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Computing vibrational effects in low-energy positron scattering from polyatomic molecules: the H2O and C2H2 stretching modes
Nuclear Instruments and Methods in Physics Research B 221 (2004) 24–29 www.elsevier.com/locate/nimb
Computing vibrational effects in low-energy positron scattering from polyatomic molecules: the H2O and C2H2 stretching modes Tamio Nishimura 1, Franco A. Gianturco
*
Department of Chemistry and INFM, The University of Rome ‘‘La Sapienza’’, Citta Universitaria, Piazzale Aldo Moro 5, 00185 Rome, Italy
Abstract We report a theoretical study and the computational results on vibrational excitations by positron impact of the polyatomic molecules CH4 and C2 H2 as examples. The study has been carried out using the body-fixed vibrational close-coupling (BF-VCC) method which is relevant for describing vibrationally inelastic processes in low-energy collisions. The interaction potential we employ to model the forces at play is described in detail and we also report our numerical techniques for solving the scattering equations. The cross-sections are obtained for the vibrational excitations of the symmetric (m1 ) and the antisymmetric (m3 ) stretching modes of the H2 O molecule and for the symmetric stretching (m1 ) mode of C2 H2 , and they are compared with simpler available approximations. 2004 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.90.+q Keywords: Positron scattering from H2 O and C2 H2 ; Vibrational excitation of polyatomic molecules; Molecular physics; Scattering theory
1. Introduction The theoretical cross-sections for the vibrationally inelastic scattering of positrons (eþ ) from water (H2 O) and acetylene (C2 H2 ) molecules are here obtained for the first time and reported in detail below. This is an extension of our work on the vibrational excitation processes that occur in sim-
* Corresponding author. Tel.: +39-06-499-13306; fax: +3906-499-13305. E-mail address: [email protected] (F.A. Gianturco). 1 Permanent address: 2-11-32, Kawagishi-kami Okaya, Nagano 3940048, Japan.
ple polyatomic molecules in the gas phase, induced either by positron [1–3] or by electron impact [4,5]. Since an incident positron is distinguishable from all the bound electrons of the molecular target, there is no effect from electron exchange a feature which needs instead to be considered in the case of electron scattering. Moreover, a positron has the additional possibility of taking up one of the electrons of the target molecule to form a positronium (Ps) atom, when the collision energy is above the threshold for the Ps formation energy (5.8 eV for H2 O and 4.6 eV for C2 H2 ). The H2 O molecule is a polar target that belongs to the C2v point group and possesses three types of vibrational normal modes, i.e. the symmetric
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.026
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
stretching (m1 ), the bending (m2 ), and the antisymmetric stretching (m3 ) modes, all of which are infrared (IR) active. Therefore, the long-range interaction with a charged projectile is expected to be effective, especially because of the molecular permanent dipole moment. The vibrational excitation processes in the present molecules, considered as having only their low vibrational energy levels populated initially, could be treated as occurring separately through its different normal modes of vibration with different fundamental frequencies and symmetries. It is therefore interesting to try to understand how the separate vibrational excitation cross-sections depend on the feature of the specific mode without considering the mixing of the modes. While, on the experimental side, total crosssections for many types of atoms and molecules have been reported (see [6,7] and references quoted therein), to the authors’ knowledge, no measurement on vibrationally inelastic cross-sections for H2 O and C2 H2 molecules by positron collisions has been reported thus far. The last few years, however, have witnessed the development of a novel technique involving a magnetized beam of cold positrons, which allowed one to carry out measurements of absolute vibrationally inelastic cross-sections for CF4 [8], CO, CO2 , H2 [9] and CH4 [10]. These data have been measured at energies as low as 0.5 eV and up to several eV, with a greatly improved FWHM value of 18 meV. In the case of electron scattering off H2 O molecules, because of remarkable developments in experimental techniques, very recently measured electron energy loss spectra with 10 meV of FWHM have revealed that in this molecule the cross-section for the m3 mode is excited much less than the m1 at low scattering energies between 0.05 and 3 eV [11]. In the case of the positron, however, because that higher resolution required is not yet available, the excitations of the m1 and m3 modes cannot be completely resolved by the current quality of the measurements, nor have been attempted even as an unresolved peak. These processes have thus stimulated the interest of theoreticians and spurred the formulation of computational methods which could provide predictions and explanations on how the phenome-
25
non is occurring at the nanoscopic level. In this paper, we therefore report our computed crosssections for the lowest excitations of H2 O molecules in both the m1 and the m3 modes, and of C2 H2 for its symmetric stretching (m1 ) mode. We further discuss their features by comparing our findings with available approximate calculations on the same excitation processes. The details of the present theory are described in the next section while the numerical techniques are presented in Section 3. In Section 4, we report the final cross-sections and discuss their behavior with reference to other available calculations. We give our preliminary conclusions in the last section of the paper. Atomic units (au) are used throughout the present analysis unless otherwise stated.
2. Theory The details of the present theory have been already reported in our previous work [2,3], so we provide here to the reader only a brief reminder of it. In order to obtain vibrational excitation crosssections for positron scattering from polyatomic molecules we need to solve the Schr€ odinger equation of the total system to yield the total wavefunction W at a fixed value of the total energy E. It should also be noted that no Ps formation channel is considered throughout the present calculations as this process is estimated to be fairly negligible at the energies we are considering. We can also assume that the orientation of the target molecule is being kept fixed during the collision since molecular rotations are usually slower when compared with the velocity of the projectile. This is usually called the fixed-nuclear orientation (FNO) approximation [12], and corresponds to ignoring the rotational Hamiltonian of the total system. Then, the total wavefunction could be generally expanded as follows: X Wðrp jRÞ ¼ rp1 ulmn ðrp ÞXlm ð^rp Þvn ðRÞ: ð1Þ lmn
Here, vn is the vibrational wavefunction of the molecule with the vibrational quantum number n ðn1 ; . . . ; nN Þ with N representing the total number of normal vibrational modes of the target
26
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
molecule. The variables R and rp now denote the molecular nuclear geometry and the position vector of the positron from the center-of-mass (c.o.m.) of the target, respectively. The unknown functions ulmn describe the radial coefficients of the wavefunction of the incident particle and the Xlm are the symmetry-adapted angular basis function [13]. In this paper, m in Eq. (1) stands for the indices (plh) collectively, and p stands for the chosen irreducible representation, l distinguishes the component of the basis if its dimension is greater than one, and h does that within the same set of (pl). After substituting Eq. (1) into the equation of the total collision system under the FNO approximation, we obtain for ulmn ðrp Þ a set of full closecoupling equations which now also include vibrational channels. These are also called the body-fixed vibrational close-coupling (BF-VCC) equations [1], ( ) d2 lðl þ 1Þ þ kn2 ulmn ðrp Þ rp2 drp2 X ¼2 hlmnjV jl0 m0 n0 iul0 m0 n0 ðrp Þ; ð2Þ
The interaction potential (V ) between the impinging positron and the molecular target is represented here by us in the form of a local potential. Thus, V is described by the sum of the repulsive electrostatic (Vst ) and the attractive positron correlation-polarization (Vpcp ) terms. For the latter term, we have employed over the years [1] the simple parameter-free model potential, introduced by Boronski and Nieminen [15], in the short-range region of rp , and have connected it smoothly with the asymptotic form of the polarization potential a0 =2r4p with a0 being the spherical dipole polarizability of the target. In addition, one can also employ a simpler computational method which has often been employed to treat vibrational excitation. It is called the adiabatic nuclear vibration (ANV) approximation (see e.g. [16]) and in it, no direct positron dynamical coupling like in Eq. (3) is active during the scattering process. The T -matrix elements are thus obtained as simple expectation values with respect to the fixed-nuclei (FN) T -matrix for each specific normal mode of the target,
l0 m0 n0
where kn2 ¼ 2ðE Envib Þ with Envib being the energy of the specific molecular vibration we are considering. Any of the elements of the interaction matrix in Eq. (2) is given by XZ dRfvn ðRÞg Vl0 m0 ðrp jRÞfvn0 ðRÞg hlmnjV jl0 m0 n0 i ¼ l0 m0
Z
d^rp Xlm ð^rp Þ Xl0 m0 ð^rp ÞX ð^rp Þ: l0 m0
ð3Þ
This method is essentially a generalization of the method proposed long ago (called the ‘hybrid theory’) for the simple case of a diatomic molecule [14]. When solving Eq. (2) under the boundary conditions that the asymptotic form of ulmn l0 m0 n0 is represented by a sum containing the incident plane wave of the projectile and the outgoing spherical wave, we obtain the K-matrix elements. Therefore, the integral cross-section for the vibrationally inelastic scattering is given by p X X lmn 2 Qðn ! n0 Þ ¼ 2 jT 0 0 0 j ; ð4Þ kn lm l0 m0 l m n where Tllmn 0 m0 n0 is the T -matrix element.
Tllmn 0 m0 n 0 ¼
Z
dRfvn0 ðRÞg Tllm0 m0 ðRÞfvn ðRÞg:
ð5Þ
3. Numerical details The H2 O molecule is placed in xz-plane and the z-axis is taken along the C2 symmetry axis with the oxygen atom on the positive side. The target wavefunctions of H2 O and C2 H2 molecules were calculated at the self-consistent field level using the single-center expansion applied to a multicenter Gaussian-type orbitals (GTOs) [17], and the basis set employed was given by the atomic and molecular orbital code GAUSSIAN 98 package [18]. The GTO basis sets we have chosen are those of D95* type which consist of, for the two molecules considered, (9s5p1d)/[4s2p1d] for oxygen, (9s5p)/ [4s2p] for carbon, and (4s)/[2s] for hydrogen. The computed wavefunction of H2 O at the molecular equilibrium geometry (1.809 au of the O–H distance and 104.5 of the molecular angle), yields the dipole moment (lz ) and the total energy as )2.299
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
4. Discussion of results The three panels shown in Fig. 1 report some of the computed quantities that are necessary to carry out the dynamical calculations described in the previous section. We describe there, in fact, the behavior of the dipole moment of H2 O as a function of the coordinate changes for the two stretching modes we are studying. The top panel reports the behavior of the dipole component along the z-axis for the case of the symmetric stretching: the equilibrium computed value ()2.299 Debye) can be compared with the experimental value of )1.85 Debye, showing a difference of about 20%. The lower two panels report the corresponding variations for the two component (along the z- and x-axis) of the molecular dipole
Dipole Moment ( Debye )
uz [ v1 ]
uzeq. = - 1.85 D ( Exp. )
-2 -2.2 -2.4
uz [ v3]
-2.28
-2.29
-2.3
0.5 uxeq. = 0.00 D ( Exp. / Cal.)
ux [ v ] 3
Debye ()1.85) and )76.0349 au, respectively: the number in brackets corresponds to the dipole experimental value. In the case of C2 H2 , the target wavefunction employed at the equilibrium geometry (C–H and C–C bond lengths are respectively 2.003 and 2.268 au) gives the total energy of )76.7992 au. The terms of the multipolar expansion for the interaction potential in Eq. (3) were retained up to l0 ¼ 24, and the scattered wavefunction of the positron in Eq. (1) was expanded up to l ¼ 12 for both the m1 and the m3 modes, and l0 ¼ 36 and l ¼ 18, in the FN approximation, for the m1 mode of C2 H2 . When solving Eq. (2) we have included the lowest two vibrational states of n ¼ 0 and 1, assuming the vibrational wavefunctions to be that of the harmonic oscillator. The above parameters were found to describe to good numerical convergence the molecular anisotropy of the interaction. In order to solve the close-coupling equations by means of standard Green’s function techniques, Eq. (2) is rewritten as an integral equation, called a Volterra equation (for details, see [19–21]). For the polarizability of a0 which appears in the asymptotic region of the Vpcp term, we first obtain the values of a0 from the D95* type wavefunctions employed, and normalize our calculations to the experimental value of a0 (9.92 au for H2 O and 28.68 au for C2 H2 ) at the equilibrium geometry.
27
0
-0.5 0.7
0.8
0.9 1.1 1 R : O - H ( angstrom)
1.2
Fig. 1. Calculated dipole moments for the m1 mode (top panel) and for the m3 (middle and bottom panels) of H2 O molecule as a function of the bond length of O–H with the molecular angle at equilibrium geometry of 104.5.
moments as a function of the antisymmetric stretching process: we see that our calculation reproduce correctly the dipole value (x-component) at the equilibrium geometry. The calculation reported in Fig. 2 show the actual computed values of the (0 fi 1) vibrationally inelastic cross-sections from their energy threshold and up to about 4 eV of collision energy. The following consideration could be made. 1. Both excitation cross-sections show very strong peaks of their cross-sections just above their respective threshold openings: the dominance of the charge–dipole coupling potential plus the polarizability effect obviously enhances the positron projectile excitation efficiency, a feature we had already detected for the IR active modes of CH4 molecule near their threshold openings [22].
28
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29 C 2H 2 +e+ v1 mode
+
Vib. CC ( n = 0 and 1 ) [v1 + v3 ] Vib. CC [v1 ] Vib. CC [v3 ] ANV ( n = 0 , 1 ) [v1 + v3 ] Dip. Born ( Cal. ) [v1 + v3 ]
0.6
0.4
0.2
0
1
2 Total Energy ( eV)
3
4
0
Fig. 2. Vibrational excitation cross-sections via the 2-state BFVCC method for the m1 (chain curve), the m3 (dotted curve), and the sum of the two modes (the solid curve) of H2 O. Comparisons are made with the calculations obtained by the ANV method (open circles) [16] and the dipole Born approximation (squares) [12]. See main text for details.
2. Both excitation processes do not show any effect from closed channels since the inelastic cross-sections given within our essentially twostate picture are seen to have converged to their final values and to change very little when additional vibrational channels are included. This result also suggests that the system does not seem to have marked Feshbach-type resonances below the threshold openings. 3. The use of the more simplified computational scheme of Eq. (5), shown in the figure by the open circles, also indicates that an adiabatic physical picture of the inelastic process is giving the same results as those from the more rigorous BF-VCC calculations down to collision energies to threshold openings, where the cusp given by the latter method is only in part reproduced by ANV calculations. 4. The simpler Born calculations [12] also indicate the importance of the more sophisticated interaction potentials we have used in our work: their calculated (m1 þ m3 ) excitation cross-sections, open squares, are smaller than the BFVCC and the ANV calculations at all the energies considered here and fail quite clearly to yield the threshold peak behaviors seen in the previous calculations.
Vibrational ( n: 0 -> 1 ) -16 2 Cross Section ( 10 cm )
Vibrational ( n: 0 -> 1 ) -16 2 Cross Section ( 10 cm )
H2O + e
0.08
0.06 ANV ( n = 0 and 1) E th ( n = 1 ) = 0.447 eV 0.04
0.02
0
1
2 3 Total Energy (eV)
4
0
Fig. 3. Vibrational excitation cross-sections via the ANV method (open circles) for the m1 mode of C2 H2 .
Since the ANV calculations turn out to be very good for realistically describing vibrationally inelastic processes, we have extended the (0 fi 1) excitation cross-section in the case of C2 H2 molecule, for which only ANV calculations for its stretching (symmetric) mode have been considered. The results of such calculations are given in Fig. 3. The first consideration one can make is about the size of the computed inelasticity: for a nonpolar molecule like C2 H2 they come out to be about one order of magnitude smaller than in the case of the polar target we have shown in Fig. 2. Furthermore, we also see that the cross-section behavior still maintains the threshold peak structure above the excited state energy opening that was shown by the H2 O target: this means that the excitation process is dominated by the polarizability behavior at energies very close to threshold and therefore tends to strongly increase for decreasing collision energies. The two-state picture is also valid in this case, since the inclusion of closed vibrational channels did not cause any relevant change in the size of the inelastic cross-sections computed in the present study.
5. Present conclusions In this work we have presented quantum calculations for the vibrationally inelastic cross-sections of a polar and a nonpolar polyatomic
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
targets, i.e. for H2 O and C2 H2 , obtained by positron impact at low collision energies. We have excluded in our treatment both the Ps formation channel and any other target excitation channel on the ground that, at the considered impact energies, those contributions are rather small [1]. The calculations included correctly the dynamical couplings between the impinging projectile and the stretching vibrations of the two kinds of molecules, limiting the present study to the symmetric and antisymmetric modes of water (m1 þ m3 ) and to the symmetric stretching mode of acetylene (m1 mode). The interaction forces were obtained from a nonempirical description of the forces at play and the quantum coupled equations were solved within the molecular frame formation of the inelastic process (BF-VCC equations). The present results clearly indicate the strong influence of structural factors on the excitation dynamics since the size of the inelastic cross-sections for the polar targets are more than a factor of 10 larger than those for the nonpolar molecule. Furthermore, the rather weak coupling between the positron and the vibrating molecules is underlined by three clear effects seen in the calculations: (i) the smaller size of the excitation crosssections when compared with similar results for electron impact excitation in the low energy region [23], (ii) the fact that the close-coupling expansion over target vibrators is essentially given to convergence by a two-state expansion and (iii) that the adiabatic simplification provided by ANV model (see previous Section 3) produces excitation probabilities which are practically the same as the BF-VCC results unless the collision energies get very close to the excitation thresholds. A more extended analysis of the vibrational excitation for all three modes of the H2 O molecule will be presented elsewhere [23].
Acknowledgements The financial supports from the Rome University Research Committee, from the European
29
Union Vth Framework collaborative project EPIC, and from the CASPUR Computing Consortium are gratefully acknowledged. One of us (T.N.) also thanks the CASPUR Consortium for the awarding of a research fellowship.
References [1] F.A. Gianturco, T. Mukherjee, T. Nishimura, A. Occhigrossi, in: C.M. Surko, F.A. Gianturco (Eds.), New Directions in Antimatter Chemistry and Physics, Kluwer Academic Publishers, Chapter 24, 2001, p. 451. [2] T. Nishimura, F.A. Gianturco, Nucl. Instr. and Meth. B 192 (2002) 17. [3] T. Nishimura, F.A. Gianturco, Phys. Rev. A 65 (2002) 062703. [4] S.C. Althorpe, F.A. Gianturco, N. Sanna, J. Phys. B 28 (1995) 4165. [5] T. Nishimura, F.A. Gianturco, J. Phys. B 35 (2002) 2873. [6] M. Kimura, O. Sueoka, A. Hamada, Y. Itikawa, Adv. Chem. Phys. 111 (2000) 537. [7] M. Charlton, J.W. Humberston, in: A. Dalgarno, P.L. Knight, F.H. Read, R.N. Zare (Eds.), Positron Physics, Cambridge University Press, Chapter 2, 2001 p. 40. [8] S.J. Gilbert, R.G. Greaves, C.M. Surko, Phys. Rev. Lett. 82 (1999) 5032. [9] J.P. Sullivan, S.J. Gilbert, C.M. Surko, Phys. Rev. Lett. 86 (2001) 1494. [10] J.P. Sullivan, S.J. Gilbert, J.P. Marler, L.D. Barnes, S.J. Buckman, C.M. Surko, Nucl. Instr. and Meth. B 192 (2002) 3. [11] M. Allan, O. Moreira, J. Phys. B 35 (2002) L37. [12] Y. Itikawa, Int. Rev. Phys. Chem. 16 (1997) 155. [13] P.G. Burke, N. Chandra, F.A. Gianturco, J. Phys. B 5 (1972) 2212. [14] N. Chandra, A. Temkin, Phys. Rev. A 13 (1976) 188. [15] E. Boronski, R.M. Nieminen, Phys. Rev. B 34 (1986) 3820. [16] M. Cascella, R. Curik, F.A. Gianturco, J. Phys. B 34 (2001) 705. [17] F.A. Gianturco, A. Jain, Phys. Rep. 143 (1986) 347. [18] M.J. Frish et al., Gaussian 98, Revision A.7, Gaussian Inc., Pittsburg, PA, 1998. [19] W.N. Sams, D.J. Kouri, J. Chem. Phys. 51 (1969) 4809. [20] T.N. Rescigno, A.E. Orel, Phys. Rev. A 24 (1981) 1267. [21] T.N. Rescigno, A.E. Orel, Phys. Rev. A 25 (1982) 2402. [22] T. Nishimura, F.A. Gianturco, Europhys. Lett. 59 (2002) 674. [23] T. Nishimura, F.A. Gianturco, J. Phys. B, submitted for publication.
Computing vibrational effects in low-energy positron scattering from polyatomic molecules: the H2O and C2H2 stretching modes Tamio Nishimura 1, Franco A. Gianturco
*
Department of Chemistry and INFM, The University of Rome ‘‘La Sapienza’’, Citta Universitaria, Piazzale Aldo Moro 5, 00185 Rome, Italy
Abstract We report a theoretical study and the computational results on vibrational excitations by positron impact of the polyatomic molecules CH4 and C2 H2 as examples. The study has been carried out using the body-fixed vibrational close-coupling (BF-VCC) method which is relevant for describing vibrationally inelastic processes in low-energy collisions. The interaction potential we employ to model the forces at play is described in detail and we also report our numerical techniques for solving the scattering equations. The cross-sections are obtained for the vibrational excitations of the symmetric (m1 ) and the antisymmetric (m3 ) stretching modes of the H2 O molecule and for the symmetric stretching (m1 ) mode of C2 H2 , and they are compared with simpler available approximations. 2004 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.90.+q Keywords: Positron scattering from H2 O and C2 H2 ; Vibrational excitation of polyatomic molecules; Molecular physics; Scattering theory
1. Introduction The theoretical cross-sections for the vibrationally inelastic scattering of positrons (eþ ) from water (H2 O) and acetylene (C2 H2 ) molecules are here obtained for the first time and reported in detail below. This is an extension of our work on the vibrational excitation processes that occur in sim-
* Corresponding author. Tel.: +39-06-499-13306; fax: +3906-499-13305. E-mail address: [email protected] (F.A. Gianturco). 1 Permanent address: 2-11-32, Kawagishi-kami Okaya, Nagano 3940048, Japan.
ple polyatomic molecules in the gas phase, induced either by positron [1–3] or by electron impact [4,5]. Since an incident positron is distinguishable from all the bound electrons of the molecular target, there is no effect from electron exchange a feature which needs instead to be considered in the case of electron scattering. Moreover, a positron has the additional possibility of taking up one of the electrons of the target molecule to form a positronium (Ps) atom, when the collision energy is above the threshold for the Ps formation energy (5.8 eV for H2 O and 4.6 eV for C2 H2 ). The H2 O molecule is a polar target that belongs to the C2v point group and possesses three types of vibrational normal modes, i.e. the symmetric
0168-583X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2004.03.026
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
stretching (m1 ), the bending (m2 ), and the antisymmetric stretching (m3 ) modes, all of which are infrared (IR) active. Therefore, the long-range interaction with a charged projectile is expected to be effective, especially because of the molecular permanent dipole moment. The vibrational excitation processes in the present molecules, considered as having only their low vibrational energy levels populated initially, could be treated as occurring separately through its different normal modes of vibration with different fundamental frequencies and symmetries. It is therefore interesting to try to understand how the separate vibrational excitation cross-sections depend on the feature of the specific mode without considering the mixing of the modes. While, on the experimental side, total crosssections for many types of atoms and molecules have been reported (see [6,7] and references quoted therein), to the authors’ knowledge, no measurement on vibrationally inelastic cross-sections for H2 O and C2 H2 molecules by positron collisions has been reported thus far. The last few years, however, have witnessed the development of a novel technique involving a magnetized beam of cold positrons, which allowed one to carry out measurements of absolute vibrationally inelastic cross-sections for CF4 [8], CO, CO2 , H2 [9] and CH4 [10]. These data have been measured at energies as low as 0.5 eV and up to several eV, with a greatly improved FWHM value of 18 meV. In the case of electron scattering off H2 O molecules, because of remarkable developments in experimental techniques, very recently measured electron energy loss spectra with 10 meV of FWHM have revealed that in this molecule the cross-section for the m3 mode is excited much less than the m1 at low scattering energies between 0.05 and 3 eV [11]. In the case of the positron, however, because that higher resolution required is not yet available, the excitations of the m1 and m3 modes cannot be completely resolved by the current quality of the measurements, nor have been attempted even as an unresolved peak. These processes have thus stimulated the interest of theoreticians and spurred the formulation of computational methods which could provide predictions and explanations on how the phenome-
25
non is occurring at the nanoscopic level. In this paper, we therefore report our computed crosssections for the lowest excitations of H2 O molecules in both the m1 and the m3 modes, and of C2 H2 for its symmetric stretching (m1 ) mode. We further discuss their features by comparing our findings with available approximate calculations on the same excitation processes. The details of the present theory are described in the next section while the numerical techniques are presented in Section 3. In Section 4, we report the final cross-sections and discuss their behavior with reference to other available calculations. We give our preliminary conclusions in the last section of the paper. Atomic units (au) are used throughout the present analysis unless otherwise stated.
2. Theory The details of the present theory have been already reported in our previous work [2,3], so we provide here to the reader only a brief reminder of it. In order to obtain vibrational excitation crosssections for positron scattering from polyatomic molecules we need to solve the Schr€ odinger equation of the total system to yield the total wavefunction W at a fixed value of the total energy E. It should also be noted that no Ps formation channel is considered throughout the present calculations as this process is estimated to be fairly negligible at the energies we are considering. We can also assume that the orientation of the target molecule is being kept fixed during the collision since molecular rotations are usually slower when compared with the velocity of the projectile. This is usually called the fixed-nuclear orientation (FNO) approximation [12], and corresponds to ignoring the rotational Hamiltonian of the total system. Then, the total wavefunction could be generally expanded as follows: X Wðrp jRÞ ¼ rp1 ulmn ðrp ÞXlm ð^rp Þvn ðRÞ: ð1Þ lmn
Here, vn is the vibrational wavefunction of the molecule with the vibrational quantum number n ðn1 ; . . . ; nN Þ with N representing the total number of normal vibrational modes of the target
26
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
molecule. The variables R and rp now denote the molecular nuclear geometry and the position vector of the positron from the center-of-mass (c.o.m.) of the target, respectively. The unknown functions ulmn describe the radial coefficients of the wavefunction of the incident particle and the Xlm are the symmetry-adapted angular basis function [13]. In this paper, m in Eq. (1) stands for the indices (plh) collectively, and p stands for the chosen irreducible representation, l distinguishes the component of the basis if its dimension is greater than one, and h does that within the same set of (pl). After substituting Eq. (1) into the equation of the total collision system under the FNO approximation, we obtain for ulmn ðrp Þ a set of full closecoupling equations which now also include vibrational channels. These are also called the body-fixed vibrational close-coupling (BF-VCC) equations [1], ( ) d2 lðl þ 1Þ þ kn2 ulmn ðrp Þ rp2 drp2 X ¼2 hlmnjV jl0 m0 n0 iul0 m0 n0 ðrp Þ; ð2Þ
The interaction potential (V ) between the impinging positron and the molecular target is represented here by us in the form of a local potential. Thus, V is described by the sum of the repulsive electrostatic (Vst ) and the attractive positron correlation-polarization (Vpcp ) terms. For the latter term, we have employed over the years [1] the simple parameter-free model potential, introduced by Boronski and Nieminen [15], in the short-range region of rp , and have connected it smoothly with the asymptotic form of the polarization potential a0 =2r4p with a0 being the spherical dipole polarizability of the target. In addition, one can also employ a simpler computational method which has often been employed to treat vibrational excitation. It is called the adiabatic nuclear vibration (ANV) approximation (see e.g. [16]) and in it, no direct positron dynamical coupling like in Eq. (3) is active during the scattering process. The T -matrix elements are thus obtained as simple expectation values with respect to the fixed-nuclei (FN) T -matrix for each specific normal mode of the target,
l0 m0 n0
where kn2 ¼ 2ðE Envib Þ with Envib being the energy of the specific molecular vibration we are considering. Any of the elements of the interaction matrix in Eq. (2) is given by XZ dRfvn ðRÞg Vl0 m0 ðrp jRÞfvn0 ðRÞg hlmnjV jl0 m0 n0 i ¼ l0 m0
Z
d^rp Xlm ð^rp Þ Xl0 m0 ð^rp ÞX ð^rp Þ: l0 m0
ð3Þ
This method is essentially a generalization of the method proposed long ago (called the ‘hybrid theory’) for the simple case of a diatomic molecule [14]. When solving Eq. (2) under the boundary conditions that the asymptotic form of ulmn l0 m0 n0 is represented by a sum containing the incident plane wave of the projectile and the outgoing spherical wave, we obtain the K-matrix elements. Therefore, the integral cross-section for the vibrationally inelastic scattering is given by p X X lmn 2 Qðn ! n0 Þ ¼ 2 jT 0 0 0 j ; ð4Þ kn lm l0 m0 l m n where Tllmn 0 m0 n0 is the T -matrix element.
Tllmn 0 m0 n 0 ¼
Z
dRfvn0 ðRÞg Tllm0 m0 ðRÞfvn ðRÞg:
ð5Þ
3. Numerical details The H2 O molecule is placed in xz-plane and the z-axis is taken along the C2 symmetry axis with the oxygen atom on the positive side. The target wavefunctions of H2 O and C2 H2 molecules were calculated at the self-consistent field level using the single-center expansion applied to a multicenter Gaussian-type orbitals (GTOs) [17], and the basis set employed was given by the atomic and molecular orbital code GAUSSIAN 98 package [18]. The GTO basis sets we have chosen are those of D95* type which consist of, for the two molecules considered, (9s5p1d)/[4s2p1d] for oxygen, (9s5p)/ [4s2p] for carbon, and (4s)/[2s] for hydrogen. The computed wavefunction of H2 O at the molecular equilibrium geometry (1.809 au of the O–H distance and 104.5 of the molecular angle), yields the dipole moment (lz ) and the total energy as )2.299
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
4. Discussion of results The three panels shown in Fig. 1 report some of the computed quantities that are necessary to carry out the dynamical calculations described in the previous section. We describe there, in fact, the behavior of the dipole moment of H2 O as a function of the coordinate changes for the two stretching modes we are studying. The top panel reports the behavior of the dipole component along the z-axis for the case of the symmetric stretching: the equilibrium computed value ()2.299 Debye) can be compared with the experimental value of )1.85 Debye, showing a difference of about 20%. The lower two panels report the corresponding variations for the two component (along the z- and x-axis) of the molecular dipole
Dipole Moment ( Debye )
uz [ v1 ]
uzeq. = - 1.85 D ( Exp. )
-2 -2.2 -2.4
uz [ v3]
-2.28
-2.29
-2.3
0.5 uxeq. = 0.00 D ( Exp. / Cal.)
ux [ v ] 3
Debye ()1.85) and )76.0349 au, respectively: the number in brackets corresponds to the dipole experimental value. In the case of C2 H2 , the target wavefunction employed at the equilibrium geometry (C–H and C–C bond lengths are respectively 2.003 and 2.268 au) gives the total energy of )76.7992 au. The terms of the multipolar expansion for the interaction potential in Eq. (3) were retained up to l0 ¼ 24, and the scattered wavefunction of the positron in Eq. (1) was expanded up to l ¼ 12 for both the m1 and the m3 modes, and l0 ¼ 36 and l ¼ 18, in the FN approximation, for the m1 mode of C2 H2 . When solving Eq. (2) we have included the lowest two vibrational states of n ¼ 0 and 1, assuming the vibrational wavefunctions to be that of the harmonic oscillator. The above parameters were found to describe to good numerical convergence the molecular anisotropy of the interaction. In order to solve the close-coupling equations by means of standard Green’s function techniques, Eq. (2) is rewritten as an integral equation, called a Volterra equation (for details, see [19–21]). For the polarizability of a0 which appears in the asymptotic region of the Vpcp term, we first obtain the values of a0 from the D95* type wavefunctions employed, and normalize our calculations to the experimental value of a0 (9.92 au for H2 O and 28.68 au for C2 H2 ) at the equilibrium geometry.
27
0
-0.5 0.7
0.8
0.9 1.1 1 R : O - H ( angstrom)
1.2
Fig. 1. Calculated dipole moments for the m1 mode (top panel) and for the m3 (middle and bottom panels) of H2 O molecule as a function of the bond length of O–H with the molecular angle at equilibrium geometry of 104.5.
moments as a function of the antisymmetric stretching process: we see that our calculation reproduce correctly the dipole value (x-component) at the equilibrium geometry. The calculation reported in Fig. 2 show the actual computed values of the (0 fi 1) vibrationally inelastic cross-sections from their energy threshold and up to about 4 eV of collision energy. The following consideration could be made. 1. Both excitation cross-sections show very strong peaks of their cross-sections just above their respective threshold openings: the dominance of the charge–dipole coupling potential plus the polarizability effect obviously enhances the positron projectile excitation efficiency, a feature we had already detected for the IR active modes of CH4 molecule near their threshold openings [22].
28
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29 C 2H 2 +e+ v1 mode
+
Vib. CC ( n = 0 and 1 ) [v1 + v3 ] Vib. CC [v1 ] Vib. CC [v3 ] ANV ( n = 0 , 1 ) [v1 + v3 ] Dip. Born ( Cal. ) [v1 + v3 ]
0.6
0.4
0.2
0
1
2 Total Energy ( eV)
3
4
0
Fig. 2. Vibrational excitation cross-sections via the 2-state BFVCC method for the m1 (chain curve), the m3 (dotted curve), and the sum of the two modes (the solid curve) of H2 O. Comparisons are made with the calculations obtained by the ANV method (open circles) [16] and the dipole Born approximation (squares) [12]. See main text for details.
2. Both excitation processes do not show any effect from closed channels since the inelastic cross-sections given within our essentially twostate picture are seen to have converged to their final values and to change very little when additional vibrational channels are included. This result also suggests that the system does not seem to have marked Feshbach-type resonances below the threshold openings. 3. The use of the more simplified computational scheme of Eq. (5), shown in the figure by the open circles, also indicates that an adiabatic physical picture of the inelastic process is giving the same results as those from the more rigorous BF-VCC calculations down to collision energies to threshold openings, where the cusp given by the latter method is only in part reproduced by ANV calculations. 4. The simpler Born calculations [12] also indicate the importance of the more sophisticated interaction potentials we have used in our work: their calculated (m1 þ m3 ) excitation cross-sections, open squares, are smaller than the BFVCC and the ANV calculations at all the energies considered here and fail quite clearly to yield the threshold peak behaviors seen in the previous calculations.
Vibrational ( n: 0 -> 1 ) -16 2 Cross Section ( 10 cm )
Vibrational ( n: 0 -> 1 ) -16 2 Cross Section ( 10 cm )
H2O + e
0.08
0.06 ANV ( n = 0 and 1) E th ( n = 1 ) = 0.447 eV 0.04
0.02
0
1
2 3 Total Energy (eV)
4
0
Fig. 3. Vibrational excitation cross-sections via the ANV method (open circles) for the m1 mode of C2 H2 .
Since the ANV calculations turn out to be very good for realistically describing vibrationally inelastic processes, we have extended the (0 fi 1) excitation cross-section in the case of C2 H2 molecule, for which only ANV calculations for its stretching (symmetric) mode have been considered. The results of such calculations are given in Fig. 3. The first consideration one can make is about the size of the computed inelasticity: for a nonpolar molecule like C2 H2 they come out to be about one order of magnitude smaller than in the case of the polar target we have shown in Fig. 2. Furthermore, we also see that the cross-section behavior still maintains the threshold peak structure above the excited state energy opening that was shown by the H2 O target: this means that the excitation process is dominated by the polarizability behavior at energies very close to threshold and therefore tends to strongly increase for decreasing collision energies. The two-state picture is also valid in this case, since the inclusion of closed vibrational channels did not cause any relevant change in the size of the inelastic cross-sections computed in the present study.
5. Present conclusions In this work we have presented quantum calculations for the vibrationally inelastic cross-sections of a polar and a nonpolar polyatomic
T. Nishimura, F.A. Gianturco / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 24–29
targets, i.e. for H2 O and C2 H2 , obtained by positron impact at low collision energies. We have excluded in our treatment both the Ps formation channel and any other target excitation channel on the ground that, at the considered impact energies, those contributions are rather small [1]. The calculations included correctly the dynamical couplings between the impinging projectile and the stretching vibrations of the two kinds of molecules, limiting the present study to the symmetric and antisymmetric modes of water (m1 þ m3 ) and to the symmetric stretching mode of acetylene (m1 mode). The interaction forces were obtained from a nonempirical description of the forces at play and the quantum coupled equations were solved within the molecular frame formation of the inelastic process (BF-VCC equations). The present results clearly indicate the strong influence of structural factors on the excitation dynamics since the size of the inelastic cross-sections for the polar targets are more than a factor of 10 larger than those for the nonpolar molecule. Furthermore, the rather weak coupling between the positron and the vibrating molecules is underlined by three clear effects seen in the calculations: (i) the smaller size of the excitation crosssections when compared with similar results for electron impact excitation in the low energy region [23], (ii) the fact that the close-coupling expansion over target vibrators is essentially given to convergence by a two-state expansion and (iii) that the adiabatic simplification provided by ANV model (see previous Section 3) produces excitation probabilities which are practically the same as the BF-VCC results unless the collision energies get very close to the excitation thresholds. A more extended analysis of the vibrational excitation for all three modes of the H2 O molecule will be presented elsewhere [23].
Acknowledgements The financial supports from the Rome University Research Committee, from the European
29
Union Vth Framework collaborative project EPIC, and from the CASPUR Computing Consortium are gratefully acknowledged. One of us (T.N.) also thanks the CASPUR Consortium for the awarding of a research fellowship.
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