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Dipole moment of ternary homonuclear systems
Volume 21, number 3
DIPOLE
1 September
CHEMICAL. PHYSICS LETTERS
MOMENT
OF TERNARY
WOMONUCLEAR
G. ADAM and J. KATRIEL Nuclear Research Cer~rre - Ncgev, P.0.5. 9001. Beerdhera.
1973
SYSTEMS
Israel
Received 15 May 1973 Revised manuscript received 22 June 1973
The dipole moment of a system of three -qua! atoms was considered. A constrained vartitioml electron model system wns performed. The results xe beheved to represent the qualitstive futures pole, which is the domimnt short- and intermediate-range contribution.
The experimental observation oftranslational absorption induced by ternary collisions of identical atoms is presently under consideration [ 11. The spectrum of this absorption depends on the behaviour of the three-atom dipole moment in a manner which is completely analogous to that carefully studied for binary collisions between different species [2]. The ternary dipole moment is of further interest due to its being a direct manifestation of three-body interactions. The role of the three-body potential has been investigated in connection with the s:ability of raregas and alkali-halide crystals [3]. The extraction of the three-body component of the potential is, however, complicated by its being superimposed on the additive two-body potential. The two-body dipole moment for identical atoms is necessarily zero, thus enabIing a direct observation of three-body effects through the associated dipole moment. In analogy with binary systems, the first investigations of the three-body potential [4] and dipole moment [5] were carried out by a multipole expansion perturbation theory. The leading terms of the long-range asymptotic expansions obtained are homogeneous of degree 9 and IO, respectively, in inverse internuclear distances, compared to the 6th power low binary potential and the 7th power low binary dipole mcment. As is well known, the multipole expansions and associated perturbation theory treatment forms a valid procedure only at sufficiently large internuclear distances. On the other hand, an independent-particle treatment
tre;ltmcnt of a oneof the ovcrIap di-
may be expected to be adequate at short and intermediate internuclear distances. This follows from the fact that the dipole moment is the expectation value of a one-body operator. Hartree-Fock-type caiculations for diatomics carried out by Matcha and Nesbet [IS] were found to be essential!y in agreement with corresponding experimental results [7], over the overlapping ranges of internuclear distances. The aim of the present study is to obtain a dipolemoment function for ternary homonuclear systems at short and intermediate internuclear distances, which are expected to contribute most of the absorption intensity. It was considered of primary concern to establish a sufficiently simple and rapid procedure which would enable further use of the resu!ts in molecular dynamics type calculations. The procedure empIoyed and the results obtained should: therefore, be considered a simplit?ed model which is realistic in a qualitative sense rather than an accurate study of any specific real system. The experimental situation is simulated by a neutral system consisting of one electron moving in the field of three clamped nucIei each charged with +&. The nuclear geomet:y is described in fig. 1, The electronic wavefunction $ = C, exp(-aAri)
was optimized for q,, etry. A straightforward
+ CB exd-o,ri)
(Ya, CYCat each nuclear geomoptimization of the Linear co451
klumc
21, number
CHEMICAL
3
PHYSICS
1
LETTERS
September 1373
El
Fig. 1. The m~clcu
gwmctry.
efficients by the solution of the corresponding secular equation might be envisaged at this stage. Such a course results, however, in a misrepresentation of the behaviour of the electronic charge distribution and the resulting dipole moment, at large internuciear distances, as is clear by consideration of the dissociation products. In order to secure the desired behaviour of the electronic charge distribu;i.Jn at large internuclear .distsnces the linear coefficients C,, C,, C, were constrained so 2s to produce equal populations on the three nuclei. The Mulliken.population criterion [S] was employed. The method consisted of imposing one third of the electronic charge on each center by determining the coefficients from the set of equations C&,
f CACBSAB + c,c$&
+c,c,s,, c3BB c& L
Fig. 2. The potential tions.
energy
for isosceles
trianrglx
configtua-
tances (e.g., R = R, = R2 = 3 au) is equilateral. At larger values of R the minimum with respect to 19is shifted towards smaller angles, exhibiting the formation of a bound diatomic species. The corresponding values of the dipole moment, which is directed along the bisector of 0, are presented in fig. 3. The dipole moment vanishes at 0 = 60” and at 0 = 180” as expected. The behaviour of the dipole moment as a function of R at 0 = 0” (diatomic), 30’ and at the negative peak is presented in fig. 4. In the case of the negative peak, the angle at which this peak occurs is also presented. The magnitude of the ternary dipole is essentially
= 113 ,
+ CBCCSUC = l/3 ,
p 0.6 + CACCSAC + cn c,s,,
1
= 113 ,
s,,, s,,
etc., being the overlap integrals. The neutralization of each atom secured by irnposing this constraint cornpenj::tes for the inherent limitati0n.s of the one-electron model. In view of the restrictions imposed on the applicability of the results by the crudeness of the model system consid;;ed, one gets a somewhat safer comparison with real systems by considering not the absolute behaviollr but rather the behaviour of the three-bodl, model system compared to that of the analogous heteronuclear twobody @stem obtained by the choice of coordinates R, =Rzand0=0’. Fig. 2 describes the energy for the various geomctrie%-‘of ari isosceles triangular nuclear configuration. The equilibrium geometry at small internuclear dis‘,452: .. : ‘,
0.L
0.2
0
-0.2
iv F$,_ 3. The dipole
ic
Im’ moment
as a function
‘. -_.., ,‘,.
_ .’
,,.’
..
.,
:.
200” of 8.
0
Volume 21, number 3
CHEM!CAL PHYSICS LETTERS
1 September
1973‘
simplicity of ‘Lhemodel presented shou!d make molecular dynamics studies of the ternary collision and of the resulting translationa absorption feasible. The potential and dipole functions obtained are certainly realistic in their essential fzaiures, thus prcviding a meaningful framework for the discussion of the translational absorption spectrum to be expected as well as of the experimental results, when they become available.
extreme
References [ll S. Weiss, Chem. Phys. Letters 19 (1973) $I. J. Phyr 39 <1961) VI J.D. Poll and J. van Kranendonk,Can. Fig. 4. The dipole moment as a function of R. similar to, though somewhat smaller than the binary one. The weakness of the absorption associated with ternary collisions is therefore a consequence of the relative rarity of such collisions and not of the smallness of the dipole moment involved. It should be emphasized in conclusion, that the
189. [31 L. Jansen, Advan. Quantum Chem. 2 (1965) 119. I41 B.hi. Asilrod and E. Teller, J. Chem. Phyr It (1343) 299. ljl C.B. Gray and B.W.N. Lo, preprint. [Cl R.L. Marcha and R.K. Ncsbet, Phys. Rev. L60 (L967) 72. (71 E. Bar-Ziv and S. Weirs, Chem. Phys. Letters 19 (L973) 148. iSI R.S. hlulliken, J. Chcm. Phys. 23 (1953) 1833.
DIPOLE
1 September
CHEMICAL. PHYSICS LETTERS
MOMENT
OF TERNARY
WOMONUCLEAR
G. ADAM and J. KATRIEL Nuclear Research Cer~rre - Ncgev, P.0.5. 9001. Beerdhera.
1973
SYSTEMS
Israel
Received 15 May 1973 Revised manuscript received 22 June 1973
The dipole moment of a system of three -qua! atoms was considered. A constrained vartitioml electron model system wns performed. The results xe beheved to represent the qualitstive futures pole, which is the domimnt short- and intermediate-range contribution.
The experimental observation oftranslational absorption induced by ternary collisions of identical atoms is presently under consideration [ 11. The spectrum of this absorption depends on the behaviour of the three-atom dipole moment in a manner which is completely analogous to that carefully studied for binary collisions between different species [2]. The ternary dipole moment is of further interest due to its being a direct manifestation of three-body interactions. The role of the three-body potential has been investigated in connection with the s:ability of raregas and alkali-halide crystals [3]. The extraction of the three-body component of the potential is, however, complicated by its being superimposed on the additive two-body potential. The two-body dipole moment for identical atoms is necessarily zero, thus enabIing a direct observation of three-body effects through the associated dipole moment. In analogy with binary systems, the first investigations of the three-body potential [4] and dipole moment [5] were carried out by a multipole expansion perturbation theory. The leading terms of the long-range asymptotic expansions obtained are homogeneous of degree 9 and IO, respectively, in inverse internuclear distances, compared to the 6th power low binary potential and the 7th power low binary dipole mcment. As is well known, the multipole expansions and associated perturbation theory treatment forms a valid procedure only at sufficiently large internuclear distances. On the other hand, an independent-particle treatment
tre;ltmcnt of a oneof the ovcrIap di-
may be expected to be adequate at short and intermediate internuclear distances. This follows from the fact that the dipole moment is the expectation value of a one-body operator. Hartree-Fock-type caiculations for diatomics carried out by Matcha and Nesbet [IS] were found to be essential!y in agreement with corresponding experimental results [7], over the overlapping ranges of internuclear distances. The aim of the present study is to obtain a dipolemoment function for ternary homonuclear systems at short and intermediate internuclear distances, which are expected to contribute most of the absorption intensity. It was considered of primary concern to establish a sufficiently simple and rapid procedure which would enable further use of the resu!ts in molecular dynamics type calculations. The procedure empIoyed and the results obtained should: therefore, be considered a simplit?ed model which is realistic in a qualitative sense rather than an accurate study of any specific real system. The experimental situation is simulated by a neutral system consisting of one electron moving in the field of three clamped nucIei each charged with +&. The nuclear geomet:y is described in fig. 1, The electronic wavefunction $ = C, exp(-aAri)
was optimized for q,, etry. A straightforward
+ CB exd-o,ri)
(Ya, CYCat each nuclear geomoptimization of the Linear co451
klumc
21, number
CHEMICAL
3
PHYSICS
1
LETTERS
September 1373
El
Fig. 1. The m~clcu
gwmctry.
efficients by the solution of the corresponding secular equation might be envisaged at this stage. Such a course results, however, in a misrepresentation of the behaviour of the electronic charge distribution and the resulting dipole moment, at large internuciear distances, as is clear by consideration of the dissociation products. In order to secure the desired behaviour of the electronic charge distribu;i.Jn at large internuclear .distsnces the linear coefficients C,, C,, C, were constrained so 2s to produce equal populations on the three nuclei. The Mulliken.population criterion [S] was employed. The method consisted of imposing one third of the electronic charge on each center by determining the coefficients from the set of equations C&,
f CACBSAB + c,c$&
+c,c,s,, c3BB c& L
Fig. 2. The potential tions.
energy
for isosceles
trianrglx
configtua-
tances (e.g., R = R, = R2 = 3 au) is equilateral. At larger values of R the minimum with respect to 19is shifted towards smaller angles, exhibiting the formation of a bound diatomic species. The corresponding values of the dipole moment, which is directed along the bisector of 0, are presented in fig. 3. The dipole moment vanishes at 0 = 60” and at 0 = 180” as expected. The behaviour of the dipole moment as a function of R at 0 = 0” (diatomic), 30’ and at the negative peak is presented in fig. 4. In the case of the negative peak, the angle at which this peak occurs is also presented. The magnitude of the ternary dipole is essentially
= 113 ,
+ CBCCSUC = l/3 ,
p 0.6 + CACCSAC + cn c,s,,
1
= 113 ,
s,,, s,,
etc., being the overlap integrals. The neutralization of each atom secured by irnposing this constraint cornpenj::tes for the inherent limitati0n.s of the one-electron model. In view of the restrictions imposed on the applicability of the results by the crudeness of the model system consid;;ed, one gets a somewhat safer comparison with real systems by considering not the absolute behaviollr but rather the behaviour of the three-bodl, model system compared to that of the analogous heteronuclear twobody @stem obtained by the choice of coordinates R, =Rzand0=0’. Fig. 2 describes the energy for the various geomctrie%-‘of ari isosceles triangular nuclear configuration. The equilibrium geometry at small internuclear dis‘,452: .. : ‘,
0.L
0.2
0
-0.2
iv F$,_ 3. The dipole
ic
Im’ moment
as a function
‘. -_.., ,‘,.
_ .’
,,.’
..
.,
:.
200” of 8.
0
Volume 21, number 3
CHEM!CAL PHYSICS LETTERS
1 September
1973‘
simplicity of ‘Lhemodel presented shou!d make molecular dynamics studies of the ternary collision and of the resulting translationa absorption feasible. The potential and dipole functions obtained are certainly realistic in their essential fzaiures, thus prcviding a meaningful framework for the discussion of the translational absorption spectrum to be expected as well as of the experimental results, when they become available.
extreme
References [ll S. Weiss, Chem. Phys. Letters 19 (1973) $I. J. Phyr 39 <1961) VI J.D. Poll and J. van Kranendonk,Can. Fig. 4. The dipole moment as a function of R. similar to, though somewhat smaller than the binary one. The weakness of the absorption associated with ternary collisions is therefore a consequence of the relative rarity of such collisions and not of the smallness of the dipole moment involved. It should be emphasized in conclusion, that the
189. [31 L. Jansen, Advan. Quantum Chem. 2 (1965) 119. I41 B.hi. Asilrod and E. Teller, J. Chem. Phyr It (1343) 299. ljl C.B. Gray and B.W.N. Lo, preprint. [Cl R.L. Marcha and R.K. Ncsbet, Phys. Rev. L60 (L967) 72. (71 E. Bar-Ziv and S. Weirs, Chem. Phys. Letters 19 (L973) 148. iSI R.S. hlulliken, J. Chcm. Phys. 23 (1953) 1833.