Joshipura, K N

Limbachiya, Chetan

Thakar, Yogesh

Vinodkumar, Minaxi

IACS Institutional Repository

Indian J. Phys. 73B (2), 245-252 (1999)I J P  B— an in ternationa l journalAdditivity approximations for molecules and radicalsK N Joshipura, Minaxi Vinodkumar1, Yogesh Thakar2 and Chelan Limbachiya1D epartm ent oi Physics, Sardar Patel U niversity, Vallabh V i d y a n a g a r - 3 8 8  120, India ' V P & R P T P Science College, Vallabh V idyanagar-388 120, India 2 M N College, V isnagai-384 315, N orth G uiarat, India ’ P S Science C ollege, Kadi-3X4 715, North G uiarat, IndiaA b s tr a c t  Various add itiv ity  app ro x im atio n s used in lite ra tu re  to ca lcu la te  e lectronim pact total cross sections for m olecules and ladicals are highlighted The sim ple additivity rule for the e -  m o le c u le  TCS is m odified and theoretical com parison is m ade w ith othci such a ttem pts The present m odified  add itiv ity  rule alongw ith  single cen tre  expansion  o f charge density appears to be the best by far foi several m olecules exam ined Specific results on the TCS of CH 4, S ill4, T7,, H ,S and C.H, have been presented vis-a-vis other experim ental and theoretical data at energies above 50 eVK e y w o rd s  A dditiv ity  rules, optieal po ten tia l, e lec tron  sca tteringP A C S  N o . 3 4 .8 0  -i1. IntroductionIn this paper we highlight various addilivity approximations frequently used for molecules and radicals, and use them to present our specific results on electron scattering cross sections, with selected molecules. When free atoms combine to form a molecule or a neutral radical, the number of atomic electrons adds up exactly, while the other static and dynamic properties do not add up simply. One can however make an effective rule of addition of atomic properties to obtain the corresponding molecular properties. Miller 11 ] has discussed simple prescription to obtain molecular polarisability as a sum of the polarisabilities of constituent atoms or group of atoms. Bader [2] prescribed a simple addition of atomic charge-densities to obtain the molecular charge-density. Approximate additivity rules are also used to represent various cross sections of electrons scattered by diatomic and polyatomic molecules. Deutch etal [3] have followed this approach for total cross sections of electron impact ionisation. We [4,5,6] started with coherent superposition of atomic scattering amplitudes to derive for the total (complete) cross sections for e~-  molecule collisions. This so called additivity rule (AR) has also been used by Jiang etal [7] and by others, with a limited success. The AR is a simple high energy approximation and has been modified by us [5,6] and by Jiang et al [8 J under different considerations.©1999IACS246 K N Joshipum ct alThe additivity rules are simple and attractive but they should be made more realistic otherwise they tend to overestimate the cross sections. In view of the basic approximations we have considered here the impact energies £  > 50 e V , where a large number of inelastic channels are open. It is therefore reasonable to define total elastic cross section Qel and total inelastic cross section Q {so that the sum of the two is the total (complete) cross section Qr at a given energy. Further at high energies, the rotational and vibrational excitation channels arc weaker and have not been considered here, (see also refs. |5,6|). Thus in this paper we have presented our total (complete) cross sections <27at 50-2(X)0 eV for the targets CH4, SiH4, F,, H-,S and These arc specifically chosen to test our improved additivity methods outlined in Section 2 below. For methane molecule a vast amount of comparative data is available [9a, 9b], while for other targets like the data available 110] is limited. Wc could not find comparable results for e -  £, scattering, but the 18-clectron targets have been mutually compared here against the background of the isoelectromc Ar atom. The modifications over AR result into belter accords with measured data.2. Theory and calculationsAs discussed in [4-6|, wc consider high-energy short wave-length collisions of an electron with a molecule or a neutral radical, for which the scattering amplitude approximates as the sum of that of its constituent atoms. Use of optical theorem then yields the following simple expression for the total (complete) e~ - molecule cross section Q1 (M).QT( M ) ^ Q Kr ( A ) .l \^( I )In this additivity rule (AR),/i is the number of atoms in the molecule and Q^n (A) is thetotal (complete) cross section fon-th atom. Eq. (1) is loo simple to be real. We modified this AR by separating the long-range polarisation potential considered at the direct molecular level. The resulting modified additivity rule (MAR), vide |4-6| reads as follows.QT( M ) ^ Q sli(i) + Qrj M l  (2)where the first term on RHS is a sum of the atomic TCS QSR calculated in short range complex potential and the second term Qf)nl is obtained through the molecular polarisation potential,viz.VAr,k)=-a (yrl  *42(r +r ) (3)Here, k is the incident momentum in a.u., and a{) and A are the molecular average spherical polarisability and the effective excitation energy respectively. The MAR, eq. (2) is better than the AR, as it employs molecular properties and A (or I, the ionisation energy). Tocalculate QSR for each atom in the target molecule wc construct a short range complex potentialAdditivity approximations for molecules and radicals 247which includes static, exchange and absorption model potentials. Well known and reliable models, vide [4-6,7,81, derived from an accurate atomic charge-density p(r) arc employed in the optical potential calculations [ 11) to obtain the total elastic cross section Qd (E ) and the total inelastic cross section Qmd (E). Thus, we have for each constituent atome ^ ( £ )  = G ,,(£ )+ (2 mW(£,). (4)The cross section Qmel includes electron impact total atomic Ionization cross section Qhm (Ef) along with all other electronic excitation cross sections. The Qmvl approaches QJtm at high enough energies, so that from cqs. (2) and (4) one can write an additivity rule for electron impact ionisation, vide {3,61 asnQ n J W ^ Q ^ J a , ) ,  (5)a-1where a  labels the constituent atoms.Thus the molecular total ionisation cross section Q/on(M) is approximated as a sum of atomic total Ionisation cross sections QJon(a). This approximation is reasonably good only at high enough energies. Deuteh etal [31 have introduced weighing factorsf i a ) in the following manner in order to make the above additivity rule (eq. 5) realistic :n<«>• (6)a=IThis modification succeeds in bringing a closer agreement of calculated QJtm (M) to experimental data in many cases [3]. However, the factors/(a) are static, i.e. they depend on structural properties while the cross section Qhm (a) are dynamic or energy dependent quantities. Hence a modification like cq. (6) is difficult to justify theoretically. Jiang etal [8] have made a similar attempt to express the molecular TCwS in the following additivity approximation :Qmt^ Q mg (*i> (7)where on the RHS the first term QMG is the molecular TCS obtained by weighing over the TCS of atoms or group of atoms in the molecule by geometrical or structural arguments. Further A(k) is an energy dependent empirical factor, introduced to contain the overestimation of calculated molecular cross-sections. The A(k) is so chosen as to reduce eq (7) to the AR (cq. 1) at high energies [8J.Rather than trying to correct the additivity rules for the cross-sections as above, one can try to work at the basic level of electronic charge-density, which is an important input in these calculations. The charge-density of a molecule AB may be approximately written [2] as a sum of the atomic charge-densities pA and pB, i.e.Pab = Pa + Pb ' <*>where pA and pB are free-atom charge-densities and eq. (8) ignores the overlap or the bonding m the molecule. Now at this stage we consider a hydride molecule AH and expand the H -  atom charge-density pH at the heavier atom (nucleus) A, following Watson [12]. The expansion248 K N Joshipura et aIinvolves the A-H bond length R as the parameter, so that we can approximate single-centre spherical charge-density of the molecule AH aspAn(r ' R) ^ P A ^ r'> + P n (r;R )- <9>We propose to improve here the simple addition of atomic charge-densities, by considering the bond charge introduced on the atoms A and H, in the process of molecular bonding. The formation of the bond re-adjusts the electronic charge on the partner atoms and the same can be calculated by molecular HF charge-densities. We have adopted the bond- charges as tabulated by Bader [2J. Thus ifq(A) in a.u. is the bond-charge acquired by atom A, then the corresponding amount 1 -  q(A) is shifted from the H-atom so that to obtain the molecular charge-density a realistic addition of atomic charge-densities must be done in the following manner.pAlAr ' K) =’mA)+q{A)~. p j r )  +'  N(H)-q(A)~N(A) • A N{H). p„{r ; R) , ( 10)where N(A) is the number of electrons in the free atom A and N(H) = 1 for the free H -atom. The bond-charge q{A) e.g. -0.535 au on O-atom in OH radical has been adopted from [2]. Let us note that as one might expect, eq. (10) when integrated, yields exactly the total number of electrons m the molecule N(AH), viz.,N(AH)=N(A)+N(H).  (11)Eq. (10) offers a better approximation compared to eq. (9).Now, the single-centre spherical charge-density p A/J(r\R)  can be used to construct the complex optical potential for the e -  AH system, to determine the various total cross sections. This approach is more reliable than even the MAR, eq. (2), because the former unlike the latter includes the valence -bonding effects through eq. (10). The single-centre (SC) approach has been adopted in studying the electron collisions with CH, NH, OH and HF in [5J. This method can be coupled with MAR to derive an effective additivity rule for larger polyatomic molecules like C2H6. in this target, two CH3 groups arc bonded with each other via C-C bond which is somewhat larger than its C-H bond. Hence, it is reasonable to assume high energy electron scattering to occur through two scattering centres considered at the two carbon atoms inC2H6.For this purpose a single centred charge-density p G(r, Rc_.n ) for each CH3 group is constructed as followsP G (r,Rc_H) = f t . p c { r) + n. f f r p „ ( r ; R(. ^  ). (12)In eq. (12)ft and/w arc bond-charge factors for C and H atoms in each of the C-H bonds, as defined in eq. (10) and n=3 shows the number of H atoms in the CHy group. Now, we can use Pq (r» rc-H ) as inPut 1° construct the complex optical potential for the electron scattering from each of the two scattering centres in the C2H6 molecule. The total molecular scattering amplitude is the sum of the amplitudes from the two groups of atoms, in this approximation. II Qsr is the total cross section for short-range potentials and a =  1,2 denotes the two scatteringAdditivity approximations for molecules and radicals 249centres then proceeding as in eq. (2), we can express the total electron-molecule cross section asQr (M) = ^ Q SR(a) + Qiml(M). {13)or=lIn this additivity rule, called the MAR-SC, the second term is calculated again from a direct molecular polarisation potential, as in eqs. (2) and (3), while QSR is calculated by startingWith ).Finally for a polar molecule with dipole moment D, one can define a high-energy grand total cross sectionQtotW) = fir W  + Qn» E, >• (14)where, the second term shows the first-Born rotational excitation cross section due to the long- range dipole potential of the target. We have found [5, 6] that gradually above 100 eV impact energy, where the above approximations arc reliable, the dipole and other anisotropic contributions become unimportant.3. Results, discussions and conclusionsVarious additivity rules for e -  molecule collisions as discussed above have been employed by several workers in the recent years to provide theoretical comparisons to experimental data. In g e n e ra l ,  the simple rule of adding atomic cross sections to obtain a molecular cross section as in eqs. (1) and (5), overestimates at low and intermediate energies and it tends to be closer to the experiments only at sufficiently large impact energies. The simple AR is a useful first approximation not only for the measured data but also for assessing improved approximations like those in eqs. (2), (6), (7) and (13). We have shown earlier [4-61 that the MARcq. (2) fares marginally better than the AR for molecules like CO, NO, C 02, NO?, N20 , NH v H2S, OCS, S 02 etc., above 50 eV. OurTCS obtained in the SC approach for** ~HF scattering 151 lie below the molecular orbital calculations [ 13) at intermediate energies. All the additivity rules discussed here are more or less consistent with the measured TCS at energies progressively above 100 eV, but the MAR-SC, eq. (13), seems to be physically more realistic as it incorporates the actual molecular properties like the bond-lengths, ionisation energy and polarisability.T a b ic  1. Total (co m p le te ) cross sec tio n s in 10 ,h cm : , for e lec tro n  co llis io n s w ith  C H 4 m oleculesEnergy (eV ) Present R esults M A R-SC eq. (13)T heore tica l data Jiang cl, a l *E x p erim en ta l D ata Zecca e l a l  [9a]5 0 12.71 17 .80 12 .308 0 10 .59 13.22 10 30100 9 .3 8 1 1 4 0 9 0 03 0 0 4 .8 9 5 02 4 .7 65 0 0 3 46 3 .2 2 3 .1 8(3 .1 3 )* *7 0 0 2 .7 2  ^ 2 .3 3 2 .4 9* Using AR [7] * ' G arcia and M anero [9b]250 K N Joshipura et alNow in the present paper our calculated total (Elastic + Inelastic) cross sections QTon molecular targets CH4, SiH4, F2, H2S and C2H6 have been examined against various experimental results, at impact energies Ei > 50, cV . In table 1 we have chosen sample energies to exhibit the present Qr vis-a-vis theoretical values of Jiang et. al. |7] in the AR and the measurements of Zccca et. al. [9a] as well as Garcia and Manero [9b].The present Qr values shown here are calculated through the MAR-Sc, eq. (13). Below 300 eV the AR results [7] are higher than the measurements [9a, 9bJ. Our theory shows a better accord with the experiment from about 100 cV onwards. It is physically reasonable to apply the single-centre expansion method, vide cq. ( 10) to nearly spherical molecules like CH4 and SiH4. The drawbacks of the AR due to valence-bond effect and multiple scattering in the molecule are overcome in the MAR-SC. Our results on the collisions of electrons with Silane molecules are given in table 2. As expected here too, our {^values obtained through eq. (13) are in better accord with the experimental data of Zccca etal [ 14], while the AR results [7] arc on the higher side, especially below 100 cV. However, it is not clear (Table 2) why the quoted results of [7] are lower than the measured data above 100 eV.Table 2. Total (com plete) cross sec tions in 10 ,ft c m \  for e lec tro n  co llis io n s  w ith  S iH x m oleculesEnergy (eV) Present results M AR-SC eq (13)T heo re tica l da ta  Jiang e l a l  *E x p erim en t data  Zecca e t a l  f 14]50 21 39 24  63 -80 18 74 18.96 16 70100 16 75 16.51 14 703 0 0 8 45 7 62 7 925 0 0 5 89 5 1 1 5 .5 27 0 0 4 60 3 76 4 14* Using AR [7]The improvement given by Jiang etal [8] through eq. (7) does not apply to molecules like CH4 and SiH4. Our modifications i.e. MAR and MAR-SC are rather more general and realistic.Now, we have also selected typical 18-electron targets Fr  H2S and C2H6 for applying our improved methods. Figure 1 shows the total (clastic + inelastic) cross sections for these molecules, at incident energies 50-2000 cV. Also shown here for reference arc the experimental TCS [15,16] of the isoelectronic Argon atom. The 18-electron molecules considered here differ from e^ch other in no. of atoms within, and also in terms of bondlengths, polarisabilitics etc. H2S is of course a weakly polar target, but at the energies of the present interest, the rotational excitation contribution is not important [6J. The TCS ofV* - F 2 collisions calculated in AR, are the lowest of all the other targets, in Figure 1. The TCS of C2H6, on the other hand are relatively the highest. We have inserted in Figure 2, the experimental data points of Sueoka and Mori [17] as well as Szmytkowski and Krzysztofowicz [ 10] on e~ -  C2H6 system. Our MAR-SC calculations on this system tend to agree with the measured data from about 200 eV onwards. In the case of H2S molecule we have not shown any other data, for the sake of clarity of the diagram, but our theoretical results show a similar accord [6] with the relevent experiments. One notices from theAdditivity approximations for molecules and radicals 251figure that the H,S cross sections tend to merge with those of isoelectronic Ar, above 300 eV. [Iven at 2000 eV there are significant differences among the TCS of all the |8-clectron targets studied here. It may be noted that the calculated cross sections of SiH4, an 18-electron molecule not covered in Figure 1, arc close to that of the Ar atom, beyond 200 eV or so.F ig u re  1. Present results .................  F , . ......................H,S • - • • • •  C \H , Experim entaldata Ar Atom □ □ □ □ Ref [ I5 | A A A A Ref 116] , C \H h molecule o o o o o o Ref [10] ****** [17]This paper thus examines various additivity rules for e~ -  molecule cross sections currently used in literature. We have introduced modifications in the simple AR to obtain reliable cross sections at intermediate and high energies. The M AR-SC, vide eq. (13), is based on a reasonably good representation of molecular charge-distribution. It gives reliable results above 100 eV, for a variety of molecules examined here and elsewhere by the present authors. We now intend to apply it to still larger polyatomic molecules in general and hydrocarbons in particular.R e fe r e n c e s(1] K J M iller ./ Am  C hem  Soc. 112 8533 ( IW 0 )12] R F  W Bader A tom s in M olecu les  (U K C larendon) (1990)13 ]  H  D e u t s e h , T  D  M a r k ,  V T a o ' o v S f r y ,  K  B e c k e i ,  C  C o r n e l i s e e n .  L  C e s p i v a  a n d  V  B o n a e i e  -  K o u i c c k y  h it  J. M ass. S p e c tro m . 137 77 (1994)[4] K N Josh ipura and P M Patel ./ P h ys  B29 3925 (1996)15] K N Josh ipura and M inaxi V inodkuinar P h ys Letts. A 224 361 (1997)16a 1 K N Josh ipura and M inaxi V inodkuinar Z P h \s  D41 133 (1997)16b] K N Joshipura and Minaxi Vinodkuinar 1C PE A C  XX Conference, (Vienna, Austria). Book ot abstracts p. W E 0 7 1  (19 9 7 )17] Y Jiang, J Sun and L Wan Phys. Rev. A 52 398 (1995)18] Y Jiang, J Sun and L Wan J. P h ys  B30 5025 (1997)252 K N Joshipura et a lfVaJ A Zecca, G Karwasz. R S Brusa and C S/mytkowski J Plm B24 2747 (1991)|9h) G Garcia and F Manero Plm Rev A57 1069 (1998)[l()| C Szmylkowski and A M Krzysztolowicz, ./ P lm  B28 4291 (1995)[11) C J Joacham Quantum Collision t h e o r y  (Amstcrdom North Holland) (1983)112) G N Watson Tlteoix oj Bcsvel Finn tu rn  (London Cambridge University Press) p 368 (1958) ( H | A Jam and K L Baluia P lm  Rev A45 202 (1992)[14] A Zccca. G Karwasz and R S Brusa P lm  Rev A45 2777 (1992)[15] C Szmytkowski and K Maciag /V iva S i i  54 271 (1996)[16] A Zccca, S Oss, G Karwasz, R Grisenti and R S Brusa ./ Plm . B20 5157 (1987)117] () Sueoka and S Mon J P lm  B19 4035 (1986)

Additivity Approximations for Molecules and Radicls

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Additivity Approximations for Molecules and Radicls

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