An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients

An improved simple model for the van der Waals potential based onuniversal damping functions for the dispersion coefficientsK. T. Tang and J. Peter Toennies Citation: J. Chem. Phys. 80, 3726 (1984); doi: 10.1063/1.447150 View online: http://dx.doi.org/10.1063/1.447150 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v80/i8 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients

K. T. Tang8) and J. Peter Toennies

Max Planck Institut fur Striimungsforschung, Bottingerstr. 4-8, D 3400 Gottingen, Federal Republic of Germany

(Received 17 March 1983; accepted 8 December 1983)

Starting from our earlier model [J. Chern. Phys. 66, 1496 (1977)] a simple expression is derived for the radial dependent damping functions for the individual dispersion coefficients C Zn for arbitrary even orders 2n. The damping functions are only a function of the Born-Mayer range parameter b and thus can be applied to all systems for which this is known or can be estimated. For H( IS )-H( IS ) the results are in almost perfect agreement with the very accurate recent ab initio damping functions of Koide, Meath, and Allnatt. Comparisons with less accurate previous calculations for other systems also show a satisfactory agreement. By adding a Born-Mayer repulsive term [A exp( - bR )] to the damped dispersion potential, a simple universal expression is obtained for the well region of the atom-atom van der Waals potential with only five essential parameters A, b, C6 , Cg, and CIO' The model has been tested for the following representative systems: Hz 3 I, Hez, and Ar z as well as NaK 3 I and LiHg, which include four chemically different types of van der Waals interactions for which either very precise theoretical or experimental data is available. For each system the ab initio dispersion coefficients together with the well-known parameters E and R m were used to determine A and b from the model potential. With these values the reduced potentials were calculated and found to agree with the experimental potentials to better than 1 % and always less than the experimental uncertainties. Some of the implications of the new model are discussed.

I. INTRODUCTION

The direct ab initio calculation of van der Waals potentials involves very complex time-consuming CI calculations, which have only been successfully carried out for a few systems such as H( IS )-H( IS) and He-He. 1 Essentially the difficulties arise from the errors inherent in the subtraction of the enormous energies of the separated atoms from the only slightly different energy of the interacting atoms. Because of this problem a perturbation calculation would appear to provide a better approach. In the long range region of zero overlap of charge distributions second-order perturbation calculations yield the dispersion series with constants C Zn' At intermediate distances in the well region, however, the effect of overlap and symmetry also has to be included. Z Unfortunately, the correct antisymmetrization of the wave function leads to formidable and as yet unsurmountable formal problems.3 At small distances the first-order Coulomb and exchange energy, which are accounted for in a SCF calculation, dominate the potential.

The recent availability of very accurate experimental potential data from molecular beam scattering experiments4

and spectroscopy, which have been carefully analyzed taking into account all relevant bulk data,5 and the availability of accurate long range dispersion constants6 has stimulated the application of comparatively simple models of the van der Waals potential. In 1973 one of the present authors 7 proposed a model in which the Born-Mayer repulsive potential [A exp( - bR )] with parameters from SCF calculations was

-) Pennanent Address: Department of Physics, Pacific Lutheran University, Tacoma, WA 98447.

added to the first three terms ofthe asymptotically (R-ct)) correct ab initio dispersion series

VIR ) = Vscp(R ) - ~ - ~ - C IO• (1)

R 6 R g RIO

Surprisingly this crude model was able to predict the experimentally available rare gas dimer well parameters to within a few percent. This observation has stimulated additional work by several groups in which Eq. (1) has been modified to take account of the effect of charge overlap on the dispersion potential in the region of the potential well.

Before discussing these recent refined models we first review briefly the history of ab initio calculations of damping functions, which have been introduced to account for charge overlap. In this work, largely restricted to Hz+ and Hz, the effect of exchange symmetry has been neglected. Following Hirschfelder,8 we will refer to that part of the dispersion energy, which neglects exchange as the polarization dispersion energy. The damping functions/zn (R ) for the individual dispersion coefficients are thus defined by

00

V~1.disp, (R ) = - Lhn(R }C~~R - Zn. (2) ,,>3

This modification of the dispersion potential goes back to Brooks9 (1952), who pointed out that the usual R -1 expansion is divergent for all R, but in a mathematical sense is an asymptotic expansion to the dispersion.9

\a) Brooks, Dalgarno, and Lewis,lOand Jansen 11 have attributed this unphysical behavior to the fact that the convenient R - 1 expansion of the electrostatic potential of two arbitrary charge distributions, used in the customary perturbation calculation of the

3726 J. Chem. Phys. 80 (8),15 April 1984 0021-9606/84/083726-16$02.10 @ 1984 American Institute of Physics


K. T. Tang and J. P. Toennies: A model for the van der Waals potential 3727

dispersion coefficients, is only valid at internuclear separations where the charge distributions do not overlap. The appropriate electrostatic expansions for overlapping distributions are known,12 but their introduction greatly complicates the theory.

In a first attempt at obtainingf6(R ), Brooks9 simply neglected the electrostatic potential in the region where the charge overlap is greatest. As we shall see later on, his result of an incomplete gamma function of order 2n = 6 comes quite close to the formula derived in this paper. In another simple attempt Frost and Woodson13 in 1958 used an argument similar to that used in the present work and also obtained an incomplete gamma function of order six to dampen the C6 term. Cusachsl4 also suggested that an incomplete gamma function should be used to dampen the dispersion and induction terms. In this connection it is interesting to note that Musher and Amos 15 in 1967 were able to show that in general the damping functions should be a product of a polynomial in R times an exponential in R, a form which is also compatible with a gamma function.

The first ab initio numerical calculations were presented by Murrel and Shawl6 in 1968 for H2 and greatly improved upon by Kreek and Meath 17 in the following year. Koide in 1976 introduced an elegant formulation and applied it to H( IS )-H( IS). 18 This approximate method was applied by Krauss and Neumann19 and co-workers to a number of rare-gas dimers. In 1981, Feltgen used an elaborate semiempirical scheme to estimate the parameters of the damping functions, also based on incomplete gamma functions, for H(IS)-H(IS)20 and He2.21 This important paper also contains a comprehensive review of most of the literature ~n damping functions.

Shortly after Feltgen's paper, Koide, Meath, and Allnate2 reported the first nearly exact calculations of the H( IS )-H(IS) nonexpanded dispersion energies. Their damping functions differ significantly from the earlier work and those of Koide. As we shall see shortly, this elegant calculation of Meath and co-workers has been of great benefit to the present study. Thus we now have a very good understanding of the polarization dispersion potential of one system H2 32:.

Another independent approach to the problem of damping functions is of some interest in connection with our work. Using a many-body formalism, Jacobi and Csanak23

were able to incorporate inelastic electron-atom scattering Born amplitUdes, modified to take account of overlap along the lines, suggested by Lassettre24 to derive a closed-form expression for the damped C6 term. According to Feltgen their result is consistent with that of Battezzatti and Magnasco,25 who used a different approach.

The problem of damping also occurs in the induction interaction between a charged particle and an atom: 1/2 a q;/R 4, where a is the static polarizability of the atom andq. is the charge. This was first studied for the interaction of a single valence electron, which overlaps with the outer electrons as it comes close to the atomic core. The first detailed calculations were made by Bethe27 and Reeh.28 In more simple models Callaway29 and Temkin30 use damping functions very similar to gamma functions. In yet another area, similar corrections are introduced in a more ad hoc way in modem

pseudopotential calculations of intermolecular forces of excited atoms.31-35

In addition to the polarization dispersion energy other terms are also expected to contribute to the long-range van der Waals potential. Murrell and Shaw,36 Musher and Amos,15 elaborating on an earlier study by Murrell, Randic, and Williams,37 have discussed the decomposition of the entire potential into a number of additive terms.38

V = VSCF + V~ll.diSP' + V;ntra + Vexdisp.

+ V~~h.diSP. + V~ll.disp. + higher-order terms, (3)

where the terms are in order of expected decreasing importance. 39 The third term Vintra refers to a correction in the potential energy as a result of the change in the intra-atomic correlation within the separate partners as they approach each other. For H2 it is, or course, zero. The fourth term takes account of the effect of exchange symmetry on the dispersion potential. These two terms are usually determined by comparison with an exact calculation and thus are not restricted to second order. Next we encounter an additional attractive term coming from a perturbation calculation, which is referred to as the spherical dispersion energy V~~h.diSP' .40 For H2 this term has been calculated by Koide, Meath, and Allnatt,22 who find that it is only important at distances somewhat less than the location of the well of the 3,I potential. At R = 3 a.u. the spherical dispersion term is about 35% of V~ll.disp .. Feltgen20 has analyzed and discussed the contribution of most of these terms to the H2 3,I and He2 potentials. The undamped third-order terms V~l.disp, have been accurately calculated for a number of systems including H2,41 He2' Be2, and the heavier rare gases.42,43

For systems other than H2 3 I much less is known about the various contributions. Vintra has been calculated by Reinsch and Meyer for He2 and at R m contributes about 13% of V SCF ,44 and for Ne2 about 12%.45 There is some evidence that Vintra may vary from one system to another. For example, in an early study Das, Wagner, and Wahl report values for H-rare gas systems which are only a few percent of V SCF and are positive at small distances but tum negative in the well region.46 In a more extensive study of He-H2 Meyer, Hariharan, and Kutzelnigg47 find that Vintra

is always positive. However, the individual contributions from He and H2 differ significantly even in sign. In H2 this contribution depends on the orientation, The maximum total contribution was, however, only about 4% of V SCF in the vicinity of Rm'

The fourth term Vex disp. has been extensively studied by Chalasinski, Jeziorski, and co-workers.48 For He2

49 and Ne/o it contributes 6%-8% of V SCF at Rm , corresponding to about 3% of the total dispersion potential. Chalasinski et al. also find that this term converges only slowly with perturbation order. 51 In the case ofBe2 this term is also positive and amounts to about 5% to 10% ofthe total dispersion term.52

In the HF dimer the effect is also about 10% ofthe &;persion potential. 53 Thus, on the basis of these few studies we estimate that the contributions from the third and fourth terms will be positive and amount to between 10% to at most about 20% of V SCF in the well region. Finally we caution that the decomposition of the potential into additive terms

J. Chern. Phys., Vol. 80, NO.8, 15 April 1984


3728 K. T. Tang and J. P. Toennies: A model for the van der Waals potential

and the above interpretation is an approximation, which is only expected to be valid for not too small internuclear separations.

The insight gained from these theoretical studies have thus led to modifications of the simple model of Eq. (1). The aim of these refinements has been to obtain an accurate description of the potential, which is simple and still reasonably universal, while relying on as few ab initio parameters as possible. In one modification of Eq. (1), Hepburn, Scoles, and Penco,54 and Ahlrichs, Scoles, and Penc055 account for charge overlap by mUltiplying the dispersion series by a single semiempirical damping function as in the Buckingham Corner potential. More recently in 1982, Scoles and coworkers56 have since extended their semiempirical model to incorporate a separate damping of the individual terms of the dispersion series, and in this way they are better able to account for chemical differences of the interacting atoms. Another important model using ab initio data is that of Ng, Meath, and Allnatt.57 A single semiempirical damping function, similar to that of Hepburn et al. is applied to the first three terms in the dispersion series. However, since the repulsive interaction is approximated by a simply calculated Coulomb integral this model contains one adjustable parameter, which is determined by comparison with experimental data.

The most elaborate model has been put forth by FeJtgen and applied successfully to H2 3.1,' 20 and He2.21 This model takes explicit account of additional terms in the potential discussed previously. The disadvantage ofFeJtgen's model is that it is not easily generalized for predicting the potentials of other systems. Most recently, Varandas and Brandii058

have used a semiempirical damping function for the individual dispersion terms to model the alkali dimer 3.I potentials with good success.

To take account of damping, Tang and Toennies59 in 1977 introduced an additive correction term, derived from a simple semiclassical Drude model as well as a cutoff function to limit the number of terms in the dispersion series. Their expression thus contained three terms:

V(R) = VscF(R) - I!2n(R) C2n n;;.3 R 2n

+M(b 2 - ~ )exP( - bR), (4)

where the SCF repulsive potential is again fitted to a BornMayer [A exp( - bR )] form. The second term is the second order dispersion series with the asymptotic R -independent coefficients C 2n , and the!2n , which equal one except for the smallest term, serve to cut off the series at the smallest term to avoid the asymptotic divergence of the dispersion series as suggested by Dalgarno and Lewis.1O The third term is the overlap correction term, which depends only on the repulsive range parameter b and a parameter M, which is a function of easily accessible properties of the interaction partners. The!2n (R) and the third term act together to dampen the dispersion series at small internuclear distances in the vicinity of the potential well.

This later model was shown to successfully predict the potential shape in the well region and the van der Waals well parameters of 12 atom-atom pairs including several rare gas

homonuclear and heteronuclear dimers H-He, H-Ar, and three alkali-rare gas systems. The same model was later extended to predict the anisotropy of the van der Waals potentials for H2 interacting with He, Ne, Ar, Kr, and Xe60,61 as well as for He-N2.62 For the He-H2 and Ne-H2 systems it was even possible to predict the bond distance dependence also in the repulsive region and in this way calculate the full potential hypersurface.63 In all cases the agreement with the available theoretical or experimental data was within the errors of the input data and/or the experiments. Note that in contrast to the previous work none of the parameters were fitted or adjusted.

In the previous work it has not been possible, however, to obtain a simple universally applicable analytic expression for the individual damping functions. The aim of the present study has been to modify our earlier semiclasssical potential model by using realistic boundary conditions to extend the model to smaller internuclear separations. The result is a damping function which satisfies the requirements of simplicity and universality and extends the range of applicability of the previous potential model.

The paper is organized as follows: First, we derive an expression for the universal damping function. This is tested by comparison with the most important previous calculations of damping functions. A further test is provided by comparison of the improved model potential with the best theoretical and experimental potentials. For H2 3.1,' all the input data is available and it is found that the SCF A value is too small by about 14%; the difference being due to additional terms in Eq. (3). For the systems He2 and Ar2, NaK 3.1,' and LiHg the correct values of A and b are estimated from the well parameters € and Rm and ab initio dispersion coefficients and the reduced shapes are found to agree very well with the best experimental fits. The paper closes with a discussion and assessment of the limitations of the model.

Throughout we have used atomic units, where 1 a.u. energy = 27.211 eV, = 2.1947X 105 cm- l

, = 3.1578X 105

K; and 1 a.u. distance = 0.529 17 A. The conversion factors for the dispersion coefficients are as follows: C6 :O.597

o 6 0 8 0 10 eV A /a.u.; Cg :0.167 eV A /a.u.; C IO:O.OO4 68 eV A /a.u.; C J2:O.0131 eV AI2/a.u .

II. A UNIVERSAL DAMPING FUNCTION In the potential well region the damping of the long

range potential in our earlier model [Eq. (2)], is dominated by the semiclassical Drude-model term. 59 This term was derived by using a classical perturbation calculation to take into account the additional shift in frequencies of the effective atomic harmonic oscillators due to the repulsive BornMayer potential. Since in the framework of the harmonic oscillator model these frequencies enter into the expressions for all dispersion coefficients this correction accounts, at least implicitly, for the overlap to all orders in the dispersion potential. Thus the last two terms in Eq. (4) can be equated with the damped dispersion energy:

_ ± C2;n + M (b 2 - l:!!...-)exP( - bR ) n;;.3R R

00 C2n = - Lhn(R )R 2n'

n;;.3 (5)

J. Chem. Phys., Vol. 80, No. 8,15 April 1984



where n' denotes the largest value of n in Eq. (4), for which /2" = 1. By neglecting terms with n > n' on the right side64

Eq. (5) can be solved forf2n:

I' = 1 _ m2n (b 2R 2n _ 2bR 2n -I)exp( - bR), (6) J2n C

2n

where 1::;3 m2n = M. Since the m2n are not uniquely specified, we can only assume that the general form off2n is

h.n (R ) = 1 - P2n (b,R )exp( - bR ), (7)

where P 2n is a polynomial of order 2n. We recall that this general expression is consistent with the predictions of Musher and Amos. 15 However, Eq. (7) and the related model of Frost and Woodson13 are the only approximations that require that the factor in the exponent is equal to the range parameter of the Born-Mayer potential.

In the next step we impose the following realistic boundary conditions onf2n (R):

f2n(R )-+1, R-+oo (8)

and

f2n(R}-+O + O(R 2n + I), R-+O. (9)

The secondary boundary condition requires that all derivatives ofj2n (R ) up to order 2n must be zero at R = O. This is equivalent to requiring that the dispersion energy of each term in the asymptotic expansion is zero at R = 0 and has a finite slope at R = O. In fact, it is known from the calculations of Koide,18 in agreement with Richardson65 and with the work of Bowman, Hirschfelder, and Wahl,66 that for H(IS)-H(IS) all the terms are equal to zero, except 2n = 6, 10, 14, etc.; all of which are expected to be finite but small at the origin. This constant can be included in the derivation, but is neglected here since its effect is negligibly small.67

These two boundary conditions determine the coefficients of P 2n (R ) uniquely with the final result:

(

2n (bR )k) f2n(R) = 1 - L -,- exp( - bR ).

k=O k. (10)

The right-hand side of Eq. (10) is the incomplete gamma function of order 2n + 1. Our derivation is most similar to that of Frost and Woodson forf6' 13 For the dispersion potential they used only the C6 term and requiredf6(Cy'R 6) to go to a finite value at R = O. Their gamma function was of one order smaller, but was also depending on bR. Finally we emphasize that the new damping function is entirely consistent with our earlier model in that the dominant term in both damping functions is the factor exp( - bR ). However, the new expression is easier to use since it depends only on band, moreover, accounts for the damping ofthe C6 term at small distances which had to be introduced in an arbitrary manner into the earlier model. Since the factor b describes the range ofthe overlap it is physically reasonable to expect it to enter into the damping function in a direct way.

The most discriminating test of Eq. (10) is provided by the recent accurate ab initio calculations of Koide, Meath, and Allnatt (KMA) for H( IS )-H( IS). 22 For comparison we have considered two slightly different values of b. In his study of the H( IS )-H( IS) potential Feltgen used the value b = 1.620 a.u. which he obtained from the average value of the slopes of the attractive l.Ig and 3.Iu SCF potentials of

Bowman, Hirschfelder, and Wahl66 in the region between R = 4 and 8 a.u. The purely repulsive 3.Iu SCF potential has a slighly different slope of 1.67 a.u. in the appropriate well region between R = 6 and 9 a. u. Since we are interested only in the 3.I potential this value was used in the comparison.

Figure 1 compares Eq. (10) with all the terms, 2n = 6··· (2) ... 20, calculated by KMA. The agreement in all cases is remarkably good, especially, when one considers that even these ab initio damping functions are expected to have small errors. The agreement is best for the most important terms with 2n = 6, 8, and 10, while for the higher-order terms the f2n = 0.5 values of the model damping functions are at internuclear distances which are larger by 2% for 2n = 12 and in the worst case by 7% for 2n = 20.

Figure 2 shows a further comparison with other more approximate damping functions for 2n = 6 and 8, also for H( IS )-H( IS). The present damping functions are in much better agreement with KMA than all previous approximations. The gamma function of order 2n, proposed by Frost and Woodson, is shifted to smaller R by 14%, whereas the frozen orbital results, one of the approximations considered by Feltgen,z° and Koide's approximation yield functions which are shifted to larger R by about II %. These latter approximations show similar deviations for 2n = 8.

Figure 3 shows a comparison of the present damping function predicted for He2 with ab initio predictions of Jacobi and Csanak23 based on a many-body formalism and with the damping functions of the HFIMD model20

•21 devel

oped by Feltgen.7o The agreement with the former for one of the two suggested average reciprocal radii a of the ground and excited states is remarkably good. The agreement with Feltgen's approximation appears to be not so good, which has to do largely with the fact that his damping functions also correct for higher-order effects. In Fig. 3 two semiempirical damping functions are also compared to our model.

0.5

o o 2.0 10.0 12.0 140 160

FIG. 1. Comparison of the model damping functions for 2n = 6 (2)···20 with the latest most accurate calculations of Koide, Meath, and Allnatt (Ref. 22) for H-H. For the model calculations the value b = 1.67 a.u. was used. The agreement is very good for 2n = 6, 8, and 10. For larger values of 2n deviations amounting to a shift in R by 2% for 2n = 12 and up to 7% for 2n = 20 are observed.

J. Chern. Phys., Vol. 80, No.8, 15 April 1984



1.0

0.5

';1" " .'1' "{i' //f

Ii t' 1/ {/ //1,1

/i l/

j § II II

II II // if

/,f l/ I il t /1 h ,.1

j /

/ II I Ii

I /1 / .1/

/ /I / //

,,/ .~/

f/

H - H

Frost-Woodson (958) present

Feltgen (981) KOide (1976)

FIG. 2. Comparison of the model damping functions for 2n = 6 and 2n = 8 for H2 with some earlier approximate expressions. Whereas the simple model of Frost and Woodson (Ref. 13) yields a 2n = 6 damping function, which is shifted to smaller values of R, the approximations of Koide (Ref. 18) and Feltgen (Ref. 20) yield damping functions shifted significantly to larger values of R.

"",,, ~ ./ O~~~~'~ __ Yr~~~~"~-J~~~~~ __ -L __ ~~L--L __ ~ __ L-~ __ -L~

10 2.0 3.0 10 2.0 3.0 4.0 5.0 6.0 70 aD 9.0 100 110 12.0 130 li.n 15.0 16.0

R [a.u.l

The damping functions of Douketis, Scoles, Mordetti, Zen, and Thakkar56 and the semiempirical formula proposed by Varandas and Branda058 both also agree very well with the present functions and those of Jacobi and Csanak and Feltgen.

Unfortunately, accurate damping functions are not available for other systems. The only comparisons possible are with the Koide type calculations of Krauss, Neumann, and Stevens for He2,19 Ar2,68 and Xe2.69 The published data on multipoledamping functionsx (LA,LB;R ) were converted to damping functions using the following expressions:

16 = X(I,I;R), Is = ~lK(2,I;R) + X (I,2;R)] (11)

and

1.0

0.9

0.8

0.7 2n = 6 2n= 8

0.6 f2n

0.5

0.4

110 = 2C(I,3)X(I,3;R) + C(2,2)X(2,2;R). 2C(I,3) + C(2,2)

(12)

In Fig. 4. these converted damping functions are compared to our model function. The b values needed for the present model were determined from a best fit of the available experiments as discussed in the next section. For H2, He2, and Ar2' these values are in good agreement with the several available SCF calculations for each system. From Fig. 4 we observe that there are significant deviations, which are all in the same direction as found for H( IS )-H( IS) between Koide's and the more accurate KMA· results for H2. This suggests that Koide's approximation leads to damping functions shifted to larger distances and that the present results may very well be

He-He

OJ ----- Feltgen (1982)

FIG. 3. Comparison of the model damping functions for 2n = 6, 8, and 10 for He2 with available approximations. The agreement of the model with the many-body theory of Jacobi and Csanak for a = 2.89 a.u. and with the approximate quantum chemical calculations ofFeltgen is reasonably good. The semiempirical approximations of Va rand as and Brandao and Douketis et al. are very similar to each other and to the present model.

0.2

0.1

0.0 0.0 4.0

............ Jocobi-Csanok (1975)

_._._. Vorondos-Brondoo (1982)

-- present

-- Douketis et 01. (1982)

6.0 8.0 10.0 12.0

J. Chem. Phys., Vol. 80, No.8, 15 April 1984



0.8

0.6

0-'

0.2

0.8

0.6

f2n

0.'

0.2

1.0

0.8

0.6

f2n

0-'

0.2

0.0 L...-~'-_~_~_...L.-_..L-_~_...J 0.0 2.0 40 6.0 8.0 10.0 12.0 ".0

R [a.u.1

FIG. 4. Comparison of model damping functions with calculations (0 and e) by Krauss, Neumann, and Stevens (Refs. 19, 68, and 69) based on Koide's method. In the model (-) the following values of b were used: HeHe:b = 2.40; Ar-Ar:b = 1.90; and Xe-Xe:b = 1.60.

closer to the true values than those based on Koide's approximation. Thus we conclude on the basis of the limited available data that the present damping functions are superior to all previous approximations for H( IS )-H( IS) and H~He and appear to give better results than the available approximations for the heavy rare-gas dimers.

III. THE NEW MODEL POTENTIAL

A further critical test of the universal damping functions is provided by using them in connection with a poten-

tial model to compare with experimental potentials. The new improved model potential is obtained by adding a' BornMayer repulsive term to the damped dispersion potential:

VIR ) = A exp( - bR )

co [ ( 2n (bR )k) ] C - L 1 - L -,- exp( - bR ) 2;n' n;.3 k=O k. R

(13)

where A and b are effective values of the repulsive potential. As we shall see shortly, A will in general be greater than A SCF' and b will be nearly equal to b SCF' where A SCF and b SCF are the Born-Mayer parameters fitted to the SCF potential V SCF = A SCF exp( - b scFR ). The reduced form isobtained by setting x = R IRm and U = VIR )IE, where V(Rm)= -E.

U(x) =A * exp( - b *x)

00 [ 2n (b*X)k ]ct - L 1- L ,exp( -b*x) --i-, (14) n;.3 k=O k. x

where the quantities with asterisk denote the following reduced potential parameters:

A * =AIE,

b* = bRm,

C C* __ 2_,,_ 2" - R 2,,'

E m

(15a)

(15b)

(15c)

Equation (13) contains as potential parameters, A, b, and C6,

Cg, C IO' C12' etc. Of these, C6, Cg, and C IO are available for well over 70 different combinations of atoms. Most of the available data is summarized in the extensive tables of Tang, Norbeck, and Certain.6 Fortunately, it is not necessary to have accurate information on the higher-order terms since these can be estimated, using the semiempirical recursion relationship

(C2,,+2 )3

C2,,+4 = --c;:- C2,,-2' (16)

presented in our earlier work.60,71 We have tested this rela

tionship in the case of H( IS )-H( IS) and have found it to predict the theoretical values for CI2> C)4t and CI6 from C6,

Cg, C IO and successive use ofEq. (16) to better than 4% with the largest error for CI6 (see Table 1).72

Because of the extreme precision of the data for the several systems to be discussed below and the sensitivity of the model the determination of the parameters A and b, to be used in the model, however, poses a more serious problem. As we shall see shortly, wherever accurate SCF data is available it is usually not repulsive enough to provide a good fit to the experiments indicating nonnegligible repUlsive contributions from the other terms in Eq. (3). Such contributions were not noticed in the earlier models presumably because of a cancellation of errors and also because the data, available at that time, was not sufficiently accurate. The use of the Born-Mayer parametrization of the effective repUlsive potential, however, does not appear to be a serious source of error. This probably can be explained by the fact that we are only interested in the region of internuclear distances near R m , which is quite narrow since the potential rises very steeply.




TABLE I. Summary of the potential parameters used in evaluating the H2 3 I potential. The numbers in parenthesis are powers of ten.

Ab initio dispersion constants C6 6.499" 6.499' ~ 1U~ 1U~ C lO 3.286(3)' 3.286(31" CI2 1.215(51" 1.198(51b

C,. 6.061(61" 6.025(6Ib

C'6 3.938(81" 4.183(8Ib

E 2.05( - 51" 2.05( - 5( ~ 7N ~W

Predicted Born-Mayer parameters A 9.30d 9.36d

(7.97Ie

b 1.664d

(1.6661' 1. 666d

a Exact ab initio values from Koide, Meath, and Allnatt (Ref. 221. The values of Bell are in agreement to four significant figures.

bValues estimated, using the recursion relationship [Eq. (161J. C Exact ab initio values from Kolos and Wolniewics (Ref. 771. d Predicted from the input data using the formula in Appendix A from the

data listed above. e Best fit in the range 6-9 a.u. of the SCF data of Bowman, Hirschfelder, and

Wahl (Ref. 661.

Because of the difficulty in determining a priori A and b values we shall use Eq. (13) in conjunction with the known dispersion coefficients to calculate effective values of A and b from the latest experimentally established values of E and R m,. The formulas used for this inversion are given in Appendix A. The errors involved especially in the case of A depend very sensitively on the errors in Rm .73 Fortunately, for many of the systems discussed here the errors in Rm are only of the order of 0.1 %. This means, nevertheless, that the errors in A are of the order of 10%. The predicted much stronger dependence of A on small changes in E and R m ,

compared to the exponential factor b means that the effective value of A will differ most from the value from SCF calculations, whereas b will be nearly the same as the SCF predictions. Once A and b have been determined in this way, then the reduced form is used to predict the shape of the potential. In the following we will compare such "experimental" A and b values with SCF data to determine the size of the additional repulsive terms. Also we will compare the predicted shapes with the experimental shapes.

IV. COMPARISONS WITH OTHER ACCURATE POTENTIALS

A. H2 3~

This is the only system for which an exact ab initio potential is available within the Born-Oppenheimer approximation. Moreover, all the dispersion coefficients up to 2n = 20 are available from several essentially exact ab initio calculations,22.74 which for the first five terms agree in the first four significant figures. Since we have already tested the damping functions for this system and found them to be very accurate, the comparison of the entire potential with the ab initio potential provides a very critical test of the BornMayer parametrization of the repulsive contributions.

Table I summarizes the input data and the predicted

...., ::j c:i ..... >. ~ <II c CII

.g c

1!! If

10-3

, , , , • , , , , , ,

10-4 , , , • ,

SCF)',

10- 5

• Bowman etal.

present " (b= 1.666a.u.1 , , , ,

• , , Rm " , , , , ,

"' • , , , 10-6L-__ ~ ______ ~ ______ ~ ______ ~ ___ '~

6 7 8 9

R [a.uJ

FIG. 5. Comparison of the H2 3 I SCF potential calculated by Bowman, Hirschfelder, and Wahl (Ref. 661 with the repulsive potential determined via the present model from the dispersion coefficients and the ab initio E, Rm values. The slopes of both potentials are nearly identical; but the semiempirical A value is larger by 17% (see Table II-

values for A and b as well as the corresponding fits ofthe SCF data. Figure 5 compares the repulsive potential predicted from the exact values of E and R m with the SCF data reported by Bowman, Hirschfelder, and Wahl.66 The comparison reveals that the predicted repulsive potential lies parallel to the SCF potential, but is shifted upwards by 16.7% compared to the SCF potential. This is direct evidence that for H2 3 ~ the additional repUlsive terms contribute an exponential repulsion given by 1.33 exp( - 1.666 R ). This can be compared with Feltgen's analysis20 in which he finds an additional repulsion from Vex.disp. [see Eq. (3)] of 0.888 exp( - 1.620R ). Thus the difference between these two determinations is due largely to the third-order dispersion contributions, explicitly considered by Feltgen. Thus the results of both these models are in good agreement. They can be compared with the 1974 analysis of Kolos,75 who used a larger basis set than in Bowman's previous related study.76 For H2 3~, Kolos found that neglect of second- and higherorder exchange and third- and higher-order polarization interactions leads to a potential, which at R = 8 a.u. was too negative by about 13%. At Rm the repUlsive potential is about 1/2 of the attractive potential (see Fig. 7), but opposite in sign. Thus the repulsive part of the potential is nearly equal in magnitude to the total potential and Kolos' result therefore implies that the repulsive potential is too small by about 13% relative to the correct potential. The difference between our best-fit repulsive potential and the ab initio SCF potential is 14% (relative to the best-fit potential) in good agreement with Kolos. Kolos finds that the neglect of higher-order dispersion terms leads to only a 3% more negative potential. Thus our model is consistent with the most accurate calculations for this system.

With the predicted effective Born-Mayer repulsive pa-




2

VIR) [10-5a.u.l

o

- I

-2

liD

H - H 3r ~

1 ..____--\ ~" /-

J~'''O' 7.0 8.0 9.0 10.0 110 12.0

R [a.u.l

FIG. 6. Comparison of the potential predicted by the model, using effective A and b values (see Table I) with the ab initio potential of Kolos and Wolniewicz (Ref. 77). The agreement is better than 0.5% in the attractive region.

rameters Eq. (13) was used to calculate the absolute potential for H2 3~. The predicted curve is compared with the exact ab initio results of Kolos and W olniewicz 77 in Fig. 6. The agreement is seen to be excellent and the differences in the negative potentials is smaller than 0.5%. In the repulsive region at R = 6.8 a.u. the differences are greater and the ab initio point is below the predicted curve by less than 10%. This apparently large discrepancy corresponds to a shift in R of only 0.014 a. u. or a relative error in Ro (V (Ro) = 0) of only 2X 10-3 which is entirely consistent with the expected errors in the fit of the SCF data and can be reduced considerably by using a three-parameter fit of the SCF data.

Figure 7 shows the relative magnitude of the contributions to the total van der Waals potential of H2 3 ~ as a func-

1,0

0.9

0,8 total c: repulsion_ Q :; 0,7 .0 .;:

'E 0

0,6 u .. 0.5 .~ c O.~ 0;

a:: 0,3

0,2

0.1

o 0,5 1,0 1.5

~otal dispersion

H2 'E

Relative Contributions to the total

van der Waals Potential

2Jl

X=R/R m

2.5 3,0

FIG. 7. The relative magnitudes of the contributions of various terms to the total vander Waals potential are plotted as a function of R. Note that in fact the repulsive and attractive contributions are of opposite sign and that only the relative absolute contributions are shown.

tion of R. The contribution from the total attractive secondorder damped polarization dispersion energy and the contribution from the leading term are seen to fall off rapidly in the vicinity of the well region. This decrease is compensated for by the increase in the absolute contribution from the repulsive potential. At Ro the contributions from the repulsive and attractive potentials are, of course, equal and each contributes 50% to the total, which at this point is zero. At Rm (x = 1.00) the attractive contribution is about 2/3 and almost exactly twice the repulsive contribution of 1/3. Note furthermore that the attractive potential still has appreciable contributions in the steeply rising repulsive region. In using Fig. 7 it must be kept in mind that the signs of the repulsive and attractive contributions are in fact opposite.

B.Hez

He2 is probably the atom-atom system which has been studied most extensively theoretically as well as experimentally. Two very accurate refined potentials based on the combined evaluation of both bulk and molecular beam scattering data have been recently proposed by Aziz and co-workers 78

and Feltgen and co-workers.21 This work and related studies of the other rare-gas homogeneous and heterogeneous dimers is summarized in Aziz's latest review.79 Having only four electrons in closed shells He2 is also ideally suited for a theoretical study. Probably the best ab initio study is the unpublished CI calculation of Liu and Maclean. 80

Table II summarizes the input data, the predicted values for A and b as well as the corresponding fits of the SCF data and lists also the reduced parameters. Figure 8 shows a comparison of the Born-Mayer repulsive potential determined from € and Rm with the SCF results off our different

TABLE II. Summary of the theoretical and model-predicted potential parameters for He2.

Ab initio dispersion constants

1.461a

14.1l' 184.0"

3.239(3)b 7.703(4)b 2.473(6)b

Experimental well parameters E 0.340( - 4)"

(0.342( - 4))d (0.334( - 4))<

Rm 5.622c (5.607)d (5.613)'

Predicted Born-Mayer parameters A 22.16

(l9.58)f b 2.388

(2.399),

• Calculated by Koide, Meath, and Allnatt (Ref. 83). bEstimated from the recursion relation [Eq. (16)]. c Determined from molecular beam scattering experiments by Feltgen et al. (Ref. 21).

d Determined from a best fit of many experiments by Aziz et al. (Ref. 78). 'Determined from differential cross sections by Burgman et al. (Ref. 84). fFrom best fit of all the SCF data plotted in Fig. 8.




:i c '-'

c c CII

~

\

10-5

\ He - He

\ \

\ \

\

~ \ \

\

\ , \

\ \ q

present \ \ \ '~( b = 2388 a.u.)

SCF/q ,

• Bertoncini-Wahl (1973)

o Mc Laughlin-Schaefer (1971) D Liu- Mc Lean (1973) 6 Dacre (19791

6 R [Q.u.]

7

FIG. 8. Comparison of the H~ SCF potentials calculated by Bertoncini and Wahl (Ref 85). McLaughlin and Schaefer (Ref. 86). Liu and McLean (Ref. 87). and Dacre (Ref. 88) with the repulsive potential determined from the dispersion coefficients and the € and Rm values of Feltgen et al. (Ref. 21). The slopes ofthe SCF and semiempirical potentials are nearly identical but the semiempirical A value is larger by 13% (see Table II).

calculations; all of which lie on a common straight line in a log plot. Here again we note that the slopes of the two curves are virtually identical, but that the experimental repUlsive potential is larger than the SCF potential by about 13% in rough agreement with Feltgen's compilation, the difference can be traced mainly to the damping function used by him (see Fig. 3). From the differences in Fig. 8 the total effective correction is given by 2.6 exp( - 2.39 R ). This difference in the repulsive potential is expected to be due largely to an intra-atomic correlation correction which, according to Feltgen, for He2 is twice the exchange dispersion term. Chalasinski and Jeziorski48,81,82 have calculated the exchange dispersion term and their points can be fitted to an exponential with slope similar to that of the SCF repUlsion. Thus these established additional repulsive contributions are predicted to have about the same exponential R dependence as the SCF repulsion in agreement with the present findings.

With the predicted effective Born-Mayer repulsive parameters Eq. (14) was used to calculate the reduced potential shown in Fig. 9. The shape obtained in this way deviates so little from the previous determinations that it can only be observed by calculating the differences

LlU(x) = U(x) - Urer!x),

1.2

1.0

O.B

0.6

~Iw 0.4

" :g 0.2 =>

:§ a ~ 8. 0.2

.., 0.4 ..

u

" ~ 06 It:

O.B

1.0

1.2 OB

1 • I

: IIU(xl.present I

I IIU(xl.Feltgen et a1.(1982) I

Hp - He

,-< II U Ix). Burgmann .. , a1.11976)

,-~.j .. , ...... >.. -------....... _---

U(xl. Aziz et a1.11979)

1.0 12 1.4 1.6. 1.8

Reduced dis\ancp x = R/Rm

0.05

0.04 :g => <l

0.03 '" :§ 0.02 C ..

'0 c.

0.01 .., .. u

" 'C ~ C

- 0.01 .. .. ~ a;

-OD2 .0 .. u c

- 003 .. ~

-004 6

-0.05 2.0

FIG. ? The heavy solid line shows the He2-reduced potential (left ordinate) of AZIZ et al. (Ref. 78) as a function of the reduced distance. The other curves cOI~pare the difference potentials U(x) - U(X)Aziz in reduced units (right ordmate) of the present model (-) with the most recent accurate potential of Feltgen et al. (Ref. 21) ( ... ) and an earlier determination based on the MSVSV model by Burgmann et al. (Ref. 84) (---). The present potential shows nearly the same deviations to the Aziz et al. potential as the accurate Feltgen et al. potential.

w~ere Urer!x) is a reference potential. Ll U(x) is also shown in Fig. 9, where, as reference potential, we have used the potential ?f Aziz and co-workers 78 instead of the more recent potential ofFeltgen and co-workers21 since the former has been fitted to an analytic expression. It is gratifying to see that our model potential differs from the Aziz potential by < 0.3% in the region of attractive forces and that overall the deviations of our model potential follow quite closely those of the more recent Feltgen potential. From these comparisons we can conclude that our model potential is certainly as close to the true potential as the best previous empirical and semiempirical potentials.

C. Ar2

More effort has probably gone into the experimental determination of the Ar2 potential than for any other system_ The investigations have made extensive use of gas phase virial coefficient data, transport coefficients, spectroscopic vibrationallevels, solid and liquid state data, as well as a large number of beam scattering measurements. Because of its relatively large reduced mass and deep well the beam data is, compared to the very quantum mechanical system He2' especially sensitive to the potential shape. The 1971 Barker Fisher Watts (BFW) potential89 has now been replaced by the 1977 Aziz and Chen potential. 90 In this latter work the HFD potential model, suggested by Ahlrichs et 01.,55 was used. More recently additional potentials have been suggested by Koide et 01.,91 using the one-parameter model of Ng, Meath, and Allnatt,57 and by Douketis et 01. 56 These are very similar to the potential of Aziz and Chen and, according to the recent comprehensive analysis of Aziz, 79 have no significant advantages in predicting any of the observed properties.

Table III summarizes the input data, the predicted values for A and b, as well as the corresponding fits of the SCF




TABLE III. Summary of the theoretical and model-predicted potential pa- 10- 1 .--------------------...,

rameters for Ar2.

Ab initio dispersion constants Co 67.2" C. 1.61O(3)b CIO 4.270(4)" C'2 1.254(6)C C'4 4.074(7)C C'6 1.466(9j<

Experimental well parameters E 4.54( _4)d

(4.54( - 4))< (4.52( - 4)(

Rm 7.1Od

(7.095)" (7.097)'

Predicted Born-Mayer parameters A 385.08

(328.85)8

b 1.917 (1.918)8

"Co and CIO are the mean values of the upper and lower bounds, given in Tang, Norbeck, and Certain (Ref. 6).

bThis C. is greater than the mean value, obtained as in footnote a as suggested by the work of Colbourn and Douglas; J. Chern. Phys. 65, 1741 (1976). C. was chosen such that the relative increase in C6, C., and CIO is consistent with the average behavior of the other rare gas dimers. The value listed is still less than the upper bound of Tang et al. (Ref. 6).

CEstimated from the recursion relation [Eq. (16)]. d Reference 90. < Reference 91. fReference 56. 8 From best fit of SCF data, plotted in Fig. 10.

data and lists also the reduced parameters. Figure 10 shows a comparison of the Born-Mayer repulsive potential, determined with the model potential from the experimental values of Aziz and Chen for E and R m with several ab initio SCF calculations. The dashed line in Fig. 10 connects the points calculated by Ahlrichs et 01.55 and Wahl,93 which appear to be more accurate than the calculations of Wadt.94 As observed for H2 3.I and He2 the slope of this curve is nearly identical to that of the empirical curve although the SCF errors are larger here. The difference potential is given by 56.2 exp( - 1.918R) and amounts to 17.1% ofthe SCFpotential. We are not aware ofany ab initio calculations of the intra-atomic correlation and exchange dispersion corrections for Ar2. It is interesting to note that the relative correction is of about the same size as for Hez and H2 3.I.

Figure 11 compares the predicted potential shape of the present model with the Aziz and Chen potential. In the attractive region the largest differences in reduced potentials are only of the order of 0.003 or about 1.5% of the total potential energy in the region of x ~ 1.4. For comparison we see that these differences are much smaller than those for the more elaborate BFW potential. The large deviations in the difference potential in the repulsive region are much smaller when compared to the actual potential, which is rising rapidly with decreasing distance. Finally, it is interesting to compare Fig. 11 with Fig. 7 of our earlier paper,59 where we compared our previous model with the BFW potential. The difference between the new potential model and the BFW

10-4

, , '. , , , SCF~~

Ar - Ar

best fit '\: ' present , ,

'0 \

\

\ ,

• Ahlrichs et at. (1977)

o Wahl

o Wadt (1979)

\

D

10-5L---S~------6L-----~7L-----~8~~

R[a.u.J

FIG. 10. Comparison of the Ar2 SCF potentials calculated by Ahlrichs et al. (Ref. 55), Wahl (Ref. 93), and Wadt (Ref. 94) with the repulsive potential determined from the dispersion coefficients and the empirical E and Rm values of Aziz and Chen (Ref. 90). The mean slope of the SCF potentials is nearly identical to that of the semiempirical potential, but the semiempirical A value is larger by 17% (see Table III).

potential and the previous potential and BFW potential are very similar. This is expected since the differences in both models are rather small in the long range region. In other words the potential shape for Ar 2 predicted by us in 1977 has

12 0.05

1.0 0.04 Ar- Ar ~

0.8 ~ llU(xl. present 0.03 <J

~w 0.6 C/l

:2 0.02 C

0.4 i 3 I 0.01 ~ 0.2 ..., ~ I ..

v .-.. ~ a :::J C r ............................... ." ..

\ ~

(5 - 0.2 I /' c: a.

\ / --<'AU (xl, Barker-Fisher--om

~ ." I .. -0.4 Watts (19111 Jl v

I _/' -0.02 :::J ." .. -0.6 I .. a:: v

-0.03 c:

I ~ -0.8 .. ~ U(xl. Azjz and Chen (19111 ~ -1.0

I -0.04 is I

-1.2 -0.05 0.8 10 1.2 1.4 1.6 1.8 2.0

Reduced distance x = R/Rm

FIG. 11. The heavy solid line shows the Ar 2-reduced potential (left ordinate) of Aziz and Chen (Ref. 90) as a function of reduced distance. The other curves compare the difference potentials U(x) - U(X)Aziz in reduced units (right ordinate) of the present model (-) and the empirical potential of Barker, Fisher, and Watts (Ref. 89) (---). The shape of the present potential agrees very well with that of Aziz and Chen (Ref. 90).




since been confinned by the recent elaborate data evaluation of Aziz and Chen.

D.NaK3.l"

The 3 ~ state of the alkali dimers offer another chemically different class of systems with which to test the new model. Until recently very little was known about these potentials. This has changed with the observation oflaser-induced ftuorescencefrom thed 3111 stateintothea3~ + stateinNaK by Breford and Engelke.95

-97 Ordinarily, spin conservation

forbids transitions to the triplet state from the ground singlet state; however, here the laser-excited D III potential is strongly coupled to the nearby d 3111 potential, enabling the ftuorescence transitions to occur. Thus Breford and Engelke were able to detennine an RKR potential up to high energies very close to the dissociation limit for NaK 3 ~ +.

Table IV summarizes the input data used to predict the shape of the potential. Unfortunately in this case we have no SCF data with which to compare the effective A and b values. Figure 12 shows a comparison of the predicted shape in the well region with the RKR potential. The overall agreement is very good over the entire range. However, a close examination of Fig. 12 reveals that for x < 1 the fitted potential lies somewhat above the RKR potential. This disagreement is consistent with that observed previously in the other systems and suggests that the Born-Mayer fit of the repulsive potential may have to be modified by including another constant in the repulsion. As a further check on our model we can also compare our predicted repulsive potential for the 3 ~ + state with that detennined spectroscopically from the observation of "forbidden" structured emission from the v = 12 vibrational state of the D II state into the repulsive part of the a 3 ~ + potential.98 The experimental potential was fitted to

V(R) = Vo + (2.24 ± O.036)exp( - O.8414R);

6.9 <R <9.6 a.u. (17)

TABLE IV. Summary of the theoretical, empirical and model-predicted potential parameters for NaK a 3 I +.

Ab initio dispersion constants ~ MI~r C. 2.2905(5r CIO 2.468(7)" C'2 3.015(9t C'4 4.l75( Il)b C'6 6.555(13)b

Experimental well parameters E 0.9263( - 3)C Rm 10.300"

Predicted Born-Mayer parameters A 3.829 b 0.7573

Experimental Born-Mayer parameters A ~~±QOO~ b 0.84l4d

"Mean values of upper and lower bounds, given by Tang, Norbeck, and Certain (Ref. 6).

bEstimated from the recursion relation [Eq. (16)]. c Reference 95. d Reference 98.

0..2

DO.

~",-D.2

" x -0.4

=> "0 ~ -0..6 CI> (5 Co

a! -0.8 u ::l "0 CI>

a:: -1.0.

-1.2 0..6

No - K 31

.-' ~

.---.--/-----, /"

i / " I I /---present . l

I--RKR'B~f"d """ .,,,,,0.. '"'" 1 I i " I \ " \ / " . \ / 'v'

10. 18 2.2 2.6 3D


FIG. 12. Comparison of the Na-K 3I-reduced potential predicted by the model using effective A and b values (sse Table IV) with the experimental RKR potential of Breford and Engelke (Ref. 95). The overall agreement is very good. Some small discrepancies are noticeable; the repulsive region (x::::::0.8) and near x:::::: 1.4.

where Vo is the potential energy above the ground vibrational state. This potential can only be a rough approximation in this range since it includes the point Ro. In this region it also disagrees with the RKR potential proposed by the same group. With this in mind the agreement is satisfactory (see Table IV) since the slope of Eq. (17) is quite similar to our Born-Mayer repulsive part.

There is also a slight disagreement in the long-range region, which is of the order of 1 % of the total potential energy for 1.3 S x S 1.8. A comparison of predicted and measured vibrational levels differs only by S 3 cm - I. This cannot be attributed to the repulsive potential since its contribution is much smaller in this region. Possibly the discrepancy comes from small inaccuracies in the higher-order dispersion coefficients. Another possible explanation is that a higher lying 3ll state distorts the 3~ potential downwards in this region.99 The RKR data extends out to internuclear separations of 32 A and the agreement with theory in this region (not shown in Fig. 12) is of course very good. Thus overall it is very gratifying to see that the theory is also able to predict the quite different shape of this potential compared to that of H2 3~, He2' and Ar2•

E. LlHg

As a final test of the new model we have chosen the system LiHg for essentially two reasons: (1) With two electrons in the outer shell the I So Hg-mercury atom fonns with the 2S1/2-Li atom a 2~ + van der Waals bond, which has a different multiplicity from the previous examples. LiHg also shows an anomalous behavior both with respect to shape and




TABLE V. Summary of the theoretical, empirical and model-predicted p0-

tential parameters for LiHg 2 ~ + •

Ab initio dispersion constants C6 4.43(2)' C. 2.54(4)' CIO 1.84(6)" C12 1.68(8)b C'4 1.95(IO)b C' 6 2.85(12f

Experimental well parameters E 0.417( - 2)" Rm 5.60"

Predicted Born-Mayer parameters A 2.246 b 1.012

• Calculated from polarizabilities and effective energies listed in Maeder and Kutzelnigg (Ref. 102). These results are in excellent agreement with values listed in the thesis of Maeder (Ref. 103).

bEstimated from the recursion relation [Eq. (16)]. cReference 100.

magnitude of the well parameters compared to the other alkali-mercury systems. (2) The potential has been studied by two independent sets of molecular beam experiments 100,101

and reasonably good agreement was obtained, The input data and the predicted Born-Mayer con

stants of the effective repulsive potential are summarized in Table V. Figure 13 compares the predicted potential shape with the measured potential. Once more the agreement is very good, even though this potential has the widest bowl of

1.2

1.0

0.8

li - Hg

0.6

:Elw > D.4

" :B :J 02

;; ~ II 0 Q.

1: ·0.2

u

j ·0.4 D::

-0.6 \ Buck et al. (1974)

-0..8 V -1.0.

-1.2,,::--;:;;-:::---;;;:--:,::---:,:---:_":-~_'--~~_~--' 0.6 0.7 0..8 0..9 10. 1.1 .2 1.3 '4 1.5 1.6 17 1.8 1.9

Reduced distance x = RlRm

FIG. 13. Comparison of the LiHg-reduced potential predicted by the model using effective A and b values (see Table V) with the experimental potential of Buck et 01. (Ref. 100). The agreement is good except in the long-range region where small errors in the experimental potential and the dispersion constants may be apparent.

all those investigated. The greatest discrepancies are found in the region of x~ 1.5. In this region we cannot rule out some errors in the experimental potential. Moreover, part of the discrepancy may be due to slight errors in the dispersion coefficients.

v. DISCUSSION

In the present paper we have derived an expression for a universal damping function for the dispersion coefficients. The damping function was tested first by comparing where possible with previous ab initio calculations. By adding a simple Born-Mayer repulsive term to the damped dispersion potential we obtained a simple model for the van der Waals potential in the well region. With this model we can predict the potential well shapes for specific systems and compare these with the most accurate presently known potentials. The excellent agreement found provides additional evidence for the accuracy of the universal damping function expression.

Figure 14 compares the calculated reduced potentials for the three systems which show the greatest differences in shapes Ar2 1.2', NaK 3.2', and LiHg 2.2'. Of these, Ar2 shows the narrowest bowl and is quite similar to H2 3.2', which is therefore not shown. In fact, the simple Lennard·Jones (12-6) potential is quite similar to the Ar2 and the other rare gas dimer potentials. The NaK 3.2' system shows a wider bowl, and the LiHg system shows the widest bowl of all the systems. The present model successfully predicts these differences from the ab initio dispersion coefficients and established well parameters. Thus the model can help us to understand what factors determine the differences in shape. For this purpose all the reduced parameters for the five systems studied are collected in Table VI. First we observe that the reduced dispersion coefficients reveal quite a different convergence behavior for each system at the minimum (x = 1). For LiHg the coefficients increase steadily for all n, whereas for NaK 3.2' this occurs for n ;;;.4, for H2 3.2' and He2 the coefficients decrease for n<.7, and for Ar2 for n<.9. In the bottom half of the table we show the damped contributions. Here we see that the values are much more similar among the different systems and show a satisfactory convergence. This is, of course, the effect desired from the damping functions.

Next we can examine the relative contributions from the attractive and repulsive potentials. At the minimum the contributions from both parts in reduced units are related by

(18)

From the values of.2' Uatt (1) listed at the bottom of Table VI, which are all greater than about 1. 7, we see that for these representative systems Urep (1) ~ 0.7, the exact magnitude depending on the chemical nature of the system. From similar arguments we can understand how the different shapes of the various species come about. The derivations of the equations in Appendix A for calculating A and b from E and R m

are easy to interpret in terms of reduced energies. In addition to Eq. (18) we have a second condition at the minimum given by




1.2

1.0

0.8

0.6

~w "

0.4

~ ::J 0.2

:2 C 0 GI '0 a. -0.2

" GI U ::l -0.4 "

- -:::::.'"; .... .... ..............

...... .<U-H9 ....

GI Q:

-0.6

-0.8

-10

-1.2 0.5 10 1.5 2.0 2.5


FIG. 14. Comparison of the potential shapes of different systems which represent three different types of van der Waals bonds. The model successfully predicts the different shapes which are due largely to differences in the reduced e: values.

d d dx Urep(x)lx= I - dx ~ Uatt(x)lx= I = O. (19)

Equations (18) and (19) indicate, as stated earlier, that the magnitude and slope of the attractive potential at the minimum determines A and b. Since the behavior of the attractive potential is largely determined by C: and to a lesser extent by C: and C To, these parameters determine the entire

TABLE VI. Comparison of the reduced parameters.

H23I He2 Ar2

q 1.388 1.361 1.152

ct 0.434 0.461 0.547

CTo 0.188 0.172 0.288

CT2 0.112 0.096 0.167

Cr. 0.093 0.072 0.108

CT. 0.098 0.073 0.077

cr. 0.171 0.107 0.061 A* 4.57(5) 6.52(5) 8.483(5) b* 13.02 13.43 13.618 J:(l)q 1.352 1.334 1.131

fW)ct 0.390 0.423 0.506

ITo(I)CTo 0.141 0.135 0.229

IT2(I)CT2 0.060 0.056 0.101

IT.(l)CT. 0.030 0.027 0.042

fr.(I)CT. 0.016 0.014 0.016

IT. (l)er. 0.008 0.007 0.006 ,. Li!.(I)Q" 2.00 2.00 2.03 2"

shape. From Table VI we observe that C: for Hz, He2, and Ar2 are about the same and on the average 1.3, whereas for NaK, C: = 2.17, and for LiHg C: = 3.36 following the trend in the observed width of the potential bowl (see Fig. 14). As pointed out previously, C6 and Cg are presently available for a large number of systems.

An important result of this work is the observation that the repulsive contributions of the exchange dispersion, intraatomic correlation and higher-order dispersion terms have the same R dependence as the SCF repulsion:

v.rue - (VSCF + V~~l.diSP') = a.A SCF exp( - bR ), (20)

where Vtrue is the true potential. The proportionality factor a for H2 3~, Hez, and Arz has been found to vary between 0.13 and 0.17. This has been rather well confirmed by additional comparisons of the other rare-gas dimers and H-rare gas systems for which we get a weighted average of about 0.19. As noted in Sec. III the uncertainties in the factor a in Eq. (20) are expected to be large because of the large influence of small errors in E6 and Rm and thus this result must presently be treated with considerable caution. More work is needed to extend our semiempirical understanding of this correction. Thus at the present time the model can be used as an instrument to predict the absolute potential curve from ab initio SCF and dispersion coefficient data only if we assume that a constant correction factor for A SCF holds for, say, all systems of a given type. In our previous work we were not able to discern any corrections to the SCF Born-Mayer parameters. On reexamination of our previous model [Eq. (4)] we now realize that the errors made in damping the longrange coefficients to a large extent compensated these additional repulsive contributions. Indeed, by using only the first three undamped dispersion terms as in the early model [Eq. (1 I], the neglected contributions to the long-range and shortrange potentials compensate each other rather well. Moreover from the error discussion of Sec. III we also find that A

NaK 3I LiHg

2.166 3.364 1.936 6.001 1.962 13.53 2.253 38.52 2.935 138.6 4.333 630.1 7.253 3619.0

4138 566.1 7.809 5.739 1.435 1.183 0.737 0.762

0.325 0.443 0.124 0.241 0.042 0.126 0.013 0.059 0.004 0.034

2.68 2.85




need not be known too accurately, but that b is of prime importance.

Thus we feel that we now understand much more clearly why our earlier models were so successful. In view of the quantum chemical theories discussed in the Introduction and Feltgen's analysis20 we are also convinced that our model is on a reasonably good theoretical footing. The new model offers the following principle advantages over the earlier work and over most of the other models used to predict or fit van der Waals potentials.

(1) The model expression is of a simple closed form and analytic. Derivatives exist everywhere to all orders. Moreover, the expression is mathematically an entire function and can thus be easily used in theories requiring an analytic extension into the complex plane such as theories of the predissociation of van der Waals complexes. 104

(2) All the five parameters (A, b, C6 , Cg, and CIO)' which are of principle importance, have a simple physical meaning and in part are available from ab initio theory. Each term can thus either be estimated using well studied procedures or determined by a fit to experiment. Thus the interplay between experiment and theory is direct. Moreover, the attractive and repulsive contributions are clearly separated.

(3) The model is easily adapted to deal with mixed systems. Using very accurate combining rules separately for the repulsive lO5

-107 and attractive parts,60 we have been able to

predict potential functions for a large number of mixed raregas and H-rare gas systems without introducing any new data other than that from the homogeneous dimers.108

(4) Finally the model offers a logical starting point for more elaborate or compact models which can be adapted to the quality of the available data or the predictive requirements.

ACKNOWLEDGMENTS

We are grateful to W. M. Meath, W. Meyer, B. Jeziorski, and other colleagues for valuable discussions and comments at the Gordon Conference on Atomic and Molecular Interactions at Wolfeboro in early August 1982 where this work was first presented. We have also benefited from an extensive exchange of ideas with R. Feltgen, R. Ahlrichs, W. Kutzelnigg, and H. D. Meyer during the preparation of the manuscript. We thank F. Engelke for making the Dissertation of E. J. Breford available to us and valuable discussions on the NaK potential. Finally we are most grateful to R. Feltgen and R. O. Watts for critically reading the manuscript.

APPENDIX A

To find A and b we assume that R m , E, C6, Cg, ... are all given. Then we use the reduced form of the potential Eq. (14). At the minimum x = 1 the reduced potential provides us with two equations from which to obtain A and b:

x= I:U= -1 (AI)

and

dU x= 1:-=0.

dx

Next we differentiate U with respect to x and get

dU

dx

from which we obtain

(A2)

(A3)

A * = 2: (eb * _ I (b *)k) 2~ C!n _ 2: (b *)2nC !n' n>3 k=O k! b n>3 (2n)!

(A4)

Next we substitute A * from Eq. (4) into the reduced form of the potential and from Eq. (1) we get

U(l) = 2:(1 - e- b* I (b *t)( 2~ - I)C!n n>3 k=O k. b

- 2:e-b*(b*)2n C!n = - 1. (A5) n>3 (2n)!

This equation has to be solved for b *. This is done most expediently by an iterative procedure on a computer. After b * is found, A * is determined from Eq. (4) and the process repeated until self-consistent values are obtained.

I For recent reviews of CI calculations see P. Claverie, in From Diatomics to Biopolymers, edited by B. Pullman (Wiley, New York, 1978); A. van der Avoird, P. E. S. Wormer, F. Mulder, R. M. Berns, Top. Curro Chern. 93,1 (1980); P. Schuster, Angew. Chern. 93, 532 (1981).

2For a discussion of these problems see H. Margenau and N. R. Kestner, Theory of Intermolecular Forces, 2nd ed. (Pergamon, Oxford, 1971), p. 170.

3For a clear discussion of this problem see p. 1221f of P. R. Certain and L. W. Bruch, inMTP International Review of Science, edited by W. B. Brown (1972), Vol. 1, Series I, p. 113. An approximate calculation for He2 has been discussed by I. K. Snook and T. H. Spurling; J. Chern. Soc. Faraday Trans. 271,852 (1975).

4For a recent review see G. Scoles, Annu. Rev. Phys. Chern. 31, 81 (1980). 'G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham, Intermolecular Forces (Oxford University, Oxford, 1981).

6K. T. Tang, J. M. Norbeck, and P. R. Certain, J. Chern. Phys. 64, 3063 (1976).

7J. P. Toennies, Chern. Phys. Lett. 20, 238 (1973). sJ. O. Hirschfelder, Chern. Phys. Lett. 1, 325, 363 (1967). 9F. C. Brooks, Phys. Rev. 86, 92 (1952); (a) For a recent discussion see R. Ahlrichs, Theor. Chim. Acta (Berlin) 41, 7 (1976).

lOA. Dalgarno and J. T. Lewis, Proc. Phys. Soc. London Sect. A 69, 57 (1956).

ilL. Jansen, Phys. Rev. 110,661 (1958). 12R. J. Buehler and J. O. Hirschfelder, J. Chern. Phys. 83, 628 (1951). 13 A. A. Frost and J. H. Woodson, J. Am. Chern. Soc. SO, 2615 (1958). 14L. C. Cusachs, Phys. Rev. 125, 561 (1962). "J. I. Musher and A. T. Amos, Phys. Rev. 164, 31 (1967). 16J. N. Murrell and G. Shaw, J. Chern. Phys. 49,4731 (1968). I7H. Kreek and W. J. Meath, J. Chern. Phys. SO, 2289 (1969). ISA. Koide, J. Phys. B 9,3173 (1976). 19M. Krauss and D. B. Neumann, J. Chern. Phys. 71,107 (1979).




2°R. Feltgen, J. Chern. Phys. 74, 1186 (1981). 2lR. Feltgen, H. Kirst, K. A. Kohler, H. Pauly, and F. Torello, J. Chern.

Phys. 76, 2360 (1982). 22A. Koide, W. J. Meath, and A. R. Allnatt, Chern. Phys. 58, 105 (1981). 23N. Jacobi and Oy. Csanak, Chern. Phys. Lett. 30, 367 (1975). 24E. N. Lassettre, J. Chern. Phys. 57,4357 (1972). 25M. Battezzati and V. Magnasco, J. Chern. Phys. 67, 2924 (1977). 26L. C. Cusachs, Phys. Rev. 125, 561 (1962). 27H. Bethe, Handbuch Phys. 24, 1 (1933), see page 334ff. 2sH. Reeh, Z. Naturforsch. Teil A IS, 377 (1960). 29J. Callaway, Phys. Rev. 106, 868 (1957). 30 A. Temkin, in Case Studies in Atomic Collision Physics II, edited by E. W.

McDaniel and M. R. C. McDowell (North Holland, Amsterdam, 1972) p. 406ff.

3'R. E. Olson, Phys. Rev. A 6,1031 (1972). 32C. Bottcher and A. Dalgarno, Proc. R. Soc. London, Ser. A 340, 187

(1974). 33C. Bottcher, T. C. Cravens, and A. Dalgarno, Proc. R. Soc. London, Ser.

A 346, 157 (1975). 340. Peach, J. Phys. B 11, 2107 (1978). 35J. C. Bellum and D. A. Micha, Phys. Rev. A 18, 1435 (1978). 36J. N. Murrell and O. Shaw, J. Chern. Phys. 46,1768 (1967). 37J. N. Murrell, M. Randic, and D. R. Williams, Proc. R. Soc. London Ser.

A 284,566 (1965). 3SSome authors also include an inter-intra coupling term. In the present

notation this term is included in the dispersion coefficients. wFor a recent review see W. Kolos; III Proceedings of the Fourth Interna

tional Conference on Quantum Chemistry, 1982. 4°Note that the induction dispersion energy, which includes only excitation

on one of the atoms, whereas the ordinary dispersion energy takes account of simultaneous excitations on both atoms [see L. C. Cusachs, J. Chern. Phys. 38, 2038 (1963); H. C. Louquet-Higgins; Proc. R. Soc. London Ser. A 235,537 (1956)] is included in the second-order SCF potential.

4'y' M. Chan and A. Dalgarno, Mol. Phys. 14,101 (1968). There is a sign error in this paper, see D. E. Stogryn, Mol. Phys. 22, 81 (1971).

42A. I. M. Rae, J. Phys. B 8, 983 (1975). 43For other recent references see the articles listed under Ref. 68 in Ref. 20

of the present paper. 44E. A. Reinsch and W. Meyer (private communication); referred to in R.

Feltgen, Ref. 17. 45E. A. Reinsch and W. Meyer (private communication, 1976). More exact

calculations have been carried out in 1982. 460. Das, A. F. Wagner, and A. C. Wahl, J. Chern. Phys. 68, 4917 (1978). 47W. Meyer, P. C. Hariharan, and W. Kutzelnigg, J. Chern. Phys. 73,1880

(1980). 4S0. Chalasinski and B. Jeziorski, Mol. Phys. 27, 649 (1974). 490. Chalasinski and B. Jeziorski, Mol. Phys. 32, 81 (1976). 5°0. Chalasinski, Mol. Phys. (to be published). 5'0. Chalasinski, B. Jeziorski, J. Andzelm, and K. Szalewics, Mol. Phys. 33,

971 (1977). 520. Chalasinski and B. Jeziorski, Theor. Chim. Acta (Berlin) 46, 277

(1977). 530. Chalasinski, Chern. Phys. Lett. 86, 165 (1982). 54J. Hepburn, O. Scoles, and R. Penco, Chern. Phys. Lett. 36,451 (1975). 55R. Ahlrichs, R. Penco, and O. Scoles, Chern. Phys. 19, 119 (1977). 56c. Douketis, O. Scoles, S. Mordetti, M. Zen, and A. J. Thakkar, J. Chern.

Phys.76, 3057 (1982). 57K. C. Ng, W. J. Meath, and A. R. Allnatt, Chern. Phys. 32,175 (1978). 5sA. J. C. Varandas and J. Brandao, Mol. Phys. 45, 857 (1982). 59K. T. Tang and J. P. Toennies, J. Chern. Phys. 66,1496 (1977). For errata

see J. Chern. Phys. 67, 375 (1977); 68, 786 (1978) . .oK. T. Tang and J. P. Toennies, J. Chern. Phys. 68, 5501 (1978). 6'K. T. Tang and J. P. Toennies, J. Chern. Phys. 74, 1148 (1981). 62p. Habitz, K. T. Tang, and J. P. Toennies, Chern. Phys. Lett. 85, 461

(1982). 63K. T. Tang and J. P. Toennies, J. Chern. Phys. 76, 2524 (1982). 64Note that the new damping function [Eq. (10)] will automatically cut off

the terms for n > n' . 65D. D. Richardson, J. Phys. A 8, 1828 (1975). 66J. D. Bowman, Jr., J. O. Hirschfelder, and A. C. Wahl, J. Chern. Phys. 53,

2743 (1970). 67Several studies have indicated that in fact the energy of the first second

order dispersion term is finite at the origin. We can take this into account by using the following expression for f.:

With a = 0 this is identical to Eq. (10) with 2n = 6, but with aofO the asymptotic value of.t;;Co/ R 6 is - a b 6Co/6! asR-+O. According to Koide (Ref. 31) this value is about - 0.0085 corresponding to an a of about - 0.04. Thus the contribution of this term is small, compared to one. In

fact, it is absorbed in the repulsive potential since

_bR(bR t C6 aC6b 6 -bR ae --- = ----'"--e .

6! R6 6!

6SM. Krauss and W. J. Stevens, Chern. Phys. Lett. 85, 423 (1982). 69M. Krauss, W. J. Stevens, and D. B. Neumann, Chern. Phys. Lett. 71, 500

(1980). 7°R. Feltgen (private communication). 7' A different method for estimating Cs and Cw dispersion coefficients has

been suggested on the basis of the harmonic oscillator model of van der Waals coefficients by Thakkar and Smith [A. J. Thakkar and V. H. Smith, J. Phys. B 7,1321 (1974)] and has been extended to C'2 and C'4 by Douketis et 01. (Ref. 56). Another method for extrapolating dispersion coefficients to high order has been suggested by Feltgen (Ref. 20). On the basis of comparisons with H2 we feel that Eq. (16) provides the best approximate procedure.

72We do not recommend using the relationship Eq. (16) for terms beyond 2n = 20 because of the accumulation of errors, which may lead to an incorrect divergence of the series. We have so far not found a system for which terms with 2n > 20 make a noticeable contribution.

73The average value of the upper and lower bounds of the theoretical estimates was used throughout. The errors in A and b were estimated for H(IS)-H(IS) by changing all the dispersion coefficients C6 , Cs, etc. by + I % and - 1 %. The resulting relative changes in A were + 10% and - 10%; the changes in b were + 0.9% and - 0.9%, respectively. A 1 %

increase in Rm , however, leads to a 100% increase in A, but only a 5% increase in b. Moreover, a 1 % increase in to leads to a 10% increase in A and a I % increase in b.

74R. J. Bell, Proc. Phys. Soc. 87,594 (1966). 75W. Kolos, Int. J. Quantum. Chern. Symp. 8, 241 (1974). 76J. D. Bowman, Jr., Ph.D. thesis, University of Wisconsin, 1971, Theoreti

cal Chemical Institute, Report No. WIS-TCI-463. 77W. Kolos and L. Wolniewicz, Chern. Phys. Lett. 24,457 (1974). 78R. A. Aziz, V. P. S. Nain, J. S. Carley, W. L. Taylor, and O. T. McCon

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21. 83A. Koide, W. J. Meath, and A. R. Allnatt, J. Phys. Chern. 86,122 (1982). 84A. L. Burgman, J. M. Farrar, and Y. T. Lee, J. Chern. Phys. 64, 1345

(1976). S5p. J. Bertoncini and A. C. Wahl, J. Chern. Phys. 58, 1259 (1973). s6D. R. McLaughlin and H. F. Schaeffer III, Chern. Phys. Lett. 12, 244

(1971). 87B. Liu and A. D. McLean, J. Chern. Phys. 59,4557 (1973). ssp. D. Dacre, Mol. Phys. 37, 1529 (1979). R9J. A. Barker, R. A. Fisher, and R. O. Watts, Mol. Phys. 21, 657 (1971). 90R. A. Aziz and H. H. Chen, J. Chern. Phys. 67, 5719 (1977). 9'A. Koide, W. J. Meath, and A. R. Allnatt, Mol. Phys. 39, 895 (1980). 92E. A. Colburn and A. E. Douglas, J. Chern. Phys. 65,1741 (1976). 93 A. C. Wahl (unpublished), referred to in Table IV of Douketis et 01. (Ref.

56). 94W. R. Wadt, J. Chern. Phys. 68, 402 (1978). 9'E. J. Breford and F. Engelke, J. Chern. Phys. 71, 1994 (1979). 96D. Eisel, D. Zevgolis, and W. DemtrOder, J. Chern. Phys. 71, 2005 (1979). 97E. J. Breford, Dissertation, University of Bielefeld, 1982. 9sE. J. Breford and F. Engelke, Chern. Phys. Lett. 53, 282 (1978). 99We thank F. Engelke for calling our attention to this possibility.



K. T. Tang and J. P. Toennies: A model for the van der Waals potential

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IOIR. E. Olson, J. Chern. Phys. 49, 4499 (1968). 102Mlle. F. Maeder and W. Kutzelnigg, Chern. Phys. 42, 95 (1979). 103Mlle. F. Maeder, Thesis, Paris 1979.

I04S._1. Chu, J. Chern. Phys. 72, 4772 (1980). lOsT. L. Gilbert, J. Chern. Phys. 49, 2640 (1968). I06F. T. Smith, Phys. Rev. A S, 1708 (1972). 107H. J. Bohrn and R. Ahlrichs, J. Chern. Phys. 77, 2028 (1982). 108K. T. Tang and J. P. Toennies (to be published).


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An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients