The role played by electronic correlation of target on the vibrational excitation cross sections of H2 by electron impact

Journal of Molecular Structure (Theochem) 541 (2001) 51±57

www.elsevier.nl/locate/theochem

The role played by electronic correlation of target on the vibrational excitation cross sections of H2 by electron impact M.-T. Lee a,*, R. Fujiwara b, K.T. Mazon b, M.M. Fujimoto c a

Departamento de QuõÂmica, Universidade Federal de SaÄo Carlos, 13565-905, SaÄo Carlos, SaÄo Paulo, Brazil b Departamento de FõÂsica, Universidade Federal de SaÄo Carlos, 13565-905, SaÄo Carlos, SaÄo Paulo, Brazil c Departamento de FõÂsica, UFPR 70900-910, Curitiba, ParanaÂ, Brazil Received 26 June 2000; accepted 2 September 2000

Abstract We report a systematic study on the role played by electronic correlation of target on vibrational excitation processes of molecules by electron impact. More speci®cally, cross sections for vibrationally elastic and inelastic electron±H2 collisions are reported in the 1.5±40 eV range. In our study, the electron±molecule interaction potential is derived using both the nearHartree±Fock and the con®guration interaction target wavefunctions. The body-frame vibrational close-coupling equations are solved using the method of continued fractions (MCF). Our study has shown that the MCF is a very ef®cient method for solving such scattering equations. In addition, the calculated results have shown that the electronic correlation effects of target signi®cantly in¯uence the calculated vibrational excitation cross sections at low incident energies and for the excitations leading to high-lying vibrational levels. Nevertheless, these effects are not relevant at higher incident energies. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Electronic correlation; Vibrational excitation; Electron impact

1. Introduction During the last two decades, there have been considerable activities on theoretical studies of electron impact vibrational excitations, particularly for diatomic molecules [1±8]. Nevertheless, most of those studies were carried out using uncorrelated Hartree±Fock (HF) target wavefunctions. It is wellknown that HF-level calculations are unable to provide accurate potential energy curves for diatomic molecules, mainly for large internuclear distances.

* Corresponding author. Fax: 155-16-2608350. E-mail address: [email protected] (M.-T. Lee).

One thus expects that the introduction of electronic correlation effects of a target into calculations may signi®cantly in¯uence the calculated cross sections, essentially for the transitions leading to high-lying vibrational levels. In this work, we report a systematic study on the role played by electronic correlation effects of a target on the calculated vibrational excitation cross section of H2. Vibrationally elastic and vibrational excitation …v ˆ 0 ! v 0 ˆ 1; 2; 3† cross sections are calculated using both the near-HF and the con®guration interaction (CI) wavefunctions in the 1.5±40 eV energy range. We expect that the present study might provide insights on the dynamics of electron impact vibrational excitation of molecules.

0166-1280/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(00)00740-5

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M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

2. Theory and calculations Since the aim of the present study focuses on the vibrational motion of molecules, the body-frame vibrational close-coupling (BFVCC) approach [2,4], which exactly incorporates vibrational effects, is applied. In addition, the rotational levels of the H2 molecule are treated as essentially degenerated. Therefore, the laboratory-frame (LF) differential cross sections (DCS) averaged over the molecular orientations, for vibrational excitation from an initial vibrational level v to a ®nal level v 0 , is expressed in the jt basis representation [9] as: ds k X 1 uBjt 0 …v ! v 0 † ˆ f dV k0 j m m 0 …2jt 1 1† mt m t t

t

C~ ˆ S~ 1 G~ 0 U~ C~ ;

t

0 2

0

£…v ! v ; k0 ; kf ; r^ †u ;

(1)

where ~jt ˆ ~l 0 2 ~l is the transferred angular momentum during the collision, m 0 t and mt are the projections of jt along the laboratory and molecular axis, respectively. The k0 and kf are the momenta of the incoming and the outgoing electron, respectively. In Eq. (1), Bjmt t m 0 t …v ! v 0 † are coef®cients of the jt-basis expansion of the LF vibrational excitation scattering amplitudes and are given by X …21† m all 0 mm 0 …ll 0 0mt ujt mt † Bjmt t m 0 …v ! v 0 ; V 0 † ˆ t

l 0 lm 0 m

 …ll 0 mm 0 ujt m 0t †Ylmt …V 0 †;

…2†

where the dynamical coef®cients a ll 0 mm 0 for the transition from an initial vibrational state uvl to a ®nal state uv 0 l are related to the partial-wave components of the vibrational excitation transition matrix elements as 0

0

all 0 mm 0 …v ! v † ˆ 2…1=2†p‰4p…2l 1 1†Š 0

0

0

…1=2† l 0 2l

£ kkf lm; v uTuk0 l m ; vl:

uv 0 l, kv20 the kinetic energy of the scattered electron in Rydberg, and Uv 0 v the vibrational excitation interaction potential operator. In the present work, a set of coupled equations are solved using the method of continued fractions (MCF). The MCF, originally proposed by HoraÂcÏek and Sasakawa [10] for single-channel electron±atom scattering, has been extended by our group to treat actual electron±molecule scattering problems. Recently we have satisfactorily applied the MCF to the calculation of elastic and electronic excitation cross sections for electron scattering by H2 in the low- and intermediate-energy range [11,12]. Eq. (4) can be converted into a Lippmann± Schwinger integral equation, in matrix form

i

…3†

In the present work, the reactance K matrices were calculated by solving the BFVCC scattering equations, viz. X …7 2 1 kv20 †C v 0 ˆ Uv 0 v C v ; …4† v

where C v 0 is the wavefunction of the scattering electron associated with the target vibrational state

…5†

where CÄ is the solution of Eq. (4) in matrix form, SÄ is a diagonal matrix which represents a set of solutions of the (N 1 1)-electron unperturbed SchroÈdinger equation with elements Svv being simply the plane Ä 0 is waves associated with the vibrational state uvl. G also a diagonal matrix which represents the unÄ is the potential perturbed Green's operator and U operator matrix. The application of MCF consists basically of de®ning the nth-order `weakened' potential operator U~ …n† as U~ …n† ˆ U~ …n21† 2 U~ …n21† uS~ …n21† l…A~ …n21† †21 kS~…n21† uU~ …n21† ; …6†

Ä matrix is de®ned and the nth-order correction of D through the relation: D~ …n† ˆ B~ …n† 1 A~ …n† ‰A~ …n† 2 D~ …n11† Š21 A~ …n† :

…7†

Here A~ …n† ˆ kS~ …n† uU~ …n† uS~ …n† l

…8†

B~ …n† ˆ kS~ …n21† uU~ …n21† uS~…n† l;

…9†

where uS~ …n† l ˆ G~ P0 U~ …n21† uS~…n21† l;

…10†

where G~ P denotes the principal-value perturbed Green's operator. The scattering KÄ matrix is related Ä matrix via with the D ~ K~ ˆ 2kD:

…11†

M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

53

Fig. 1. DCS for vibrationally elastic electron±H2 scattering at: (a) 1.5 eV; (b) 3.5 eV; (c) 6 eV; and (d) 10.8 eV incident energies. The solid lines represent the calculated results with CI target wavefunctions; dashed lines the calculated results with HF target wavefunctions; full circles the measured data of Linder and Schmidt [22]; and open circles the measured data of Brunger et al. [23].

It is expected that U~ …n† de®ned in Eq. (6) becomes weaker and weaker with increasing n. As a result, the iterative procedure can be stopped after some steps when the desired convergence is achieved. The converged KÄ matrix would correspond to what can be obtained through the exact solution of the scatterÄ matrix can be ing Eq. (5). In practice, the nth-order D obtained using the Eq. (7) by setting D…n11† ˆ 0: Repeating the procedure of Eq. (7), one obtains backÄ . The nth-iteration KÄ wardly D~ …n21† ; D~ …n22† ¼D~ …1† and D matrix is calculated via the Eq. (11). The transition matrix is given by T~ ˆ 2

2K~ : ~ …1 2 iK†

…12†

The electron±molecules interaction potential for low-energy vibrational excitation is formed by three

main components, viz. ~ ˆ Vst …~r ; R† ~ 1 Vex …~r ; R† ~ 1 Vcp …~r ; R†: ~ Vint …~r ; R†

…13†

~ is the electrostatic The static potential, Vst …~r ; R†; term arising from Coulomb interactions between the projectile and the nuclei and electrons of the target. In this study, the R-dependent Vst was derived exactly from the target wavefunctions. These wavefunctions were calculated for 12 internuclear distances varying from 0.8±3.0 a.u. The calculations were carried out using both the HF self-consistent-®eld (SCF) method as well as the CI method. These wavefunctions were constructed with a 5s/3p uncontracted cartesian Gaussian basis set of Huzinaga [13] augmented by three s- (a ˆ 0.04, 0.015 and 0.005) uncontracted functions. With this basis set, the calculated SCF and CI energies for the ground state H2 at the

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M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

Fig. 2. DCS for vibrational excitation …v ˆ 0 ! v 0 ˆ 1† of H2 by electron impact at: (a) 1.5 eV; (b) 3.5 eV; (c) 6 eV; and (d) 10.8 eV incident energies. The symbols used are the same as in Fig. 1.

equilibrium internuclear distance (1.4006a0) are 21.133022 and 21.171083 a.u., respectively, to be compared with the corresponding values of 21.1336 a.u. [14] and 21.1744 a.u. [15]. The second term in the RHS of Eq. (13) is the exchange contribution arising from the antisymmetrization requirement on the electron±H2 wavefunction. Rigorously accounting for a nonlocal energydependent exchange contribution into vibrational excitation studies is very dif®cult. In the present study, we employ an approximate free-gas local exchange potential based on the familiar Hara's model [16] and modi®ed by Morrison and Collins [17]. The essence of the so-called `tuned freeelectron-gas exchange' (TFEGE) potential is to treat the quantity I(R) in the local momentum as a theoretical parameter. Nevertheless, the determination of I(R) is based entirely on theoretical considerations. For instance, at each internuclear distance and for

each incident energy, Pthey are determined from the adjustment to the 2 g eigenphase sum obtained in exact static-exchange calculations. The Vcp appeared in Eq. (13) is the correlation± polarization potential arising at short range from bound-free many-body effects and at long range from induced polarization effects. Strictly speaking, the Vcp is also nonlocal and depends on the scattering energy. In the present work, this contribution is approximated by a parameter-free local potential as prescribed by Padial and Norcross [18]. The published R-dependent dipole polarizabilities of H2 [19] are used to describe the asymptotic form of the Vcp. In this study, the exchange potential and Vcp are also generated using both the CI and HF R-dependent electronic densities. Finally, in order to calculate the vibrational excitation interaction potential matrix elements Uv 0 v, the vibrational wavefunctions f v(R) were calculated

M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

55

Fig. 3. Electron impact DCS for vibrational transitions: (a) …v ˆ 0 ! v 0 ˆ 0†; (b) …v ˆ 0 ! v 0 ˆ 1†; (c) …v ˆ 0 ! v 0 ˆ 2†; and (d) …v ˆ 0 ! v 0 ˆ 3† in H2 at 4.5 eV incident energy. The symbols used are the same as in Fig. 1 except open squares, which represent the experimental results of Wong and Schultz [24].

using the numerical method of Cooley [20] from the RKR potential curve of the electronic ground state H2 [21]. These wavefunctions were calculated in a 701-point grid, covering the 0:8 # R # 3:0 a:u: range. The R-dependent interaction potentials de®ned in Eq. (13) calculated using both the CI and HF target wavefunctions were interpolated over the same grid and the integral over this internuclear distance grid was evaluated using the Simpson's rule. The number of vibrational states that must be included for solving the BFVCC scattering Eq. (2) depends on the incident energy, especially on whether the scattering is resonant or nonresonant. In the energy range covered herein, it required four vibrational states to converge the reported elastic and excitation cross sections to 5%.

3. Results and discussion Fig. 1(a)±(d) shows the calculated DCS for the vibrational elastic …v ˆ 0 ! v 0 ˆ 0† electron±H2 collisions at 1.5, 3.5, 6 and 10.8 eV incident energies along with some experimental data available in the literature [22,23]. In general, there is a very good agreement between the calculated and measured data in the entire energy range covered herein. In addition, the vibrationally elastic DCS calculated using the CI and HF wavefunctions agree very well with each other indicating that for this target, the in¯uence of the electronic correlation on the elastic electron scattering is not important. In Fig. 2(a)±(d) we present our calculated vibrational excitation …v ˆ 0 ! v 0 ˆ 1† DCS for electron±H2 scattering at the same incident energies

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M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

Fig. 4. ICS for: (a) vibrationally elastic electron±H2 scattering; and (b) vibrational excitation …v ˆ 0 ! v 0 ˆ 1† of H2 by electron impact. The solid lines represent the calculated results with CI target wavefunctions; dashed lines the calculated results with HF target wavefunctions; shortdashed lines the calculated results of Snitchler et al. [29]; full circles the measured data of Linder and Schmidt [22]; open diamonds the measured data of Shyn and Sharp [26]; stars the experimental results of Srivastava et al. [25]; full squares the experimental results of Ehrhardt et al. [27]; and open triangles the experimental results of Gibson [28].

as in Fig. 1. Selected experimental results available in the literature [22,23] are also shown for comparison. In general, there is a good qualitative agreement between the calculated and measured data. The quantitative agreement is also good except for incident energy of 3.5 eV, where our calculated data signi®cantly overestimates the excitation DCS. These discrepancies are probably due to the use of the FEG local exchange potential in our calculation.

Further, it is clearly shown in Fig. 2(a)±(d) that the electronic correlation effects of the target signi®cantly in¯uence the calculated excitation cross sections at 1.5 eV. Nevertheless, it is relatively unimportant for incident energies above 3.5 eV. In Fig. 3(a)±(d) we show the calculated DCS for vibrational transitions …v ˆ 0 ! v 0 ˆ 0; 1; 2 and 3) in H2 by electron impact at 4.5 eV, respectively. As expected, the in¯uence of the electronic correlation

M.-T. Lee et al. / Journal of Molecular Structure (Theochem) 541 (2001) 51±57

effects of target are more important for high-lying transitions than for low-lying transitions. Comparison with experimental results [22,24] has shown very good agreement for the v ˆ 0 ! v 0 ˆ 0 and 1 transitions. For the transitions leading to v 0 ˆ 2 and 3, only qualitative agreement is seen. Since the high-lying vibrational excitations have been very little investigated, more systematic studies, both theoretical and experimental, are clearly of interest. In Fig. 4(a) and (b) we compare our calculated integral cross sections (ICS) for the …v ˆ 0 ! v 0 ˆ 1† and …v ˆ 0 ! v 0 ˆ 1† electron±H2 scattering with some available experimental data [22,25±28]. The calculated results of Snitchler et al. [29] are also shown for comparison. In general, our calculated ICS for the vibrationally elastic scattering are in good agreement with the available experimental data as well as with the calculated results of Snitchler et al. [29]. In addition, the good agreement between our ICS calculated both with HF and CI wavefunctions reinforces that the electronic correlation effects of the target are not relevant for vibrationally elastic electron±H2 scatterings. As expected, our calculated ICS with HF and CI wavefunctions for the …v ˆ 0 ! v 0 ˆ 1† transition show signi®cant differences at low incident energies. For this transition, our CI and ICS agree only qualitatively with the experimental data of Ehrhardt et al. [27]. Quantitatively, our calculation signi®cantly overestimates the cross sections near the region of maximum. The use of FEG local exchange potential is again attributed to these discrepancies.

Acknowledgements This research was supported by Brazilian Agencies: Conselho Nacional de Desenvolvimento Cientõ®co e TecnoloÂgico (CNPq), FundacËaÄo de Amparo aÁ Pesquisa do Estado de SaÄo Paulo (FAPESP), FINEP-PADCT and CAPES-PADCT. R.F. thanks

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