dimensional irreducible representation (IR) of the point group of the molecule in question. Conversely, the only possible degeneracies of eigenvalues are given by the dimension of the IR's. On the basis of the above discussion, one can now construct the symmetry-adapted wave functions (SAWF), which have the symmetry properties of an IR. If the point group of the molecule has g elements/~n with c IRs F/(of dimension li each) and the character (sum of the diagonal elements of the representation) of the operation R, in F/can be represented by xi(Rn), then, with each IR, there is associated a projection operator [86]

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

360

= z_,

g n=l

xi(Ro)*Ro

(7)

such that, given any arbitrary wave function ~ which is not an eigenfunction of N and which does not have symmetry properties appropriate to IR, the projected function li

g xi(Rn)

nn~b

(8)

belongs to the IR F/. 0i is called the SAWF. In other words, if an arbitrary function 0 could be expanded in terms of the complete set of known eigenfunctions On of Ygthen the effect of/~i on ~bis to remove from this expansion all terms not belonging to the IR F/. For example, the spherical harmonics Y'~(O, ok) form the basis of a (2l + 1)-dimensional IR of a three-dimensional full rotation group. Altman [87] and Altman and Cracknell [88] have discussed methods to generate symmetry-adapted basis functions of many point groups. Sometimes it is convenient to work with real spherical harmonics [87, 89]. In order to represent the molecular states and orbitals, we follow the nomenclature of Herzberg [47]. In brief, the capital letter A is used to represent one-dimensional IR with character + 1, while for the character - 1 the letter B is used. For a two-dimensional IR, the letter E is used, while a three-dimensional IR is denoted by letter T (or F). Suffixes 1, 2, 3 . . . are used to distinguish different IR's with the same dimensions. The subscripts g or u differentiate between even (gerade) and odd (ungerade) representations of the same class if inversion symmetry also exists. Lower case letters (a, b, e, t, etc.) are used to represent individual orbitals with the same symmetry properties. For example, the ground state of the water molecule is represented by 1 2 2 2 2 2 a I 2 a 1 3 al 1 b I 1 b 2 (1A1) ; here the five orbitals are of al (three in number), b~ and b 2 types, i.e., orbital a~ transforms according to A~ IR, b~ according to B 1 IR, and so on. When the main interest is in nonresonant scattering, for instance, the general energy dependence of the total cross sections could be qualitatively provided by approximate electronic densities which include only the description of outer electrons [90]. These semiempirical (INDO) methods have been applied only to simple linear molecules and although they treat the outer electrons explicitly, the two-center repulsion integrals are neglected to simplify calculations. The resulting wave functions usually predict equilibrium geometries fairly well but fail to yield accurate electronic energies. Since the overall shape of electron densities is obtained fairly rapidly, one might hope, in principle, to use them for explanatory studies over wide ranges of nuclear geometries and at several values of collision energies before focusing on specific features of the experiments that need a more sophisticated approach. In a similar vein, another semiempirical model for chemically bonded systems is provided by the "muffin-tin" model. Here bound electrons are assigned to one of two distinct volumes surrounding the nuclei [91]. The inner regions are given as atomic-like spheres around each nucleus, while the bonding electrons are confined to an outer sphere which surrounds the molecule as a whole. Usually, the atomic spheres touch each other. The appeal of this model is obviously in its simplicity, although the quite unphysical discontinuities which appear in its total electron densities often force users to introduce smoothing functions that complicates calculations without necessarily improving the physical picture [92]. Here again results for resonant scattering are often rather inaccurate [93]. Attempts at some empirical corrections will be discussed below, when specific applications to polyatomic systems are examined.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

361

Finally, fully ab initio wave functions for all the electrons of the target can be used by carrying out the representation of the electron density either at the single-determinant level with an essentially converged expansion over analytic functions (self-consistent-field (SCF) results of the Hartree-Fock (HF) or near-HF quality) or by including the largest possible amount of bound-electron correlation energy via extensive expansions over several target configurations [83] (multiconfiguration (SCF) approach). Naturally, the latter approach is likely to produce electron densities of very high quality although the problem of determining the behavior of their response functions (i.e., the (N + 1)-electron optimized densities) becomes correspondingly very complex. No test for polyatomic systems, however, has been carried out at this level of accuracy, although some preliminary results o n C H 4 look very promising, as we will discuss later. It therefore appears that the overall sensitivity of computed cross sections to the quality of the target charge distribution is still a rather uncharted territory of numerical experiments, where only one or two paths have been traced for systems such as H 2 and N 2 [94]. In a general calculation of molecular wave functions, the molecular orbitals are written in terms of LCAO (linear combination of atomic orbitals) centered at each atom of the molecule. This kind of multicentered target wave function is very inconvenient to use in any e--molecule scattering equation. It is, therefore, more convenient if all the quantities, such as the bound and the continuum orbitals, interaction potentials, etc., are expanded around a single center, normally the center-of-mass (COM) of the system. Then, the scattering equations for the continuum electron function will be similar to those for electron-atom scattering and will become computationally more tractable. The main problem in this single-center-expansion (SCE) approach is obviously the representation of orbitals on the off-center nuclei as it may demand a large basis at the expansion center. However, for pure structure calculations the earlier difficulties of evaluating multicenter integrals for more than two or three nonaligned centers also led to significant work on the calculation of single-center target wave functions for diatomic as well as polyatomic molecules, although the molecular properties derived from single-center target wave functions are of only mediocre quality by today's standards [95].

3.2. Interaction forces It is rather instructive to examine the forces at play between electrons and molecules as separate CC' quantities originating from a total, given interaction potential for each scattering channel, c, VTOT. If in eq. (1) the total wave function is expanded in terms of all possible target states (rotational, vibrational and electronic), and then the antisymmetrized operator is applied to ensure obedience to the Pauli principle, the resulting coupled integro-differential equations will automatically include all types of acting forces without any approximation. However, such a natural expansion is still beyond the limits of present-day computers even for the simplest target, the hydrogen molecule. The scattering equations Will contain nonlocal and often complex terms that will be very complicated to handle numerically. Therefore, although all forces should be accurately included in the total interaction potential (at least at the level of accuracy provided by the chosen target electron density description) one often resorts to cc' semiempirical methods to treat the different components of VTOT, which are at the outset written in the following way: cc' VTOT(r, r ,t . R, E) = vS[ (r; R) ~c' +

ex rrOPTz Kc~'(r, r'; R) dice,+ Ucc, tr, r ,t . R, E),

(9)

where r is the scattering electron coordinate and r' represents collectively the coordinates for all the bound electrons. R represents the nuclear positions for each chosen molecular geometry and E is the

362

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

collision energy. V~cx is the local, diagonal static interaction for channel c and K~ derives from the well-known requirement that the total wave function must satisfy the Pauli principle: it therefore describes the diagonal, non-local exchange interactions. The optical potential Uc°PT contains the non-local, energy-dependent description of the adiabatic and nonadiabatic target response to the perturbing field due to the incoming electron. In general, useful expressions for Uc°PT can be obtained from linear-response theory [96] and many-body perturbation theory [97], but in most current applications to polyatomics, it is essentially represented as a local, energy-dependent polarization potential which could be conveniently divided into a short-range part, VSoR and a long-range asymptotic form, vLRAF --Pol " It is immediately obvious that the above three contributions present different levels of complexity when one is interested in their computation from first principles. The local, diagonal static interaction simply represents the undistorted electron density of the target molecule as probed by the incoming electron treated as a point charge with zero kinetic energy. It is usually the easiest contribution to obtain from the isolated target wave function; it therefore carries the same quality of information as that already built into the chosen molecular charge distribution. For polyatomic targets, as we will see below, mostly very simple levels of approximation have been employed thus far and very few treatments get at all close to a reliable HF-type of target description [98]. The latter method, on the other hand, provides a very good description of the static potential and the various multipole moments. The traditional bottlenecks in most treatments of electron scattering from molecules, diatomics, and polyatomics, have been the exchange and the optical (correlation) potentials; the main reason for this being their non-local, non-separable forms within a coupled-channel (CC) description of the multichannel problem [6]. As a consequence, most of the recent work has aimed at obtaining the exchange interaction (a short-range interaction within the volume of the target electron density) in a separable form by using a local, energy-dependent formulation at a semiclassical level [99], of the Hara-freeelectron-gas-exchange approximation (HFEGE) with the added orthogonalization constraints [100, 101], or by using a separable expansion over a set of L 2 functions [102, 103]. For nonlinear polyatomic molecules no exact calculations of exchange interaction have been carried out thus far (except the one by Lima et al. [104]). The few examples have been limited to the HFEGE-plus-orthogonalization [101] or to the semiclassical, energy-dependent approximation [105]. The general agreement found with experiments, however, appears to suggest that most of the attractive exchange interaction within the short-range region is correctly accounted for by the above models, although a more quantitative assessment of their reliability vis ~ vis exact exchange kernels awaits future calculations. Polarization effects in the scattering of charged particles arise from correlation between the motions of the projectile and the target electrons. Rigorously, these effects manifest themselves as virtual electronic excitations of the target through the influence of closed electronic channels [6]. Their optical potential formulation yields an energy-dependent, nonlocal contribution which needs to be obtained from some suitable expansion of the total wave function via the target electronic states and pseudostates [4]. However, most calculations for polyatomic targets have resorted instead to separable, adiabatic forms that may or may not carry any correction for energy-dependent effects. In the following we report more specifically on what has been done thus far for such molecules. 3.2.1. The static potential Gianturco and Thompson [106] first generated a SCE static potential surface between an electron and a nonlinear polyatomic molecule belonging to any point group. This interaction is given by pure Coulombic forces between positively (nuclei) and negatively (electrons) charged particles:

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

M Z, Vin~= - ,=,E Ir Zlil

~ -~- =

1 I r - rjl

363

(10)

where rj are the coordinates of N bound electrons and R i of M nuclei. All these coordinates are referred to the molecular (or body-fixed, BF) frame of reference. Naturally the interaction, eq. (10), is molecular orientation dependent, which gives rise to a torque on the molecule so that rotationally inelastic transitions may take place. If the scattering is electronically elastic, then a static potential can be obtained by averaging V~,t over the ground state target wave functions at a fixed molecular geometry, V(r) = (~oll~intl~o)

(11)

where the integration is carried out over all target electron coordinates. For "closed-shell" polyatomic molecules, V(r) transforms according to the one-dimensional totally symmetric A 1 IR of the given molecular symmetry point group [107]. For example, in the case of the H20 molecule, V(r) belongs to the A1 IR of C2v point group. For a closed-shell system, the total SCF wave function of the target is given by a single Slater determinant of one-electron functions qbi(ri; R) (which can be taken to be real) and given by a truncated expansion over an optimized combination of analytical functions pj(r~):

dPi(ri; R) = ~ Cij(R ) pj(ri) . J

(12)

For pj(ri), one can choose either a Slater or a Gaussian type function. Once the one-electron functions (12) are known, the static term (11) can be computed easily for each required value of r. For a closed-shell target, ~00is given by 1

~J0- (z!)l/2 eae~''''F'Tr ~b,(1) q~(2)... ~b~.(Z).

(13)

Here e is the usual antisymmetrizer taking values +1, -1 and 0 and a sum is carried out over the repeated indices a, f l , . . . , 7r, where each index runs over all ground state orbitals of the molecule. Each ~b is taken to be real and expanded over the COM of the molecule in terms of symmetry adapted functions which transform according to one of the IR's of the molecular point group. We write ~b as: dPa(r)

= E r -1 Ul~hj(r) X~J(r)

(14)

J

where we have not shown explicitly the dependence on R for simplicity of notation. In eq. (14) the X functions are linear combinations of spherical harmonics [Appendix A] and belong to the /z/th component of the pith IR, with hj distinguishing between different bases with the same (pi~ilj). The X functions form a basis for the IR of the given molecular point group as already discussed in the literature [108]. Thus the X function can be defined as (dropping the suffix j) +1

XV(1) = E b p",hmS~m(i) m=O

where the real spherical harmonics are given by [106]

(15)

364

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

S~,,(i) = (

(2/+ 1)(/- m)! 1/2 PT'(cos 0) fP(~) 2 rr--O-7 n~ i-(-~-7-L o ) )

(16)

with p = - 1 and fl(~b) = cos m4),

f-l((h) = sin m~b

for m >- 0.

The b coefficients of eq. (15) are discussed, and some are tabulated, in the literature [109]. Equation (11) can now be written as

V(r) =

,o12 I E

J r - rj[ -1 drldr2 • . . d r N - ~ Zilr_Ri[ 1.

j=l

(17)

i=1

Defining the one-electron charge density as

Po(rl) .

. .3 f I.,ol2 dr2dr

"dr N 2 ~ [~b( r ,)12,

(18)

ot

where the factor of 2 appears due to spin integration and the a sum being over each doubly occupied orbital, we see that P0 belongs to the totally symmetric one-dimensional IR (A1) of the molecular point group [110]. Expanding P0 in terms of A 1 symmetry functions,

Po(r)= E - 1r p,h(r) XthA 1 (~),

(19)

lh

we can rewrite eq. (17) as V(r)

(20)

= Vel(r ) + Vnuc(r )

where the electronic term is given by

Vel(r) = f po(rl)Ir -

rl]-'

dr1,

(21)

and the nuclear contribution by M

V, uc(r) = - ~ Zilr-Ri[ -1

(22)

i=1

Using eqs. (15) and (18), the charge density in turn can be written as

po(r) = ~ r-: &M(r) sA~

(23)

LM

where

pLM(r) = 2 ~.. U~nilirni(r)Unjtimj(r) S~iiPiS~JPjSMLA1 di ot t]

(24)

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

365

(for the angular integrations, see appendix A). The first term of eq. (20) can now be evaluated by writing

Ve,(r) = E VLM(r) sMAI(/~)

(25)

LM

where the coefficients 6/~M are given by 417- f rL< 6tM(r) - (2L + 1) PLM(rl)~ dr1"

(26)

r>

Similarly, the nuclear term can also be expanded in terms of A 1 symmetry-adapted functions, writing A

Ir- R I-' = E r< ~. r>

47r Y~(1) Y~*(fCi)

(27)

2A + 1

with r< (r>) being the smaller (greater) of r and R;. For any R i 0, we simply have a contribution of -Zi/r only, whereas for any group of nuclei with the same Ri = Rj and Z i = Zj (and different from zero), eq. (22) is given by =

Vnuc(r)= Zi ~] ~] AA. 4rr r~< saAj(i) , ,.->o 2A + 1 rA>+1

(28)

where Aho =

A~,. = ( - l y ' V~ C~, ;

# >0, p = 1

; jz>o, p = - i

AA~, and M

C~,. = ~ V~(/~j),

(29)

j=l

where j runs over all nuclei with the same Rj. Outside the molecular charge distribution, i.e., in the asymptotic region, the static potential can be written in terms of multipole moment tensors:

V(r) = ~ VLh(r) X LA1 h ( rA ) Lh

(30)

where the first few terms are due to permanent dipole, quadrupole, octupole, etc., moments of the isolated molecule at the fixed nuclear geometry. Gianturco and Thompson [106] have given such single-center static potential surfaces for several polyatomic molecules (CH 4, NH3, H20 ) and compared them with multicenter calculations. They found that agreement is excellent when the nuclear contribution is included exactly, rather than being

366

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

truncated at the same I value as the electronic term (one should note here that for large values of l, the nuclear contribution dominates over the electronic one in such a way that beyond that l value, the electronic terms can be neglected altogether). The more efficient multicenter approach turns out to be very cumbersome when treating the dynamics and has not been applied thus far to nonlinear systems.

3.2.2. Polarization potential As mentioned earlier, the difficult task of correctly computing the polarization potential from a polyatomic target has been most often circumvented in the literature by making use of the fact that, in the asymptotic region, the response of the molecule in an electric field is measured by the value of its isotropic (%) and anisotropic (a2) polarizabilities. This means that at large distances, the polarization interaction can be exactly represented by - [ % + azP2(cos 0 ) ] / 2 r 4 whereas within the molecular charge density region, the form of this force is very hard to predict. A simple phenomenological remedy has been to propagate the asymptotic form smoothly via a cut-off function involving an adjustable parameter r e, Vpol(r) = - \ ~ r 4 + -2r- 4 P2(cos 0)] fc(r, rc)

(31)

where several forms of re have been tried [111,112]. The use of cut-off type potentials has been much criticized [113-115]. Gibson and Morrison [115] have recently summarized the earlier work on the use of such approximate polarization potentials. The parameter r c is chosen by tuning theoretical quantities to the observed ones. In recent years there have been many attempts to incorporate nonadiabatic effects by going beyond the cut-off procedure [115], although most of these have been for diatomic targets; the same discussion should however be valid also for polyatomic targets. For polyatomic systems Jain and Thompson [116] first used a parameter-free polarization potential and succeeded in reproducing structure in the cross sections; for example, Ramsauer-Townsend (RT) minimum in e-CH 4 scattering, the 2 eV B 2 shape resonance in e-H2S elastic and vibrational excitation processes [117]. In their model, Jain and Thompson [116] calculated second-order energy from the first-order distorted wave function by employing the method of Pople and Schofield [118]. This method is not very efficient when calculating molecular polarizabilities, and therefore, Jain and Thompson were forced to scale their new polarization potential via the experimental value of the polarizability, thus partially correcting the shortcomings of their model. At the same time, Abusalbi et al. [105] calculated the e-CH 4 polarization interaction in the semiclassical local kinetic energy approximation and also using an ab initio, extended basis set to generate adiabatic and nonadiabatic polarization potentials. Their semiclassical local kinetic energy potential is defined as the difference between the real part of the exact optical potential and the static-exchange potential [119]. Both their potentials differ significantly (in shape and magnitude) from that of Jain and Thompson [116] at short distances. Note also that unlike Jain and Thompson's polarization potential, which is defined in a frame of reference fixed to the molecule, Abusalbi et al. define their potentials in a space-fixed (SF) set of coordinates. It is also worth noting here that the above approximate polarization potentials are used along with approximate local exchange interaction to be discussed next. In some cases, therefore, such an ad hoc polarization term may also unwittingly compensate for exchange effects, if exchange is treated

F.A. Gianturco and A. .lain, The theory of electron scatteringfrom polyatomic molecules

367

approximately or neglected altogether. It has also been observed [120] that an accurate parameter-flee polarization potential may require an exact treatment of exchange interaction. In any model-exchange calculation, the polarization has to compensate for shortcomings of the exchange approximation and therefore the consequent errors in the treatments coming from both the above contributions could easily lead to incorrect results, as recently discussed for the "rr resonance in the CO molecule [121]. The crucial part of the polarization interaction clearly appears to be the one corresponding to the short-range correlation between the motions of the electronic projectile and that for the bound electrons of the target molecule. As shown by the previous discussion, in the case of polyatomic molecules only empirical methods or simple models have thus far been tested the experimental findings and they have often provided very different "shapes" for the correlation potentials. A clear example is shown in fig. 4, where two different polarization potentials are plotted for e--NH 3 scattering: the empirical cut-off form (dashed curve) provides a much stronger interaction in the short-range region, while the more satisfactory, parameter-flee form [122] clearly yields a much weaker form of correlation. Although the use of the latter form produces computed cross sections that are in better agreement with the available experimental data, the approximate treatment of exchange contained in both calculations still casts strong doubts on the reliability of these methods. Finally, another parameter-flee polarization potential that has been recently introduced for atoms [123] and diatomic molecules [124] is currently being tested for polyatomic targets [125,126] and seems to provide rather satisfactory results in spite of its simplicity. The method is essentially based on the flee-electron-gas (FEG) model for the total electron density, whereby the correlation energy can be given as an analytic function of the molecular charge density in the high- and low-density limits, with a simple interpolation form in the intermediate region [127, 128]. Since such a polarization potential does not take into account the long-range interaction, one simply joins the two forms where they cross near the boundary of the target charge distribution. Such crossings are essentially independent of the target system [124], while the overall effect of such potentials on scattering observables has so far been rather satisfactory, albeit not fully explained as to details of operation. The simultaneous use of approximate

vis~vis

k

"• wo

I

i

I

i

c0.75fi

o "6

t I

"~

j

tO

I

',\

X t

',,

I

',

I

i

" GT(rc=0"81)

Qso I

025

i

"

j

02

i

AT"

1.0

\,,

,..

30

~0 r (a. u.)

Fig. 4. Comparison of the spherical components of the computed, parameter-free polarization potential (AT) and of the empirical cut-off potential (GT) for electron-NH3 interaction (from ref. [122]).

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

368

exchange interactions is in fact making harder to discern, within the FEG model, the relative reliability of each functional form that is essentially based on the quality of the chosen charge density p(r) (see below).

3.2.3. Exchange interaction For diatomic targets (and also linear polyatomic systems) there has been significant progress in including this interaction exactly [102, 120] and in principle, the same techniques can be extended for use in nonlinear cases. This has only been accomplished in part [104] because of the increased numerical problems, which, even for diatomic molecules, are not yet completely solved. There are several excellent review articles that have discussed this interaction in detail albeit only for diatomic targets [4, 129-132]. We intend to briefly repeat here only what pertains to polyatomic molecules. Due to the numerical complexity of the exact treatment of exchange, several attempts have been made to model it through simple localization. We would first discuss these model-exchange potentials and their application to polyatomic targets. Three kinds of model exchange potential have been tested for polyatomic molecules in the past few years. The first is the semiclassical exchange approximation [133-136] used for electron-molecule scattering at intermediate as well as at low energies and recently employed for e-CH 4 scattering [105]. The semiclassical exchange potential is calculated via the following expression: Vex(r , k 2) = ½[2k 2 - V~,(r)] - ½ { [ 2 k 2 - V,,(r)] 2 + 47rpo(r)} '/2 .

(32)

Thus the semiclassical exchange is energy-dependent (k 2) and depends on the unperturbed charge density of the molecule. Again, such a local potential transforms to an A 1 representation of the molecular point group. It can be expanded in terms of symmetry-adapted functions of A 1 IR: Vex(/', k2) =

Z Vth ex (r, k 2) X,hA1 ( r^)

(33)

lh

and the corresponding coefficients vt~(r) can be evaluated using eq. (24). The second kind of model exchange potentials depend on the FEG nature of the noninteracting Fermi gas of electrons [4]. Hara [100] first introduced this simplified exchange treatment into electron-molecule scattering and since then it has been used widely for diatomic and polyatomic systems [129, 130, 101]. The Hara version of exchange potential (HFEGE) is finally derived as [100, 101], vHFEGE, x

2 kv(r)(1

1-7/2

in l+T/ ]

(34)

where kF(r ) = (37r z Po(r)) 1/3 ,

~(r) = (k 2 + 2 I + kv)2 , / 2 / k v ,

(35)

and I is the ionization potential of the bound system. It can be seen from eqs. (34) and (35) that as r--->~ the numerator of 7? should be k, and not (k:+ 21) 1/2 as in eq. (35). This has led to several alternative forms of the HFEGE exchange potential, namely

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules •(r)

= ( k 2 q-

369

k 2~1/2/1¢ F/

(36)

"'~F

or, a semiempirical form, in which I is kept as an adjustable parameter. However, such a tuned form of the HFEGE potential is not numerically convenient because of its energy dependence. The form (36) is too strong and erroreously may compensate for polarization effects at very low energy, thus giving rise to effects such as the RT minimum. Morrison and Collins [129] have tested such a tuned HFEGE potential in e-H 2 scattering. Dill and Dehmer [137] have used a rather simplified form through standard X, techniques, namely Xt~(r) = -(3/270 Vex

-x[3 3p(r)] 1/3

(37)

where the proper value of the parameter a x is not known for e-molecule problems and therefore v has to be treated as an adjustable parameter. In the third kind of model-exchange approximation recently used in polyatomic systems, the orthogonalization technique [106, 111] is also included along with the HFEGE potential. In this method, an additional inhomogeneous term is introduced in the scattering equation: /3

As ~bt~(r)

(38)

where the As are Lagrange multipliers to enforce orthogonality of continuum orbitals orbitals ~bt~(r) belonging to the same IR, i.e., the integral

fobS(r) F(r) dr

F(r) to

bound

(39)

is made to vanish. Salvini and Thompson [101] and later Jain and Thompson [116, 117, 138, 139] investigated this and found that for e--polyatomic molecules a successful exchange treatment is one in which the orthogonalization method is also incorporated along with the local HFEGE term. The introduction of orthogonalization leads to an inhomogeneous scattering equation, but since this term is of short-range character, the final radial equation can still be solved fairly easily [140, 141]. Orthogonalization could in principle be considered an exact technique, whenever a similar bound function within each molecular IR corresponds to each continuum orbital, to yield the correct extra HartreeFock orbital within the static-exchange model. All the above approximate techniques have been employed along with approximate polarization interactions and therefore they still need to be checked against exact exchange calculations. Unlike standard techniques of calculating exchange terms exactly through static-exchange equations [132], a separable form of the exchange kernel has been found quite promising and easy to implement numerically [102, 103]. For polyatomic systems the extension is quite straightforward except that storage problems on a computer become excessively large. The exchange kernel K(r, r') can be expanded as

K(r, r') = ~ x.(r) y. x.(r') • t~

(40)

Then, one can choose an orthonormal basis of Gaussian or Slater functions [X~] that gives a diagonal representation of the exchange potential with eigenvalues y~. To obtain the final scattering equations

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

370

(see section 4.2), we generate single-center separable exchange terms by projecting each chosen basis function onto a symmetry-adapted function belonging to the IR in question, i.e. X.,h(r) = r

(41)

dF x . ( r )

which can be evaluated in closed form [102, 103]. Note here that the separable form of exchange needs to be evaluated for each symmetry element separately. Its energy independent form makes it . ptt [r,~) are available only for linear systems [143[. numerically convenient. Programs to compute X~tht Convergence should be checked with respect to a, l and the radial span r. For linear systems, one can obtain proper convergence within as few as 10 values of a and l and around 10 Bohr radii of radial distance. However, the same number of l values for nonlinear systems may pose serious numerical problems, while the strongly directional nature of the molecular orbitals of the bound systems requires a completely different analysis of the numerical efficiency of such methods.

3.3. Wave functions for the bound nuclei Under the BOA, the Schr6dinger equation for the nuclear motion (rotational and vibrational) reduces to

{-

zNz.,

+ ZNNE iR -= ,12 + e'(RN)- e

}

N(RN)=0,

(42)

where E is the total energy, the summation over N and N' runs over the number of atoms in the molecule and E'(RN) is the potential field due to the bound electrons. The wave function for the bound nuclei ON(RN) in polyatomic molecules is the subject of many textbooks and we shall discuss only those aspects pertinent to the present article [46, 47, 144-146]. Recently, Thompson [10] has discussed, albeit briefly, the vibrations of polyatomic molecules within a general review of vibrational excitations of diatomic molecules by collisions with electrons. Since the vibrational period is much smaller than the rotational period and nuclei make many oscillations before any appreciable rotation of the molecule, we further assume that the vibrational and rotational motions can be separated and treated independently. It should be noted here that such an approximation may be less accurate for polyatomic molecules [46, 47] and for higher vibrational and/or rotational levels; however, the error introduced should not be significant for the present simple small molecules and low-lying vibrational and/or rotational states. Equation (42) can now be decomposed into the sum of rotational Hamiltonian Y(rot for a non-vibrating (rigid) molecule and a vibrational Hamiltonian for a non-rotating molecule, i.e.,

ffffrot =

1

^2

^2

^2

(Jx/I x + Jy/Iy q- Jz/Iz),

(43)

and ~avib =

1 ~

3N-6 E i=1

^2

~2

Pi + Aiei •

(44)

Here (ix, J~, Jz) are the components of the total angular momentum operator j of the molecule and (Ix, Iy, I z) are three principal moments of inertia along the axes (x, y, z) fixed in the molecular frame.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

371

The ~)i are the normal coordinates which are related to the components of Si, the internal displacement coordinates of the atoms. /5 is the canonical momentum operator defined as

Pi = ~I'/30i

(45)

with T=~

1

3N-6

~

~2 Qi.

(46)

i=1

Therefore, the eigenfunctions ~ON(RN)of eq. (41) can now be written as a product of the eigenfunctions of ~rot and ~vib. In eq. (44) we have kept only the quadratic term in Q~ but higher-order terms must be used to get more accurate wave functions. Thus, a simplified form of (44) will give rise to uncoupled harmonic oscillator wave functions, i.e., 3N-6

~0vib= ~I ~Ov,(Oi),

(47)

i=1

and to the eigenenergy, 3N-6 Evi b =

£ Evi , i=1

(48)

with Evi = (v i + ½)~.

(49)

Here ui (i = 1, 2 , . . . , 3N - 6) is the frequency of the ith normal mode and the harmonic oscillator wave function for this normal mode is written as q~o,(Q,) = [3']/Z/Trl/22V'(v,)!] 1/z exp(- 3~Q~/2)

nvi(T]/2Qi)

(50)

where 3~ = 4zr2vi and Hn(x ) is the Hermite polynomial of degree n. For diatomic molecules, there is only one normal mode (stretching) Q1 = x/-gR, where R is the internuclear separation and/~ is the reduced mass of the molecule. The above harmonic picture is true only for low vibrational levels. The vibrational levels of a polyatomic molecule are denoted by (v l, v 2 , . . . , U3N_6) , which are ordered so that the modes appear in the sequence according to the group theory. For example, the three normal modes of a water molecule are represented by (vl, 02, v3). The (100), (010) and (001) are termed as the fundamental levels (i.e., those levels for which all the quantum numbers are zero except the one with value 1). Levels of the type (020) (003), e.g., in which only one mode is excited with more than one quantum number (v/> 1), are called overtones. When two or more quantum numbers are excited, the modes are known as combination levels. We will be concerned only with fundamental levels. We now consider briefly the symmetry properties of the vibrational wave function. Several standard texts discuss this topic in detail [46, 47, 144]. The symmetry operations in standard group theory notation also give information on the degeneracy of the normal modes. While the point group

372

F.A. Gianturcoand A. Jain, The theory of electronscatteringfrom polyatomic molecules

representation is particularly easy to apply for a rigid molecule, for a real rotating, vibrating molecule, a more accurate method involves the conservation of molecular symmetry with respect to feasible permutation of nuclei [147]. Let us take as an example the H20 molecule. If both the OH bonds are equal (as in the case of symmetric and bending modes), then the nuclear configuration belongs to the C2v point group; otherwise, in the case of an asymmetric mode, it transforms according to the C s point group. The three normal modes, namely the symmetric (100), the bending (010), and the asymmetric (001), should transform according to one of the four IR's A~, A 2, B 1 and B 2. It follows from simple group theory techniques [144, 148] that the modes (100) and (010) belong to the totally symmetric IR A~ and the asymmetric stretch mode (001) belongs to the B 2 IR. (It should be noted here that we are considering the H20 molecule in the y - z plane; if one takes it into the x - z plane, the corresponding IR for the (001) mode would be B 1.) In other words, the nine-dimensional (3N, here N = 3) representation F, will contain the following vibrational representation (after taking out three rotational IR's and three translational IR's) Fvib = 2A l + B 2 .

(51)

To obtain selection rules for a vibrational excitation, we consider the properties of the eigenfunction under the molecular point group. For example, in the C2v point group, the application of a symmetry operation multiplies the normal coordinates Qi by ---1; therefore, the ground state wave function which contains only quadratic terms in Qi, is left unchanged by each operation. This implies that the ground state function 0v(000) belongs to A 1 IR. However, the wave function for the fundamental levels transforms according to Q1 itself. This means that 0v(001) belongs to B 2 while $~(010) and ~Ov(100) both belong to A1, as mentioned earlier. In general, from symmetry considerations, transitions are allowed from the ground to that excited state whose wave function has the same symmetry species as at least one of the dipole moment components/zx,/Xy and/z~ [148]. We now discuss the symmetry properties of the rotations in a general polyatomic molecule. Polyatomic molecules can be classified into three types of rotations according to their moments of inertia (/~, Iy, Iz) in all the three directions: (a) the spherical top, with I x = Iy =/~ (e.g., CH 4, SF6); (b) the symmetric top, with I x =Iy ~ I z (e.g. NH3); (c) the asymmetric top, with I x ~ l y ¢ I z (e.g. H20, H2CO). The rotational eigenfunctions of a spherical top are trivial and can be written in terms of quantum numbers J (total rotational angular momentum), K (component of J along SF axes) and M (component of J along BF axes). But the total energy depends only upon J. Since K and M each have (2J + 1) different values for each J, the spherical top's rotational levels are (2J + 1)Lfold degenerate. If the nuclear spin of each H atom is ignored, e.g., as in the case of CH 4, then the spherical top's eigenfunctions belong to the A 1 IR for all values of J [48]. For a symmetric top, the eigenfunctions are also characterized by the same three rotational quantum numbers ( J K M ) and can be defined as [149-151]: ( 2J + l ] 1/2 = C-8-U- /

1,

(52)

where ~ is the well-known rotation matrix and (a/33,) are three Euler's angles. For an asymmetric top, K is no longer a good quantum number; therefore, the eigenfunctions of an

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

373

asymmetric top are not trivial. (The subject of an asymmetric top has been fully described in the literature [152-156].) The asymmetric top eigenfunctions can be expanded in terms of symmetric top functions, namely J

~jM,(a/3Y) =

E K=-J

g~ K

(53)

where, in place of K, we have introduced a pseudo-quantum number r, such that - J -< ~--< J, and have assigned r = - J to the lowest energy level, z = - J + 1 to the next lowest level, and so on. The expansion coefficients a JzK are determined by diagonalizing the asymmetric top Hamiltonian of eq. (43) with the eigenfunctions of eq. (53). Alternatively, it is expedient to employ a basis set of (2J + 1) linear combinations of ~bjx s M, which are adapted to the symmetry properties of the asymmetric top. An asymmetric top belongs to the four-group V(E, C 2, C y, C~), under which the eigenfunctions ¢~KMeither are invariant or change sign only. Here E is the identity element and C2 is a two-fold rotation axis about (x, y, z) fixed in the s molecular frame. The corresponding IRs of the V-group are A, B x, Byand B z; ~bjKg belongs to one of these IR's. We choose $SKM to be (54)

~SKM,(afl),) = 1 (g6K , + s

with A = 0 or 1. Under the symmetry operations of the four-group V, the SJKM~ transforms as (55a)

s

E~jKM,~ = ~IJKMA x

S

(

(55b)

l~J+AdtS

C2¢JKMA = \ - - x ]

"~YJKMA

y S |'~J+K+AdtS C2~] JKMA = (--~t l WJKMA z

s

= (-1) K

(55c)

S

(55d) s

In table 1, we have shown the relationship between the symmetry representation of $Jl(g, and the parities of K and J + A. The Hamiltonian in the new basis set therefore factors into four blocks corresponding to the four possible IRs of the group. Natanson et al. [157] have discussed this problem with respect to permutation and permutation inversion operations on the rotational Hamiltonian; the final results are basically the same and give similar selection rules. Table 1 Parity of

Symmetry representation S

K

J + A

of ~bjxM,

e* e 0 0

e 0 e 0

A Bz Bx By

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

374

In order to evaluate eigenvalues

E jr

and the expansion coefficients a~:, J~ we write

s

(56)

Hrott~JKMA = E H K sI , ; , a ~ S Ks, M , ~ • K' •

s

s

s

Hence the symmetrized mamx HKK,Asplits into two submatrices H 0 and H 1 corresponding to A = 0 and 1, respectively. Further, each submatrix can be factorized according to the parity of K. Finally we would have four submatrices E + ( A = 0 , K = e v e n ) , E - (A= 1, K = even), O + ( A = 0 , K = o d d ) , O (A = 1, K = odd). The matrix elements of the H matrix are given by [156]

K21+

CK 2

(57)

HK, K-+2= HK_*2,K = }(B - A)[(J ~ K ) ( J ;- K - 1)(J -+ K - 1)(J + K + 2)]

(58)

HKK = ½(A + B)[J(J + 1) -

where A (= ½Ix), B (= ½Iy), C (= llz) are the three rotational constants of the asymmetric top. The rank and the IR of each submatrix are shown in table 2. Finally, the secular equation to be solved for each case can be written as E

-

=0

(59)

K

where the M-matrix elements are those of the submatrices E +, E-, O + and O - A specific program (ASYMTOP [158]) was written in order to evaluate expansion coefficients a~" and the energy levels E j, for general asymmetric top molecules. 3.4. Electron motion versus nuclear motion

In addition to the well-known BOA, in which all the occurring motions, electronic, vibrational and rotational, are considered as uncot~pled under certain conditions, we can also examine this issue with respect to the incoming electron. We can further divide the problem into the scattering of electrons by polar (e.g. H20, NH 3, SO 2, etc.) and non-polar (e.g., CO 2, CH 4, SF6, etc.) systems, which differ quite significantly both in the physics and in the computational ease of generating cross sections. The subject of electron-polar molecule scattering has been previously discussed in detail [6-9] with respect to its applications in many applied sciences and engineering [159]. To solve eq. (1) in an ab initio manner, one should expand the total wave function q~Tin terms of all possible target states with proper coupling with the incoming electron states. In this natural approach, the rotational and the vibrational Hamiltonians are correctly included in the final radial equations. Table 2 Rank a n d l R o f E +,E , O ~ a n d O IR

Rank

Matrix

A

Parity of K

J (even)

J (odd)

J (even)

J (odd)

E+ E O* O-

0 1 0 1

even even odd odd

A B~ By Bx

B~ A Bx B~

~J + 1 ½J ½J ½J

~(J+l) ½(J- 1) ½(J+ 1) ~(J + 1)

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

375

However, this is not very practical because of a large number of the so-called open channels that are immediately introduced by the nuclear Hamiltonians presiding over rotations and vibrations (~rot and ~vib)" Even for the simplest case, i.e., the H 2 molecule, a full ro-vibrational close-coupling calculation is not computationally feasible. For polyatomic systems, even the well-known rotational close-coupling formulation of Arthurs and Dalgarno [160] may be very slow to converge and the inclusion of vibrational channels will make the problem too cumbersome to be solved by present-day computers. There are, however, other simplifications that can be introduced in formulating the theory without losing much of the physics or much of the necessary insight into the process. Let us consider the fact that in the vicinity of the molecule, the incoming low-energy electron feels a very strong attractive force and consequently the collision time becomes so short that there is no appreciable nuclear motion (rotational and/or vibrational) during the collision. In other words, the coupling between the incident electron angular momentum and the rotational and/or vibrational states is too weak to be significant at most energies. One can therefore neglect the rotational and vibrational Hamiltonians altogether and treat their energy spectrum as totally degenerate. Such is the philosophy of the so-called "adiabaticnuclei (AN) approximation", which is a counterpart of the BOA in the isolated molecular case. If one considers that the typical collision time r c is of the order of ~ 1 0 -16 sec, which is much less than the rotational (r R - 10 -12 sec) or the vibrational period (r v ~ 10-14 sec), one can then calculate the FN scattering amplitudes as a function of internuclear coordinates and only after that perform angular and/or radial quadrature in order to obtain the rotationally and/or vibrationally inelastic scattering amplitudes. In some situations it may happen that r c "~ r R but r c =rv; this means that the vibrational motion can no longer be treated adiabatically. Such situations arise, for instance, in the resonant vibrational cross sections. In spite of its promising simplicity and wide applications, there are still several physical cases where the AN approximation is no longer valid. For example, near rotational and/or vibrational thresholds, close to shape and core-excited resonances,* and in the case of polar molecules, where the presence of long-range forces makes the coupling still strong at large distances from the target. It is worth mentioning that the validity of the BOA implemented when obtaining the target bound electronic states is still assumed to hold even when the extra electron spends large amounts of time around the target molecule. Thus one can say that, in general, nonadiabatic effects stemming from the coupling of electronic and nuclear degrees of freedom can be neglected in treating low-energy electron-molecule collisions, while great care should still go into selecting the most realistic and efficient basis for expanding the initial and final molecular ro-vibrational states involved in the energy transfer processes. For diatomic polar targets some of the formulae that have been obtained within the AN scheme are particularly useful and can be extended to polyatomic targets: see, for example, the closure formulae of Dalgarno and Crawford [161], the angular frame-transformation model [162], the energy-modified AN scheme [163] and the scaled adiabatic-nuclei-rotation theory, particularly suitable near threshold [164]. Chandra and Temkin [165-167] introduced a hybrid approach, in which the nonresonant symmetries were treated by the AN approximation, while in the resonant symmetry the nuclear motion was considered properly; they applied this theory to the vibrational excitation of the N: molecule, where a well-known rrg resonance dominates. No further extension to polyatomic molecules has ever been attempted along those lines. A more rigorous theory has been formulated by Chang and Fano [168] (see also Fano and Dill [169]) * This limitation holds for narrow shape resonances with widths smaller than 1-2 eV.

376

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

in which the radial region is partitioned depending on the physical situations. For example, near the molecule (short distances) a BF frame of reference theory is more appropriate, while at large distances, the nuclear motion should be considered in a SF frame. Solutions for the outer and the inner regions may then be joined smoothly at a chosen boundary. To see in detail the implications of this so-called frame transformation (FT) theory is not a simple task and, except for a few diatomic cases, it has never been employed for the majority of the molecular systems of interest.

4. The scattering equations In the previous sections we have primarily discussed the general features of the forces at play in any electron-molecule scattering process in order to get an efficient and physically transparent way of relating the structural properties of a target molecule to a unified picture of the whole electrostatic interaction. The dynamical effects of the nuclei that are introduced by including the previously defined terms, d~ro t and ~°vib, in the total Hamiltonian represent one of the major hurdles in the electron-molecule collision problem. As mentioned earlier, we need to set up our scattering equations in the most efficient way. It is therefore clear that different frames of reference could provide, in each different situation, the most elegant way to treat and simplify the collision problem.

4.1. Frames of reference Previous reviews [6, 7, 170, 171] have given a fairly complete picture of the various frames of reference, or rather of the various schemes for translating between them. Although their main emphasis has been on diatomic molecules, the basic ideas remain the same for polyatomic targets. The present discussion will thus be brief. Assuming that the FN scheme is valid, the impinging electron's angular momentum vector ! couples (just as the bound electrons) to the main molecular axis of symmetry R and therefore the internal product A -- l. R yields a well-defined quantum number for the system. Whenever long-range forces are not too important, the spatial region within which A represents a collision constant can extend up to the asymptotic region where the S-matrix can be defined [6]. In this case, the point group symmetry of the BOA molecular electronic wave function leads to the decoupling of the scattering equations into subspaces of each given projection A (diatomics) or of each given IR (polyatomics) to which each l projection belongs for that target molecule. In the presence of long range forces (such as the dipole potential), an FN-defined scattering matrix leads to divergent total cross sections since each angular momentum l no longer couples to R in the asymptotic region. Therefore, after determining a suitable value of the relative distance between the electron and the molecule, one must re-introduce the contribution from the nuclear Hamiltonians. If one considers a spherical top rigid rotor for simplicity of notation, this means that we also have the Euler angles as variables for the nuclei and therefore a good quantum number will be the total angular momentum J = l + j where j is the rotational angular momentum vector, and the inversion parity r/= (_)t+j of the whole system [160]. Different choices can however be made for the set of axes where both nuclei and electrons move during the collision. Such choices correspond to different frames that can in turn help to choose further approximations (less drastic than the FN approximation of above) depending on kinematic conditions.

F.A. GianturcoandA. Jain,Thetheoryof electronscatteringfrompolyatomicmolecules

377

A fixed frame in the laboratory (SF) uses an angular basis of eigenvectors of 12,j2, j2, JR (JR is the projection of J along R) and r/ which are useful to express asymptotic decoupling since the radial interaction appears on the r.h.s, of the corresponding coupled equations and vanishes outside the range of action of the electrostatic interaction. On the other hand, one can choose a rotating frame, held fixed with its main axis along the molecular principal axis (MF), whereby the angular basis of eigenfunctions is obtained from the eigenfunctions of 12, Ig, j2, JR and r/'= (-1)~r/, which is particularly useful to introduce approximations when In, i.e., l" R, is only approximately considered a good quantum number (e.g. the FN approximation) [168]. Finally a third possible choice is one in which the rotating frame has one axis that always points at the incoming projectile during collision [172] (BF frame). This selection requires an angular basis of eigenvectors of j2, JR, j2, JR and parity 77 which is particularly good to describe dynamical couplings where JR can be approximately considered a constant during the motion [173, 174]. For a linear target the possible choices of above can be summarized as follows: ISF basis) = YtjJm( rA, R~) = ~ (lmt; Im][JM) YT't(i) • Y~J(R) mlmj

(60a)

[MF basis) = Y[aM(i,R)= Y a ( i ) (\ 2J J, ^ ~ +/ 111/2 ~MA(R)

(60b)

. ~ y ~ ( R ) ( 2J + 1]~/2 ~ t a ( / ) Iav basis) = Yja,M (r,g) = \~1

(60c)

where the orientation vectors of the incoming electron (i) and main molecular axis (i~) have different references in each case and (... [... } are the coupling coefficients. One can also move from one reference frame to another by a unitary transformation applied at some preselected e--molecule distance rt: (61)

ISF) = u + {[MF) }u.

Their practical implementation, however, has been attempted in only a few diatomic cases [175-177] and never for a nonlinear polyatomic molecule.

4.2. Coupled expansions We first set up our scattering equations in the BF frame of reference. Here, the Schr6dinger equation of the electron-molecule system reads as, 1 2+

+

Vint(r)

-

E) b(r, ri) = O,

(62)

where the operators in the bracket are, respectively, the kinetic-energy operator of the scattered electron, the unperturbed molecular target Hamiltonian, the interaction energy between electronmolecule complex and the total energy of the system. The above equation depends parametrically on the nuclear coordinates R. The total wave function ~bcan be expanded such that

~b= sg ~ @i(ra,r2,...,rz) Fi(r)+ ~ ~i(rl, r2,...,rz+a)ai i=1

i~l

(63)

F.A. Gianturco and A. Jain, The theory of electron scattering [rom polyatomic molecules

378

where ~i are the target electronic states and any other suitable pseudostate that can represent the target response function to the polarization of the molecule due to incoming electron. The continuum function (scattered electron) Fi(r) describes the motion of the scattered electron in molecular state i and ~/is the usual antisymmetrization operator• To include correlation effects, one must also add some correlation functions sci, with a i as the weighted factors• Let us assume that the center of the e--molecule system rests at the COM of the molecule (since the incoming electron is very light compared to the heavy molecule, this approximation is perfectly valid). The spin of the scattered electron is coupled with the total spin of the molecule to form an eigenstate of S 2 and its projection along the symmetry axis (say, S~) corresponding to the total spin quantum number S and its substate M s which are constants of motion during collision. Each of these eigenstates can be expanded over the set of angular functions according to the prescriptions of eqs. (60), which provide the basis of an IR of the molecular symmetry point group. Such a new set of SAWF was discussed in section 3.1 and can be used for both the continuum and bound orbitals. In this representation, eq. (63) becomes op~SMs "~- ~ ~ T~p~,SMst,. -1 p~s ~ c.p,SMs(r r2, _ ,_p~s ilh \°1' r2, • • " , t z , r, o')r Eel h (r) + bi \ 1, " " • , tZ+l]tli i=1 Ih

(64)

i=1

where the meaning of the various symbols has been defined in section 3 and tr is the spin variable for the continuum electron. Each of the component functions q~ in eq. (64) is defined as follows: d')Pl'tSMs

2 -~-

--Pil~iSiMsiz c~ i trl, r:,

Ms;ims~

i 1/2 S rz) g l hPtZ( r ) r l ( ~1, msi ) (-,S ~MsimsiMs ^

(65)

where ~7's are electron spin functions for the free electron and C's are the Clebsch-Gordan (CG) coefficients. The bound orbitals ~bi describe the Z-electron target wave function used in expansion (63), each transforming like a particular IR of the molecular point group and with a specific total spin eigenstate. The main molecular symmetry axis defines the M s direction. As a standard technique, we now project the Schr6dinger equation (62) onto the channel functions ~p~,SMs and onto the symmetry adapted correlation functions ~ p~'sMs. We obtain the following set of ilh infinite coupled integro-differential equations for each scattered electron channel function (p/~S):

r

2

3t- k i f i l h (r)

=2 2

~ [Vi,h, p~s Crh'(r) + With, p~srrh'(r) + 0 p"s (-, • ith, rrh',tr~lF ,J p~s rrh',r)

i'=1 l'h'

(66)

The direct potential matrix vp,S -- ilh, i'l'h' defines the coupling between two channel functions through the operator of eq. (10). The symmetry-adapted coupling scheme of eq. (64) simplifies the BF equations greatly. For each electronic eigenstate of the target, one can write the corresponding form of the potential originating from this state. A typical diagonal term, for example, can be written as V P i ' i s ( r ) = Z ViSlh(r) XP/~'is(~) lh

which means that in eq. (66) the direct potential matrix elements are written as

(67)

F.A. Gianturcoand A. Jain, The theory of electron scatteringfrom polyatomic molecules

plxS Vilh, i,t,h,(r) 6pp,~lzlt'

379

=

= Z ViLH(r LH

) f "~lh ~rptxS~crPiIxiSyp'I'L'S d~ ~ L . "~t'h'

(68)

where the angular integration can be performed easily (appendix A). The second term on the right-hand side of eq. (66) is the exchange term and can be solved approximately as discussed in section 3.2.3. The correlation energy aPe's V ilh, i'l'h' arises from the elimination of the m equations involving the correlation functions (~). A well-known procedure for doing this is to diagonalise the electronic part of the (Z + 1)-electron Hamiltonian matrix within the space of the correlation functions, thus yielding a set of eigenfunctions ~ p~'sMs with the corresponding eigenvalues e [178-180]. There are, however, practical problems in choosing an efficient and realistic set of m functions to properly represent correlation at short range, as we have already discussed in section 3. Finally, the full set of the integro-differential equation (66) provides one with a set of linearly independent solution vectors within each ( p l z S ) subset and with a dimension index lh. To further distinguish this set, we introduce the suffix i T h ' and rewrite the radial solutions as Fp~,S ith. i'rh'(r) • If we include only the electronic ground state of the target molecule, then the summation (and therefore the index i) over i' can be dropped in eq. (66). When dealing with closed-shell targets the total molecular spin S is zero and therefore the indices S and M s can be omitted altogether for this special case (which, however, involves most of the molecules studied so far). Now, to obtain the expression for the scattering amplitude we proceed as follows [181]: we form a general solution f~'(r) in terms of the linear combination of the independent solutions, i.e. f~Z(r) = ~ a p~, i Fi/ptl.(r), i

(69)

(note that i(j) refers to lh), which yields the total scattered electron function (70)

F(r) = E r-~ aiPJZFp~,(r) Xf~,(p) . ijpp.

From the standard boundary conditions of the scattering theory, the F(r) has the asymptotic form, ikr F(r)r.~

eik. r + f(/~. t~) e___,

(71/

r

with/~ and ~: being the initial and final directions of the continuum electron. With S-matrix boundary conditions, the function F(r) should satisfy: F(r) r--,= ~ k -'/2

~ aP~[exp{-i(krpl.~lhl'h'

½Z'Tr)}6hh,6 w -- exp{i(kr - ~l i , rt)}S,hp,' t'h'] XP'~'(i) •

(72)

We can determine the a-coefficients in the SF frame of reference by carrying out a simple rotation and equating the ingoing and outgoing parts of eq. (71) with those of the incident plane wave. The a-coefficients are found to be aPh" --

2zr ilX~'h~(/~) ik

(73)

380

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

Note that in deriving eq. (73) we have made use of the following completeness relation, E VT'*(f,) VT(i) = E Xf~"(k) Xf2(i ) . m

pp.

Finally, the scattering amplitude in eq. (71) in the BF frame of reference is determined to be f(f~" i ) = E

2"17"

- ~ X f ~ ( f c ) Xrh,(r), pp. ^

:l-l'[

c, pp.

b'th, rh' - ~trcShh, ) .

(74)

lhl'h ' PP.

In actual calculations, it is convenient to work with real boundary conditions, thus extracting the real K matrix which can be written in terms of the S matrix of eqs. (72) and (74) as S = (1-iK)/(l

+iK).

The scattering amplitude (eq. (74)), can be transformed into the SF frame of reference through the following rotation relations, pp. ^ Xrh,(r)

=

E /~PP. l'h'm'

rAm, ( O ~ ] ~ ) Y~,(i')

m 'A

(75)

and

x72(f,) E =

(76)

'

mA'

where the prime coordinates (i',/~') refer now to the SF coordinate system. Since/~' and/~ are along the z direction, eq. (76) reduces to X~(/¢)

=

E -PP. ( 2 l + 1 ~ 1 / 2 l blhm\ 4~" / ~0m(O~J~')/)'

(77)

m

Using eqs. (75)-(77), the SF scattering amplitude is given by f(

i. r".,

aft'y)

=

2 lhml'h 'm'

{7r(2/+ 1)} ~/2 i'-"" gP" gP" V~,(i') ik ~ Ibm v l ' h ' m '

App.

l'

X ~Am(O~) ~lorn(Ol[~'~)(S \

Pp. lh, l'h'

-- all,ahh, )

(78)

It should be noted here that the sum is now over both the positive and the negative values of m. The new/~'s are related to the old b's by the following relations: /~,~ - (--1)m ei~(q-1)J4b m =b m _

q

m>0 m=O

i~-(q 1)/41..

X/2 ~

ul,.i

m<0

(79)

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

381

where q =---1, depending upon the particular IR. The transformation of the angular part of the scattering amplitude of eq. (78) into the other frames of reference defined in section 4.1 can be performed by using eqs. (60a-c) defined in that section for the angular part of the corresponding S matrix. 4.3. R-matrix m e t h o d

A general description of the R-matrix theory has been previously discussed in several places that report the latest references on the subject [6, 171,178, 182-185]. We therefore repeat here only a brief account of its significance in the present context. In fact, the R-matrix approach is a general mathematical device to transform the solutions of a given problem from one region to another. In e--molecule scattering theory, the two physically separated regions (the inner, of short-range character, and the outer, of long-range character) defined by the electron-molecule distance are treated in a different way in accordance with the different physics involved in each region. The basic concepts are the same as those discussed previously in section 4.1 on frame transformation. In the inner region, the total wave function of the e--molecule system is expanded, as in eq. (63), in terms of bound and continuum orbitals. The bound orbitals vanish at the chosen boundary while the continuum ones are non-zero at the boundary and propagate further in the asymptotic region. The partitioning of the configuration space in the R-matrix treatment, broadly speaking, is analogous to that of frame-transformation theory, but is more geared to a quantitative treatment of the problem. The "inner-region" is in fact the "core-region", where the physics comes from the solution of a (Z + 1) electrons bound-state molecular structure calculation. Standard molecular-structure codes can be adopted and modified in order to evaluate the R matrix at about the boundary of the molecular charge density, although the determination of the right boundary surface is still a matter mostly decided by educated guesses. It is convenient to define the R matrix with respect to "inner-region" solutions at a spherical ~ s i'h'r(r) . of eq. (69), the R matrix boundary r = r,. Thus, in terms of each BF continuum solution F pih~, can be written as [6, 178] Fib I (r) = ~ R p~s th.t'h' l'h'

d F,','h'(r) p.S ~r

/ ra

p.s -- - - Fi,rh.(ra) ra

•

(80)

As discussed in earlier applications of R-matrix theory [6, 178-185], the choice of the parameter b is quite arbitrary and therefore it can be set at zero. To the best of our knowledge, there has been no application of R-matrix treatment to e--polyatomic molecules. Since the calculations in the inner region can be treated within the BOA, the complicated nature of rotations and vibrations of polyatomic molecules does not produce the main obstacle while the breaking of the cylindrical symmetry in nonlinear systems causes most of the difficulties when treating electron-electron interactions involving bound and continuum molecular orbitals. 4.4. Dividing angular space

Both the frame transformation or the R-matrix theory demand a partition of the r space, which is often an arduous task. In an alternative approach (that still has some limitations), suggested by Collins and Norcross [162], the orbital angular momentum space is partitioned in terms of low l, intermediate l,

382

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

and high l contributions. Since this quantum number appears in the scattering equation as the centrifugal term with the familiar form l(l + 1)/r 2, and influences the scattered function significantly at small r as well as at large r, its low and high values have different effects on the final results depending on their relative importance with respect to the main electrostatic interaction. The higher partial waves do not easily penetrate into the inner region and therefore can be treated in a completely different way than that used for the more penetrating, low-/partial waves. The determination of low-/and high-/ values is again based on some empirical testing for each problem, but simple approximations such as the Born approximation are quite helpful in judging angular-space boundaries. For example, as I increases, the centrifugal barrier becomes so strong that the distortion of the scattered function can be described as mostly due to long-range interaction as given in simple theories such as the Born approximation or the distorted-wave (DW) theory. This is a great computational advantage as one sees that, after a given number of partial waves, the close-coupling K-matrix elements become identical to either those from first-order theory (intermediate l) or those from the very simple Born approximation (high 1). This type of treatment is essential for polar molecules where, due to the often strong and long-range dipole potential, a very large number of partial waves (sometimes a few hundred) contribute significantly to the cross section, particularly near to the forward direction. It therefore becomes easier to explore further the proper use of first-order theories after accurate calculations have been carried out for small partial waves. The treatment of Collins and Norcross [162] was in the SF frame of reference for higher partial waves. However, Siegel et al. [186, 187] employed this theory in a more efficient way by treating the "intermediate-/" partial waves in the BF frame of reference. When only the dipole interaction potential is used, the calculation of the S matrix for "intermediate l" becomes very simple. Since the final S matrix constructed from various methods is for all the necessary S-matrix elements, it is appropriate to call this the "hybrid S-matrix" approach (see fig. 5). This kind of hybrid approach is not only computationally economical but seems to be an essential tool for treating molecules with a permanent dipole moment. For moderately polar molecules (D - i a.u.) partial waves with maxima I up to 6 (or 10) are sufficient for close-coupling calculations. Beyond that the Born K-matrix elements give

Iow-I region

i I I

t k

~lose-Co uplin~ ~alculations

I

I l

Intermediate- I region m ultipole (e.g. dipole) in teractions only

h i g h - I region f i r s t Born a p p ro xima tio n

Fig. 5. Partition of angular momentum space in the hybrid S-matrix calculation approach (see text for details).

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

383

the correct contributions within a few percent. In a variety of polyatomic polar molecules ( H 2 0 , NH 3, H2S ) Jain and Thompson [110, 138, 139] have used the angular-frame-transformation technique to obtain well-converged cross sections in the energy range 0-10 eV. This theory was further improved for linear molecules by Norcross and Padial [188] in their so-called multipole-extracted-adiabatic-nuclei (MEAN) approximation, where the well-known closure formula of Crawford and Dalgarno [161] was employed in a more general way. It would be interesting to make a comparative study of the two schemes, i.e., angular and the radial frames; even for the simple diatomic molecules such an investigation may be fruitful. From a physical point of view, the two spaces (r and l) are not separated but rather strongly inter-related. Thus, it is only out of numerical expediency that one can use either the radial-frame transformation or the angular frame transformation, with the proviso that both approaches must yield the same final scattering amplitudes.

4.5. Numerical methods to solve the scattering equations The end product after solving our basic scattering equation (66) is to generate the K matrix (or S or the T matrix) from proper boundary conditions on the scattered electron function F(r). In a single channel problem, this is equivalent to determining the phase shifts, which in turn can be used to obtain the various cross sections. There are several excellent review articles and texts [189-191] that deal with the issue of solving second-order differential equations and therefore we do not need to give details here of such techniques. As pointed out earlier, the SCE approach makes the numerical procedure analogous to e--atom (ion) scattering. The basic equation to be solved for either electron-atom (ion) or -molecule can then be rewritten from eq. (66) in a more general and simple way as, d dr 2

1(l + 1) ] r ~ + k 2 Fm(r) = ~ (Vvt,,(r) + W,,r,(r) Ft,r,(rl).

(811

1°

The basic procedure to solve eq. (81) and to determine the reaction matrix K can be briefly sketched as follows: from the regular boundary conditions at the origin, a set of linearly independent solutions F(r) are matched, at a convenient radial boundary with the correct asymptotic solutions that are known exactly. The K matrix should be independent of the matching radius. Chandra [141], for example, has given details of this procedure and written a computer code that can be used, in general, for coupled differential equations such as eq. (81). This code requires another program that can generate the asymptotic solutions [192]. In the past decade or so, there have been many advances in the calculation of accurate asymptotic solutions and recently Norcross [193] has reviewed this progress for diatomic molecules and atomic processes. An alternative approach is to solve eq. (81) via integral-equation methods [194-198]. In this method, the coupled second-order integro-differential equations are converted into a set of coupled integral equations. The resulting linear-algebraic equations can be solved easily through standard computer codes. This method is more stable, and the exchange term can even be treated almost exactly without complicated computational problems (see ref. [132] for more discussion). This technique may not necessarily require an asymptotic code in addition to the propagation code, and is easily adaptable to single-center expansions such as those used thus far to treat polyatomic targets. In the cases discussed in the literature, however, only the former integrator has been employed (see examples below).

F.A. Gianturcoand A. Jain, The theory of electronscatteringfrom polyatomic molecules

384

5. The computational models In the previous section, we discussed how to set up the scattering equations and some of the numerical techniques to solve them. In practice, one has to choose or devise a different approach depending on the situation. For example, a SCE method may be very difficult to apply on systems like SF6, due to the requirement of a large number of terms in the expansion; near threshold (vibration and/or rotation) scattering will require a proper coupling of nuclear quantum numbers, therefore, a BF approach along with the AN approximation may not be a suitable one. In the following section, we will try to describe the few computational models employed in the past few years to determine e--molecule scattering parameters for polyatomic targets.

5.1. The continuum-multiple-scattering (CMS) approach One of the simplest, and most approximate, methods which has been used to study electron scattering by polyatomic molecules is the continuum multiple scattering X method (CMSX~), originally designed to treat photoionization problems and recently extended to treat electron collisions [199-205]. In the electron scattering case, the continuum electron is treated as moving in a spherically averaged molecular field which is divided into an inner region, chosen from the electron densities of the isolated component atoms, and an outer region encompassing the whole molecule. When the inner spheres are made to overlap by amounts dictated by their atomic-like properties, and the Slater's transition state method (STSM) [92] is employed to produce some form of short-range electron density polarization, then reasonable values for the molecular electron affinities (EA's) were found for several polyatomics [205]. Two additional empirical criteria are then used to cut off the incorrect long-range behavior of the exchange potential (that would overestimate Coulombic contributions to the interaction) and to hinge on the correct TrLRAFt Veo ~ tr),x so that overall attractive behavior for the long-range interaction is ensured. One variant of the above prescriptions also involves the use of a local exchange approximation [100] to treat the exchange interaction both in the inner region and, in a modified way, in the outer region [206-2081. Thus we can define three potential regions: I, N atomic spheres; II, a spherical region surrounding all the N atomic spheres in a closed-packed manner; III, outside region for large values of r. For simplicity, the potential in region II is assumed to be constant and the region III potential to be spherical (however, one can also use a non-spherical term with little additional effort [200]). The total wave function of the scattered electron may be written

(82)

~/total = I/tI + ~tll + ~JIIl

where each term is a solution of the corresponding potential region with appropriate boundary conditions. In region I, the wave function q~ is the sum of wave functions on each atomic sphere, i.e., N i

m ^

~l = 2 Oi = £ Aim f t(kri) Yt (ri) i= l ilm

where fi(kri) is the solution of

(83)

F.A. Gianturcoand A. Jain, The theory of electronscatteringfrom polyatomic molecules

l(l+ 1) (A/r2(A/ i [- rl\dri/ i\dri ] + ~ r i2 +Vi(ri)_k 2]ft(kri)=O i

385

(84) i

which satisfies the condition ri .ft(kri)= 0 at the origin. The coefficients At,~ are determined from the continuity of the wave function at the boundary of the sphere. The form of On is chosen easily from the continuity of the total wave function at each spherical boundary: ~//II = £ A~m jl(kTo) Y?(ro) "~-£ ~ Bl'm' ~Tr(kr,) lrn i I'm'

yrrn'(ri)^

(85)

where ]t and rh are the usual spherical Bessel functions, and r 0 is the radius of region II. In region III, the wave function ~/III must be continuous at r 0 and obey asymptotic boundary conditions. Therefore, ff/III = £ [Al~mf~'t(kro) + B~ g~lI(kro)] YT(/~o) , lm

(86)

may be chosen from the correct boundary conditions. Here, f and g are the regular and irregular solutions of eq. (84) with V~replaced by Vm. In order to obtain the physical scattering parameters, all three solutions are matched at various spherical boundaries and the normalized K matrix can be obtained by solving a set of linearly independent equations [202]. Once the K matrices are determined, the cross sections are easily computed by standard techniques [see below]. Because of its strongly modeled form for both the interaction potential and the target electron densities, the above method is rather difficult to analyze as to its real capability to describe the elementary features of the force fields that control the collisional processes under examination. It can, however, provide some qualitative trends of behavior along series of complicated molecules that are still out of reach for the more ab initio methods. Its application to the Ramsauer-Townsend minimum of CH4,for instance, has been rather poor in yielding its qualitative position as well as in providing total differential cross sections for the resonance region around 8-10eV. However, its application to the interpretation of resonances as observed in electron transmission spectroscopy [209] has been a rather promising direction for using the CMSX, model as a rapid tool for systematically analyzing data.

5.2. SF scattering via model potentials When differential, total and partial inelastic cross sections need to be computed for comparison with experiments, the corresponding scattering amplitudes naturally have to be produced in the SF frame of reference. In e--molecule calculations, however, the non-local nature of the correct exchange kernel requires a further integration at each step of propagating solutions, a difficulty that can be more easily overcome when the multichannel scattering integrodifferential equations are given in the BF frame of reference. If, however, model forms are used for the interaction potential contributions described in section 3.2, then the multichannel problem can be cast into its SF formulation along the lines followed for the scattering of, say, atom-molecule systems (e.g. see ref. [191]), where only energy-independent, local, BOA adiabatic potentials are usually employed. For nonlinear, polyatomic systems, this has been done for CH 4 only, and only at one collision energy [105], while similar procedures have been used for linear polyatomics such as CO 2 [210] and HCCH [211]. In the earlier discussion of the various possible frames of reference, we have underlined the different

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

386

physical regimes that allow us to disregard either the whole nuclear motion or its slower part involving molecular rotations. Under those conditions for which the nuclear motion is strongly coupled with the incoming electron partial waves, the full Hamiltonian of eq. (2) needs to be used, with the correct contributions from eqs. (43) and (44). In the following, we derive SF scattering equations for general nonlinear asymmetric top systems first for the rotational excitation and then for general ro-vibrational transitions. Chandra [212] has also given a similar derivation for asymmetric top molecules. Of the several Hund's cases [213] which consider the influence of electronic angular momentum (l) on the rotation of the molecule, we are concerned here with those in which the coupling between I and the rotational angular momentum vector (j) is very strong. The Schr6dinger equation between an incident electron and an asymmetric top molecule in the SF frame of reference can be recast as [6, 214] JM JM (--lv2 -}- V(12, r) - k 2 + ~rot)~,tj~./(r) = Ej, ~0j,, (r)

(87)

where 12 denotes the three Euler angles, Ej, are eigenenergies of the asymmetric top rotational states [see eq. (59)], Y(rot is the rotational Hamiltonian, k 2 is the incident electron energy, and JM

G,(r)

r- 1

=

E fj,,,,, ' j,t(r)

j'T'

Y F,'l'(12'

(88)

i).

The angular functions of eq. (88) are the result of coupling between l and j, such that [see eq. (60)]: YiJ,~(12,/) = E C(jlJ; mjmtM ) (~#~,¢(a) Ytm,(i)

(89)

mjrn I

where mj and m t are respectively the projection quantum numbers of j and I along the SF axis. In order to rewrite eq. (87) in terms of radial functions fJ(r) we project this equation onto the angular functions of eq. (88) to obtain:

+ k2

m J'~'

1(I + 1) r2

l

j

fj,,(r)

J fj,~,,,(r) J = 2 E (yj,J~t ( a , i)lV(a, r)]yi,~,,,(a,i)) " JM

(90)

j'r'l'

with 2 j,~, = 2[Ej~ - Ei,~, l . k ff,

(91)

The radial function fJ of eq. (90) satisfies the following boundary conditions,

fJ (r) r'~o 0 fJ(r) r~~ k J~. -l/2J'~'Lv, [~ ' ~ ,6t/, e x p { - i ( k j -

½/Tr)} - S j,~'t', J j~t exp{i(kj~, j,~,r - ½/'7r)}].

(92)

To obtain the scattering amplitude, we proceed by writing first the total wave function for the electron-molecule system:

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

1ft = " L.J " L.J

387

(93)

A JM jT! f/J JM jT!

JM jT!

which will have the following asymptotic form, 1Jf -

r~oo

L

r -I

JMj'T'jT

k j--;:,~2jTA:T7[ 5jj' 5TT , 5/1' exp{ -i(kjTr - !l1T)}

/I'

(94) The total wave function

1ft

is the sum of the incident and the scattered wave functions (95)

where the incident plane wave in the (jr)th channel is exp(ik form:

jT

r)~jmjT with the following asymptotic

•

(96) Since there should be no incoming wave, equating the coefficients of exp( -ikjTr) in eqs. (94) and (96) we obtain,

L

JMjTj'T' 1/'

kj~,I;~T' A::: 5jj' 5TT , 5/1' Y:~I'(n, f) exp( !il1T)

Li +

1 1

= :

v1T+1 C(jll; mjOM) y:::(n, f) expOil1T) .

(97)

jT JM!

If one now uses the orthonormality relation,

then eq. (97) reduces to the following, well-known form:

A::: = i!+lC(jlJ; m j OM)«21 + 1)1T/kjT )1/2

in the (jT) channel

=0

otherwise.

(98)

With the help of eqs. (94) to (98), we finally get: "'scat

'-::00 '~" Vk. JT

1

.' k. r

j'T " JT

JT

-1

exp

(Ok

1 j'T', jT •

) '" ~(.",

r ~ mj

J'

J r m j ~ Jrm j 0

) -

'" j'miT'

where the scattering amplitude f for the transition jTm j ~ j'T' m; is given by

(99)

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

388

V-~ it-v V ~ + 1 C(jlJ; mjOM)

f( j'r'mj <--jT"mj) = JMll'm l,

,

x C(j'l'J; m/, mt,,M ) Ti,,,t,.j,t

(100)

and the T-matrix is defined in terms of the S matrix as, T = 1 - S.

(101)

The expression obtained here for the scattering amplitude is quite similar to that for the diatomic case [212] except for the new quantum number r which occurs for asymmetric top molecules. (Note that for nonlinear spherical top molecules there is no extra r quantum number.) In the above formulation, we have neglected exchange interaction which can be included either as a model potential, and simply added to V in eq. (90), or exactly by properly antisymmetrizing the total wave function ~V. The resulting coupled equations are obviously given by a more complicated expression although the general indexing of each scattering channel after the expansion of eqs. (88) and (89) remains essentially the same. In eq. (100), a sum over JM implies that a sufficient number of (JM) basis functions needs to be included to get convergence up to a predesigned level of accuracy. One of the important differences comes from the rapid increase in the number of rotational levels per cm- 1 that occurs in most nonlinear polyatomics as opposed to the simpler, more "quantized" diatomic molecules. The direct consequence of this strange reduction in the energy spacing between expansion eigenstates is that convergence problems naturally become more severe, especially as the collision energy increases to values of a few eV and more.

6. Computed dynamical observables In sections 4.2 [eq. (78)] and 5.2 [eq. (100)] we derived expressions for the scattering amplitude in the BF and the SF frames of reference respectively. Once the K- or the T-matrix is known from scattering calculations, several physically observed quantities (differential, total and momentum transfer cross sections) can be obtained for elastic and inelastic (rotational, vibrational and electronic excitations) processes. The processes of photoionization and photoexcitation can also be obtained if seen as half scattering problems in which there is no incoming wave. In this section we give a brief account of other important processes also, such as dissociative attachment (DA), associative detachment (AD) and resonant scattering.

6.1. Elastic scattering To obtain elastic differential cross sections (DCS), we need to average ]fl 2 (where f is the elastic BF scattering amplitude [eq. (78)] referring to SF coordinates via three Euler angles) over all orientations of the molecule, i.e. do'_ 1 f dO 87r 2 ]f[2dO.

(102)

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

389

Now from eq. (78), eq. (102) can be written explicitly dtr dO

--

1 87r2k 2

~ i t-r ( - i ) i-r [(2• + 1)(2/÷ 1)] 1/2 ( S l hp/z , l'h'

v..,, !

f

~ lhrh b l'h'm' ~ ' l ' f ~ ' r h ' - - l ' V

fi~

-- ~ll'~hh ' ) ( S [ ~ , [,f~, -

~'m'~

8~,6~h,)*

(103)

*'dO

where the sum runs over lhm lhrh l'h 'th' l'h'm' nd ptz fill Carrying out angular integrations and using some angular-momentum algebra, we finally write eq. (102) as do = ~ A L PL(cos 0) d---~ L

(104)

where PL(cos 0) are the Legendre polynomials and the A L coefficients are found to be

A L = T k1- ~ ( 2 L + I ) 2 ~ ×

(bib m-p~

p~'

il-- 1,

(-i)r-r((2l+l)(2l'+l)(2f+l)(2f'+l))~'~(0~

~e~t,h,m,)~tbe~" T:;'

gP-~ ~'*(--1)'~+'~'(ml

f

0r ~)(~ ~ ~) L

r

r f --rfl

(105) In eq. (105) the sum is over lhm lhrh l'h'm' l'h'rh' plx and fi/L The total (integral) (o-i) and the momentum transfer (O'm) cross sections follow immediately from eq. (104): o"I =

f dor ~ sin 0 dO d~b = 4,'rrAo

(106)

f do" (1 - cos 0) sin 0 dO d~b = 4zr(A 0 - 1A~).

(107)

and O'm =

Thus, the computation of o-~ and trm becomes very simple once the low-order A L (L = 0 and 1) coefficients are obtained. The expressions for A 0 and A 1 c a n be derived easily from eq. (105) in a closed form. For example, A 0 is given by

1

Z

A° = 4k 2 lhm

~_~p# Ibfh"m Tt~,t,h,.-th,.

2 (__]'~m+m"

, ~,

•

(108)

l'h'm" pt~

This is, strictly speaking, the way to obtain a rotationally summed cross section. In practice, however, it can be efficiently computed by sums over magnetic substates as reported in section 6.2.1. 6.2. Rotational excitation We first discuss the rotational excitation in the AN approximation [215]. The scattering amplitude for the transition i--> f is given by

390

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules = ( i l f ( / ¢ "r~", a / 3 7 [ f )

f(i--~f)

(109)

,

where (i I and (fl are respectively the initial and the final rotational eigenfunctions. In the following we give expressions for the rotational excitation cross sections (DCS, 0-1, O'm) for the asymmetric, symmetric and spherical top molecules, separately. 6.2.1. The asymmetric top First we rewrite the SF frame of reference scattering amplitude [eq. (78)] in the following way: 27r .1 f(/~'i'; afl7)= -~- ,h,,,,'h',,,' E ,

l'

np~

8'7'r 2

[(2l + 1)(21' + 1)]1/2 q YT,(i')

× IbP~' Tpu~

t

s'q' lh, l'h' ~pu l'h'm'J~ I~ll'm'*n "glmodtSq*

lhm

(110)

where s (s') simply stands for (-1) m ((-1) m') and the function ~b is defined as I~tlmm'

\'~2

/

"~

(~m'm(

~ l ) " ~ 12~m,_rn(Ol~l)) ,

= ( 2 / + 1 t 1/2 \

8,7'/'2 /

m>O m=O

Dim'; ,

(111)

(note that v = -+1). Using symmetry-adapted rotational eigenfunctions (see section 3.3), eq. (110) can be recast in the following form: 27r f(JrM + J'r'M') = =~

X

~

Ihml'h'm' npp.

.t-t' 1

87r2 [(2• + 1)(21' + 1)] '/2 q YT'(i')

-p~ plz -pl.t v' dts'q' , j/tsq , u, {blh m T,h,,, h, b,,h,m, } Z aK, '~ aK',' "~' Jf q'1'K'M' "Fl'm'n "VlrnO I~JKM dO KK' pp'

(112)

that holds for any general rotational transition J r M ~ J'r'M'. Jain and Thompson [138] have discussed selection rules for the transition JrM---> J'r'M' and found that transitions are allowed only between symmetric (even r) states or between antisymmetric (odd r) states; this leads to Ar = 0, -+2, +-4,.... Since magnetic substates (M, M') are not easily resolved, the corresponding DCS for the transition Jr---> J'r' is written by summing over final substates M' and averaging over initial substates M, i.e., d~r k' 1 ~'' (2J + 1) If(JrM---~ J'r'M')12 d ~ (Jr--+ J'r') = --~ MM

(113)

where the final wave vector k' is related to the initial k by the energy balance equation: 2k '2 = 2k 2 + Ej, - Ej,~,

(114)

and the energy levels Ej~ are those defined earlier. Finally, we can expand the DCS in terms of Legendre polynomials,

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

391

do- (Jr----*J'C)= --k k' ~' AI~(Jr---> J'C)PL(c°s O) dO

(115)

L

where A L coefficients can be obtained from eqs. (112) and (113) by making use of standard angular-momentum algebra. We thus obtain the final expression for each of the above, inelastic coefficients:

Aa(Jr---> J'r')= (2J' + 1)(2L + 1)(-1) L ~] i,_r (-i)f_ r ((2l + 1)(2/' + 1)(2/+ 1)(2[' 4k2 u'H'

+

1)) 1'2

J+J'

X(oi ot ~)(~ ~ L) o ~]

(-1)~(2j+I)W(ll'I[';jL)M/~m'M[~) * (116)

~]

mjm; j = l J - J ' l

where M J it' mj =

2 mm'hh' plx

-

m m'

b l'h'm'

(117)

lh, l'h' "~'Jr, J'r' '

and W(...) are the Racah coefficients [214], while the G coefficients are given by: (a) When K, K' > 0 1 ±

~'1

J~" ax, a Jr' r,' , , ( - l l r ' [ ( r J

J' -K'

" mJ,)+(--1)"(-~

" -r'J ' mJ,)

J' m J , ) + (--1 ) ~+~' ( - K' r'J' ~,)] • +(--1)~'(KJ K'

(118a)

(b) When K = 0, K' > 0 aJm]

j,,,,~'

Jr J,~, (_I)K,[(~ " 1~ g'l,' ~ aooaK,~, 0 - rJ' mJ,)+(-1)~'(0J g" t ~,)]

,

(l18b)

(c) When K > 0, K' = 0 1 GG, =__=y ar~aoo ,,

"~/2~Kv

[(/~ J'o

m',)+(--ay(_,,

' "o mJ,)]•

(118C)

(d) When K = 0 , K ' = 0 J~-

J'~-'

j

J'

-

GJSj:,, , = ao0 ao0 (o o m~,).

(l18d)

The program EROTVIB [216] has been written to evaluate the A L coefficients of eq. (116).

6.2.2. The symmetric top By employing symmetric top eigenfunctions in eq. (108), the expression for the DCS can be derived from the following relationship: do" (JK--* J'K') = k ! S' 1 d---~ k ~M' (2J + 1) If(JKM"~

J'K'M')[2

(119)

392

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

where 2k '2 = 2k 2 + E~K - Ej,~,

(120)

and the (JK) energy level is defined as

Ej~: = AJ(J + 1) - (B - A)K:

(121)

where A and B are the two rotational constants of a symmetric top. Following a similar algebraic manipulation as that for the asymmetric top molecules given in the previous section, we obtain the following equality: kt

d~ (jK___~j,K,) = ~LAL(JK___>j,K,)PL(COSO ) dO -k

(122)

where the new inelastic AL(JK + J'K') coefficients are now given by:

AL(JK--* J'K') =

( 2 J ' + 1)(2L + 1)(-1) c ~ i,_r(_i)~_r 4k 2 it'll'

x((21+1)(2[+1)(21' +1)(2[' +1))1/2(io ot 16)(~ ~ Lo) J+J'

Z

x

(-1)J(2]+l)W(ll'[[';]L)(~

-K]'. ~j)2M{7"M]--m'*lr . . .

(123)

i=lJ-J'l and the M matrix is here redefined as:

M~,, Z " "=

b- ~ m ( - ~

"

m'

" b-P~'

]mj)

l'h'm'

T p~ lh, l ' h '

'

(124)

mm'hh' pl~

It is easy to see from eq. (124) that transitions for which K - K' ¢ m' - m are forbidden.

6.2.3. The spherical top For spherical top molecules, the DCS can be obtained from those of the symmetric top by summing over all K' and averaging over all K. Thus, d~r (j__. j , ) = k' 1 ~ I f ( J K m ~ J'K'M')I 2 dg2 k (2J + 1) 2 I,;K'MM'

(125)

where, 2k '2 = 2 k z + E j - E r

(126)

and

= BJ(J+ 1),

(127)

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

393

where B is the rotational constant of the spherical top. From eq. (125) it is now easy to find that do. dO

(J--->J') =

k' T

~ AL(J--->J') PL(cos O)

(128)

L

where

AI~(J--->J')= ( 2 J ' + 1)(2L + 1)(-1) L ~ i'-" ( - i ) r-r ((2l+ 1)(2[+ 1)(2l' + 1)(2[' + 1)) w2 4k2(2j + 1)

trfr

J+J'

x ( oi ot oL)(r ~ L) o

~]

(-1)iW(I,I',[,[';jLIM{S'M[r '*

(129 /

y=lJ-J'l

and the definition of the M matrices is the same as that given in eq. (124). In a similar way one can obtain expressions for the DCS, 0-t and o"m from the SF scattering amplitude [eq. (100)]. (Note that the final values of the absolute cross sections must be the same in both the BF and the SF frames of reference.) In the SF frame of reference, the DCS for the rotational excitation Jr--+ J'z' is written as [160]:

do. (Jr-+J'r')= 4 k( -21( 2) "J-+' l ) ~ =o AL(Jr-+J'r')PL(c°sO) d---~

(130)

where A L ( J r - - + J'g't) = Z

Z(IJllt4; JL) Z(IJII'4;

J ' L ) Tj,rq,,2 ~ ,~, T,,~,,,h ~ ,~t' * ,

(1311

JIJ2ll '

[[,

with

Z(abcd; ef) = (-1)(1/2)(Y-"+c)((2a + 1)(2b + 1)(2c + 1)(2d +

1)) 1'2

C(acf'~000) W(abcd; ef). (132)

It is now easy to write expressions for the total and the momentum transfer cross sections. For example, the integral cross section is given by <(Jr--,

~r E (2Jl + l ) l T- ,J1, , , , , , t = (2J + 1)k = ,,t,,

2

•

(133)

The transformation relations between observables in the two frames have been extensively discussed in the literature [191,217].

6.3. The vibrational excitation In the adiabatic-nuclei theory we can use the Chase formula [215] for the vibrational excitation. For exciting a vibrational level from state lv) to state Iv'), we could adopt the same kind of formulae for

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

394

the DCS, o-i, a m, etc. as those set forth in the previous section for pure rotational excitation or for the elastic scattering. The T matrix is now replaced by the following matrix element: r , ,"", ,,h, (v; v ' ) = ( v r

",,h,(Q31v '

(134)

.

The normal coordinate-dependent T matrix is first obtained by computing the scattering amplitude at various Qi values. In a proper vibrational-close-coupling (VCC) theory, the scattered function is, however, projected onto a complete set of vibrational wave functions, i.e. (neglecting rotational Hamiltonian) (135)

fth( r, Qi) = ~ f th(r) flPv(Qi) o

and the final scattering equations to be solved are given by: da ~rr2

+ 1) + k 2] -l(l -r2 o f~h(r) = 2 E

{Vthvv'' ,,h,(r ) + W vv' v' ,h ' rh,(r)} f t,h,(r)

(136)

l'h'v'

where k2o= k 2 - E v

(137)

Vth,~'h,(r)= (vlVth, t,h,(r; Qi)lV')

(138)

and

oo' and the exchange term Wlh" l'h' is similarly obtained. Equations (135) and (136) refer to BF coordinates. To date there have been no calculations on a polyatomic system using eq. (136) to directly determine vibrational coupling with the continuum radial coefficients f~h(r) that are produced by such equations. The only attempts have been carried out for the stretching modes of CO2, although they should be considered, strictly speaking, extensions of linear diatomic methods: Thirumaiai et al. [210] examined the (001) mode at 10 eV of collision energy, while Morrison and Lane [312] studied the (100) mode excitation between 0.2 and 2.0 eV. In addition to the above mentioned methods that can be used to obtain vibrational excitation cross sections, several empirical and semiempirical approaches have been proposed in recent years. In the main they have dealt with diatomic molecules, especially the well-studied case of the vibrational structure of e-N 2 scattering. Such methods have been well documented in several reviews ([218] and references therein), including a recent one by Kazanskii and Fabrikant [7]. Very recently Wadehra [219] has prepared a fairly extensive article on resonant vibrational excitation and DA processes. It is however limited to diatomic targets only. The effective range theory has also been employed [220] to study vibrational excitation processes, although for diatomic targets only. McCurdy and Turner [221] have presented a time-dependent formulation of the boomerange model to treat electron-molecule collisions: they studied the propagation of a wavepacket that moves on the complex potential surface of the metastable ion. Possible extension of their work to polyatomic systems is also briefly examined by these authors [221].

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

395

6.4. Electronic excitation Unlike elastic, rotational and vibrational excitation processes, an electronic transition in a molecule is more complicated to investigate theoretically. Very crude approximations such as the Born (and other Born-related variations), distorted wave and eikonal, do not predict reliable cross sections at energies below about 100 eV (in fact these lowest-order theories can be appropriate only at high energies (E - 1 keV)). Semiclassical theories, such as the impact-parameter model are even less reliable because of the inherent incorrect physics. The cross sections obtained from all such zeroth-order models are in general larger by a factor of two to five with respect to experiments and the shape of cross sections (in particular DCS) is never obtained correctly. More accurate theories, such as the close-coupling method, have recently been tried to study the simplest case of H 2 [222, 223] and N 2 [224]. Except for the hydrogen molecule and for some attempts on the N2 molecule, there are no satisfactory theoretical studies of electronic excitation of molecules (diatomic or polyatomic).

6.5. The transport of electrons in gases In a more general sense, when electrons are made to travel through molecular gases all the dynamical observables discussed in this section are in principle amenable to observation. Most studies, however, have focused on the low-energy behavior of such electrons in several molecular gases, as they are known to have direct application in explaining the details of processes occurring in gas discharges, in diagnosing plasma characteristics of molecular gases, in gas laser kinetics and upper-atmosphere ionizations. Many excellent reviews and topical volumes have provided extensive coverage of this area in recent years and the reader is referred to them [77, 78, 225, 226] for more detailed information. When the electron energy increases, a correspondingly larger number of inelastic channels become open and the relevant final processes therefore begin to compete with each other, thus making the customary two-term solution of the BE no longer realistic for treating the relevant phenomena. Quite a number of papers have therefore discussed the mathematical details of the transport coefficients (which characterize the transport properties of electrons in gases) and the ways of improving their computed values by going beyond the two-term approximation [227-234]. The basic parameters describing the transport coefficients are the drift velocity vd, the electron mobility ~t, the temporal growth constant v and the diffusion (longitudinal or transverse) coefficient D. Their analysis in relation to the required cross sections has been given in many places (e.g., see refs. [231, 232]) and will be only briefly summarized here. At the start of the calculations a set of cross sections is assumed and the Boltzmann equation is thus solved to obtain the electron energy distribution function at several values of E/N and T. Here E is the electronic field strength, N the gas number density and T the gas temperature. The final transport coefficients are then computed by a quadrature of the distribution functions over E/N and T, multiplied by the chosen set of cross sections pertaining to the processes included in the modeling of the electron motion through the gas. Such calculations are iterative in the sense that each set of computed values for the transport coefficients is compared with the experimental findings and the initial guess for the needed cross sections is then readjusted to yield better agreement after each iteration. Obviously, a knowledge of the input cross sections becomes crucial to the efficient and successful conclusion of the self-consistent procedure. The study of transport properties therefore provides an important test for the quality of calculated cross sections that can be used to start the fitting procedure. On the other hand, the process of averaging over E / N and T does blur many of the detailed features of the numerous cross sections needed for each calculation, especially

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

396

since the weighting effect from distribution functions often shifts around the relative importance (and therefore the perceived need for improvement) of different cross sections. When one considers a swarm of electrons travelling through a molecular gas under the influence of an electric field E, one can define a density distribution function f (r, v, t), with r and v being the position and velocity vectors respectively, that can be directly computed by solving the following integro-differential equation:

c)f_j + V "Vrf-- e E "Vv f = B(r, 13, t) dt m

(139)

where B provides the electron distribution function rate of change in time as due to collisions. It is also called the collision operator. The distribution function f is related to the electron number density by the relation:

f

f(r, v, t) dv = n(r, t).

(140)

If one now considers the special case in which the electron swarm is in thermal equilibrium with the gas, the integral expression for D, # and the rate coefficient K become

(2),121-j~ f

D=\m /

~

fM(e)de

(141)

0

/z = -

( 2 ) '/2 e f e d ~/ ~-~ O'm(e) de [fM(e)] de

(142)

0

(2tlJ2i

K = ,~/

e ~rr(e) fM(e) de

(143)

0

where O'm(e) is the momentum transfer cross section, o'r(e ) the reaction cross section, fM(e) is the Maxwellian electron energy distribution function at any temperature T, and e and m are the electronic charge and mass respectively. Several polyatomic gases have been studied under the conditions of validity of the above equations and the corresponding transport observables have been obtained for CH 4 [79, 80], CO 2 [233] and for several halogenated hydrocarbons [234].

6.6. Resonant scattering Collisional resonances provide one of the most important mechanisms through which electrons interacting with molecules can effectively transfer energy to the available molecular degrees of freedom. In a general sense, they do not require any special theoretical technique other than those already described in the previous sections. It is however of interest here to touch on some aspects of e -molecule resonances as they have provided, from the very beginning, a unified framework for the interpretation of a wide variety of phenomena that were experimentally observed. Resonances in all the processes of the previous paragraphs can lead to enhancement of an order of

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

397

magnitude or more in vibrational excitation and electronic excitation cross sections. They also provide one of the prime mechanisms for dissociative attachment (see below). Structurally, resonant scattering also provides information on metastable negative ions, negative electron affinities, sequential orbital energies of unbound orbitals of different symmetries and general features of doubly excited electronic states. They also represent a quantum mechanical phenomenon and therefore their theoretical treatment provides a clear way of testing dynamical treatment which are only classical or semiclassical in nature. An important recent theoretical advance has come from the calculation of resonances by pseudo-bound state techniques such as complex scaling or stabilization [235-238], whereby dynamical processes are treated as much as possible via well established bound state, structural techniques without having to explicitly resort to the usual scattering boundary conditions. In molecular photoionization and autoionization, resonant processes have shown up as exit channel, half collisional resonances. Recent advances in availability of synchrotron sources and proliferation of high-resolution vacuum spectroscopy apparatuses are creating greater interest in resonances because of their major role in interpreting the observed features of a great variety of polyatomic systems [239]. Finally, electron-molecule resonances are also responsible for the observed structures in inner-shell electron energy-loss spectra in the energy regions around the core ionization threshold of one of the molecular atoms. They obviously act as a final state interaction with the neutral target and are the main mechanism currently invoked to explain the features observed in X-ray absorption spectra above threshold and before the EXAFS diffraction patterns [240, 241]. Very recently, resonant effects have been dramatically shown in low-energy (0.2 to 30 eV) electron scattering from multilayer films of simple diatomics, condensed on a metal substrate at near 20 K, whereby the isolated molecule resonances are combined with phonon energy loss in the bulk structure of the layers [242, 243]. 6. 7. Dissociative attachment processes

When low-energy electrons form negative ion complexes with the interacting molecules through the process of being captured in a negative ion resonant state, one of the energetically accessible decaying channels is given by the DA pathway: e- + A,Bm--. (AnBm)* A* + (B~)*

(144)

where an asterisk indicates a molecular excited state (rotational, vibrational, or electronic). The fragments A n (or Bin) may be composed of one or more atoms, i.e. molecules in their ground or excited state. This area of research has been the subject of several recent review articles covering both the theoretical and experimental aspects of the problem [244-246]. As stated before, most of this discussion is limited to diatomic molecules. From the experimental point of view, one can measure several quantities such as the total cross section, rate constant, kinetic energy distribution of the DA fragments, and angular distribution. There have been, therefore, many measurements on the O'DAof the dissociation products and of the energy and angular distributions of the fragment negative ions of polyatomic molecules (see ref. [244]). For example, in e - - H 2 0 scattering, H-, OH- and O- ions have been detected and total cross sections analyzed as a function of incident electron energy. Molecules such as H20 , D20 , SF6, CO2, CH4, CD4,

398

F.A. Gianturcoand A. Jain, The theory of electron scatteringfrom polyatomic molecules

NH3, ND3, NF3, HzS, D2S , OCS, CS2, N20, NO 2, SO:, HCN, and other cases including more complex polyatomic molecules and halocarbons have been discussed at length in ref. [244] and need not be repeated here. All updated information (to 1984) on these systems have been presented there in tabular form for almost every polyatomic molecule studied so far thus making our work much easier to complete, as we shall talk only about the most recent experimental data on DA processes and theoretical attempts to understand them. Unfortunately, theoretical progress is not very satisfactory for polyatomic gases, although some simplified resonant models have recently been tried in order to obtain some information on DA cross sections. In a general way, O'Malley and Taylor [247] formulated a theory for the angular distribution of fragment negative ions in which the rearrangement scattering amplitude was written as for the fragment

A-, /A (I(~) ~" f X *(R) Fvj(R ) ~ (/~rl~el Iff)0(j~e+ ) dR

(145)

where X(R) is the final nuclear state (describing dissociation) and ff)r the resonant wave function. Fvj(R) is the initial rotational-vibrational wave function, ~b0 the initial target ground state, ~b~+the nonresonant electronic wave function, and Yfet is the electronic Hamiltonian. Note that the resonant (~br) and the nonresonant (~b0~b~ +) parts are coupled by Y(e~.In the frozen rotation approximation (as assumed in ref. [247], in some cases this restriction was removed [248]), the angular part of x(R) is replaced by a delta function 6(/~-/~) with the convention that R points toward the final atomic negative ions. The theory of O'Malley and Taylor [247] is well suited to cases where spherical harmonics are almost the correct eigenfunctions of the direct scattering, a fact that may not be true for polar molecules. Teillet-Billy and Gauyacq [248] reformulated then the theory of O'Malley and Taylor to deal specifically with polar molecules. Instead of the usual spherical harmonics, they expressed the DCS for DA in terms of dipolar angular modes [249]. Results for the angular dependence of DA cross sections were presented for H - / H 2 0 and H-/HzS and compared with the corresponding measurements. Agreement between theory and experiment was especially good for the DCS maxima around 90°, which turned out to be produced quite accurately by these model calculations. It is now clear, however, that although there is a wealth of experimental data available for the DA cross sections on a large number of polyatomic gases [244], much more detailed theoretical work should be carried out in this direction as only very simplified approaches have been tried thus far and with rather limited success.

7. Specific examples Although the theoretical and computational studies of electron scattering from polyatomic molecules have been rather limited in size and scope, there are indeed several examples where numerous aspects of the collisional process have been analyzed, theoretically and experimentally, in recent years. Because of the general applications to the study of gas discharge phenomena, interstellar medium kinetics, and chemical laser modeling, most studies have focused on a relatively small number of molecules such as CH4, H20 , SF6, NH3, H2S, Sill 4, CO 2 and a few others. In this section we have therefore chosen to present the experimental and theoretical data collected on the above molecules in the past five years or so, in order to assess how accurately current theories are reproducing experimental findings and what observables still present the greatest challenge to theorists.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

399

7.1. CH 4 This molecule has been one of the best studied examples of polyatomic targets, as observations on collision processes were begun as early as the 1920's and the early 1930's. Like some of the rare gases, e.g. argon, it exhibits a minimum in the low-energy total cross sections (RamsauerTownsend (RT) effect) and a broad maxima around 7-8 eV, thus presenting a region of strong increase of cross section values between 0.4 and -5.0eV. Apart from being the simplest structure for an enormously large class of organic molecules, C H 4 is present in our interstellar dust and is one of the primary constituents of the atmosphere of Uranus and Neptune [250]. Moreover, an accurate knowledge of e - - C H 4 cross section data is also required to optimize the operating characteristics of diffuse discharge switches [251,252], where methane is often needed because of its high drift velocities at low values of E/N. In the region below 15 eV Rohr [39] has summarized all the experimental work done on this system up to the late 1970's, while above that collision energy experimental data have been extensively compared with theoretical model calculations [253-254]. It is only in the past five years, however, that there has been a resurgence of activity on this molecule that has produced new high-quality measurements for its total cross sections [255-266] and for its rotationally [29, 266] and vibrationally [41,261, 267] inelastic processes. One of the most recent reviews of e - - C H 4 experimental data has been presented in tabular form by Curry et al. [44]. The first theoretical treatment of electron-methane elastic scattering was~carried out by Buckingham et al. [268], who calculated a model self-consistent field for bound electrons in C H 4 and employed it to treat the scattering process, without inclusion of exchange and polarization effects. Because of the importance of the above terms of interaction, these early calculations did not manage to correctly reproduce experiments in the region below - 1 0 eV. It should be noted here that at the time of the first calculation there were already five experimental studies available on the angular distribution in e - - C H 4 scattering [269-273] and three measurements on the total cross sections [274-276]. After an interruption of about 35 years, swarm experiments were performed on CH 4 in order to obtain momentum transfer and inelastic cross sections [277-281] in the very low-energy region (E -< 1 eV). Tice and Kivelson [282] used the cyclotron resonance technique to measure e - - C H 4 momentum transfer cross sections as a function of electron velocity. In 1976, Gianturco and Thompson [112] made the first attempt at an ab initio level on the e - - C H 4 elastic scattering process. They calculated an accurate static potential of near Hartree-Fock quality by using single-center wave functions for the 1A1 state of CH 4 and the effects of exchange were included via the orthogonalization technique [107], while polarization of the target was treated through a parameter ensuring a correct cut of the asymptotic potential. Gianturco and Thompson adjusted this model potential to reproduce observed structures (the RT effect) in the total cross section while later on the same authors [181] extended their previous work to include DCS and momentum transfer cross sections by employing a bigger basis set to describe the target molecule. No other calculation existed at this time on e - - C H 4 elastic scattering except for the multiple scattering Xa potential employed by Varga et al. [208] for the total elastic cross sections in the range 0-16 eV. By employing a model exchange potential (HFEGE) and a phenomenological polarization potential Varga et al. obtained qualitative results (e.g. the position of the RT minimum around 1.2 eV as compared to the experimental value around 0.4 eV). To further improve the model of Gianturco and Thompson, Jain and Thompson [116] attempted to employ a potential in which no adjustable parameter was involved. In brief, they used an HFEGE e -CH 4

400

F.A. Gianturco and A. Jain, The theory of electronscatteringfrom polyatomic molecules

potential plus orthogonalization for treating the exchange interaction, while the polarization potential was determined by using the method of Pople and Schofield [118] and the approach of Temkin [284]. The static potential was obtained in the same manner as described in ref. [106]. The new results of Jain and Thompson [116] agree well with the experimental data available at that time since their DCS at 3, 5, 7.5 and 10 eV were in reasonable agreement with the data of Tanaka et al. [256] and Rohr [288]. The parameter-free model for the polarization interaction appeared to be working at very low energies also, although it produced there a minimum in the cross section at much lower energy (around 0.2 eV) when compared to previous findings. However, it was in excellent agreement with the (then) most recent measurement by Barbarito et al. [257]. In fact, they had obtained the RT minimum position around 0.2 eV in contrast to all earlier experimental values of about 0.4 eV. More recent experiments [259,265] have found the position of the RT minimum to be once more around the same old values of 0.4 eV and not shifted to lower energies as measured by Barbarito et al. It therefore appears that the parameterfree model of Jain and Thompson is still too weak to correctly yield the RT minimum position. Figure 6 displays various experimental and theoretical total cross sections for e - - C H 4 scattering in the RT minimum region. In this figure we have not included the CMSX, calculation of Varga et al. [208] and Tossell and Davenport [283], because in this region their calculations are in serious disagreement with experiment. A static-exchange-plus-accurate-polarization calculation could obviously provide further insight into discrepancies between theory and experiment and current work along these lines is certainly needed (e.g. see refs. [125] and [126]). A typical angular distribution from scattering of electrons off C H 4 molecules at 10 eV is shown in fig. 7, where experimental and theoretical data are compared over a wide range of angular values. Almost all theoretical models appear to reproduce satisfactorily the general shape of the measured points. It should be noted, however, that both the small-angle and the large-angle regions are those which suffer the most from experimental errors. Very recently, a fully ab initio calculation was carried out by including static and exchange potentials only [104]. The authors applied the Schwinger multichannel scattering theory to e - - C H 4 collisions in --- T--

1

- T - T ]

., 5 F / jFerch

I

-7--]-

-

T

- T T -

I

o

ol , 85j

[ ~

bO

~

--F-I-

•

/

!

/

- - ° 1 ~/ - I/" Ja,n, T h o m p s o n /

Oianturco Thomp- /

0.5

1.0

1.5

E l e c t r o n energy, E(e V) Fig. 6. Total cross sections (computed values and experimental values) for electron scattering from CH 4 molecules around the Ramsauer-Townsend minimum. See text for meaning of references (taken from ref. [259]).

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

401

I

E to Q c) 30

'Ii AI

tlG-

Jclln

104 . . . .

A b u s o l b l et eL

Thompson

46

~:~

Curry el ol.

266

0

MOiler et ol.

256 289 260

0 [i] •

Tanoka el at. Rohr Newell el al.

e--CHz

// 62.0

~

c~

1.0

,

10

1

50

~

I

90

_ _ ,

__

I

i

_

130

Scattering Angle (deg) Fig. 7. Angular distributions obtained from theory and experiments for e -CH4 collisions at 10 eV. The continuous and dashed lines refer to calculations, while the various symbols indicate experimental points.

the energy region between 3 and 20 eV. In spite of the lack of polarization effects, their study is very interesting in that it allows one to try and estimate, over a wide range of relative energies, the varying importance of polarization effects. Although their total cross sections were found to be still far from the experimental values, the corresponding angular distributions suggested that polarization effects should be most important at low collision energies and for small-angle scattering. An example of their findings, and a comparison with both previous theories and experiments, is shown in fig. 8. One can clearly see that in the high-energy domain both theoretical models fare rather well in reproducing the measured distributions over the whole angular range, thus indicating the reduced importance of polarization forces as electrons probe more closely the molecular charge distribution for higher collision energies. Only recently, measured data on the rotational excitation of C H 4 have become available in the literature [266]. However, before the appearance of the experimental data, Jain and Thompson [138] and Abusalbi et al. [105] performed state-to-state cross section calculations for the rotational excitation of C H 4 by electron impact. While the former authors presented their data for the whole range of collision energies between 1 and 15 eV, the latter group only showed computed inelastic cross sections at 10 eV. The purely elastic ( 0 o 0) DCS of ref. [138] turned out to compare very well with the measured values of Mfiller et al. [266], while for the excitation processes like the (0--, 3) and (0---*4) transitions the theory predicts much lower cross sections than those given by experiments. The shape of both distributions, however, is quite similar over the whole range examined. For the data at 10 eV, the two

402

Gianturco and A .

F.A.

!

E 10-

,.t~ t..J

Jain,

The theory of electron scattering from polyatomic molecules

i

•

L

,,

,,-i-° ,¢=-

r" 0

,m

..4..i_j tlJ

t13 6

--

0 t,L..J rio .4.--

t--" tl)

t-.

tlJ

q-.

2

O

0

3o

6o 9o 120 Iso Scattering Angle (deg)

~80

Fig. 8. Computed and measured angular distributions for e - C H 4 scattering at 20 eV. The continuous line refers to static-exchange calculations [104] and the dashed one to model values from ref. [254], while the open circles and crosses are measurements from resp. ref. [256] and ref. [348]. (Figure reproduced from ref. [104].)

available theoretical calculations [105, 138] agree closely in shape, while the inelastic cross sections of Jain and Thompson are smaller than those of Abusalbi et al. Such cross sections, on the other hand, show a generally fiat behavior over most of the angular region: after transferring a rather large amount of angular momentum to the target molecule, the outgoing electron is left with very little "torque". This means that the effect of d- and f-wave components can be considered as largely negligible in treating the excitation process, thus simplifying the subsequent angular distributions. In conclusion, qualitative and quantitative differences still exist in the rotationally inelastic range of cross sections between theories and available experiments. In both areas, however, the accuracy of the relevant data still needs to be improved a great deal before a final comparison can be realistically made. There are no published theoretical data on the vibrational excitation of C H 4 by electron impact while some experimental data are available for the excitation of u13 and uz4 modes of this molecule [38-44,267]: from these experimental results it appears that the d-wave dominated broad resonance of the elastic channel has some effect on the angular distribution of vibrational excitation channels.* Our discussion so far has been limited to the 0-10 eV energy range. However, experimental studies have been carried out up to a few hundred electron volts and an updated summary of earlier experimental and theoretical work is given by Jain [253, 254]. Above the 10-15 eV range, usual close-coupling calculations become very difficult to perform: Szabo and Ostlund [285] have employed the first Born approximation using CNDO molecular wave functions, while Dhal et al. [286] used the eikonal approximation. Jain [253,254] proposed instead a spherical model in which an optical potential (consisting of static, exchange and polarization potentials derived from target wave functions) was * Recent unpublished results [350] for the symmetric stretching mode show that the differential cross sections have the oscillatory nature expected of a d-wave resonance effect.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

403

numerically calculated to yield the needed partial wave analysis. His results for DCS, integral and momentum transfer cross sections, turned out to be in good agreement with experiment in the energy range which goes from 20 eV to about 800 eV. 7.2. S F 6 By current theoretical standards, the S F 6 molecule is a very large assembly of atoms containing 70 bound electrons, and therefore very difficult to handle by ab initio computational methods. In addition, because of the rather large nuclear charges of the fluorine atoms away from the center-of-mass, the single-center approach described before is very unlikely to work, as a very extended basis of STO functions on the sulphur would be needed to realistically represent charge distributions and cusp behavior at the F atoms. It is therefore not surprising that very little theoretical work has been done on this system which has not resorted to drastically approximate models. On the experimental side, it was found that above 1 eV the cross sections are dominated by several shape resonances [287-290], while at very low energies the attachment of slow electrons to S F 6 has a very high cross section, the average at an energy between 0 and 0.1 eV being about 2 x 10-14 c m 2 [291]. This value is of the order of the physical limit set by the electron wavelength when only s-wave 14 2 scattering is considered (i.e. 2 x 10- cm at 0.05 eV). Further observations with electron beams of fairly well defined energy showed a narrow peak of less than 100 meV width as the energy goes to zero, without however being able to establish a finite width for it [292]. Model calculations have been carried out on this system within the FN approximation for the dynamics, and by further simplifying the interaction potential via the use of the CMSX~ approach [293]: the authors found that the alg , t~u, t2g and eg channels all exhibit resonance features at energies around 2.1, 7.2, 12.7 and 2710 eV respectively. Considering that the experimental results of Kennedy et al. [287] were quite close to the theoretical prediction, it was surprising to find out that the complete disregard of nuclear dynamics worked so well for such a complex molecule: one would normally expect, in fact, that to repeat these calculations with a further averaging over ground vibrational levels should markedly change the shape and features of the computed curves. Similar FN calculations were also carried out by Benedict and Gyemant [207] who could not obtain, however, the same good results already found by Dehmer et al. [293]. They presented results between 10 and 60eV, and the position of their computed resonances were all consistently lower than experiments. The overall shape of the elastic cross section computed in ref. [293] is shown in fig. 9. It may be reasonable to surmise that the approximate CMSX~ approach is not effective. Dynamical calculations from the Hartree-Fock wave function and using the first Born approximation were carried out as a function of momentum transfer by Pulay et al. [294], but no calculations have ever examined this system within a rigorous treatment of both the interaction and the dynamics of the collision process. Model calculations of the attachment process for S F 6 target were recently made with via a slight modification of the effective range theory (ERT) employed for diatomic targets [295,296]. Although no specific features of either the interaction potential or the scattering process were introduced in the model, it provides a first attempt at coupling nuclear dynamics with electron-molecule interaction, thus associating s-wave resonances to nonadiabatic effects induced by the "breathing" nuclear motion of the fluorine atoms. A great deal of work remains to be done on this very interesting molecule since so many experimental studies have been carried out over a broad range of collision energies.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

404

t~O-

g

. . . . . .

20-

/flu

10

20 KE(eV)

30

40

Fig. 9. Comparison of measured and computed total and partial cross sections for e -SF 6 scattering. The dashed curve is the absolute value measured by Kennerly et al. [287]. The partial computed cross sections for the resonant channels are labeled according to molecular symmetries (taken from ref. [293]).

7.3.

H20

Due to its obviously great importance in many fields of research, including atmospheric physics and radiation physics, the water molecule has been extensively studied both in the laboratory and with theoretical models. The FBA provided the earliest, simplest possible model which could be used to gain some knowledge of its cross sections for various processes [250,297-298]. Fabrikant further studied the problem via the approximate dynamics of effective range theory [299, 300], while Gianturco and Thompson [181] made the first attempt at calculating e - - H i O elastic cross sections from ab initio methods. Later on, the same model was extended to compute integral cross sections and rotational excitation cross sections [116, 138]. The FN approximation leads to divergence in the forward direction when the target is a polar molecule (see section 4). The latest calculations therefore employed the angular frame transformation theory (see section 4.4) to avoid this problem [116, 138]. Several measurements were carried out for the total cross sections [301,302]. As far as the rotational excitation is concerned, the theoretical treatment of interaction and dynamics with ab initio methods [138] yielded computed values for the (00~10) inelastic cross sections that were in satisfactory agreement with the measurements of Jung et al. [27]. It is not clear, however, whether or not one can correctly compare the two sets of data, as the experiments were analyzed within the so-called high-J approximation, an approximation which is not employed in the theoretical model. Very recently, Danjo and Nishimura [303] measured elastic cross sections for the e-H20 system at 4-200 eV in a cross beam method. In the late 1970's several experiments were carried out on the vibrational excitation of H20 [45,304, 305] for which the corresponding cross sections are characterized by a strong threshold peak and a broad feature around 6 to 8 eV. The FBA calculations [306-309] were found to be inadequate to reproduce the above structure in the cross sections, while a more rigorous approach was used by Jain and Thompson [139], who started from first principles and employed the usual Chase approximation [215] to study the excitation of symmetric and bending modes in the energy range from threshold up to 10 eV.

F.A. Gianturco and A. ,lain, The theory of electron scatteringfrom polyatomic molecules

405

In fig. 10, the calculations of Jain and Thompson [139] for the bending-mode excitation cross sections are plotted at several energies and over a rather wide angular distribution. In spite of the AN approximation and of the treatment of nuclear motion as being mode-uncoupled and harmonic, one finds that these first ab initio calculations on the vibrational excitation of a polyatomic system reproduced quite satisfactorily the experimental data. Since in the asymmetric mode the point-group symmetry changes, no calculations were carried out for that case with the model of ref. [139]. The effective range theory, however, was still used by Fabrikant [300] to provide some illustrative behavior of the DCS at 0.61 eV. Fujita et al. [310] used the Glauber approximation to obtain angular distributions of electrons scattering off H20 at higher energy values (i.e. 50, 100 and 200 eV). As one might expect, agreement with experiments was found to be acceptable only at small angles, while for 0 > 30 ° the differences between experiment and the theory are of several orders of magnitude, with the overall shape not being reproduced at all. At even higher collision energies, in the range of a few keV, it turns out that the FBA calculations of DCS as a function of momentum transfer [285] are in fair agreement with experimental findings, as one is clearly outside the region of interaction where complicated chemical forces dominate the collision process [311]. Trajmar and Hall [351] suggest that the DA process in H 2 0 is related with a vibrational mechanism of excitation that goes via a B 1, Feshbach-type resonance, which is not included by the simple calculations of ref. [139].

xlO-18

"To_ tn12.0 I ~

9.0

8.0

x

tO

: ~ 6.0 - / L,¢ OJ

~ 3.0 -

leV

0 L_

6o -I,--

!\

g 4.0

\x I

I

I

2.0

I

x~

4eV

2.0

~'x~x_x~

I

i

I

I

1

I

I

~ x

3.0- ×

2.0 \

6eV

3.0

- L

xN

8eV

2.0

x\

x'~-X~x~

o

1

x

°~

1.0

2.0 ~,~x_ x

t~.0 3eV

¢_

t,..O

2eV x

x

I

6.0

x\

do lb

-

0

x ,,.,.-,~

' 40

x

1.0

' 8'0 120

Angle (deg) Fig. 10. DCS at various collision energies of vibrational excitation of H20 via the (000) + (010) transition. The computed curves are from ref. [139], while the experimental crosses are from ref. [305].

406

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

7.4. CO 2 If one wants to use the SCE approach for this molecule, one quickly discovers that a very large number of terms are needed to correctly describe the nuclear and electronic regions around the oxygen atoms. Therefore, even if the number of atoms and the cylindrical symmetry might suggest that the electronic structure for CO 2 should be fairly simple to describe, as no permanent dipole moment is present, any quantitative agreement with observations has turned out to be quite elusive. Moreover, vibrational excitation of the bending mode is a very important process that immediately requires a different symmetry for its electronic structure, hence rapidly complicates the ab initio approach even if one is prepared to use a very extended expansion. Progress on the theoretical work for e - - C O 2 elastic and rotationally inelastic scattering has been reviewed up to 1980 by N. Lane [6] and extended up to 1984 by Trajmar et al. [32]. In the following we are therefore reporting only briefly the most important features of this system. The integral elastic cross sections exhibit a well-known % shape resonance in the 3-8 eV region. Either by solving the correct coupled equations [312-314] or via the use of the C M S X model [315], the general shape of the resonant feature can be reproduced fairly well through the use of empirical polarization potential. In the energy region away from the resonance, however, there are significant differences between experimental and computed cross sections, the latter being very sensitive to vibrational averaging procedures especially in the region of the resonance [315]. At energies beyond the resonance position, elastic and ro-vibrationally inelastic calculations have been carried out at 10 eV [316] and at Ecolt->-20eV [317]. In these calculations semiempirical MO methods were used to generate the target wave functions, no coupling was allowed between vibrational modes, and the Chase [215] integral was carried out via a three-point calculation for the T-matrix elements; fair agreement was found for elastic DCS while errors of up to 30% were found in the integral and vibrationally inelastic cross sections. At higher energies, the o-~ and o-m were computed within a two-potential model of a multiple scattering approach that based its initial choice of potential from previously used molecular wave functions [318]. In this energy region (20 to 1 500 eV) agreement between theory and experiment was found to be generally good, with the exception of the intermediate angular region (60° to 100°) below 100 eV: the dip in the computed DCS was deeper than the experimental one. One possible explanation for the discrepancy stems from the lack of highly anisotropic terms in the chosen interaction terms, which are obviously very important in controlling at high energy the angular distributions [319] over large 0c~ ranges. The problem of the low-energy behavior of the vibrational (100) excitation of CO 2 has intrigued several researchers for many years [320, 321] and has recently been studied by solving the FN coupled equations via different forms of exchange and polarization interactions [312, 322,323]. Such lengthy but approximate calculations indicate that a O-gthreshold peak should exist as induced by a virtual state with a large negative scattering length. 7.5 H2S

The H2S molecule carries great importance in molecular astrophysics as a common constituent of interstellar molecular clouds [324] and as an expected component of cold circumstellar envelopes [325]. Laboratory measurements for H2S exhibit considerable structure in both their differential and integral cross sections, elastic and vibrationally inelastic ones [326]. Data from dissociative attachment measure-

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

407

ments show the presence of peaks around - 2 eV in the HS- production [327], around 5.5 eV in the Hproduction and at about 10 eV in the S- production [328]. Spurred by the experiments of Rohr [326], Gianturco and Thompson [181] carried out the first ab initio calculation on the elastic scattering of electrons off H2S targets. By adjusting the parameterdependent polarization potential to the observed maxima in the cross sections, they obtained two close resonances of Al-symmetry at 3 and 6 eV and a B2-symmetry resonance around 5 eV. Later on, the use of a parameter-free model polarization potential with the same wave function as before [117] produced qualitative agreement with Rohr's measurements, as shown by the vibrationally elastic (rotationally summed) cross sections in fig. 11. One sees there that the 2 eV shape resonance is clearly reproduced by theory at the angles shown in experiments, quite a satisfactory accord in view of the approximate features which are still present in the theoretical approach. One can qualitatively distinguish three broadly different regions in the e--H2S cross sections: (i) the low-energy (<1 eV) threshold peak, (ii) a broad shape resonance of B 2 symmetry around 2 eV and (iii) a very broad maximum around 6-7 eV. Theoretical studies of the rotationaUy inelastic processes for a few of the lower-lying states (J = 0 to J' = 2) have been carried out in the energy region from 0 to 10 eV by Jain and Thompson [117]. They found that, except for the dipole channel (the 00~10 transition) the allowed rotational channels

20

20°

4.0°

8

vl ~ 15 ~

.

N"~ 10 "-D

6

-

4.

L.;

W°

5-

•

2

t/) tO

i

i

i

t

80 °

60 ° QJ

(n 3 I

30

0

•

1

2.0

.

-

~

1.0

Q.I L.

(:LI

,

I

,

=

+

I

,

I

100°

[

I

i

I

=

l

120°

1.0!

-

tl.0

0.5

. i

!

I

3

5

I

7

9

I

!

3

5

7

9

EnergyleV) Fig. 11. Rotationally summed (vibrationally elastic) DCS for e-H2S scattering as a function of collision energy at selected scattering angles. Tile computed curves are from ref. [117] while the experimental points are from ref. [326].

408

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

( 0 0 ~ 00, 0 o~ 2_2, 00 ~ 20 and 00 ~ 22) clearly exhibit enhancements around 2 eV, thus identifying the B 2 symmetry as the state chiefly responsible for this resonance in all channels. The vibrationally inelastic DCS for the present molecule are also strongly dominated by the 2 eV structure in the B 2 channel, as shown in fig. 12. Since Jain and Thompson [117] did not calculate near-threshold cross sections, they could correctly employ the AN theory for collisions above 1 eV and therefore consider the correct appearance of distinct maxima at around 2 eV at all angles (they consider only the symmetric and the bending modes). It should be mentioned here, however, that the experiments interpreted the isotropic nature of their vibrationally inelastic DCS between 20 ° and 120° as due mainly to an Al-symmetry resonance, while the calculations of ref. [117] obtained their agreement only via the B 2 symmetry. 7.6. NH 3 In spite of its great importance in many areas of molecular physics and fundamental chemistry, very few studies have been carried out on this molecule. Earlier calculations essentially employed the Born

o

40

1.0 ¢..

E •.5-°

__£

e-

J

I

~ _ _ J ~

I

60 °

2I?

o

u'l o

1.0

eI

L.

1 - -

I

~

.

_

_

~

I

'~

120 °

r-n

1.0

I

2

3

4.

Energy

.5

6

(eV)

Fig. 12. Vibrationally inelastic DCS for the e-H2S case at 40 °, 60 ° and 120 ° computed for the (000)--, (100) excitation process at various collision energies. The solid curves report the computed values of ref. [117], while the circles are the experimental data of ref. [326].

17.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

409

approximation [329,330], while only recently a close-coupling study was carried out on the momentum transfer cross sections via a parameter-free model [122] that produced very good agreement with several experimental data as indicated in fig. 13. More recently, Sueoka and coworkers [264] have measured for the first time the total cross sections for electron-NH 3 scattering from low (-1 eV) to high energy (few 100 eV). In the high-energy domain model potential calculations were carried out by Jain [332], who used a very simple spherical potential plus dipole interaction and obtained encouraging results in the small-angle region (2°-10°) for energies of 300, 400 and 500 eV: the comparison was made with the experimental data of Harshbarger et al. [333]. More calculations include the use of the FBA to obtain high-energy DCS as a function of momentum transfer [285] and the calculation of momentum transfer cross sections by using the effective range theory [299]. 7. 7. Sill 4 It is only in recent years that the Sill 4 molecule has attracted the attention of workers in the area of electron scattering because of its important applications in plasma chemistry. To study the properties of electron beams traveling through the silane gas has therefore acquired substantial relevance both from the theoretical and the experimental standpoints. Low-energy cross-section data have been analyzed by swarm experiments [43,334-335] but substantial differences appear between different analysis of experimental data. The e--Sill 4 scattering cross sections exhibit a shape resonance feature measured earlier around 2 eV [336] and more recently around 3 eV [264].

i,\

1043 "\. " ' \ ".~\

NH 3

e -

\._ "X\. '

\"N. . %•• ~ , . ".

"\.\\.. -

"\o~%..

•

"",e

\.

o E

10- 1 6 // 10 -2

i

i

i

i

111 1 0 -1

a

i

i

a Inlll

I

I

I

"\

I I I1

10 °

Energy (eV) Fig. 13. Integral momentum transfer cross section for the e - - N H 3 system as a function of collision energy. The theoretical data are given by the four different curves [122], while the filled circles are the experimental points from ref. [331].

410

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

Model calculations using the C M S X method were carried out by Tossell and Davenport [283], who were able to get qualitative agreement with the transmission experiments of ref. [336]. They also found that the shape resonance is caused by the T 2 symmetry and the computed total cross sections are almost flat beyond 4 eV. A very recent study (using the same model as in ref. [116]) was carried out for the integral cross sections from 1 eV up to 10eV [337]. The authors observed the RT minimum around 0.1 eV and a marked shape resonance around 3 eV, in agreement with the more recent measurements [264]. Since their RT minimum occurred at too low energy, they proceeded to tune the empirical potential to get the right position and then used the same potential for all contributing symmetries: they found then the resonance position at 2 eV. If they, however, adopted a more flexible approach and employed their usual parameter-free potential for all symmetries except that for the A 1 symmetry, which controls the RT minimum, then their computed cross sections exhibited a maximum around 3 eV and the corresponding ~rm cross section, turned out to be in better agreement with the swarm data [43]. Figure 14 summarizes the above analysis and clearly indicates that further measurements are needed to solve this riddle, as theory is not sufficiently free of parameters to do the job unambiguously. 7.8. Other studies on polyatomics

Electron scattering from hydrocarbons has necessarily a long history if one considers the first member of the series, methane, as we have discussed in section 7.1. Data for other molecules in the series were only measured in recent years: DCS for C2H 2, C2H 4 and C2H ~ [338], ionization cross -15

10

e*-Si totAL

~

--..,

H~

o2 / ' \

cross sZCrlONS(A#

///

. "~, X×xx

10 ~

",

.~/tjlll

×x

co £b Q:: --4 tO - )

e-(JT PotentialJ

....

e- (J T Potential, tunedJ

......

e-fJ T+G T h y b r i d )

x x xx e*[JT Potentiol) • Q OSueoke's

01 0.01

i 0.1

Electron

measurements

I 1

E n e r g y (e V)

Fig. 14. Integral cross sections computed with various potential forms for Sill 4 (from ref. [337]). The JT potential is a parameter-free potential for all symmetries, while the GT potential is a parametric potential "tuned" to the RT minimum of this system. The filled circles are the experimental points of ref. [264].

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

411

sections of C2H 6 [339] and its dissociation cross sections [340]. Very recently, extensive measurements have been carried out on the total cross sections of several hydrocarbons (ethane, propane, propene, cyclopropane, n-butane, isobutane and 1-butene) over a rather large range of collision energies (from 10 eV up to 400eV) using both electron and positrons as projectiles [341]. If we consider the C2H 2 molecule, we see that it is isoelectronic with N2, CO and HCN and shows like those systems a IIg shape resonance around 2.6 eV. In spite of its not being any larger than the well-studied N 2 and CO cases, however, only a few limited studies exist for C2H 2. At a collision energy of 10 eV, Thirumalai et al. [211] presented elastic and rotationally inelastic cross sections via an ab initio model potential. Their calculated cross sections were much larger than those for the N 2 case that were computed with much the same model by the same research group: thus illustrating how the greater spatial extension of C2H 2 markedly increases the corresponding interaction region, hence the relevant cross sections. Further calculations on the C2H 2 molecule were carried out via the CMSX~ approach [342] for total cross sections from 0 to 5 eV (note that the data shown in fig. 1 of that reference need to be divided by a factor of 2). For the higher region of collision energies, calculations were carried out for the same system between 100 and 1 000 eV by Jain et al. [343] using a two-potential coherent scattering model and were found to be in satisfactory accord with existing experimental data. The vibrational excitation of C2H 2 has been recently studied experimentally by Kochem et al. [344] who also made a theoretical analysis of the relevant modes. Of the five existing normal modes, only the infrared active v3 (asymmetric stretch) and v5 (asymmetric bending) are strongly excited near their thresholds and within the shape resonance region (2-3 eV). A simple theoretical model has been formulated by Chang [345] to treat vibrational excitation of polyatomic linear (or nearly-linear) molecules. This approach determines the scattering amplitude for the direct excitation process, for the shape resonance region and then the interference effects between these two regions. As an example, fig. 15 shows the experimental DCS for the v5 mode of C2H 2 at 1.6 and 2.6 eV, together with the corresponding calculations via Chang's model [345]. Similar calculations at lower collision energy (0.235 eV) show that the Born cross sections reproduce very closely the experimental curves. As the energy is increased, however, the DCS for the v5 mode excitation markedly depart from the Born behavior. It should be mentioned here that the theoretical curves (solid line) in fig. 13 are fitted to the experimental data, with the fitting parameters being provided by the v5 resonance decay amplitude and its phase relative to the direct dipole amplitude. Another linear polyatomic target that has recently received attention from the theoretical point of view is the HCN system, for which cross sections have been computed from very low collision energies up to and beyond the resonance region (from 6 x 10 -4 eW to 11.6 eV) [120]. A parameter-free model was used for the interaction, where the static and exchange contributions were treated exactly and the polarization form was taken from the FEG correlation model discussed earlier (section 3.2.2). The same model has been recently extended [346] to study the behavior of the various interaction forces as the internuclear distances are stretched in the molecule. Several interesting features of the collision process are interpreted in terms of the corresponding behavior of the potential energy curves: some intermediate HCN- states are invoked to explain the observed behavior. N20 is another system, isoelectronic to CO 2 and linear in its ground state, which has been studied only experimentally [352]. The major structure in the total cross sections at 2.3 eV has been interpreted as due to the excited 2A1 or 2X+ states [353,354], but the entire resonant structure below 1 eV may be due to the 21-I and 2X÷ N2O- states. No theoretical results allow us to decide between these states. An example of large molecular targets studied for total cross sections by model calculations seems to

412

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

×104o ---

- - fo

(a) E0=1.6eV ~ bending mode

O.6

O3 IE

OA.

tO

0.2

ow .,4--

I1J

\ ,

15

ffl

45 75 105 Scattering ongte (deg)

0 x 10-16 U

~'i~.

(b)'VE0= 2.6ing eV mode sbend

0.30

t'GJ I-(I; ~-

0.15 ~"~'-----4..__{ ~ o I

15

I

~"

"" I

~5 Scattering

{ I

I'

oo r

75 105 angte(deg)

Fig. 15. DCS for the u5 mode in C2H 2 at 1.6 and 2.6 eV (from ref. ]344]). The chain curve shows the Born results while the solid curves give the data from the full theory [345].

be that of the 1,3,5-trifluorobenzene together with other haloderivatives of ethylene, analyzed via the CMSX, method [347]. It is fair to say, however, that the whole area of large molecules is still in its infancy as far as theoretical treatments go. We still need to check the general validity of our less elaborate models not only by assessing their capacity to successfully fit experiments, but also by controlling how well the various simplifications in them reproduce the quantities obtained from more rigorous theories. In other words, benchmark calculations on the smaller, more popular, polyatomics appear to be the only way to test all the theoretical ingredients analyzed in the present report in order to proceed more confidently in generating the simpler models to be applied to larger molecules.

8. Conclusions

In the previous sections we have tried to present a fairly detailed picture of progress in the past few years in theoretically treating the many processes that occur when a "slow" electron interacts with a gaseous, many-atom molecule. The examples and their discussion demonstrate that a great deal of work still needs to be done in developing a comprehensive and efficient computational method that can deal with electron-molecule collisions from thresholds to 100 eV of collision energy. The apparently easiest part of the interaction between partners, i.e. the "undistorted" charge distribution from the molecular electron density, still requires the setting up of reliable, ab initio wave

F.A, Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

413

functions for several values of the whole nuclear geometries. The effort required to do this increases roughly with the fourth power of the number of basis functions and, if one wants to generate an "augmented" basis set for a better description of the second-order linear response terms (polarization and correlation), a rather extended basis of L 2 functions becomes necessary. For large molecules and for vibrationally inelastic cross sections, usually needed over a range of collision energies, this procedure rapidly gets out of hand; thus the effort to refine or extend more approximate descriptions of target electron densities seems a more appealing alternative for the near future. The results already shown by the single-center-expansion approach are certainly encouraging. Exchange and polarization interactions have been the traditional stumbling blocks in e--molecule calculations, and they are obviously even more difficult when nonlinear polyatomics are involved. A simplifying approach, which writes down a separable expansion of the full Feshbach optical potential using L 2 basis-set expansion and configuration interaction methods, and then solves sets of coupled integral equations by the introduction of Gauss quadrature of the resultant set of algebraic equations, has been very promising when applied to diatomic targets (see section 3.2.3). It may well be the best approach for polyatomic molecules as well, although more rigorous, benchmark calculations on the more familiar, simpler diatomics are needed before its real quality can be established with certainty. The use of FEG models for exchange and, recently, for correlation effects (see section 3.2.2) provides a much simpler approximate prescription for handling molecular targets. As discussed in the previous examples (section 7), because it is capable of yielding qualitative and quantitative agreement with experiments, it seems a useful procedure for complex targets, provided a way can be found to obtain molecular electron densities with an acceptable degree of accuracy (i.e. from ab initio approaches which do not carry empirical treatment for their two-electron interaction, as is often the case in the so-called semiempirical methods). Here again, the use of the SCE method has proven to be fairly accurate and rather simple from the computational viewpoint, although most effective for a limited class of polyatomics only. If efficient ways could be found to extend it to more general types of molecules, it can provide a very realistic tool for investigating rovibrational excitation processes. Finally, only during the past year have attempts been made to produce a rigorous and exact treatment of e--polyatomic (i.e. CH4) molecules [104]. Although still in their infancy, these approaches are obviously essential to progress toward model calculations with some real theoretical understanding of the reasons why specific approximations work, or identifying the mutual-cancellation effects that result in reproduction of experimental observables, and, thus, deceptively imply proof of goodness of a model. Thus the anticipation is that the next few years will see progess on fully ab initio methods for simple polyatomics, together with the parallel extension to several, often complicated, cases of effective model treatments that catch most of the relevant physics in the process.

Acknowledgements One of the authors (A.J.) would like to express his gratitude to Dr. D.W. Norcross for providing the facilities for writing the major part of this work during his stay in Boulder. F.A.G. also thanks Professor P.G. Burke for his hospitality at Queen's University, Belfast, during the summer of 1984 when this report was begun. He also acknowledges the support of the NATO Scientific Division for the award of a Senior Fellowship that allowed the completion of the present work. Both authors sincerely thank the JILA Scientific Reports Office for preparing the first draft of the manuscript and particularly Ms. L. Volsky for her expert assistance in its editing.

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

414

This work was funded in part by the U.S. Department of Energy (Division of Chemical Sciences) and by the Italian National Research Council (CNR).

Appendix A: Real spherical harmonics

(S~Q(O, dp))

In our notation, the usual spherical harmonics are defined as (2L + 1)(L-[MI)! ~/2 e 'M* Y~(O,&)=((-1)u+lil)~/2\ ~r(-~71~/-~. ) PILMI(cosO)

(A1)

where L = 0, 1 , 2 , . . . ; M = L, L - 1 , . . . , - L and pILMt(cos0) are associated Legendre polynomials. The phase convention adopted here is that of Condon and Shortley. Various properties and integrals with the above YLM are given by Weissbluth [145]. A real spherical harmonic (RSH) is defined as

sMO(o, &) = 1(0 + 1, Q - 1) (-1) M ~1 (yM + Q (_I)M yzM)

(A2)

where (a, b) means a + ib. Note that when M = 0, the RSH reduces to a normal spherical harmonic. Also, since S~/°'s are all real, it follows that SIffQ

: (SL

MQ , )

.

By carrying out some complicated but straightforward algebra, we can deduce the following properties of RSH's:

(1)

SM''(OL,,,4,)SM22OL2

A[ L

1/2

(2El

+ 1)(2L2

cLI

+ 1)

where A is defined in the following conditions (a) M 1 ¢ 0 ; M 2 ~ a 0 ; M = M I + M 2 1

L1

L2

L

A=-2-~(QI+Qz-Q~Qz+I) CM1 M2 m (b) M ~ # 0 ; M 2 # 0 ; M = M ~ - M

A = ~ (1 Q , + Q z - Q a Q z (C) M 1 # 0 ; M z # 0 ; M = M ~ - M A=~

1

2>0 L1 L2 L +1) Q2(-1) M2CM1 -M2 M

2<0 L1

L2

L

(01 + Q 2 - QaQ2+ 1) Q1 (-1) M~ CM~ M2 IMI

(d) M I # 0 ; M z # 0 ; M = 0

L2

o

g sIMI(Q1Q2)[O ~ )

o

L

,,

(A3)

F.A. Gianturcoand A. Jain, The theory of electron scatteringfrom polyatomic molecules

A = ( - 1 ) MC~'

415

L~ L -Mr 0 ~(21,O2

(e) all other cases A

L1

L2

%

M2

L

f

J sMIQI(o ~)) sM2Q2(o ~)) sin 0 dO d~b = L1 \ ~ L2 \ ~

(2)

(3)

~L1, L2

~M1,M2

QI, Q2

(A4)

f SM1QI(O L1 k ~ ~)) sMQ( O, 4 ) sM2Q2(o L2 k 4 ) sin 0 dO d~b

[ ( 2 L 1 + 1)(2L 2 + 1)] ',2

= A[

J

(A5)

L2 o /~ o ~O, OlO

Co

where the A coefficients are defined as: (a) M 1 # 0;

M z # 0; M

1

A= -~

= M 1+ M2 L1

L2

L

(QI + Q z - Q'Qz + I) CM, M2 M

(b) M 1 ¢ 0; m 2 ¢ 0; M = M 1 - M 2 > 0 1

L1

A = ~ - ~ (Q1 + Q2 - Q,Q2 + 1) Q z ( - 1 ) m2 CM1

L2

L

_M2 m

(c) M I # 0; m 2 # 0; M = M 1 - M 2 < 0

1 A=_~(Q,+Qz_Q,Qz+I)

QI(_I)MIC~]~

I~z c -M2 [MC

(d) M I # 0 ; M 2 # 0 ; M = M 1 - M 2 = O A =(--1)M1

CM11

L2 -M 1

L 0

~QI, Q2

(e) all other cases

A=CLd~

t2 m M2

M

(4) Some of the RSH and their products: (

(a)

3

]1/2

S~±1=-\-~/

sin0 f(-1)

.~,1+1 3 ( 5 ~1/2 (b) oz = - ~ \ 1 - ~ / sin20 f(+l)

(A6) (A7)

416

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

1 ( 1 5 ] 1/2 (C) 822-+ = \6--~/t (1-- COS20) f(+--2) .~,1+_1 -- 1 (d) SI ±1 ~1 ~

01 S0

~

1

(A8)

01 3 S2 + ~

(A9)

21 82

where fl + m) = cos(m~b); f ( - m) = sin(m~b).

Appendix B: First Born K-matrix elements for C3v and C2v-point groups using asymptotic part of the interaction In the FBA, the Born K-matrix elements are evaluated from: K p"

lh, l'h'

= - 2 k J j,(kr) jt,(kr) r 2 d r ( X f ~ I ~AH vx.(r) X AI.[Xt,hp., ).

(B1)

0

For the C2v point group, B .¢ol + ~B'

E vAH(r)Xa% = - - S~ 1 "~- ~ u 2 AH r r

r

21

C sOl

S 2 + --

r

3 +''"

(B2)

and for the C3v group A B 01 C S01 C ' $ 3 - 1 Z v x r t ( r ) X ~ = - - S°, 1 + ~ S 2 + 3 q- ~ 3 aH r r r r

+....

(B3)

Here the coefficients A, B, B', C, C' are related with the dipole (D), quadrupole (Q°2, Q~) and octupole (O °, ~2~) moment tensors, respectively, i.e., A = -(47r/3)~/ZD

B = -(41r/5)1/2Q~ B' = -(4-tr/ 15) 1/2Q22

(B4)

C = -(4~'/7) 1/2~2°3 C' = -(48¢r/35)1/20~. We list in the following, some of the explicit expressions for the FBA K-matrix elements for the few lowest moments of the interaction. (i) Dipole moment: Putting A = H = 1, vll= - D / r 2 and xA11 = (47r/3)1/2S~ ' ' we obtain,

F.A. Gianturco and A. Jain, The theory of electron scatteringfrom polyatomic molecules

417

oo

KPUth,rh' = 2 D ( a T r / 3 ) l / 2 k

J j,(kr) jr(kr)drJ S 7 q

5°11 s r ' q ' d r

(B5)

.

0

Performing radial and angular integrations, we get finally, sin[(l-l')rc/2]

Ke'*th,,'h'= 2D3 v ( 2 / + 1)(21' + 1) l(7713 -/-7077] ) (-1)mC'o

l'

l

l

o o Cm

,,

-m

~

O"

(B6)

Note that the dipole K-matrix elements are energy independent. (ii) Quadrupole moment: Using eqs. (B2) [or (B3)] in eq. (B1) we get the following FBA K-matrix jl(kr) j r ( k r ) dr {B I~1 + B' I~ 1} r

K p'* lh, l'h' = - 2 k [~

(B7)

0

where (B8)

1~ Q = f S ? q S MQ 8 7'q' d~.

Equation (B7) can be reduced to F(s - 2)(B I°2' +

7rk th, rh'

4

B' I221)

(B9)

F(s) F[s - l - ½1 F[s - l' - ½1 '

where s = ½(l + l' + 1) + ½. Appendix C: The first Born approximation for the rotational excitation of asymmetric, symmetric and spherical top molecules

Itikawa [308-309,329-330] has given general expressions for the rotational excitation of asymmetric and symmetric top molecules by electron impact in the FBA. Here we summarize his formulae relevant to the present study. 1. Asymmetric top: For a general transition (Jr--->J'r'), Itikawa has derived expressions of the differential and the total cross sections with proper consideration of molecular symmetric properties. For the dipole transition only, the DCS becomes k' 4 D Z ( 2 J ' + 1)

do" (j¢___>J ' r ' ) :

dO

k

3q 2

I(J'r'lJr)lOll 2

(C1)

where

(J'r'lJr)t.,8 = 2 a rj¢~ a K J'¢', , ( - 1 ) K ( ~

J~: 01),

m = 0.

(C2)

K>0 v/J'

It can be easily seen that for the dipole (00---->10) transition, eq. (C1) becomes (similar to diatomic molecules)

418

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

d~ (0o__, lo ) = ~D 2 k' k '2 1 0) dO k- (k 2 + - 2kk' cos

(C3)

The momentum transfer cross section is evaluated as usual; for the above case of ( 0 0 ~ 10) transition with dipole term only, we obtain 87rD2 (1 o-m- ~ 3k

(k-k')eln 2kk'

k+k' l-k- ~ l / "

(C4)

Similarly, for the total cross section, we shall have

8¢rD 2 k + k' 0"I - ~ 3k In k'-------/" Jk-

(C5)

2. Symmetric top: Itikawa [329] has discussed the general form of the DCS for the J K ~ J'K' transition of a symmetric top:

do'B (JK--* J ' K ' ) - k' ( 2 J ' + l ) ~ dO

k

7r

~m (~S -K' J' - tm f drr2 ~m(r) j,(qr) 2

(C6)

0

with q = (k z + k '2 - 2kk' cos 0) ~2. For the dipole interaction, eq. (C6) reduces to

do" (JK---~ J'K') 4 k' D 2 ( 2 j ' + 1) - 3 k dO q2 ( ,~

s'~: ~,

)2

(C7)

which for the transition ( 0 0 ~ 10) reduces to the same expressions as above (C3)-(C5) for various cross sections. 3. Spherical top: The expression of DCS for a spherical top is written as k' 1 [ do" (JKM---> J ' K ' M ' ) = -k- 2 - ~ d r e x p ( - i q , r){~j~MIV(r)I~,K,M,)I 2 dO "

(C8)

Writing

V(r) = ~ Plm(r) Y t ( r ) ,

(C9)

lm

where, for CH4, ~m(r) = ~t(r) b-Alm1 and YV(/~) = E YV(~') @.m(a]33,), l

(C10)

n

we can have

<

I G,I ,M, > =

[(2J + 1)(2J' + 1)] 1/2 ( J, l s, E Vim(r) YV(f') j dO @M'K' @,m ~MU" 8 "/r2 lmn

(C11)

419

F.A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules

Equation (Cll) can be simplified to

(~O+Ka4lV(r)l~+,K,M,)=[(2J+l)(2J,+l)]l/Z(_l)M-r~Z,zm(r)yT(~,)(

,

--M

,' 1)( - K, ~', ~m)•

M'

(C12)

,rnn

Now, expanding exp(-iq • r) = 4~r Z -iA]a (qr) AA'

Y]'*(i) Y]'(~I),

(C13)

we write finally the expression for the scattering amplitude as

f(JKM--+ J'K'M') = 2 [(2J + 1)(2J'

+ 1)] 1/2

(-1) M-K ~ -ity~'(t~) lmn

× ( -M'

K," 'm)fdrr 2 Plm(r) jt(qr) .

(C14)

The DCS is calculated from do" (j___>j )= 1 k' ' ~, If(JKM--+ J'K'M')I 2 d/2 (2J+ 1)2 k KK'MM' '

(C15)

which can be written as do k' (2J' +1 ~ ~ 1 f r-2 ]t(qr) 2 dO (J---> J') = ~ \ ~ / (2/+ 1) dr plm(r)

(C16)

The total cross section is given by 2(2J' + 1) f o.I(J--> J') - k2(ZJ + 1) ~lm(21 + 1) -1 k+k' dq q f dr r 2 Plm(r) Jr( qr) Ik-k'l

2

•

(C17)

0

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