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A relationship between the induced dipole moment and the interatomic potential of the rare gases
CHEhfICAL
Volume 39, number 1
PHYSICS
LETTERS
1 Api1
,_
1976
.-
-.
_
A RELATIONSHIP INTERATOlkIIC
BETWEtiN POTENTiAL
THE INDUCED
DIPOLE
MOMENT
AND THE-
.-
OF THE RARE GASES
Ezra BAR-ZIV Nuclear Research Center - Negev, Beer-Sizeva. Israel Received 18 November 1975
The tempdrature dependence of the theoretical second moment of the IR absorption of the rare eases was compared to the experimental one. The comparison has been done by substituting the reduced form of the dipole, ct*. by a simple function of the reduced potential, V*. It is shown that for p* (I V* the best agreement between the theoretical calculation and the experimental
results is obtained.
The induced IR absorption spectra of pairs of rare gases are still the only sources of information on induced dipoles arising during collisions between dissimilar atoms. Induced dipoles have been evaluated from the absorption moments [I -31 and line shapes [4-61. These dipole moments are developed mainiy by
electronic overlap [7,8] during binary collision. For almost ali pairs of rare gases a number of induced IR spectra have been published recently, covering a range of temperatures 191. For these experimental results Bar-Ziv and Weiss found an empirical relation between the second absorption moment, and several molecular properties, as a function of the temperature [9] : ‘yap
=
K&/k)
(elm)rg&Y2
-
“1
JT’3’2.
(1)
Hererexp is the experimental second absorption moment; Kr is a pro ortionality constant, equal to 1.04 X 1016 cm- % s2 deg-’ Am-’ ; E is the potential well depth; k is the Boltzmann constant; m is the reduced mass; rmin is the location of the minimum of the potential; Ia2 - al I is the difference of the polarizabiliiies of the two atoms, and T” is the reduced temperature deimed by T* = kT/e, where T is the absolute temperature. Poll and van Kranendonk showed that the second absorption moment, is classically related to the intermolecular potential and the induced dipole moment by the following equation [lo] :
Here r is the interatomic distance; ,u is the dipole moment; V is the interatomic potential. Introducing reduced quantities: V* = V/E, p* = p/p,, where p. is a strength parameter, and r* = rlrti one obtains: ytheo =(42/3mc2)
gi P(2),
(3)
where I* (2) is a reduced integral, given by
I*(2) = f [
($)’ +‘s] exp(-V
*/T*
)re2 dr*. (4)
When the expression of rexp is compared to that of ram, two new empirical relations are obtained:
I* (2) = K,T+jlz
.
0)
and &j =K@Z
- ay111’2 ,
(6)
whert KI and KM are proportionality
constants. Bar-Ziv and Weiss calculated the reduced dipole by substituting into l*(2) a many parameter function for the dipole moment which was based on a comparison of I* (2) to K1P312. The shape of this reduced dipole is very similar to the shape of the reduced potential evaluated by Lee and co-workers [ 1 I] ,
‘31
CHEMICAL PBYSICS -Lk-TERS
5 :,_yolurn&39;ktmber.l : _: ._, :
.
1 April 1376
..
t 0.9
”
too
50
T
Fig. 1. The rcdu.ced dipole moment (solid line) evdiuated by Bar-
Ziv and Weiss 131 and the reduced interatomic potential (dashed line) evaluated by Lee et al. [ 111.
this is illustrated in fig. 1. It is, therefore, tempting to hypothesize that a simple relationship exists between the induced dipole moment and the interatomic potentiitl. Inorder to examine this hypothesis, P(2) was calculated using ,u* defined as a simpie function of I/*. The calculated dependence cf 1*(2) versus T'3/2 wqscompared with the one observed experimentally. TWO functions were examined: one in which p* was assumed to be proportional to V$ and another one in which p* was proportional to V* ,k* . In Gg. 2 the calculated I” (2) is plotted against T*312, using the -above two functions. It can be clearly seen that the function in which p* is proportional to V* gives
almost a straight line which fits tba experimental results. On the other hand the calculated 1*(2) dependence on y312 using l Lhe second function is not a straight line and deviates from the fiwt at high temperatures. Therefore, the preferable function is y*(r*) a V*(r*). If the reduced dipole, expressed above, is multiplied by the empirical expression of the dipole strength [3] - given by eq. (6) - one obtains-a relation between the induced dipole moment and the interatomic potential j.r(Fj= 2.26 X 105\42.-01+‘2V(r)
_
(7)
= 5. IOX 1O1'a1V2(r)and &r) =.5.10 X 10” 62 V2(r) a new expression for p(r) can
If one defines&~)
32
--
.-
150
*3/z
Fig. 2. The calculated I*(2) plotted versus T*3’2; solid line - using p*(r*) = V*(r*) and dashed line - using p*(r*) = v*(r*)/r*; dotted line - experimental behavior:
be obtained:
P2k) = &)
- PL:@) -
03)
Thus 1-(1(r)and ~z(r) are the collision induced atomic dipoles of the first and the second atoms respectively for a certain pair. This expression is, therefore, a summation rule for these atomic dipoles.
References [I] S.L. Brenner and D.A. McQuarrie, Can. J. Phys. 49 (1971) 837.
[2] R-W. Hartye, C.G. Gray, J.D. Poll and MS. Miller, Mol. Phys., to be published. [ 3 ] E. Bar-Ziv and S. Weiss, J. Chem. Phys., to be published. [4] H.B. Levine and G. Birnbaum, Phys Rev. 154 (1967) 86. [S] E. Bar-Xv and S. Weiss, J. Chem. Phys. 59 (1973) 5333, and references therein. 161 D.A. McQuarrie and R.B. Bernstein, J. Chem. Phys. 49 (1968) 1958. 171 W. Byers Brown and D.M. Whissnant, Mol. Phys. 26
(1973) 1105. [8] A.J. Lacey and W. Byers Brown, blot. Phys. 27 (1974) 1013. [9] E. Bar-Xv and S. Weiss, J. Chcm. Phys., to be published. IlO] J.D. Poll and 3. van Kranendonk. Carx J. Phys:39 (1961) 189. [Ill C.H. Chen, P.E. Siska and Y.T. Lee, J. Che‘m. Phys. 59 (1973) 601.
Volume 39, number 1
PHYSICS
LETTERS
1 Api1
,_
1976
.-
-.
_
A RELATIONSHIP INTERATOlkIIC
BETWEtiN POTENTiAL
THE INDUCED
DIPOLE
MOMENT
AND THE-
.-
OF THE RARE GASES
Ezra BAR-ZIV Nuclear Research Center - Negev, Beer-Sizeva. Israel Received 18 November 1975
The tempdrature dependence of the theoretical second moment of the IR absorption of the rare eases was compared to the experimental one. The comparison has been done by substituting the reduced form of the dipole, ct*. by a simple function of the reduced potential, V*. It is shown that for p* (I V* the best agreement between the theoretical calculation and the experimental
results is obtained.
The induced IR absorption spectra of pairs of rare gases are still the only sources of information on induced dipoles arising during collisions between dissimilar atoms. Induced dipoles have been evaluated from the absorption moments [I -31 and line shapes [4-61. These dipole moments are developed mainiy by
electronic overlap [7,8] during binary collision. For almost ali pairs of rare gases a number of induced IR spectra have been published recently, covering a range of temperatures 191. For these experimental results Bar-Ziv and Weiss found an empirical relation between the second absorption moment, and several molecular properties, as a function of the temperature [9] : ‘yap
=
K&/k)
(elm)rg&Y2
-
“1
JT’3’2.
(1)
Hererexp is the experimental second absorption moment; Kr is a pro ortionality constant, equal to 1.04 X 1016 cm- % s2 deg-’ Am-’ ; E is the potential well depth; k is the Boltzmann constant; m is the reduced mass; rmin is the location of the minimum of the potential; Ia2 - al I is the difference of the polarizabiliiies of the two atoms, and T” is the reduced temperature deimed by T* = kT/e, where T is the absolute temperature. Poll and van Kranendonk showed that the second absorption moment, is classically related to the intermolecular potential and the induced dipole moment by the following equation [lo] :
Here r is the interatomic distance; ,u is the dipole moment; V is the interatomic potential. Introducing reduced quantities: V* = V/E, p* = p/p,, where p. is a strength parameter, and r* = rlrti one obtains: ytheo =(42/3mc2)
gi P(2),
(3)
where I* (2) is a reduced integral, given by
I*(2) = f [
($)’ +‘s] exp(-V
*/T*
)re2 dr*. (4)
When the expression of rexp is compared to that of ram, two new empirical relations are obtained:
I* (2) = K,T+jlz
.
0)
and &j =K@Z
- ay111’2 ,
(6)
whert KI and KM are proportionality
constants. Bar-Ziv and Weiss calculated the reduced dipole by substituting into l*(2) a many parameter function for the dipole moment which was based on a comparison of I* (2) to K1P312. The shape of this reduced dipole is very similar to the shape of the reduced potential evaluated by Lee and co-workers [ 1 I] ,
‘31
CHEMICAL PBYSICS -Lk-TERS
5 :,_yolurn&39;ktmber.l : _: ._, :
.
1 April 1376
..
t 0.9
”
too
50
T
Fig. 1. The rcdu.ced dipole moment (solid line) evdiuated by Bar-
Ziv and Weiss 131 and the reduced interatomic potential (dashed line) evaluated by Lee et al. [ 111.
this is illustrated in fig. 1. It is, therefore, tempting to hypothesize that a simple relationship exists between the induced dipole moment and the interatomic potentiitl. Inorder to examine this hypothesis, P(2) was calculated using ,u* defined as a simpie function of I/*. The calculated dependence cf 1*(2) versus T'3/2 wqscompared with the one observed experimentally. TWO functions were examined: one in which p* was assumed to be proportional to V$ and another one in which p* was proportional to V* ,k* . In Gg. 2 the calculated I” (2) is plotted against T*312, using the -above two functions. It can be clearly seen that the function in which p* is proportional to V* gives
almost a straight line which fits tba experimental results. On the other hand the calculated 1*(2) dependence on y312 using l Lhe second function is not a straight line and deviates from the fiwt at high temperatures. Therefore, the preferable function is y*(r*) a V*(r*). If the reduced dipole, expressed above, is multiplied by the empirical expression of the dipole strength [3] - given by eq. (6) - one obtains-a relation between the induced dipole moment and the interatomic potential j.r(Fj= 2.26 X 105\42.-01+‘2V(r)
_
(7)
= 5. IOX 1O1'a1V2(r)and &r) =.5.10 X 10” 62 V2(r) a new expression for p(r) can
If one defines&~)
32
--
.-
150
*3/z
Fig. 2. The calculated I*(2) plotted versus T*3’2; solid line - using p*(r*) = V*(r*) and dashed line - using p*(r*) = v*(r*)/r*; dotted line - experimental behavior:
be obtained:
P2k) = &)
- PL:@) -
03)
Thus 1-(1(r)and ~z(r) are the collision induced atomic dipoles of the first and the second atoms respectively for a certain pair. This expression is, therefore, a summation rule for these atomic dipoles.
References [I] S.L. Brenner and D.A. McQuarrie, Can. J. Phys. 49 (1971) 837.
[2] R-W. Hartye, C.G. Gray, J.D. Poll and MS. Miller, Mol. Phys., to be published. [ 3 ] E. Bar-Ziv and S. Weiss, J. Chem. Phys., to be published. [4] H.B. Levine and G. Birnbaum, Phys Rev. 154 (1967) 86. [S] E. Bar-Xv and S. Weiss, J. Chem. Phys. 59 (1973) 5333, and references therein. 161 D.A. McQuarrie and R.B. Bernstein, J. Chem. Phys. 49 (1968) 1958. 171 W. Byers Brown and D.M. Whissnant, Mol. Phys. 26
(1973) 1105. [8] A.J. Lacey and W. Byers Brown, blot. Phys. 27 (1974) 1013. [9] E. Bar-Xv and S. Weiss, J. Chcm. Phys., to be published. IlO] J.D. Poll and 3. van Kranendonk. Carx J. Phys:39 (1961) 189. [Ill C.H. Chen, P.E. Siska and Y.T. Lee, J. Che‘m. Phys. 59 (1973) 601.