vibrational excitations of ch4 by electron impact: a...

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 2873–2887 PII: S0953-4075(02)33922-1

Vibrational excitations of CH4 by electron impact: aclose-coupling treatment

Tamio Nishimura1 and Franco A Gianturco2

Department of Chemistry and INFM, The University of Rome ‘La Sapienza’, Citta’ Universitaria,Piazzale A Moro 5, 00185, Italy

E-mail: [email protected]

Received 18 February 2002, in final form 22 May 2002Published 18 June 2002Online at stacks.iop.org/JPhysB/35/2873

AbstractWe present a quantum dynamical treatment of the vibrational excitation ofgaseous methane molecules from single collisions with low-energy electrons.The exact vibrational coupled equations are solved using the four differentnormal modes of the molecular target and the numerical bound states of eachmode. The interaction forces are treated exactly for the static effects whileexchange, correlation and polarization contributions are added via parameter-free model potentials. The equations are solved in a body-fixed reference frameusing a single-centre expansion formulation and the ensuing inelastic, partialcross sections are compared with experiments and with earlier calculations.

1. Introduction

Low-energy electron scattering from methane is of significant interest in many atmosphericand technological applications. CH4, in fact, is an important constituent in the atmospheresof Jupiter, Saturn and our own planet, where it also plays a significant role in the network ofgreenhouse gas kinetics. If one wishes to understand the fairly complex chemistry of theseenvironments one needs to know, among other things, the detailed array of several state-to-statecross sections for this gas in collision with low-energy electrons, the latter being present inthe atmosphere as produced from a variety of ionization/excitation processes. Methane is alsoan important constituent in radiation detectors such as drift chambers and diffuse dischargeswitches, where its marked Ramsauer–Townsend (RT) minimum at around 0.35 eV, and itsresonant enhancement present at higher energies (∼8 eV) cause the unusually high conductivityof this gas for low-energy electrons [1–3].

As a result of the fundamental interest from so many research areas, there has been asubstantial number of measurements and calculations of scattering cross sections for methane.1 Permanent address: 2-11-32, Kawagishi-kami Okaya, Nagano 3940048, Japan2 Author to whom any correspondence should be addressed.

0953-4075/02/132873+15$30.00 © 2002 IOP Publishing Ltd Printed in the UK 2873

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2874 T Nishimura and F A Gianturco

Table 1. Vibrational modes of CH4.

Frequency (eV)

Mode Exp.a Cal. Symmetry Activity

ν1 (symm. stretch) 0.362 0.390 A1 Ramanν2 (twisting) 0.190 0.207 E Ramanν3 (antisymm. stretch) 0.374 0.402 T2 IRν4 (scissoring) 0.162 0.180 T2 IR

a The recommended values based on experiments [37].

A fairly detailed review of both experiments and calculations has been recently presented (see,e.g., Bundschu et al [4] and Tanaka and Boesten [5]) and therefore we will not be repeating ithere. It will suffice to say that the corresponding analysis of vibrational excitation cross sectionshas been much harder to come by, both experimentally and theoretically. The theoreticalanalysis, in particular, has been carried out only a few times: by us using an approximatetreatment of the vibrational dynamics with an exact exchange contribution [6] and it was alsoattempted for the symmetric and the antisymmetric mode excitations by Rescigno et al [7]. Wefurther analysed the symmetric mode using a model treatment for several Td molecules [8] andstudied the effect of exchange interaction on the four normal-mode excitation cross sectionsfor methane [9]. None of the above studies, however, has carried out the correct dynamicalcoupling between the impinging electron and the vibrating nuclei, as they have all attempteda more approximate description of such coupling terms instead.

This work, therefore, aims at analysing the features of computed inelastic cross sectionswhen the exact quantum dynamics is included within the theoretical treatment, this being thefirst study of this type for a nonlinear, polyatomic target. The following section outlines ourtheoretical approach while section 3 presents and discusses our results in relation to previoustheoretical and experimental works. Our conclusions are summarized in section 4. Atomicunits (au) are used throughout unless otherwise stated.

2. The theoretical method

2.1. General outline

In the case of nonlinear polyatomic molecules the interaction between the incident electronand the polyatomic target, particularly its multicentred and nonspherical features, is markedlymore complicated than for linear molecules and, furthermore, we have to solve a multi-channel problem which includes the rotational and vibrational states of a polyatomic moleculewith multiple modes of vibration that have different fundamental frequencies and differentsymmetries. It is thus important to understand in detail how the vibrationally inelastic crosssections for each normal mode depend on the feature of that particular mode. The Td pointgroup has four types of such modes which describe its lower-lying vibrations, i.e. the symmetricstretching (ν1), the twisting (ν2), the antisymmetric stretching (ν3) and the scissoring (ν4)modes. Since the ν3 and the ν4 modes are infrared (IR) active and therefore an additionallong-range interaction due to the induced dipole moment plays an important role, we shallfurther analyse this feature in more detail when discussing our data (see table 1).

Taking advantage of this effect, in fact, Davis and Schmidt [10] calculated theinelastic cross sections for CH4 + e− collision by using the dipole Born approximation nearthe vibrational threshold and found sharp peaks for the ν4 IR active mode. Their resultssupported what was found in the behaviour of the measured differential cross sections recorded

Vibrational excitations of CH4 by electron impact: a close-coupling treatment 2875

by Rohr [11] and Sohn et al [12] at low collision energies, and suggested that direct excitationprocesses indeed play an important role in this system. One of the most simple and popularmethods that could be employed to obtain the inelastic cross sections beyond the Born scheme isto calculate them within the adiabatic nuclear vibration (ANV) approximation which assumesthat the motion of nuclei and of the scattering electron are adiabatically separable (e.g. see [13]).When one applies the ANV approximation to CH4 + e−, however, some corrections should bemade for non-adiabaticity because the ANV scheme is known to fail at low collision energies,in particular, in the vicinity of threshold. In fact, many studies (e.g. see [9]) have shownthat the inelastic cross sections obtained by the ANV method do not tend to zero as theyapproach each threshold of vibrational excitation. In order to avoid this drawback, Althorpeet al [6] calculated the electron scattering wavefunction within the ANV approximation andthen placed it into the off-shell Lippman–Schwinger equation: their inelastic cross section forCH4 becomes zero at the threshold, and peaks are found just above the threshold for the IRactive modes.

The present calculation is based on the fixed nuclear orientation (FNO), i.e. the rotationallysudden approximation, while the vibrational transition is treated within a rigorous close-coupling formulation [14]. We further assume that the target molecule is kept in itselectronically ground state and therefore we take into account three kinds of interaction betweenthe incident electron and the target molecule: the effects which originate from an ab initioelectrostatic interaction, a model electron exchange and a model target polarization. The lattertwo effects are thus only included through local model potentials. In our earlier work [6], wetreated the electron exchange exactly as a non-local term when solving the coupled equationsvia an iterative procedure. However, we are also interested in finding ways to compute thiseffect more easily from a computational point of view by introducing it in the form of a local,model potential without, however, losing the physical qualities of the vibrational close-couplingmethod.

Since the details of the present theory have already been recently reported in our work onpositron scattering [15, 16], we provide here only a brief reminder of it. To obtain vibrationalexcitation cross sections for electron scattering from polyatomic molecules we need to solvethe Schrodinger equation of the total system for the wavefunction �, at a fixed value of the totalenergy E. The total Hamiltonian is represented by the sum of the operators of the molecularHamiltonian, of the kinetic energy for the scattered electron, and of the interaction between theincident electron and the target molecule. The molecular Hamiltonian of this work only consistsof the rotational and vibrational parts of the more complicated, full molecular, electronuclearHamiltonian. Hence, we include no effect from the possible electronic excitations or fromother reactive and break-up processes involving our molecular target. In other words, weassume that during the scattering event the molecular electronic wavefunction is always thatof the neutral ground electronic state of CH4.

We also assume that the orientation of the target molecule is fixed during the collision, sincethe molecular rotation is usually slower when compared with the velocity of the projectile at theenergies we are considering. This is called the FNO approximation [14], and corresponds toignoring the rotational Hamiltonian. Then, the total wavefunction could be generally expandedas

�(r|R) = r−1∑lνn

ulνn(r)Xlν(r)χn(R). (1)

Here, χn is the vibrational wavefunction of the molecule with its vibrational quantum numbersn ≡ (n1, n2, . . . , nT ), where T represents the total number of different normal vibrationalmodes of the target molecule. The variables R and r denote the molecular nuclear geometryand the position vector of the scattered electron from the centre-of-mass of the target,


respectively. The unknown ulνn functions describe the radial coefficients of the scatteredelectron wavefunction and Xlν are the symmetry-adapted angular basis functions introducedearlier by us [17]. The symbol ν in equation (1) globally stands for the indices specifying theirreducible representation and those distinguishing its degenerate members.

After substituting equation (1) into the Schrodinger equation of the total system underthe FNO approximation, we obtain a set of full close-coupling equations for each ulνn(r)

function that now explicitly includes the vibrational channels. These are called the body-fixedvibrational close-coupling (BF-VCC) equations (see e.g. [16, 18, 19]){

d2

dr2− l(l + 1)

r2+ k2

n

}ulνn(r) = 2

∑l′ν ′n′

〈lνn|V |l′ν ′n′〉ul′ν ′n′(r), (2)

where kn is written as

k2n = 2(E − Evib.

n ) (3)

with Evib.n being the energy of the specific molecular vibration we are considering. We should

further note that, given the low level of vibrational excitation caused by electron impact atthe considered energies, the contributions from inter-mode couplings have been consideredas negligible. This simplification may not hold when considering highly excited molecularvibrators. Any of the elements of the interaction matrix in equation (2) is given by

〈lνn|V |l′ν ′n′〉 =∑l0ν0

〈n|Vl0ν0 |n′〉∫

dr Xlν(r)∗Xl0ν0(r)Xl′ν ′(r) (4)

where

〈n|Vl0ν0 |n′〉 =∫

dR{χn(R)}∗Vl0ν0(r|R){χn′(R)}. (5)

One should point out here that this method is essentially a generalization of the method proposedlong ago (called the ‘hybrid theory’) for the much simpler case of a diatomic molecule [19].When solving equation (2) under the usual boundary conditions, we obtain the K-matrixelements. Therefore, the integral cross section for the vibrationally inelastic scattering isgiven by

Q(n → n′) = π

kn

∑lν

∑l′ν ′

|T lνnl′ν ′n′ |2 (6)

where T lνnl′ν ′n′ is the T -matrix element.

The interaction potential between the impinging electron and the molecular target isrepresented here in the form of a local potential. Thus, the interaction is described by the sumof the attractive electrostatic (V st ), the electron exchange (V ex) and the electron correlation–polarization (V ecp) terms. To obtain the V ecp in the short range of the interaction region, wemake use of a simple parameter-free model potential (V corr ) based on an electron–electroncorrelation energy (εe−e) suggested by Carr et al [20,21] in the framework of a homogeneouselectron gas [22],

V corr (r|R) = d

dρ{ρ(r|R)εe−e[ρ(r|R)]}, (7)

where ρ denotes the undistorted electron density of the target correctly obtained from many-body calculations. In our present formulation of this model, the short-range V corr is connectedsmoothly to the asymptotic form (V pol) of its spherical component, an earlier proposal forlinear molecules [23] further expanded by us to nonlinear targets [24],

V pol(r|R) ∼r→∞ −α0(R)

2r4. (8)


Table 2. The range of internal coordinates for the four vibrational normal modes of CH4. ri : bondlength (Å) of C–Hi , θij : molecular angle (deg) of HiCHj , βAB : angle (deg) between the planesof H1CH3 and H2CH4. Vibrational displacements for the normal coordinates are given as Rs

(s = 1, . . . , 4) ((amu)1/2 Å) = ±1.

r1 r2 r3 r4 θ13 θ24 βAB

Req 1.082 1.082 1.082 1.082 109.47 109.47 90.0ν1 ±0.502 ±0.502 ±0.502 ±0.502 — — —ν2 — — — — — — ±58.18ν3 ±0.546 ∓0.546 ±0.546 ∓0.546 — — —ν4 — — — — ±51.46 ∓51.46 —

Here, α0 is the spherical component of the target polarizability and is limited to its dipolecontribution. As for the electron exchange, use is made of a semiclassical exchange (SMCE)potential adapted to nonlinear targets [25] (see also [26]) and applied to low collision energies,as discussed in our earlier work [8, 9],

V ex(r|R) = 12 [E − {V st (r|R) + V ecp(r|R)}]− 1

2 {[E − {V st (r|R) + V ecp(r|R)}]2 + 4πρ(r|R)}1/2. (9)

2.2. Numerical details

The target wavefunction of the electronic ground state for CH4 was calculated at the SCF (self-consistent field) level using our familiar single-centre expansion (SCE) method [27] applied toa multicentre Gaussian-type orbital (GTO) expansion, and the basis set employed was providedby the GAUSSIAN 98 package [28]. We have chosen GTO basis sets of type D95∗, which arethe same as employed in our recent calculation on positron scattering [29]. The terms of themultipolar expansion of the interaction potential in equation (4) were retained up to l0,max , andthe scattered wavefunction of the electron in equation (1) was expanded, with the inclusion ofthe lowest two vibrational states n = 0 and 1, up to lmax which yielded K-matrix elementsconverged within 1% (see figures 1 and 2). For the specific information on all the requiredparameters and properties for each of the vibrational modes which we study in this work,see table 3. In order to solve the close-coupling (CC) equations by means of standard Greenfunction techniques, equation (2) is rewritten as an integral equation (a Volterra equation:for details, see [30, 31]). In the case of the ν1 mode (which belongs to the A1 irreduciblerepresentation) by fixing the molecular symmetry to be that of the Td point group the ensuingrange of bond lengths of C–H was taken to vary from 0.87 to 1.6 Å in order to correctly takeinto account the effects of the nuclear displacements from the equilibrium geometry of thetarget (Req). As for the other modes, the vibrational normal coordinates Rs with s = 1, . . . , 4,were taken to vary in the range given in table 2. For the asymptotic part of V ecp, i.e. V pol ,we first obtained the values of α0 from the very large basis set (quintuple-zeta) of Dunning’scorrelation consistent type [32], and then we normalized its results to the experimental valueof α0 (17.54 au) known for the molecular equilibrium geometry.

3. Results and discussion

One of the first tests for the CC treatment of the internal excitation process is to verify, amongother things, the validity of the diabatic expansion over asymptotic vibrational states of thetarget molecules. The calculations reported in figures 1 and 2 show, in fact, the behaviour ofthe (0 → 1) excitation cross sections for all four normal modes of CH4 when different sets of


Figure 1. Computed BF-VCC inelastic cross sections for the (0 → 1) excitation processes. Theupper panel refers to the ν1 mode and the lower panel to the ν2 mode. The solid curves refer tothe two-state calculations while the open circles show the three-state calculations. The ν2 mode isshown with its multiplicity.

vibrational states are coupled during the quantum dynamics. The more extensive calculationswhich included three vibrational states per mode (n = 0, 1 and 2) are shown at specificenergies (open circles). The total symmetric mode, ν1, involves calculations where the dipolepolarizability was taken to vary with the normal coordinate (VP calculations), while for allthe other modes, because of the symmetries of the nuclear motions, we took the average value


Figure 2. The same as in figure 1 but this time for the two additional normal modes associatedwith the ν3 (upper panel) and ν4 (lower panel) excitations. Both mode cross sections are reportedwith their correct multiplicity.

of the dipole polarizability, i.e. its fixed value at the equilibrium geometry (FP calculations).The validity of such a treatment has been recently discussed in our work on positron scatteringfrom methane [29] and therefore we refer the interested reader to that discussion. All the datashown in the above two figures clearly indicate that the electron motion during the molecularvibrations is weakly coupled to the motion of the nuclei and therefore the ‘local’ perturbationof the asymptotic vibrational states of the target is rather small. This means that the simple


Table 3. Computational details of the two-state BF-VCC calculation.

Normal Point Maximum number ofmode group l0,max lmax coupled channels

ν1 Td 24 12 A1 = 22, A2 = 8, E = 28, T1 = 36, T2 = 48ν2 D2 28 14 A = 114, B1 = 112, B2 = 112, B3 = 112ν3 C2v 24 12 A1 = 98, A2 = 72, B1 = 84, B2 = 84ν4 C2v 24 12 A1 = 98, A2 = 72, B1 = 84, B2 = 84

Table 4. Computed derivatives of the spherical polarizability.

Derivative ((amu)−1/2 Å2) ν1 ν2 ν3 ν4

∂α0/∂Rs |R=Req 2.18 0.00 0.00 0.00

two-state approximation within the CC equations is sufficient to describe to convergence the(0 → 1) transition process, for which the addition of the virtual excitations into the n = 2states is seen to play a negligible role.

Because of the existence of previous calculations, and in order to further test the qualityof our present results, we report in figure 3 a comparison between recent computational datafor the two Raman active modes, the ν1 symmetric stretch (upper panel) and the ν2 twisting(lower panel), the latter shown summed over its multiplicity. The following comments couldbe made after perusing the results assembled in that figure:

(i) For the totally symmetric mode, the use of the correct polarizability dependence onvibrational motion is an important ingredient for producing the inelastic cross sections:the FP results (dot–dashed curve) are in fact smaller than the VP results (solid curve) anddiffer the most for energies near to threshold;

(ii) The simpler adiabatic treatment of the dynamics which is involved in the ANV calculationsturns out to be surprisingly close to the CC results, apart from the threshold region and inthe region of the resonant feature around 7 eV. This closeness is even more marked in thecase of the twisting excitation reported in the lower panel;

(iii) The exact exchange and the off-shell T -matrix approach from our earlier work [6] seemto provide inelastic cross sections which are surprisingly similar to the CC cross sectionsof this work in terms of size and energy dependence, while showing the most markeddifferences in the resonance region, where the impinging electron is becoming mostsimilar to a bound electron, thereby failing the implicit approximation on which ourmodified semiclassical model, SMCE, is based [25, 26].

The results which we present in figure 4 make the same type of comparison as mentionedabove, but refer now to the IR active modes of the molecule, i.e. the antisymmetric stretch (upperpanel) and the scissoring (lower panel) modes, both summed over their multiplicities. Becauseof the existence of a transition dipole moment (TDM), a type of coupling which dominatesthe scattering at collision energies near threshold, we have also carried out two further setsof calculations: one involves the use of the simpler dipole Born approximation [14] whichemploys our computed TDM (triangles), and the other corresponds to using the TDM from theexperiments, as derived from the measured IR absorption intensities [33] (squares). In orderto make our present discussion clearer, we further report in tables 4 and 5 the values of thedipole polarizability derivatives for the four modes and the computed and experimental valuesof the TDM for the IR active modes. The following comments could be made:


Figure 3. Comparison between different computations for the (0 → 1) vibrational excitation ofthe ν1 mode (upper panel) and of the ν2 mode (lower panel). The solid curves show the presentBF-VCC results while the open circles are the present ANV computations. The dashed curves arethe earlier calculations of Althorpe et al [6].

(i) For both vibrational modes the ANV calculations are very close to the CC results andreproduce very well both their size and their energy dependence: the only differenceshows up at threshold, as expected;

(ii) The use of the dipole Born approximation is not a realistic choice for calculating theinelastic cross sections at energies away from each threshold. Furthermore, the size ofeither the experimental or the computed values for the TDMs appears to make some


Figure 4. The same as in figure 3, but this time for the ν3 excitation (upper panel) and the ν4excitation (lower panel). The meaning of the symbols is the same as in figure 3. In this figurewe further show the computed cross sections obtained from the dipole Born approximation, usingeither the experimental (open squares) or the theoretical (open triangles) values of the TDMs andof the normal frequencies.

difference for the ν3 excitation but essentially no difference for the ν4 excitation. This iseasily understood from the results given in table 5, where we see that the TDM valuesare very close to each other for the latter excitation while the experiments for the formerprocess give smaller values than those given by our calculation;

(iii) The comparison with our earlier calculations [6] which used the exact exchangecontribution but an approximate dynamical coupling, shows that the only differences


Table 5. Derivatives of the induced dipole moments for the ν3 and ν4 modes.

Derivative (D (amu)−1/2 Å−1) Presenta Experimentb

∂µ0/∂R3|R3=Req 1.18 0.719∂µ0/∂R4|R4=Req 0.582 0.501

a From the present calculations without scaling.b Derived from experimental IR intensities as given in [33].

between these two sets of results show up in the resonant range of energies, where theexact exchange makes possibly the greatest difference with respect to our SMCE model,as already suggested before.

In order to evaluate more precisely the quality of the present calculations, we furthercompare them with existing experiments, which revealed the combined excitations of the(ν1 + 3ν3) and of (2ν2 + 3ν4) modes. The data presented in figure 5 refer to the (ν1 + 3ν3)excitation cross sections, where the ν3 mode is included with its correct multiplicity. Threedifferent sets of experimental data are shown: those by Tanaka et al [34], by Shyn [35] and byBundschu et al [4]. We further report our earlier calculations of the same excitations [6]. Thebroad maximum around 8 eV is due to the well known p-wave dominated shape resonance ofT2 symmetry [4]. The following comments could be made:

(i) The experimental data are not in very good agreement with each other, with the resultsby Shyn being the largest in the resonance region and Tanaka’s results being smaller inthe same region by as much as 50%. No experimental integral cross section data exist atvery low energies just above threshold and therefore we have no confirmation of the peakshown there by the calculations;

(ii) All our calculated quantities are in fair agreement with each other since both our earliercalculations [6] and the present results exhibit similar size and energy dependence.However, as mentioned before, the differences, when they exist, are largest for around8 eV of collision energy, at the top of the broad shape resonance for this system;

(iii) As expected, the largest inelastic contributions come, over the whole range of examinedenergies, from the ν3 excitation which is, on average, about four times larger than theν1 excitation. Thus, even for electron-impact excitation the presence of a non-vanishingTDM value drives the excitation process more efficiently both near threshold and in theresonance region.

Similar considerations could also be made when we examine the results associated withthe (2ν2 + 3ν4) combined excitation, presented in figure 6. The same sets of experimental dataas those in figure 5 are plotted for comparison. The combined excitation of these two modesis indeed very similar to that for the two previous ones: the energy dependence shows a broadmaximum in the resonance region and a strong peak appears at threshold. Furthermore, theoverall size of the excitation probability is comparable with that shown in figure 5. Our earliercalculations [6] indicate the (2ν2 + 3ν4) combined excitation to be slightly larger in size withrespect to the (ν1 + 3ν3) process while the experiments point to the opposite: our current CCcalculations agree better with the experimental suggestions and follow the envelopes of themeasurements rather well for both sets of excitation cross sections. In the case of the (2ν2 +3ν4)excitation, the IR active ν4 mode clearly dominates near threshold but it is comparable withthe ν2 excitation in the resonance region. It is also reassuring to see that, for this combinedexcitation, the experiments happen to also provide data at lower energy and indeed confirm thetheoretical peak shown by the present calculations. One should further note that the marked


Figure 5. Vibrational excitation cross sections (0 → 1) computed with the BF-VCC method forthe ν1 (dots), the 3ν3 (broken curve) and the weighted sum of the two modes (solid curve). Thepresent calculations are also compared with the experiments by Tanaka et al [34] (crosses witherror bars), Shyn [35] (diamonds with error bars), Bundschu [4] (full circles with error bars) andthe calculations by Althorpe et al [6] (chained curve) for (ν1 + 3ν3) modes.

Figure 6. The same as in figure 5 but for the (2ν2 + 3ν4) modes. The cross sections reported werecomputed with the BF-VCC method for the 2ν2 (dots), the 3ν4 (broken curve) and the sum of thetwo types of modes (solid curve).

peaks presenting themselves near threshold only appear for the IR active modes ν3 and ν4 andare chiefly driven by the low-energy behaviour of the dipole interaction that dominates thescattering, especially near thresholds. The computed values of all four mode excitations arereported in table 6 as a function of collision energy.


Table 6. Computed vibrationally inelastic integral cross sections (Å2) via the two-state BF-VCCcalculations for the transition (0 → 1) of the four normal modes of the CH4 molecule.

Ecoll. (eV) ν1 2ν2 3ν3 3ν4

0.2 — — — 0.3100.22 — 0.023 — 0.3800.25 — 0.036 — 0.4210.3 — 0.045 — 0.4450.4 0.070 0.055 — 0.4260.42 0.102 — 0.167 —0.45 0.115 — 0.230 —0.47 0.117 — 0.245 —0.5 0.115 0.062 0.252 0.3990.6 0.098 0.066 0.255 0.3700.7 0.082 0.069 0.250 0.3451.0 0.050 0.073 0.234 0.2881.5 0.031 0.074 0.205 0.2292.0 0.027 0.078 0.183 0.1943.0 0.041 0.098 0.170 0.1614.0 0.074 0.142 0.200 0.1575.0 0.116 0.202 0.267 0.1716.0 0.149 0.259 0.345 0.1967.0 0.164 0.293 0.406 0.2218.0 0.162 0.300 0.436 0.2409.0 0.150 0.288 0.436 0.250

10.0 0.135 0.267 0.418 0.25311.0 0.121 0.243 0.391 0.25112.0 0.109 0.221 0.360 0.245

4. Summary and conclusions

In this work we have applied the exact treatment of the quantum dynamics for the electron-impact vibrational excitation of a polyatomic molecule to the four normal modes of gaseousCH4. The interaction between the impinging electron and the vibrating target have beenobtained using an ab initio approach that models both the exchange and correlation forceswithin an SCE formulation of the scattering problem. The vibrational, coupled equationshave been solved within a molecular frame and using the integral formulation of the Volterraequations. The results obtained are in good agreement with the experimental findings for thecombined detection of the (ν1 + 3ν3) and the (2ν2 + 3ν4) groups of normal modes: the energydependence of the measured data and their relative size (on an absolute scale) are all givenrather accurately by our calculations. In the case of the ν4 excitation at threshold, our dataseem to provide the cross section structure at threshold rather close to the scanty measurementswhich are available.

The present calculations also indicate the weak nature of the perturbation induced byimpinging electrons on the nuclear motion of the target: the CC expansion over asymptoticvibrational states in fact converges rather quickly and suggests a simple two-state model tobe sufficient to describe the excitation process. Furthermore, our calculations are able toseparate the four individual contributions from the molecular normal modes and indicate theantisymmetric stretch to be the most efficient mode into which the molecule is excited byelectron impact at low collision energies. This effect is seen to hold over the whole rangeof examined collision energies, although at threshold the excitation into the scissoring mode


shows the strongest peak contribution. One of the useful results from the present study is alsogiven by the performance of the model, localized exchange interaction which we have testedin our computations and have compared with our earlier work with exact exchange [6]. Wehave found, in fact, that the approximate form of exchange markedly reduces the computationaleffort and therefore could allow the extension of this approach to larger polyatomic targets: ourrecent work on cyclopropane [36] indeed found a good accord with experiments using the sameexchange potential. In that instance, the experiments were only given, for several vibrationalinelastic process, at one scattering angle and as a function of collision energy: their shape waswell reproduced by our calculations. Finally, the comparison between exact CC results for thevibrational dynamics with the simpler adiabatic treatment of the ANV approximation showshere that the latter scheme is very good over a broad range of collision energies, it describeswell the resonant excitation region and only fails very close to threshold, as should be expected.

In conclusion, we feel that the present work provides additional evidence of the fact that thecomputation of vibrational excitation cross sections by low-energy electron impact in gaseouspolyatomics could be treated fairly realistically by the combined use of approximate localexchange models (SMCE) and of approximate dynamical coupling (ANV), thereby allowingus to extend the calculations to fairly complicated molecules with larger numbers of atomswithout exceeding current computational capabilities. However, one should also note here thatthe above is valid whenever no shape resonances are occurring during the examined vibrationalmode, as is often the case in large polyatomic targets.

Acknowledgments

We are grateful to Dr R Curik for several helpful discussions when this work begun, and toDr T Mukherjee for his early involvement with the computations. The financial supports ofthe Italian National Research Council (CNR), and of the Italian Ministry for University andResearch (MURST) are gratefully acknowledged. One of us (TN) is indebted to the Universityof Rome for the award of a postdoctoral research fellowship, and to the Max-Planck Societyfor extending his stay at the University of Rome while this work was carried out.

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vibrational excitations of ch4 by electron impact: a...

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