- Email: [email protected]
Modelling of repulsive potentials from charge density distributions: A new site-site model applied to inert gas atoms with the diatomic molecules H2, N2, O2
Chemical Physics 122 ( 1988) 337-346 North-Holland, Amsterdam
MODELLING OF REPULSIVE POTENTIALS FROM CHARGE DENSITY DISTRIBUTIONS: A NEW SITE-SITE MODEL APPLIED TO INERT GAS ATOMS WITH THE DIATOMIC MOLECULES Hz, N2, 0, Carl NYELAND Max-Planck-Institut
’ and J. Peter TOENNIES ftir Strijmungsforschung,
D-3400 Giittingen, FRG
Received 9 October 1987; in final form 29 February 1988
A recently developed combining rule for short-range inert gas atom-atom repulsive potentials based on charge overlap integrals is applied to atom-diatom potentials in a novel orientation-dependent site-site approximation. Very good agreement with available ab initio SCF calculations is obtained for the systems He-Hz, He-N,, He-O,, Ne-H,, Ne-N, and Ar-N,. New estimates are reported for anisotropic potentials for the other systems X-Y*, where X = He, Ne, Ar, Kr, Xe and Y2= O,, Nz, Hz, for which ab initio quantum chemical calculations are not available.
1. Introduction
sp=
With the advent of recently developed very accurate semitheoretical models for van der Waals potentials it is now possible to generate potentials for atom-atom and atom-molecule systems if the necessary ab initio potential parameters are available [ 1,2 1. As input data the dispersion coefficients and the Born-Mayer parameters from accurate SCF calculations are required. Whereas dispersion coefficients are now available for many systems, SCF repulsive potentials are not widely available and their calculation for systems with many electrons can be quite time consuming. In a recent publication [ 3 ] we presented a new combining rule for short-range repulsive intermolecular potentials based on an analysis of the short-range intermolecular potentials between fifteen pairs of the five inert gas atoms. It was found that the potentials VABfor the unlike systems can be determined from
where pA and pB are the undisturbed electron density distributions of the atoms A and B, respectively, and r is the intermolecular distance. The correlation parameters KABand yABwere shown to fulfill the following simple combining rules:
VAB=KAB (Splr2)YAS , where S, is the charge density fined by
integral
de-
’ Permanent address: Chemistry Laboratory III, H.C. Orsted Institute, University of Copenhagen, DK-2100, Copenhagen, Denmark.
0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
dr,
(lb)
K AB=(KAAKBB)"*,
(2)
Y.~B=~(YAA+YBB).
(3)
The coefficients K and y for the systems AA and BB are in turn obtained from the potentials and charge density distributions of the homogeneous-like systems using eq. ( 1) . The possibility of obtaining anisotropic intermolecular potentials from charge density distributions has been explored previously by Nikitin and coworkers [ 41, Start and Williams [ 5 ] and by Winicur [ 6 1. Also the electron gas method of Gordon and Kim [ 7 ] which attempts to predict the entire potential should be mentioned here. The site-dependent separation method described here is most similar to that of Clementi et al. [ 8 ] in which a potential between two large molecules is considered as a sum of atom-atom interactions where different bonding environments are accounted for. In our case the bonding environments are always the same, but we
(la) overlap
PAP0
B.V.
338
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
consider the influence of different geometrical configurations. On the other hand we restrict ourselves to the short-range repulsive part of the potentials. Moreover we make extensive use of combining rules and thereby avoid problems associated with the absolute determination of the potentials. Still another approach for three-particle potentials has been considered by Varandas and co-workers [ 9 1. They take the configurational dependence into account by adding a three-particle contribution to the contributions of independent atom-atom interactions. The present paper starts with a description of the orientation-dependent site-site approximation. The implementation of the newly derived combining rules is discussed next and used to analyze the repulsive potentials between inert gas atoms and some common diatomic molecules. Good agreement is obtained between estimates from this procedure and the available ab initio SCF calculations for the anisotropic potentials. A number of estimates of intermolecular potentials for systems for which no ab initio information is present are also given. Atomic units are used throughout: 1 au lengthz0.52917 8, and 1 au energy=27.210 eV.
2. Site-site description of charge distributions and intermolecular potentials The idea of approximating the anisotropic interaction between two molecules by considering the forces to be centered on specific sites in the molecules is an old one. In the usual approximations the sites are the nuclei in the individual atoms making up the molecules [ lo]. Recent ab initio SCF calculations of the short-range potential of N2-N2 [ 111 and 02-O2 [ 121 showed that this approximation is reasonably well fulfilled. In our approach for the atom-diatom interaction, the force centers are also at the atoms in the molecule but the interaction is assumed to depend on the direction of approach of the atom with respect to the bond axis of the molecule. Thus the site-site potentials are different for the linear ( 11) C,, and the broad-side ( I ) CZv orientations. This is illustrated by examining the charge distributions for the ground state of molecular oxygen [ 13- 15 ] shown in fig. 1a. The outermost contour, ~~6.1 x 1O-5 e- auP3 (e-
denotes the charge of one electron), is reproduced in fig. lb together with results of the orientation-dependent site-site separation for the two geometries ( II and I). It can be seen that by using different parameters for the two different orientations a perfect fit of the real contour is achieved at the two different geometries. This modification of the site-site approximation was suggested by examination of the recently published short-range intermolecular potentials for He-H2 [ 161, He-N, [ 17-191, He-O, [20], Ne-H2 [ 2 11, Ne-N, [ 22 ] and Ar-N2 [ 22 1. The charge densities of Hz, N2 and 0, obtained from SCF calculations by Cade and co-workers [ 13-l 5 ] also suggest a decomposition into separate spherical distributions centered on the individual atoms in the molecules in an orientational approach, as discussed above for OZ. The implementation of the orientation-dependent site-site approximation as used here is based on the following simple considerations. Since we will see that density distributions and the intermolecular potentials have the same general behaviour we consider for the moment only the densities. For homonuclear molecules the density distributions for specified orientations p,, (R,, ) and pI (R, ) are usually well approximated over the range of interest by exponential expressions pII (R,, ) =B,, exp( --&R,, ) , Pi
(R,
1 =B,
ew(-BIRI
(da) ) ,
Cab)
where R,, and R, are the corresponding distances from the center of mass of the molecule in the specified geometries (see fig. 2). Assuming only contributions from sites located in the nearest atom or atoms in the molecule eq. (4) can easily be expressed in terms of exponential site-site charge distributions. For the collinear ( II ) approach one obtains pg”(r)=Bsy”
exp( -/3-‘r)
,
(5)
where BY‘=B,,
exp( - $P,,6) ,
(6)
and Pi-“=P,,
*
(7)
Here 6 is the distance between the two atoms in the molecule and r is the site-to-site distance from the ap-
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
9 = 6.1. lo-’
339
e-Ibohr)-’
b1
al
Fig. 1. (a) SCF charge density contours for the ‘Z; ground state of molecular oxygen, from ref. [ 151. The outermost curve is for p= 6. I x 1O-5 e- au-‘. The charge density changes by a factor of two between the curves. (b) The charge density contour forp=6.1 x low5 e- auw3. (--) are compared with the orientation-dependent site-site tit for the II geometry (--) and with the site-site tit for the n geometry (--.-e).
proaching atom to the nearest atom in the molecule. Similarly one has for the _!_geometry, defining a site-site charge distribution p:_’ as in eq. (5 ), and assuming additivity of charges
pl ~3.21 exp( -2.46 R, ) .
p,_ (R, ) =2B:_S exp( -P:_“r) ,
Using eqs. (4)- (8) one gets
(8)
where r2= R: + 4s’. In this case it is not possible to write a simple analytical expression for B‘;’ and j?:_ but these constants can be easily obtained from fits toeq. (8). The procedure Qxemplified for the charge distribution of O2 using the SCF results given in table 1.
Assuming exponential charge distribution one obtains p,, =89.0exp(
-2.71 R,, ) .
pi-‘=4.05 exp( -2.71 r) , and Table 1 Ab initio charge distributions 6=R,
for homonuclear diatomics at
a’ p (e-au-‘)
RI, (au) b,
R, (au) b,
R
NZ
6.1(-S) 2.0( -3)
5.38 4.10
4.73 3.20
2.070
02
6.1(-5) 2.0(-3)
5.24 3.95
4.42 3.00
2.280
f-f2
6.1(-S) 2.0( -3)
4.28 2.84
4.16 2.64
1.449
System
W
(au)
‘) All charge distribution results belong to the same calculation of Wahl and co-workers [ 13-151. The results given for
Fig. 2. Geometries and coordinates used to describe the site-site orientation-dependent approximation for atom-homonuclear diatom systems.
p= 2.0( - 3) are from table I of ref. [ 141. The results given for p= 6.1( - 5 ) are obtained from figures of ref. [ 15 1.The values for R.,.given are also from ref. [ 141 concerning Nz and Oz. The R, value for Hz is from ref. [ 231. b, R,, and R, are the center of mass distances for which the charge density p is given in the second column, and the molecule has the equilibrium distance R,, as given in the last column.
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
340
Table 2 Exponential charge distribution parameters. The parameters B and p are defined in eq. (4) for molecules and in eqs. (5) and (8) for atoms System N2
Geometry
B
P
II
14.5 2.95 89.0 3.21 2.00 0.87
2.73 2.28 2.71 2.46 2.43 2.30
I
II
02
I
II
H*
_L
II
N
8.60 2.37 4.05 3.01 0.344 0.571
I
II
0
I
II
H
I
p;-“=fp,
~3.01 exp( -2.52
2.73 2.33 2.71 2.52 2.43 2.33
r) .
This example shows that the charge distribution parameters are indeed quite different for the two orientations. Parameters for NZ, O2 and H2 obtained from charge density data of refs. [ 13- 15 ] given in table 1 are listed in table 2. Also the separation of the anisotropic potentials between two like homonuclear diatomic molecules into site-site potentials is accomplished in a similar fashion. The results are given in table 3. The SCF results for the like molecular potentials are from Table 3 Site-site intermolecular tional site-site approach
potential data obtained from SCF results ‘)
in the orienta-
System
Geometry
A b’
b b’
Nz-N*
II
213 213 123 121 0.88 1.02
2.09 2.09 2.17 2.16 1.65 1.61
I 02-02
II I
Hz-H2
II I
aJ SCF information was taken from: [12]; H2-H2 ref. [24], V,,=9.62 X exp ( - 1.58 R, ) were obtained the molecular potentials of system
Nr-Nz ref. [ 111; 02-Or ref. exp(-1.65 R,,), V,=5.11 as the SCF equivalences for following the method of ref.
111. b’ The site-site potentials are given on the exponential form V=A Xexp( - br), where r is the site-site
distance.
Table 4 Correlation
parameters
%-N,, N,-N, G II-0 0,-G, HII+ H,-H,
(K, y) for atoms in diatomic
I,
K
Y
12.4 60.0 20.9 27.3 4.66 2.99
0.754 0.879 0.788 0.846 0.666 0.660
molecules a)
a) K, and y are defined in eq. ( 1). The correlation numbers for the geometry X,-X, are calculated by the combining rule from numbers obtained for the geometries X.-X ,,and X ,,-X ,,
refs. [ 11,12,24]. Numerical tests on the correctness of the site-site approximation compared to the SCF information used are easily carried out. The orientational site-site separation and the exponential fitting contribute an error of less than one percent in calculations of overlap integrals and estimates of potentials. The combining rule used in this paper contributes an error of up to ten percent for the atom-diatom systems considered as discussed in section 3.
3. Charge density overlap calculations of intermolecular atom-diatom potentials In the site-site approximation the determination of intermolecular potentials from charge density overlap of the site-site charge distributions for the geometries II and 1 separately reduces to a treatment similarly to the one considered in ref. [ 31 for mixed rare gas atomic systems. This is illustrated by considering the broadside (I ) approach for the atom-diatom case for which one obtains from eq. ( 1)
=K~~(S~~B/r2)Y~~+K~~(S,tac/r2)Y~~, (9) in which the site-site potentials are now given as functions of the site-site charge density overlap integrals. This separation can easily be extended to intermolecular potentials between molecules of any number of atoms. In ref. [ 31 we suggested the combining rule V**(s,lr2)=rT/AA(Sp/12)
~BB(qJr2)l”2
(10)
341
C. Nyeland, J. P. Toennies /Repulsive potentials from charge density distributions
for atom-atom interactions which is equivalent to the rules of eq. (2) and (3 ) mentioned in the introduction. This combining rule will now be applied to estimating site-site potentials for atom-diatom systems. The results will be compared with SCF calculations for the same potentials. Correlation parameters (K, y ) for the like systems obtained by use of the results in the tables 2 and 3 are listed in table 4 for the site-site potentials corresponding to different geometrical conditions of nitrogen, oxygen and hydrogen atoms in the molecules. Following the combining rule of eq. ( 10) one then obtains, using also data from the tables 1 and 3 of ref. [ 3 ] for the inert gases, results for the intermolecular potentials V,,(R,, ) and I’1 (R, ) for the mixed systems. These are shown as molecular potentials in tables 5-8 and in figs. 3-8 for the X-Y2 systems for which SCF results are available. Note that these SCF results were not used in establishing the parameters used in the combining rule calculations and provide an independent test of the procedure. The respective values for the overlap integrals divided by r* are also listed in tables 5-8. It is seen that the estimated values for the potentials of the mixed systems based on the combining rule usually come within loo/6 of the available SCF results. It should be mentioned that the estimated repulsive potentials for the rare gas atoms are based on the potential model of Tang and Toennies [ 11. According to ref. [ 1] these effective repulsive potentials include in addition to the pure SCF repulsion a small exchange dispersion term, which for the rare gas atoms was found to contribute an additional 17% to the repulsion. Thus the semi-empirical values for the
single-atom parameters were reduced by 8.5Ohto approximate the pure SCF repulsion. The ab initio results for H2-H2 [24] have been corrected for dispersion and damping following the method of Tang and Toennies [ 11. It is particularly interesting to observe in fig. 3 that three rather different SCF results, those of Jaquet and Staemmler for He-O2 [ 201, those of Wormer and van der Avoird for 02-O2 [ 121 and the partly experimental results for He-He from ref. [ 21 are seen to follow the combining rule (eq. ( 10) ) rather well. The same behaviour was observed for the other systems shown in figs. 4-8. Thus estimates of potential surfaces for similar X-Y2 systems may be obtained using the combining rules if charge distributions for the separate systems and the intermolecular potentials of the corresponding like systems are available. From data of tables 2 and 4 of this paper and of tables 1 and 3 of ref. [ 3 ] intermolecular potentials of all the fifteen X-Y2 systems where X=He, Ne, Ar, Kr, Xe and Y,=02, N2, Hz can easily be obtained. Final results for the Born-Mayer parameters of the potential Vi,and V, are listed in table 9. In fig. 9 these estimated Born-Mayer parameters have been used to generate via the Tang-Toennies potential model the full potential curves for I/,, and V, including dispersion and damping for Ar-N,. Also shown in fig. 9 are the potentials obtained by the multiproperty analysis of Candori et al. [25]. In ref. [ 261 it is shown that the spherical part of another potential, which is very similar to the present predictions, is in good agreement with second virial coefftcients suggesting that the present potential probably is in even better agreement with the equations of state than the experimen-
Table 5 Comparison of estimated results for He-O, with SCF calculations z Geometry
RI, or R,
S.lr’
V (comb. rule)
v (SC&
II
4
5 6 I
0.9941(-3) 0.5589( -4) 0.3252( -5) 0.1906( -6)
0.739( - 1) 0.741(-2) 0.764( -3) 0.792( -4)
0.605(-l) 0.691(-2) 0.712(-3) 0.67 (-4)
4 5 6 7
0.2875( -4) 0.2088( -5) 0.1494(-6) 0.1067(-7)
0.735( 0X37( 0.943( 0.106(
0.761(-2) 0.910( -3) 0.103(-3) 0.10 (-4)
_L
-2) -3) -4) -4)
a) R,, and R, are the center of mass distances and r the corresponding site-site distances. (See fig. 2.)
I2011
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions Table 6 Comparison
of estimated
results for He-N2 with SCF calculations V (comb.
‘)
[191)
Geometry
Rm,or R,
II
4 5 6 7
0.1521(-2) 0.8511(-4) 0.4866( - 5) 0.2856( -6)
0.895(-l) 0.938( -2) O.lOO( -2) 0.109(-3)
0.7650( - 1) 0.1048(-l) 0.1309( -2) 0.151 (-3)
0.7806( - 1) 0.1070(-1) 0.1333(-2) 0.163 (-3)
0.7723( - 1) 0.1071(-l) 0.1347(-2) 0.1547(-3)
I
4 5 6 7
0.4238( -4) 0.3415(-5) 0.2759( -6) 0.2267( -7)
0.127(-l) ‘0.151(-2) 0.180( -3) 0.217( -4)
0.1244(-l) 0.1723(-2) 0.222 (-3) 0.27 (-47)
0.1261(-l) 0.1758(-2) 0.228 (-3) 0.29 (-4)
0.1255( - 1) 0.1746( -2) 0.2262( -3) 0.2732( -4)
a1 See footnote
Table 7 Comparison
rule)
V(SCF,
1171)
V(SCF,
1181)
VWF,
to table 5.
of estimated
results for Ne-N,
and Ar-N2 with SCF calculations
a)
System
Geometry
RI, oral
S,/ r2
V (comb. rule)
V(SCF,
Ne-N,
II
5 6 7
0.1863(-3) 0.1171(-4) 0.7480( -6)
0.236(-l) 0.251(-2) 0.271(-3)
0.233( - 1) 0.284( -2) 0.327( -3)
I
5 6 7
0.7487( 0.6372( 0.5410(
-5) -6) -7)
0.372( -2) 0.434( -3) 0.505( -4)
0.362( -2) 0.461(-3) 0.56 (-4)
II
5 6 7
0.1003(-2) 0.8782( -4) 0.7791(-5)
0.105(O) 0.148(-l) 0.209( -2)
0.839( - 1) 0.139(-l) 0.212( -2)
I
5 6 7
0.4396( -4) 0.4655( -5) 0.4887( -6)
0.199(-l) 0.283( -2) 0.398( -3)
0.177(-l) 0.287( -2) 0.436( -3)
Ar-N,
‘) See footnote Table 8 Comparison
[221)
to table 5.
of estimated
results for He-H,
and Ne-H,
with SCF calculations
a)
System
Geometry
RI, or R,
&Jr2
V (comb. rule)
V(SCF,
He-H2
II
4 5 6
0.4549( -4) 0.3124( -5) 0.2196( -6)
0.547( -2) 0.753( -3) 0.107(-3)
0.618( -2) 0.896( -3) 0.121(-3)
I
4 5 6
0.1182( 0.9278( 0.7310(
-4) -6) -7)
0.336( -2) 0.517( -3) 0.799( -4)
0.416( -2) 0.597( -3) 0.807( -4)
II
4 5 6
0.9263( -4) 0.6671(-5) 0.4983( -6)
0.124(-l) 0.166( -2) 0.227( -3)
0.130(-l) 0.169( -2) 0.220( -3)
I
4 5 6
0.2466( -4) 0.2028( -5) 0.1687( -6)
0.744( -2) O.lll(-2) 0.166( -3)
0.871(-2) 0.114(-2) 0.152(-3)
Ne-H,
‘) See footnote
to table 5.
[ 16,211)
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
t\
\.
i
RI, 0 R, (a. u.) R,, , RL5.R, (o. o.) Fig. 3. Combining rule results following the orientation-dependent site-site approach for different geometries. Straight lines are combining rule results, while the points are the SCF results from ref. [20]. 10-lc
I
I
I
Fig. 5. See caption of fig. 3. The SCF results are from ref. [ 221.
tal potential of Candori et al., which was not adjusted to tit this property. There is, however, still some uncertainty concerning the correct anisotropy since the rotationally inelastic scattering data, which was not available when the multiproperty potential was determined, has not yet been analyzed [ 271.
4. Discussion
R,, , RI ta.u.1 Fig. 4. See caption of fig. 3. The SCF results are from refs. [ 17-191. They are not distinguishable on the scale shown and coincide within the area of the square symbols (see table 6).
Some small deviations between the SCF results and estimates from the combining rule are seen in figs. 3-8. These could be due to the fact that the SCF information on the charge distributions for the diatomic molecules was available only for the range R,, and R, x3-5 au and therefore had to be extrapolated to large values of R,, (I )by assuming an exponential behavior. Thus the estimated values outside the range mentioned should be taken as approximations only. Presently we cannot establish whether these deviations are due to the limited information about the charge distributions of the molecules or to the combining rule itself. Moreover it should be noted that the correlation numbers K, y are nearly independent of R as long as both the SCF potentials and charge distributions are well reproduced by the assumed exponential form. For systems where this is not the case the combining rule will have to be mod-
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions 2_
3
L
R,, , R, fa.u.1 .5, 5
I
6
I
7
1
8
RI, ’ RI (a.u.1
Fig. 6. See caption of fig. 3. The SCF results are from ref. [22].
Fig. 8. See caption of fig. 3. The SCF results are from ref. [ 2 I].
ified by introducing a simple parametric dependence on R. In ref. [ 31 the difference between the primitive combining rule
Table 9 Estimated results for geometry-dependent Born-Mayer potential parameters a)
RI, , RI (a. u.1 Fig. 7. See caption of fig. 3. The SCF results are from ref. [ lk].
System
A,,
b,,
AL
b,
He-O* Ne-O2 Ar-0, Kr-0, Xe-0, He-N, Ne-N, Ar-N, Kr-N2 Xe-N, He-H, Ne-H, Ar-Hz Kr-H, Xe-H,
682 1700 1970 2860 1670 639 1650 1870 2890 1690 14.1 36.6 49.4 73.0 49.0
2.29 2.28 2.01 1.99 1.81 2.23 2.23 1.96 1.95 1.77 1.97 2.00 1.78 1.77 1.61
44.9 135 224 346 252 63.1 174 354 547 399 5.94 15.0 28.2 40.2 30.1
2.18 2.22 1.96 1.94 1.77 2.13 2.15 1.96 1.94 1.77 1.87 1.90 1.72 1.71 1.56
a) The Born-Mayer potential parameters are defined by V,,=A,, xexp(-b,,R,,)and V,=A,exp(-b,,R,,).
C. Nyeland, J.P. Toennies /Repulsive potentialsfrom charge density distributions
345
the second derivative assuming that the parameters B and /? do not depend on geometry -
present
---
exp. ICondori
et al.1
(a2(p;~?B))~=o
=,,+pB)[(!$f+$].
-II 5
I
6
8
7
9
IO
R(a. u.) Fig. 9. Comparison of orientations obtained ters of table 9 from the imental potential [25]
the entire van der Waals potential for two with the estimated Born-Mayer parameTang-Toennies model [ I] with an experobtained from a multiparameter tit.
and the combining rule used here (eq. ( 10) ) was discussed (see especially eq. (20) in ref. [ 31). It was found that the factor FAB appearing in the result for the density overlap integral is an increasing function of the difference between the /.?parameters for the charge densities of the interacting systems. In the present study it is therefore not surprising that the largest devitations ( 20-40°h for rz 6 au) are for the systems like He-N, and He-H2 for the I geometry and for the systems like Ar-Hz and Xe-N2 for the 11 geometry, systems in which the differences in /J are largest. With the availability of analytic expressions for the charge distributions and potentials we can discuss the shapes of the lines of constant charge density or constant potential for atom-homonuclear molecule interaction. First we note that the form depends upon the distances involved and upon the parameters of the exponential distributions. For a symmetrical case one can easily show that the shapes are convex for distances sufficiently far away from the two centers and concave at smaller distances. Assuming a distribution of the form
p=p,+pB=B[exp(-BrA)+exp(-BrB)l,
(12)
which is roughly in agreement with the results of refs. [ 13-I 51 shown in fig. 1, and denoting the distance between the two atoms in the molecules 6 one gets for
(13)
Here the x axis (see fig. 1) is displaced away from the line between the two centers and x=0 corresponds to r,= rg=r. The distance r* at which the transition from convex to convace occurs is given by the condition that the second derivative vanish. One gets r*= t [$/3P+ ( &J?2d4-t82)“2] .
(14)
For example for 8~3 and 6=2 one gets r*= 3.3 for this critical configuration. For r> r* the distributions are convex, for r
This project was supported in part by the Danish Natural Science Research Council. We are grateful to K.T. Tang (Tacoma, USA) and R.O. Watts (Adelaide, Australia) for valuable discussions. Also we would like to thank J. Schiifer (Garching, Germany) for providing us with his unpublished results for the H2-H2 potential and M. Faubel (Gottingen, Germany) for computing the potentials of fig. 9.
References [ I] K.T. Tang and J.P. Toennies,
J. Chem. Phys. 80 (1984) 3726. [2] K.T. Tang and J.P. Toennies, 2. Physik D 1 (1986) 91. [3] C. Nyeland and J.P. Toennies, Chem. Phys. Letters 127 (1986) 172.
346
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
[4] G.A. Kapralova, E.E. Nikitin and A.M. Chaikin, Chem.
Phys. Letters 2 (1968) 58 1. [5] T.L. Starr and D.E. Williams, J. Chem. Phys. 66 (-1977.) 2054. [6] D.H. Winicur, J. Chem. Phys. 68 (1978) 3734. [ 7 ] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 ( 1972) 3 122. [ 81 E. Clementi, Lecture notes in chemistry, Vol. 19. Computational aspects for large chemical systems (Springer, Berlin, 1980). [9] A.J.C. Varandas, Mol. Phys. 53 (1984) 1303; A.J.C. Varandas and J. Brandao, Mol. Phys. 57 ( 1986) 387; J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley and A.J.C. Varandas, Molecular potential energy functions (Wiley, New York, 1984). [lo] J.G. Parker, Phys. Fluids 2 ( 1959) 449; C. Nyeland, J. Chem. Phys. 46 ( 1967) 63; J.R. Sweet and W.A. Steele, J. Chem. Phys. 47 ( 1967) 3022. [ 111 R.M. Bems and A. van der Avoird, J. Chem. Phys. 72 ( 1980) 6107. [ 121 P.E.S. Wormer and A. van der Avoird, J. Chem. Phys. 81 (1984) 1929. [ 131 P.E. Cade, K.D. Sales and A.C. Wahl, J. Chem. Phys. 44 (1966) 1973.
[ 141 R.F.W. Bader, W.H. Henneker and P.E. Cade, J. Chem. Phys. 46 ( 1967) 3341. [15]A.C.Wahl,Sciencel51 (1966)961. [ 161 W. Meyer, P.C. Hariharan and W. Kutzelnigg, J. Chem. Phys. 73 (1980) 1880. [ 17 ] P. Habitz, K.T. Tang and J.P. Toennies, Chem. Phys. Letters85 (1982) 461. [ 181 R.R. Fuchs, F.R.W. McCourt, A.J. Takker and F. Grein, J. Phys. Chem. 88 ( 1984) 2036. [ 191 A.J. Banks, D.C. Claryand H.-J. Werner, J. Chem. Phys. 84 (1986) 3788. [ 201 R. Jaquet and V. Staemmler, Chem. Phys. 101 ( 1986) 243. [ 2 1 ] P.C. Hariharan, private communication ( 1977). [ 221 P. Habitz, private communication. [23] K.T. Tang and J.P. Toennies, J. Chem. Phys. 76 (1982) 2524. [24] J. Schafer, private communication. [25] R. Candori, F. Pirani and F. Vechiocattivi, Chem. Phys. Letters 102 (1983) 412. [ 261 M.S. Bowers, K.T. Tang and J.P. Toennies, to be published. [ 271 M. Faubel and G. Kraft, private communication. [ 281 M. Fuchs and J.P. Toennies, J. Chem. Phys. 85 ( 1986) 7062.
MODELLING OF REPULSIVE POTENTIALS FROM CHARGE DENSITY DISTRIBUTIONS: A NEW SITE-SITE MODEL APPLIED TO INERT GAS ATOMS WITH THE DIATOMIC MOLECULES Hz, N2, 0, Carl NYELAND Max-Planck-Institut
’ and J. Peter TOENNIES ftir Strijmungsforschung,
D-3400 Giittingen, FRG
Received 9 October 1987; in final form 29 February 1988
A recently developed combining rule for short-range inert gas atom-atom repulsive potentials based on charge overlap integrals is applied to atom-diatom potentials in a novel orientation-dependent site-site approximation. Very good agreement with available ab initio SCF calculations is obtained for the systems He-Hz, He-N,, He-O,, Ne-H,, Ne-N, and Ar-N,. New estimates are reported for anisotropic potentials for the other systems X-Y*, where X = He, Ne, Ar, Kr, Xe and Y2= O,, Nz, Hz, for which ab initio quantum chemical calculations are not available.
1. Introduction
sp=
With the advent of recently developed very accurate semitheoretical models for van der Waals potentials it is now possible to generate potentials for atom-atom and atom-molecule systems if the necessary ab initio potential parameters are available [ 1,2 1. As input data the dispersion coefficients and the Born-Mayer parameters from accurate SCF calculations are required. Whereas dispersion coefficients are now available for many systems, SCF repulsive potentials are not widely available and their calculation for systems with many electrons can be quite time consuming. In a recent publication [ 3 ] we presented a new combining rule for short-range repulsive intermolecular potentials based on an analysis of the short-range intermolecular potentials between fifteen pairs of the five inert gas atoms. It was found that the potentials VABfor the unlike systems can be determined from
where pA and pB are the undisturbed electron density distributions of the atoms A and B, respectively, and r is the intermolecular distance. The correlation parameters KABand yABwere shown to fulfill the following simple combining rules:
VAB=KAB (Splr2)YAS , where S, is the charge density fined by
integral
de-
’ Permanent address: Chemistry Laboratory III, H.C. Orsted Institute, University of Copenhagen, DK-2100, Copenhagen, Denmark.
0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
dr,
(lb)
K AB=(KAAKBB)"*,
(2)
Y.~B=~(YAA+YBB).
(3)
The coefficients K and y for the systems AA and BB are in turn obtained from the potentials and charge density distributions of the homogeneous-like systems using eq. ( 1) . The possibility of obtaining anisotropic intermolecular potentials from charge density distributions has been explored previously by Nikitin and coworkers [ 41, Start and Williams [ 5 ] and by Winicur [ 6 1. Also the electron gas method of Gordon and Kim [ 7 ] which attempts to predict the entire potential should be mentioned here. The site-dependent separation method described here is most similar to that of Clementi et al. [ 8 ] in which a potential between two large molecules is considered as a sum of atom-atom interactions where different bonding environments are accounted for. In our case the bonding environments are always the same, but we
(la) overlap
PAP0
B.V.
338
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
consider the influence of different geometrical configurations. On the other hand we restrict ourselves to the short-range repulsive part of the potentials. Moreover we make extensive use of combining rules and thereby avoid problems associated with the absolute determination of the potentials. Still another approach for three-particle potentials has been considered by Varandas and co-workers [ 9 1. They take the configurational dependence into account by adding a three-particle contribution to the contributions of independent atom-atom interactions. The present paper starts with a description of the orientation-dependent site-site approximation. The implementation of the newly derived combining rules is discussed next and used to analyze the repulsive potentials between inert gas atoms and some common diatomic molecules. Good agreement is obtained between estimates from this procedure and the available ab initio SCF calculations for the anisotropic potentials. A number of estimates of intermolecular potentials for systems for which no ab initio information is present are also given. Atomic units are used throughout: 1 au lengthz0.52917 8, and 1 au energy=27.210 eV.
2. Site-site description of charge distributions and intermolecular potentials The idea of approximating the anisotropic interaction between two molecules by considering the forces to be centered on specific sites in the molecules is an old one. In the usual approximations the sites are the nuclei in the individual atoms making up the molecules [ lo]. Recent ab initio SCF calculations of the short-range potential of N2-N2 [ 111 and 02-O2 [ 121 showed that this approximation is reasonably well fulfilled. In our approach for the atom-diatom interaction, the force centers are also at the atoms in the molecule but the interaction is assumed to depend on the direction of approach of the atom with respect to the bond axis of the molecule. Thus the site-site potentials are different for the linear ( 11) C,, and the broad-side ( I ) CZv orientations. This is illustrated by examining the charge distributions for the ground state of molecular oxygen [ 13- 15 ] shown in fig. 1a. The outermost contour, ~~6.1 x 1O-5 e- auP3 (e-
denotes the charge of one electron), is reproduced in fig. lb together with results of the orientation-dependent site-site separation for the two geometries ( II and I). It can be seen that by using different parameters for the two different orientations a perfect fit of the real contour is achieved at the two different geometries. This modification of the site-site approximation was suggested by examination of the recently published short-range intermolecular potentials for He-H2 [ 161, He-N, [ 17-191, He-O, [20], Ne-H2 [ 2 11, Ne-N, [ 22 ] and Ar-N2 [ 22 1. The charge densities of Hz, N2 and 0, obtained from SCF calculations by Cade and co-workers [ 13-l 5 ] also suggest a decomposition into separate spherical distributions centered on the individual atoms in the molecules in an orientational approach, as discussed above for OZ. The implementation of the orientation-dependent site-site approximation as used here is based on the following simple considerations. Since we will see that density distributions and the intermolecular potentials have the same general behaviour we consider for the moment only the densities. For homonuclear molecules the density distributions for specified orientations p,, (R,, ) and pI (R, ) are usually well approximated over the range of interest by exponential expressions pII (R,, ) =B,, exp( --&R,, ) , Pi
(R,
1 =B,
ew(-BIRI
(da) ) ,
Cab)
where R,, and R, are the corresponding distances from the center of mass of the molecule in the specified geometries (see fig. 2). Assuming only contributions from sites located in the nearest atom or atoms in the molecule eq. (4) can easily be expressed in terms of exponential site-site charge distributions. For the collinear ( II ) approach one obtains pg”(r)=Bsy”
exp( -/3-‘r)
,
(5)
where BY‘=B,,
exp( - $P,,6) ,
(6)
and Pi-“=P,,
*
(7)
Here 6 is the distance between the two atoms in the molecule and r is the site-to-site distance from the ap-
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
9 = 6.1. lo-’
339
e-Ibohr)-’
b1
al
Fig. 1. (a) SCF charge density contours for the ‘Z; ground state of molecular oxygen, from ref. [ 151. The outermost curve is for p= 6. I x 1O-5 e- au-‘. The charge density changes by a factor of two between the curves. (b) The charge density contour forp=6.1 x low5 e- auw3. (--) are compared with the orientation-dependent site-site tit for the II geometry (--) and with the site-site tit for the n geometry (--.-e).
proaching atom to the nearest atom in the molecule. Similarly one has for the _!_geometry, defining a site-site charge distribution p:_’ as in eq. (5 ), and assuming additivity of charges
pl ~3.21 exp( -2.46 R, ) .
p,_ (R, ) =2B:_S exp( -P:_“r) ,
Using eqs. (4)- (8) one gets
(8)
where r2= R: + 4s’. In this case it is not possible to write a simple analytical expression for B‘;’ and j?:_ but these constants can be easily obtained from fits toeq. (8). The procedure Qxemplified for the charge distribution of O2 using the SCF results given in table 1.
Assuming exponential charge distribution one obtains p,, =89.0exp(
-2.71 R,, ) .
pi-‘=4.05 exp( -2.71 r) , and Table 1 Ab initio charge distributions 6=R,
for homonuclear diatomics at
a’ p (e-au-‘)
RI, (au) b,
R, (au) b,
R
NZ
6.1(-S) 2.0( -3)
5.38 4.10
4.73 3.20
2.070
02
6.1(-5) 2.0(-3)
5.24 3.95
4.42 3.00
2.280
f-f2
6.1(-S) 2.0( -3)
4.28 2.84
4.16 2.64
1.449
System
W
(au)
‘) All charge distribution results belong to the same calculation of Wahl and co-workers [ 13-151. The results given for
Fig. 2. Geometries and coordinates used to describe the site-site orientation-dependent approximation for atom-homonuclear diatom systems.
p= 2.0( - 3) are from table I of ref. [ 141. The results given for p= 6.1( - 5 ) are obtained from figures of ref. [ 15 1.The values for R.,.given are also from ref. [ 141 concerning Nz and Oz. The R, value for Hz is from ref. [ 231. b, R,, and R, are the center of mass distances for which the charge density p is given in the second column, and the molecule has the equilibrium distance R,, as given in the last column.
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
340
Table 2 Exponential charge distribution parameters. The parameters B and p are defined in eq. (4) for molecules and in eqs. (5) and (8) for atoms System N2
Geometry
B
P
II
14.5 2.95 89.0 3.21 2.00 0.87
2.73 2.28 2.71 2.46 2.43 2.30
I
II
02
I
II
H*
_L
II
N
8.60 2.37 4.05 3.01 0.344 0.571
I
II
0
I
II
H
I
p;-“=fp,
~3.01 exp( -2.52
2.73 2.33 2.71 2.52 2.43 2.33
r) .
This example shows that the charge distribution parameters are indeed quite different for the two orientations. Parameters for NZ, O2 and H2 obtained from charge density data of refs. [ 13- 15 ] given in table 1 are listed in table 2. Also the separation of the anisotropic potentials between two like homonuclear diatomic molecules into site-site potentials is accomplished in a similar fashion. The results are given in table 3. The SCF results for the like molecular potentials are from Table 3 Site-site intermolecular tional site-site approach
potential data obtained from SCF results ‘)
in the orienta-
System
Geometry
A b’
b b’
Nz-N*
II
213 213 123 121 0.88 1.02
2.09 2.09 2.17 2.16 1.65 1.61
I 02-02
II I
Hz-H2
II I
aJ SCF information was taken from: [12]; H2-H2 ref. [24], V,,=9.62 X exp ( - 1.58 R, ) were obtained the molecular potentials of system
Nr-Nz ref. [ 111; 02-Or ref. exp(-1.65 R,,), V,=5.11 as the SCF equivalences for following the method of ref.
111. b’ The site-site potentials are given on the exponential form V=A Xexp( - br), where r is the site-site
distance.
Table 4 Correlation
parameters
%-N,, N,-N, G II-0 0,-G, HII+ H,-H,
(K, y) for atoms in diatomic
I,
K
Y
12.4 60.0 20.9 27.3 4.66 2.99
0.754 0.879 0.788 0.846 0.666 0.660
molecules a)
a) K, and y are defined in eq. ( 1). The correlation numbers for the geometry X,-X, are calculated by the combining rule from numbers obtained for the geometries X.-X ,,and X ,,-X ,,
refs. [ 11,12,24]. Numerical tests on the correctness of the site-site approximation compared to the SCF information used are easily carried out. The orientational site-site separation and the exponential fitting contribute an error of less than one percent in calculations of overlap integrals and estimates of potentials. The combining rule used in this paper contributes an error of up to ten percent for the atom-diatom systems considered as discussed in section 3.
3. Charge density overlap calculations of intermolecular atom-diatom potentials In the site-site approximation the determination of intermolecular potentials from charge density overlap of the site-site charge distributions for the geometries II and 1 separately reduces to a treatment similarly to the one considered in ref. [ 31 for mixed rare gas atomic systems. This is illustrated by considering the broadside (I ) approach for the atom-diatom case for which one obtains from eq. ( 1)
=K~~(S~~B/r2)Y~~+K~~(S,tac/r2)Y~~, (9) in which the site-site potentials are now given as functions of the site-site charge density overlap integrals. This separation can easily be extended to intermolecular potentials between molecules of any number of atoms. In ref. [ 31 we suggested the combining rule V**(s,lr2)=rT/AA(Sp/12)
~BB(qJr2)l”2
(10)
341
C. Nyeland, J. P. Toennies /Repulsive potentials from charge density distributions
for atom-atom interactions which is equivalent to the rules of eq. (2) and (3 ) mentioned in the introduction. This combining rule will now be applied to estimating site-site potentials for atom-diatom systems. The results will be compared with SCF calculations for the same potentials. Correlation parameters (K, y ) for the like systems obtained by use of the results in the tables 2 and 3 are listed in table 4 for the site-site potentials corresponding to different geometrical conditions of nitrogen, oxygen and hydrogen atoms in the molecules. Following the combining rule of eq. ( 10) one then obtains, using also data from the tables 1 and 3 of ref. [ 3 ] for the inert gases, results for the intermolecular potentials V,,(R,, ) and I’1 (R, ) for the mixed systems. These are shown as molecular potentials in tables 5-8 and in figs. 3-8 for the X-Y2 systems for which SCF results are available. Note that these SCF results were not used in establishing the parameters used in the combining rule calculations and provide an independent test of the procedure. The respective values for the overlap integrals divided by r* are also listed in tables 5-8. It is seen that the estimated values for the potentials of the mixed systems based on the combining rule usually come within loo/6 of the available SCF results. It should be mentioned that the estimated repulsive potentials for the rare gas atoms are based on the potential model of Tang and Toennies [ 11. According to ref. [ 1] these effective repulsive potentials include in addition to the pure SCF repulsion a small exchange dispersion term, which for the rare gas atoms was found to contribute an additional 17% to the repulsion. Thus the semi-empirical values for the
single-atom parameters were reduced by 8.5Ohto approximate the pure SCF repulsion. The ab initio results for H2-H2 [24] have been corrected for dispersion and damping following the method of Tang and Toennies [ 11. It is particularly interesting to observe in fig. 3 that three rather different SCF results, those of Jaquet and Staemmler for He-O2 [ 201, those of Wormer and van der Avoird for 02-O2 [ 121 and the partly experimental results for He-He from ref. [ 21 are seen to follow the combining rule (eq. ( 10) ) rather well. The same behaviour was observed for the other systems shown in figs. 4-8. Thus estimates of potential surfaces for similar X-Y2 systems may be obtained using the combining rules if charge distributions for the separate systems and the intermolecular potentials of the corresponding like systems are available. From data of tables 2 and 4 of this paper and of tables 1 and 3 of ref. [ 3 ] intermolecular potentials of all the fifteen X-Y2 systems where X=He, Ne, Ar, Kr, Xe and Y,=02, N2, Hz can easily be obtained. Final results for the Born-Mayer parameters of the potential Vi,and V, are listed in table 9. In fig. 9 these estimated Born-Mayer parameters have been used to generate via the Tang-Toennies potential model the full potential curves for I/,, and V, including dispersion and damping for Ar-N,. Also shown in fig. 9 are the potentials obtained by the multiproperty analysis of Candori et al. [25]. In ref. [ 261 it is shown that the spherical part of another potential, which is very similar to the present predictions, is in good agreement with second virial coefftcients suggesting that the present potential probably is in even better agreement with the equations of state than the experimen-
Table 5 Comparison of estimated results for He-O, with SCF calculations z Geometry
RI, or R,
S.lr’
V (comb. rule)
v (SC&
II
4
5 6 I
0.9941(-3) 0.5589( -4) 0.3252( -5) 0.1906( -6)
0.739( - 1) 0.741(-2) 0.764( -3) 0.792( -4)
0.605(-l) 0.691(-2) 0.712(-3) 0.67 (-4)
4 5 6 7
0.2875( -4) 0.2088( -5) 0.1494(-6) 0.1067(-7)
0.735( 0X37( 0.943( 0.106(
0.761(-2) 0.910( -3) 0.103(-3) 0.10 (-4)
_L
-2) -3) -4) -4)
a) R,, and R, are the center of mass distances and r the corresponding site-site distances. (See fig. 2.)
I2011
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions Table 6 Comparison
of estimated
results for He-N2 with SCF calculations V (comb.
‘)
[191)
Geometry
Rm,or R,
II
4 5 6 7
0.1521(-2) 0.8511(-4) 0.4866( - 5) 0.2856( -6)
0.895(-l) 0.938( -2) O.lOO( -2) 0.109(-3)
0.7650( - 1) 0.1048(-l) 0.1309( -2) 0.151 (-3)
0.7806( - 1) 0.1070(-1) 0.1333(-2) 0.163 (-3)
0.7723( - 1) 0.1071(-l) 0.1347(-2) 0.1547(-3)
I
4 5 6 7
0.4238( -4) 0.3415(-5) 0.2759( -6) 0.2267( -7)
0.127(-l) ‘0.151(-2) 0.180( -3) 0.217( -4)
0.1244(-l) 0.1723(-2) 0.222 (-3) 0.27 (-47)
0.1261(-l) 0.1758(-2) 0.228 (-3) 0.29 (-4)
0.1255( - 1) 0.1746( -2) 0.2262( -3) 0.2732( -4)
a1 See footnote
Table 7 Comparison
rule)
V(SCF,
1171)
V(SCF,
1181)
VWF,
to table 5.
of estimated
results for Ne-N,
and Ar-N2 with SCF calculations
a)
System
Geometry
RI, oral
S,/ r2
V (comb. rule)
V(SCF,
Ne-N,
II
5 6 7
0.1863(-3) 0.1171(-4) 0.7480( -6)
0.236(-l) 0.251(-2) 0.271(-3)
0.233( - 1) 0.284( -2) 0.327( -3)
I
5 6 7
0.7487( 0.6372( 0.5410(
-5) -6) -7)
0.372( -2) 0.434( -3) 0.505( -4)
0.362( -2) 0.461(-3) 0.56 (-4)
II
5 6 7
0.1003(-2) 0.8782( -4) 0.7791(-5)
0.105(O) 0.148(-l) 0.209( -2)
0.839( - 1) 0.139(-l) 0.212( -2)
I
5 6 7
0.4396( -4) 0.4655( -5) 0.4887( -6)
0.199(-l) 0.283( -2) 0.398( -3)
0.177(-l) 0.287( -2) 0.436( -3)
Ar-N,
‘) See footnote Table 8 Comparison
[221)
to table 5.
of estimated
results for He-H,
and Ne-H,
with SCF calculations
a)
System
Geometry
RI, or R,
&Jr2
V (comb. rule)
V(SCF,
He-H2
II
4 5 6
0.4549( -4) 0.3124( -5) 0.2196( -6)
0.547( -2) 0.753( -3) 0.107(-3)
0.618( -2) 0.896( -3) 0.121(-3)
I
4 5 6
0.1182( 0.9278( 0.7310(
-4) -6) -7)
0.336( -2) 0.517( -3) 0.799( -4)
0.416( -2) 0.597( -3) 0.807( -4)
II
4 5 6
0.9263( -4) 0.6671(-5) 0.4983( -6)
0.124(-l) 0.166( -2) 0.227( -3)
0.130(-l) 0.169( -2) 0.220( -3)
I
4 5 6
0.2466( -4) 0.2028( -5) 0.1687( -6)
0.744( -2) O.lll(-2) 0.166( -3)
0.871(-2) 0.114(-2) 0.152(-3)
Ne-H,
‘) See footnote
to table 5.
[ 16,211)
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
t\
\.
i
RI, 0 R, (a. u.) R,, , RL5.R, (o. o.) Fig. 3. Combining rule results following the orientation-dependent site-site approach for different geometries. Straight lines are combining rule results, while the points are the SCF results from ref. [20]. 10-lc
I
I
I
Fig. 5. See caption of fig. 3. The SCF results are from ref. [ 221.
tal potential of Candori et al., which was not adjusted to tit this property. There is, however, still some uncertainty concerning the correct anisotropy since the rotationally inelastic scattering data, which was not available when the multiproperty potential was determined, has not yet been analyzed [ 271.
4. Discussion
R,, , RI ta.u.1 Fig. 4. See caption of fig. 3. The SCF results are from refs. [ 17-191. They are not distinguishable on the scale shown and coincide within the area of the square symbols (see table 6).
Some small deviations between the SCF results and estimates from the combining rule are seen in figs. 3-8. These could be due to the fact that the SCF information on the charge distributions for the diatomic molecules was available only for the range R,, and R, x3-5 au and therefore had to be extrapolated to large values of R,, (I )by assuming an exponential behavior. Thus the estimated values outside the range mentioned should be taken as approximations only. Presently we cannot establish whether these deviations are due to the limited information about the charge distributions of the molecules or to the combining rule itself. Moreover it should be noted that the correlation numbers K, y are nearly independent of R as long as both the SCF potentials and charge distributions are well reproduced by the assumed exponential form. For systems where this is not the case the combining rule will have to be mod-
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions 2_
3
L
R,, , R, fa.u.1 .5, 5
I
6
I
7
1
8
RI, ’ RI (a.u.1
Fig. 6. See caption of fig. 3. The SCF results are from ref. [22].
Fig. 8. See caption of fig. 3. The SCF results are from ref. [ 2 I].
ified by introducing a simple parametric dependence on R. In ref. [ 31 the difference between the primitive combining rule
Table 9 Estimated results for geometry-dependent Born-Mayer potential parameters a)
RI, , RI (a. u.1 Fig. 7. See caption of fig. 3. The SCF results are from ref. [ lk].
System
A,,
b,,
AL
b,
He-O* Ne-O2 Ar-0, Kr-0, Xe-0, He-N, Ne-N, Ar-N, Kr-N2 Xe-N, He-H, Ne-H, Ar-Hz Kr-H, Xe-H,
682 1700 1970 2860 1670 639 1650 1870 2890 1690 14.1 36.6 49.4 73.0 49.0
2.29 2.28 2.01 1.99 1.81 2.23 2.23 1.96 1.95 1.77 1.97 2.00 1.78 1.77 1.61
44.9 135 224 346 252 63.1 174 354 547 399 5.94 15.0 28.2 40.2 30.1
2.18 2.22 1.96 1.94 1.77 2.13 2.15 1.96 1.94 1.77 1.87 1.90 1.72 1.71 1.56
a) The Born-Mayer potential parameters are defined by V,,=A,, xexp(-b,,R,,)and V,=A,exp(-b,,R,,).
C. Nyeland, J.P. Toennies /Repulsive potentialsfrom charge density distributions
345
the second derivative assuming that the parameters B and /? do not depend on geometry -
present
---
exp. ICondori
et al.1
(a2(p;~?B))~=o
=,,+pB)[(!$f+$].
-II 5
I
6
8
7
9
IO
R(a. u.) Fig. 9. Comparison of orientations obtained ters of table 9 from the imental potential [25]
the entire van der Waals potential for two with the estimated Born-Mayer parameTang-Toennies model [ I] with an experobtained from a multiparameter tit.
and the combining rule used here (eq. ( 10) ) was discussed (see especially eq. (20) in ref. [ 31). It was found that the factor FAB appearing in the result for the density overlap integral is an increasing function of the difference between the /.?parameters for the charge densities of the interacting systems. In the present study it is therefore not surprising that the largest devitations ( 20-40°h for rz 6 au) are for the systems like He-N, and He-H2 for the I geometry and for the systems like Ar-Hz and Xe-N2 for the 11 geometry, systems in which the differences in /J are largest. With the availability of analytic expressions for the charge distributions and potentials we can discuss the shapes of the lines of constant charge density or constant potential for atom-homonuclear molecule interaction. First we note that the form depends upon the distances involved and upon the parameters of the exponential distributions. For a symmetrical case one can easily show that the shapes are convex for distances sufficiently far away from the two centers and concave at smaller distances. Assuming a distribution of the form
p=p,+pB=B[exp(-BrA)+exp(-BrB)l,
(12)
which is roughly in agreement with the results of refs. [ 13-I 51 shown in fig. 1, and denoting the distance between the two atoms in the molecules 6 one gets for
(13)
Here the x axis (see fig. 1) is displaced away from the line between the two centers and x=0 corresponds to r,= rg=r. The distance r* at which the transition from convex to convace occurs is given by the condition that the second derivative vanish. One gets r*= t [$/3P+ ( &J?2d4-t82)“2] .
(14)
For example for 8~3 and 6=2 one gets r*= 3.3 for this critical configuration. For r> r* the distributions are convex, for r
This project was supported in part by the Danish Natural Science Research Council. We are grateful to K.T. Tang (Tacoma, USA) and R.O. Watts (Adelaide, Australia) for valuable discussions. Also we would like to thank J. Schiifer (Garching, Germany) for providing us with his unpublished results for the H2-H2 potential and M. Faubel (Gottingen, Germany) for computing the potentials of fig. 9.
References [ I] K.T. Tang and J.P. Toennies,
J. Chem. Phys. 80 (1984) 3726. [2] K.T. Tang and J.P. Toennies, 2. Physik D 1 (1986) 91. [3] C. Nyeland and J.P. Toennies, Chem. Phys. Letters 127 (1986) 172.
346
C. Nyeland, J.P. Toennies /Repulsive potentials from charge density distributions
[4] G.A. Kapralova, E.E. Nikitin and A.M. Chaikin, Chem.
Phys. Letters 2 (1968) 58 1. [5] T.L. Starr and D.E. Williams, J. Chem. Phys. 66 (-1977.) 2054. [6] D.H. Winicur, J. Chem. Phys. 68 (1978) 3734. [ 7 ] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 ( 1972) 3 122. [ 81 E. Clementi, Lecture notes in chemistry, Vol. 19. Computational aspects for large chemical systems (Springer, Berlin, 1980). [9] A.J.C. Varandas, Mol. Phys. 53 (1984) 1303; A.J.C. Varandas and J. Brandao, Mol. Phys. 57 ( 1986) 387; J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley and A.J.C. Varandas, Molecular potential energy functions (Wiley, New York, 1984). [lo] J.G. Parker, Phys. Fluids 2 ( 1959) 449; C. Nyeland, J. Chem. Phys. 46 ( 1967) 63; J.R. Sweet and W.A. Steele, J. Chem. Phys. 47 ( 1967) 3022. [ 111 R.M. Bems and A. van der Avoird, J. Chem. Phys. 72 ( 1980) 6107. [ 121 P.E.S. Wormer and A. van der Avoird, J. Chem. Phys. 81 (1984) 1929. [ 131 P.E. Cade, K.D. Sales and A.C. Wahl, J. Chem. Phys. 44 (1966) 1973.
[ 141 R.F.W. Bader, W.H. Henneker and P.E. Cade, J. Chem. Phys. 46 ( 1967) 3341. [15]A.C.Wahl,Sciencel51 (1966)961. [ 161 W. Meyer, P.C. Hariharan and W. Kutzelnigg, J. Chem. Phys. 73 (1980) 1880. [ 17 ] P. Habitz, K.T. Tang and J.P. Toennies, Chem. Phys. Letters85 (1982) 461. [ 181 R.R. Fuchs, F.R.W. McCourt, A.J. Takker and F. Grein, J. Phys. Chem. 88 ( 1984) 2036. [ 191 A.J. Banks, D.C. Claryand H.-J. Werner, J. Chem. Phys. 84 (1986) 3788. [ 201 R. Jaquet and V. Staemmler, Chem. Phys. 101 ( 1986) 243. [ 2 1 ] P.C. Hariharan, private communication ( 1977). [ 221 P. Habitz, private communication. [23] K.T. Tang and J.P. Toennies, J. Chem. Phys. 76 (1982) 2524. [24] J. Schafer, private communication. [25] R. Candori, F. Pirani and F. Vechiocattivi, Chem. Phys. Letters 102 (1983) 412. [ 261 M.S. Bowers, K.T. Tang and J.P. Toennies, to be published. [ 271 M. Faubel and G. Kraft, private communication. [ 281 M. Fuchs and J.P. Toennies, J. Chem. Phys. 85 ( 1986) 7062.