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On the theory of intermolecular forces between inert gas atoms
Volume 26, number 2
ON THE THEORY
CHEMICAL
PHYSICS LETTERS
OF INTERMOLECULAR
FORCES
15 May 19j4
BETWEEN
INERT GAS ATOMS
J. LLOYD and D. PUGH Department
of Pure and Appked
Chemistry.
Unirersity
o]~StrathcIyde.
Glasgow GI IXL.
UK
Received 25 February 1974
The method described by Gordon and Kim and modified by Rae has been further modiIkd to restrict the self-exchange correction to nn electron gas composed of the outer shell electrons. Revised rcsuhs are presented for the interaction potentials between various pairs of inert gas atoms.
Gordon and Kim [l] have described a method for calculating the intermolecular potential energy be-
tween closed shell systems at short and intermediate separations. In their method the electronic charge density at all points is put equal to the sum of the atomic charge densities. The Coulombic interactiorl terms are
then calculated exactly within this approximation and the kinetic energy, exchange and.correlation
energies
are obtained from the theory of the free electron gas. It is proposed by Gordon and Kim that an interpolation procedure should be used to join their curves onto the dispersion force potentials calculated at larger separations- Rae [I?] has argued that this procedure is incorrect and that the potential due to dispersion forces should be added to that of Gordon and Kim at all points. He also points out that since the self-energy terms in the Coulomb interaction have been omitted a correction is required to remove the self-exchange term from the free electron gas approximation to the exchange energy. In making the latter correction Rae considers that the interacting system is equivalent to an electron gas containing N electrons where N is the total number of electrons in the atomic pair (i.e., 4 for He-He, 20 for Ne-Ne, etc.). In this note it is argued that only the outer shell electrons should be counted. The kinetic energy and exchange terms in the free electron gas energy arise from the part of the wsvefunction which ban be expressed ‘as a single determinant of plane waves. When using electron gas formulae to obtain energy terms in a molecular system the Hartree-Fock solution for the molecule is being
approximated at each point by such a plane wave determinant. The number of electrons in the electron gas, which simulates the HF solution at a given point, is therefore determined by the number of HF orbitals contributing to the electron density at that point. If pi(r) is the charge density associated with the ith occupied HF orbital, Ee&) the exchange energy density excluding self-exchange and E!!(r) the total exchange energy then,
where p(rt) is the total charge density at rl _The second term on the right is the correction for self-exchangeA very important feature of the method of ref. [l] is that the atomic energy densities are subtracted at each point in the integral over the charge cloud so that the small interaction energy is obtained directly. The entire interaction energy arises from that part of space lying between the atoms where the combined charge distribution differs from that of a single atom. Whatever may be the approximation to the molecular orbitals implied by the method of ref. [I], it is clear that the quantities p,(r) are likely td be small in the region of interest unless i refers to an orbital coxistrutted from the valence shell orbitals of the atoms. It is a reasonable assumption that, over the region which contributes to the interaction, the average vaI-’ ues of th’e pi for the valence electron states are rough281 “.
Volume 26, number 2
-1s May 1974
CHEMICAL PHYSICS LETTERS
Table 1 Exchange interaction energies for Ar2 and Kr2 in the region of the potential minimum exchange)_ All values in atomic units Kr-Kr
Ar-Ar R
E,, only valence
EC-,,all shells
present 6.0 6.4 6.8 7.0 8.0 9.0
-0.5393 -0.3071 -0.1744 -0.1314 -0.3185 -0.7853
shells present
x 1o-2 x 10” x 10” x 10” X 1O-3 x 10-4
-0.5371 -0.3062 -0.1740 -0.1311 -0.3 180 -0.7844
x x x x x x
lo-* lo-’ 10” lo-* 10-3 lo4
ly the same. The valence electrons may therefore be approxhmated as a free electrongas. It is certainly not true that the average values of pi for all orbit& in the atoms are comparable over the region of interaction. It is not therefore permissible to treat the entire assembly of electrons as a free electron gas. In table 1 we exhibit the exchange interaction energies for Ar-Ar and Kr-K.r calculated by the method of ref. [l] around the region of the potential minimum with (a) all electrons included and (b) only the
(no correction
has been ma&
for &f_
-___
E,., all shells
Eex only valence
present
shells present -0.1268 -0.7637 -0.4578 -0.3530 -0.9601 -0.2541
-0.1284 x 10-l -0.7694 x 10” -0.4617 X 10” -0.3548 x lo-* -0.9600 X lO-3 -0.2538 x 1o-3
X IO-’ X 10” x 10-2 x W2 x lO-3 x 1o-3
outer shell electrons included_ It can be seen that there is negIigible contribution to the exchange interaction from the inner shells. Similar results have been found for Ne-Ne with the Is shell included and ex/cluded. These results show that it is quantitatively correct to ignore the terms in eq. (I) in which i refers to inner shells. We have accordingly repeated Rae’s calculations for the homonuclear pairs Ne-Ne, Ar-Ar and Kr-Kr 08
06
-.
.
0.4
I\
0.2.
3 m PI ‘5 -0.2.
;
.*
I
.
;
I I t I ‘,
6.5 R,_Jaicl
Fig. 1. The Art interaction potential ir. the region of the potential well. The solid line is obtained by the present calculation. The chain broken line is that of ref. [2] and the dotted line refers to experimentally derived data 131. All values ate in atomic units.
282
-.
. . :
.
7.0
7.5 80 RW-W (a”)
8.5
9.0
Fig. 2. The Krz interaction potential in the region of the potential well. The sotid line + obtained by the present calculation. The chain broken line is that of ref. 121 and the dotted line refers to experimentally derived data [Sl. AU values are in atomic units.
Volume 26, number 2
CHEMICAL PHYSICS LETTERS
1s May 1974
Table 2 Summary of results obtained usinS a valence shell exchange potential 0
Ar-Ar Ne-Ne Kr-Kr He-Ne Ne-Ar Kr-Ne Kr-Ar He-Ar
(A)
E(X lo-r6
(A)
‘min
:SpL -_____
C&.
expt.
3.4 2.79 3.66 2.66 3.21 3.39 3.54 3.06
3.32 [1], 3.35 [3] 2.73 [11,X77 141 3.52 [ 1],3.59 [S]
3.8 3.12 4.1 2.96
3.7 [1],3.8 [3] 3.03 [l], 3.07 [4] 3.95 [1],4.0 [S] -
3.08-3.16
3.57 3.78 3.97 3.41
3.38-3.64 3.53-3.80 3.8 [l] 3.4- 3.72
CIlC. ___--_
[I]
3.41 [I] 3.09-3.32
[I]
taking 16 electrons in the electron gas in each case
when correcting for self-exchange. The theoretical values of Cg, C8 and Cl0 121, the constants in the long range terms, have been used throughout. The results for Ar-Ar and Kr-Kr are shown in figs. 1 and 2. In these cases Rae’s results produced potential minima that were appreciably too deep. Considerably improved agreement with experiment has been obtained. The Ne-Ne curve is not greatly altered by omitting the 1s electron in the self-exchange correction and the full curve is not shown here. Various parameters for the three cases are shown in table 2 along with currently available experimentally derived data [2-S]. We have also calculated potential curves for a number of heteronuclear pairs, in each case counting only Table 3 Values for the Cn dispersion force coefficients used in the
present calculation
-.----~.--~--
[ 11 [ 1J [ 11
erg)
CillC.
expt.
198 63.2 246 40.13 89.3 87.2 216 55.8
195 [l], 193 :31 63 (11, 54.7 [4] 273 Ill.276 [S] 76-111 84-115 237 [l] 29-49
[I] 111 [I]
the outer shell electrons in making the self-exchange correction. Theoretical values of C6 for these cases have been obtained from ref. [6]. To obtain estimates of C8 and Cl0 we have used an interpolation procedure. Four calculated sets of values for Cs and Cl0 (ref. [2]) are available - for the homonuclear pairs. If these quantities are plotted logarithmically against the sum of the Pauling ionic radii for the interacting atoms, a linear relationship is obtained which provides a method of interpolating for the heteronuclear sysstems. The validity of this procedure is confirmed by plotting C, values, which are available for the heteronuclear cases, in the same way, when linear plots are again obtained. The C6, C, and Cl0 values are given in table 3. Allowing for the uncertainties in many of the experimental results the parameters tabulated appear to be in reasonable agreement with experiment.
Dispersion energy coefficients (au) c6
Kr-Kr Ar-Ar Ne-Ne
He-He He-Ar He-Ne Ne-Ar Ne-Kr Ar-Kr
128 64.8 6.48 1.471 9.6 3.03 19.7 27.3 91.0
[21 [2] [2] [2] [6] [6] [6] [6] [a]
2474.0 1129.0 57.2 14.1 131.8 30.9 257.0 407.4 1698.0
References
Cl0
cs
[2] [2] [2] [2]
6.16 x 2.46 X 698.0 182.0 1995.0 389.0 4266.0 7762.0 3.89 X
104 [2] 10” [2] [2] [2]
[I] [2] [3) [4] [S]
IO4
[6]
R.G. Gordon and Y.S. Kim, 1. Chem. Phys. 56 (1972) 3122. A.I.M. Rae, Chem. Phys. Letters 18 (1973) 574. G.C. Maitland and E.B. Smith, Chem. Sot. Rev. 2 (1973) 181. C.C. M&l&, Mol. Phys. 26 (1973) 5 13. D.W. Gough. G-C. hlaitland and ES. Smith. Mol. phys. 25 (1973) 1433. G. Starkschall and R.G. Gordon, J. Chem. Phys. 54 (197i) 663.
-. : *
‘.
.:.
:.
ON THE THEORY
CHEMICAL
PHYSICS LETTERS
OF INTERMOLECULAR
FORCES
15 May 19j4
BETWEEN
INERT GAS ATOMS
J. LLOYD and D. PUGH Department
of Pure and Appked
Chemistry.
Unirersity
o]~StrathcIyde.
Glasgow GI IXL.
UK
Received 25 February 1974
The method described by Gordon and Kim and modified by Rae has been further modiIkd to restrict the self-exchange correction to nn electron gas composed of the outer shell electrons. Revised rcsuhs are presented for the interaction potentials between various pairs of inert gas atoms.
Gordon and Kim [l] have described a method for calculating the intermolecular potential energy be-
tween closed shell systems at short and intermediate separations. In their method the electronic charge density at all points is put equal to the sum of the atomic charge densities. The Coulombic interactiorl terms are
then calculated exactly within this approximation and the kinetic energy, exchange and.correlation
energies
are obtained from the theory of the free electron gas. It is proposed by Gordon and Kim that an interpolation procedure should be used to join their curves onto the dispersion force potentials calculated at larger separations- Rae [I?] has argued that this procedure is incorrect and that the potential due to dispersion forces should be added to that of Gordon and Kim at all points. He also points out that since the self-energy terms in the Coulomb interaction have been omitted a correction is required to remove the self-exchange term from the free electron gas approximation to the exchange energy. In making the latter correction Rae considers that the interacting system is equivalent to an electron gas containing N electrons where N is the total number of electrons in the atomic pair (i.e., 4 for He-He, 20 for Ne-Ne, etc.). In this note it is argued that only the outer shell electrons should be counted. The kinetic energy and exchange terms in the free electron gas energy arise from the part of the wsvefunction which ban be expressed ‘as a single determinant of plane waves. When using electron gas formulae to obtain energy terms in a molecular system the Hartree-Fock solution for the molecule is being
approximated at each point by such a plane wave determinant. The number of electrons in the electron gas, which simulates the HF solution at a given point, is therefore determined by the number of HF orbitals contributing to the electron density at that point. If pi(r) is the charge density associated with the ith occupied HF orbital, Ee&) the exchange energy density excluding self-exchange and E!!(r) the total exchange energy then,
where p(rt) is the total charge density at rl _The second term on the right is the correction for self-exchangeA very important feature of the method of ref. [l] is that the atomic energy densities are subtracted at each point in the integral over the charge cloud so that the small interaction energy is obtained directly. The entire interaction energy arises from that part of space lying between the atoms where the combined charge distribution differs from that of a single atom. Whatever may be the approximation to the molecular orbitals implied by the method of ref. [I], it is clear that the quantities p,(r) are likely td be small in the region of interest unless i refers to an orbital coxistrutted from the valence shell orbitals of the atoms. It is a reasonable assumption that, over the region which contributes to the interaction, the average vaI-’ ues of th’e pi for the valence electron states are rough281 “.
Volume 26, number 2
-1s May 1974
CHEMICAL PHYSICS LETTERS
Table 1 Exchange interaction energies for Ar2 and Kr2 in the region of the potential minimum exchange)_ All values in atomic units Kr-Kr
Ar-Ar R
E,, only valence
EC-,,all shells
present 6.0 6.4 6.8 7.0 8.0 9.0
-0.5393 -0.3071 -0.1744 -0.1314 -0.3185 -0.7853
shells present
x 1o-2 x 10” x 10” x 10” X 1O-3 x 10-4
-0.5371 -0.3062 -0.1740 -0.1311 -0.3 180 -0.7844
x x x x x x
lo-* lo-’ 10” lo-* 10-3 lo4
ly the same. The valence electrons may therefore be approxhmated as a free electrongas. It is certainly not true that the average values of pi for all orbit& in the atoms are comparable over the region of interaction. It is not therefore permissible to treat the entire assembly of electrons as a free electron gas. In table 1 we exhibit the exchange interaction energies for Ar-Ar and Kr-K.r calculated by the method of ref. [l] around the region of the potential minimum with (a) all electrons included and (b) only the
(no correction
has been ma&
for &f_
-___
E,., all shells
Eex only valence
present
shells present -0.1268 -0.7637 -0.4578 -0.3530 -0.9601 -0.2541
-0.1284 x 10-l -0.7694 x 10” -0.4617 X 10” -0.3548 x lo-* -0.9600 X lO-3 -0.2538 x 1o-3
X IO-’ X 10” x 10-2 x W2 x lO-3 x 1o-3
outer shell electrons included_ It can be seen that there is negIigible contribution to the exchange interaction from the inner shells. Similar results have been found for Ne-Ne with the Is shell included and ex/cluded. These results show that it is quantitatively correct to ignore the terms in eq. (I) in which i refers to inner shells. We have accordingly repeated Rae’s calculations for the homonuclear pairs Ne-Ne, Ar-Ar and Kr-Kr 08
06
-.
.
0.4
I\
0.2.
3 m PI ‘5 -0.2.
;
.*
I
.
;
I I t I ‘,
6.5 R,_Jaicl
Fig. 1. The Art interaction potential ir. the region of the potential well. The solid line is obtained by the present calculation. The chain broken line is that of ref. [2] and the dotted line refers to experimentally derived data 131. All values ate in atomic units.
282
-.
. . :
.
7.0
7.5 80 RW-W (a”)
8.5
9.0
Fig. 2. The Krz interaction potential in the region of the potential well. The sotid line + obtained by the present calculation. The chain broken line is that of ref. 121 and the dotted line refers to experimentally derived data [Sl. AU values are in atomic units.
Volume 26, number 2
CHEMICAL PHYSICS LETTERS
1s May 1974
Table 2 Summary of results obtained usinS a valence shell exchange potential 0
Ar-Ar Ne-Ne Kr-Kr He-Ne Ne-Ar Kr-Ne Kr-Ar He-Ar
(A)
E(X lo-r6
(A)
‘min
:SpL -_____
C&.
expt.
3.4 2.79 3.66 2.66 3.21 3.39 3.54 3.06
3.32 [1], 3.35 [3] 2.73 [11,X77 141 3.52 [ 1],3.59 [S]
3.8 3.12 4.1 2.96
3.7 [1],3.8 [3] 3.03 [l], 3.07 [4] 3.95 [1],4.0 [S] -
3.08-3.16
3.57 3.78 3.97 3.41
3.38-3.64 3.53-3.80 3.8 [l] 3.4- 3.72
CIlC. ___--_
[I]
3.41 [I] 3.09-3.32
[I]
taking 16 electrons in the electron gas in each case
when correcting for self-exchange. The theoretical values of Cg, C8 and Cl0 121, the constants in the long range terms, have been used throughout. The results for Ar-Ar and Kr-Kr are shown in figs. 1 and 2. In these cases Rae’s results produced potential minima that were appreciably too deep. Considerably improved agreement with experiment has been obtained. The Ne-Ne curve is not greatly altered by omitting the 1s electron in the self-exchange correction and the full curve is not shown here. Various parameters for the three cases are shown in table 2 along with currently available experimentally derived data [2-S]. We have also calculated potential curves for a number of heteronuclear pairs, in each case counting only Table 3 Values for the Cn dispersion force coefficients used in the
present calculation
-.----~.--~--
[ 11 [ 1J [ 11
erg)
CillC.
expt.
198 63.2 246 40.13 89.3 87.2 216 55.8
195 [l], 193 :31 63 (11, 54.7 [4] 273 Ill.276 [S] 76-111 84-115 237 [l] 29-49
[I] 111 [I]
the outer shell electrons in making the self-exchange correction. Theoretical values of C6 for these cases have been obtained from ref. [6]. To obtain estimates of C8 and Cl0 we have used an interpolation procedure. Four calculated sets of values for Cs and Cl0 (ref. [2]) are available - for the homonuclear pairs. If these quantities are plotted logarithmically against the sum of the Pauling ionic radii for the interacting atoms, a linear relationship is obtained which provides a method of interpolating for the heteronuclear sysstems. The validity of this procedure is confirmed by plotting C, values, which are available for the heteronuclear cases, in the same way, when linear plots are again obtained. The C6, C, and Cl0 values are given in table 3. Allowing for the uncertainties in many of the experimental results the parameters tabulated appear to be in reasonable agreement with experiment.
Dispersion energy coefficients (au) c6
Kr-Kr Ar-Ar Ne-Ne
He-He He-Ar He-Ne Ne-Ar Ne-Kr Ar-Kr
128 64.8 6.48 1.471 9.6 3.03 19.7 27.3 91.0
[21 [2] [2] [2] [6] [6] [6] [6] [a]
2474.0 1129.0 57.2 14.1 131.8 30.9 257.0 407.4 1698.0
References
Cl0
cs
[2] [2] [2] [2]
6.16 x 2.46 X 698.0 182.0 1995.0 389.0 4266.0 7762.0 3.89 X
104 [2] 10” [2] [2] [2]
[I] [2] [3) [4] [S]
IO4
[6]
R.G. Gordon and Y.S. Kim, 1. Chem. Phys. 56 (1972) 3122. A.I.M. Rae, Chem. Phys. Letters 18 (1973) 574. G.C. Maitland and E.B. Smith, Chem. Sot. Rev. 2 (1973) 181. C.C. M&l&, Mol. Phys. 26 (1973) 5 13. D.W. Gough. G-C. hlaitland and ES. Smith. Mol. phys. 25 (1973) 1433. G. Starkschall and R.G. Gordon, J. Chem. Phys. 54 (197i) 663.
-. : *
‘.
.:.
:.