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A Survey of Some Empirical

and Semi-Empirical Interatomic

and Intermolecular Potentials

BENJAMIN M. AXILROD

246

SI

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

IS
THE NATIONAL BUREAU OF STANDARDS

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NBS TECHNICAL NOTE 246

ISSUED OCTOBER 3, 1966

A Survey of Some Empirical

and Semi-Empirical Interatomic

and Intermolecular Potentials

Heat Division

Statistical Physics Section

Institute for Basic Standards

National Bureau of Standards

Washington, D.C., 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 40 cents

_rf^VI < \ / I

CONTENTS

Page

1. Introduction 1

2. Repulsive Potential for Atoms in S States - Buckingham — ——

1

2.1 The H + H ( E State) Interaction 2

2. 2 The H + He Interaction 3

2. 3 The He + He Interaction 4

3. Semi-Empirical Functions — Frost and Associates — 6

4. Potential for Inert Gases - Woolley 7

5. Empirical Potentials Fitted to Gas or Gas and Crystal Data 9

5. 1 Mason and Rice 9

5. 2 Whalley and Schneider 10

5. 3 Schamp and Associates 12

5. 4 Konowalow and Hirschfelder 12

5. 5 Extrapolations 13

6. Delta Function Models 15

6. 1 Frost 16

6.2 Lippincott - < 16

6. 3 Mason and Vanderslice 18

7. A Semi-Classical Model of Atomic Interaction 20

8. Acknowledgments 24

9. Appendix 24

10. References 25

11

.

List of Figures 27

in
ABSTRACT

Many empirical and semi-empirical potential

functions have been proposed or developed to represent

the pairwise interaction of atoms and molecules. The

simple pair potentials proposed usually contain parameters to be adjusted to fit certain properties, for

example, when chemical combination does not occur,

virial coefficients and similar quantities for the gaseous

state, or compressibility and heat of sublimation for the

crystalline state. In the present survey a number of

potentials proposed recently are examined to indicate

the variety of concepts employed and the suitability

of the potential functions.

Keywords: Atoms, compressibility, heat of

sublimation, molecules, potential

functions - empirical and semiempirical, transport properties,

virial coefficients

IV
A SURVEY OF SOME EMPIRICAL AND SEMI-EMPIRICAL INTERATOMIC

AND INTERMOLECULAR POTENTIALS

Benjamin M. Axilrod*

1 . Introduction

Many empirical and semi-empirical potential functions have been proposed or developed to represent

the pairwise interaction of atoms and molecules. 1 These functions usually contain parameters to be adjusted to fit certain properties, for example, when chemical combination does not occur, virial coefficients

and similar quantities for the gaseous state, or compressibility and heat of sublimation for the crystalline

state. It should be emphasized that our interest is principally in pairs or systems that do not react chemically.

A proposed potential is judged a good approximation to the true interaction if the proposed function

is successful in predicting experimental properties other than those used for obtaining the potential parameters. If a number of properties are predicted in agreement with experiment, we obviously have more

confidence in the potential.

Usually, potentials of simple functional form are sought in order to facilitate the calculation of the various properties. The latter influence has sometimes led to the use of potential forms or of atomic

models that are obviously incorrect, such as the rigid sphere atom model. Nevertheless, such incorrect

potentials have often proved useful for indicating the sensitivity of different properties to the shape of the

potential and for deriving approximations to various properties.

Some approaches are strictly empirical, others employ some classical or quantum theory concepts.

In the following sections, a number of recent papers are examined to indicate the variety of concepts

employed and the suitability of the potential functions.

Potentials given in earlier works and their application to equilibrium and transport properties in gases

are described in the treatise of Hirschfelder, Curtiss and Bird [1] . Recent surveys of empirical potential

functions for bound states of diatomic molecules, for which case spectroscopic information is the principal source of experimental data, have been made by Steele, Lippincott and Vanderslice [2] and by Varshni[3].

Since we are interested in methods that can apply or have been applied to the repulsive interaction,

the papers discussed here were selected with this viewpoint. Some, however, which describe methods for bound states were included either because of possible application to repulsive interaction or because of

the novelty of the method.

2. Repulsive Potential for Atoms in S States — Buckingham

Buckingham [4] proposed a semi-empirical function to fit the repulsive interaction of a pair of atoms

in S states for the internuclear separation R ranging from zero to the point where the potential is zero. He

emphasized "the aim is to fit the correct repulsive potential so far as it is known, not merely an incomplete theoretical approximation, e. g. , first-order perturbation energy" [4]. The function proposed was

V(R) = (Z

1 Z

2/R)p(R)exp(-aR) (1)

Theory predicts that the interaction of a pair of particles (atoms, etc.) is influenced by the presence of neighboring particles. Since this complicates the problem, as a first approximation it is usual to disregard this effect. Intuitively, this approximation is better in gases (except at high density) than in solids or liquids.

Numbers in brackets indicate literature references at the end of this paper. *Present Address: 9915 Marquette Drive, Bethesda, Maryland.

1
where Zj and Z2 are the nuclear charges of the two atoms and p(R) is a polynomial

p(R) = 1 + Pl R + . . . Pj R

3

usually of the fourth degree. The suitability of eq. (1) was indicated by noting it was consistent with the following facts: for extremely small separations, the interaction potential must be dominated by the Coulomb repulsion of the nuclei, and for larger separations theoretical calculations of the repulsion of two hydrogen or two helium atoms were found3 well represented by the product of a polynomial and an exponential, a property that might be expected to apply for other simple atomic pairs.

The arbitrariness in the choice of the constants a. and the p. was reduced by examining the behavior

of V(R) for very small R. For the latter condition, first-order perturbation calculation indicated that Ve

, the electronic energy of the system,

Ve

= V(R) - Z-^/R, (2)

was approximately

Ve

(R) = E

Q

+ E

2 R

2 + ... (Za)

where E is the energy of the "united atom" of nuclear charge Zj + Z2. Eq. (2a) together with eq. (1) yields the relations between _p_'s and_E's

Pl = a + E /(Z 1 Z

2

); p

2

= -fa

2 + E a/(Z 1 Z

2

)

p 3 = £a 3 + [E

2 +iE a

2 ]/(Z

1 Z

2

).

Thus, if E is known — the electron configurations of the interacting atoms must be correlated with the proper united atom electronic state -- this information, together with data on the repulsive potential at various separations can be used to determine the constants in eq. (1).

3 The preceding ideas were applied [4] to the interactions H + H( E), H + He, and He + He and the results are summarized in the following paragraphs.

3 2. 1 The H + H( EM

State) Interaction

The excited state (ls,2p) P of He was taken as the appropriate united atom state. Insufficient experimental evidence existed for this case so the constants .a and p* were obtained by fitting (a) the experimental united atom energy and (b) for the range 1. 5 to 6a ,~~theoretical calculations of James,

Coolidge and Present [5] (for close distances) and of Hirschfelder and Linnett [6] (for larger separations).

The potential derived to fit the above data for internuclear distances less than 6a D was, in atomic units

(e 2/a00, with R in units of a ):

V(R) = (1/R)(1 + 1. 167R + 0.039R2 + 0. 518R3 + 0. 864R4

) exp(-2. 3R). (3)

Buckingham examined the Heitler-London-Sugiura results and the Griffing-Wehner [J. Chem. Phys. 23 , 1024 (19 55)] data for hydrogen and helium, respectively.

4 Measured relative to the separated atoms as the zero of energy. Thus: the energy of the S ground state

of He is -2.903 atomic units, the er°rgy of the (Is, 2p) 3 P state is 0.770 above the former; and the relative electronic energy of the united atom E is

EQ

= E(He: ls2p 3 P) - 2E(H: ls^) = -2.903 + .770 + 1.000 = -1.133.

In the various examples, Buckingham used electronic energies from C. M. Moore, Atomic Energy Levels,

Vol. I. (Nat. Bur. of Standards, 1949).
Buckingham also gave an equation [eq. (3a), below] of the type of eq. (3), except with a fifth power

polynomial, to fit the united atom energy and to approximate the James-Coolidge-Present data and a potential VDl(R) derived with second-order perturbation theory by Dalgarno and Lynn [7]. The potential Vfjl wa s somewhat lower than that of Hirschfelder and Linnett and vanished between 5 and 6a and had a minimum

o about 7a .

V(R) = (1/R)(1 +.967R - .174R2 + .830R3 + .54R4

- .12R5 )exp(-2.1R).

Numerical values given by eqs. (3) and (3a) are listed in Table I.

TABLE I

(3a)

Repulsive Potentials for H + He Interactions (in atomic units)

R(a )

H + H(3£u) (From Table 1, reference 4) H + He

(From Table 2, reference 4)

Eq. (3) JCP & HLa Eq. (3a) Eq. (5) Scatter- ingb

VM.R. S. (Eq. (4))

0.8 0. 512 TCP 0. 529

1.0 .360 . 373 0.204

1. 5 . 190 0. 190 . 189 . 120

1.87 . 125 . 125 . 121 2.0 HL . 0750 0.0716 0.0904d

2.629 .0495 .0486

3.0 .0241 .0191 .0188 ,0194d

3.077 .0268 .0261

4.0 .00397 . 00329 .00332

4.018 .00639 .00686

5.0 .00C37 .000422 .000506

5.012 .00119 .00145

6.0 -.00030 3. 57_5c 5.65~ 5

6.011 .00021 .00025

a References 5 and 6, respectively.

k Reference 8. U A notation here and elsewhere such as 3. 57 means 3. 57 x 10 . d Calculated from equation (4).

2. 2 The H + He Interaction

The united atom (Li(ls) (2s) S) energy is available and also a potential derived from atom-atom

scattering data [8] for separations of 2. 2 to 3. 2a . In addition it appears [4] that V(R) was required to be zero for a value of R_ near 6. 8a Q

, the zero of an "exp-slx" potential obtained by Mason, Ross and Schatz[9]

for separations greater than 2a Q. The latter authors, to represent the repulsive energies they calculated

for the range 2 to 5a by a variational method and to represent also the attractive dispersion energy which

is important for larger separations, fitted their results with the equation

V(R) 4.606exp(-1.76R) - 2.94/Re (4)
The constant C = 2.94 was not fitted but was obtained [9] by applying combining rules to theoretically derived values of the dispersion force constants C^ and C for the interaction

r 4.-L. ui i ti—ti He-He

of the like pairs.

The repulsive potential obtained by Buckingham was

V(R) = (2/R) (1+.113R-2.069R

2 + 2. 089R

3 -.259R4

) exp (-2. 15R) (5)

The potential from scattering [8] and the theoretical calculations [9] are close near 3a separation

so in this region equations (4) and (5) are in good agreement. The scattering potential near 2a

is appreciably below the theoretical potential [9] so that as R decreases below 3a equation (5) yields lower values than (4). Between 5. 5 and 6. 5a where V(R) is much reduced, (less than

2 x 10 compared to greater than 0.003 'below 4a ) equation (5) yields values about two-thirds

those of equation (4). Numerical values from equations (4) and (5) are shown in Table I.

2. 3 The He + He Interaction

2 1 2 The interacting atoms are in (Is) S states and it is assumed that the united atom state is (Is) (2s) 2 as in Be ( S ). The potential of equation (1) was fitted entirely to experimental data namely V for the united atom, potentials from scattering results (for distances of 1 to 2a and 2. 5 e to 3aQ

) and

the zero of V(R) to R , the zero of empirical potentials based on analysis of equilibrium and transport

properties of helium gas.

The derived potential was

V(R) = (4/R) (1+0.265R -2. 419R

2 + 2.616R - .436R ) exp (-2. 48R) (6)

Since as R decreases from 3 to la the atom-atom scattering potential [10,11] differed increasingly

from some contemporary quantum theoretical calculations [12, 13] of the interaction (from near agreement to four times smaller) equation (6) does likewise in this range. Potential values

obtained from equation (6) and from the scattering data used are shown in Table II, and in Figures 1 and 2.

Subsequently Buckingham and Duparc [14] made a quite accurate quantum mechanical calculation of the He + He interaction for Jhe range zero to 0.8a

Q

. The electronic energy V was

in good agreement with a value at 0.4 A calculated with different wave functions by Ransll [15]. A semi-empirical potential, equation (6a), to improve on equation (6) was dgrived by fitting: the experimental united atom energy; a theoretical value by Bingel's [16] method (and with BD wave

functions) of E the coefficient of R in equation (2a) for V

g

; recent atom-atom scattering data [17] for separations of 1.8 to 2.8a

o

5In reference 4 use is made of the Lennard-Jones (12-6) potential of de Boer and Lunbeck [R. J. Lunbeck, Thesis, Amsterdam (1951)], and an (exp-6, 8) potential [R. A. Buckingham and R. A. Scriven, Proc. Phys. Soc. A65 , 376 (1952)] which had zeros at 4.84 and 4. 94a

Q

, respectively.

^Reference 16 gives explicit expressions for E, and for E_,

is extended, in terms of the electron charge densities of 1

the coefficient of R if equation (2a) the united atom.
TABLE II

Repulsive Potentials for He + He Interaction (in atomic units)

Buckingham (1958 Buckingham and Duparc (1962) Recent Scattering (S) and

Theoretical

(T) Values.

R(a ) o 0.8

1.0 1.04

1.4

1.89

2.0

2.8

3.0

4.0

Equation (6) 0.567

.344

.189

.0814

. 0120-3

,942

Scattering

.331

3

„18i

a

.0958

3

.om

b

Equation (6a)

0.939

.621

. 309

.0958

.0148

.00879

.454"

Q Scattering

.0959

.0176

fc^VtluP'

1.516 0.918T

.537S

.406T

.233S

.129S

.120T

•

a Reference 10. " Reference 11. c Reference 17.

S. from reference

a. u. , an equati

T

! 18; from ref Dn in refere ncerence 19, computed from V(R) = e 19 for range 0. 5 to 1. 0A°. 7.036 exp [-3. B46R(A) ]

and the zero of V(R) in equation (1) which was adjusted to be at 4. 90a

Q

, "roughly in accord with

other empirical evidence". The equation obtained was

V(R) = (4/R) [1 + .611R- 2.268R

2 + 4. 033R

3 -. 736R ] (6a)

exp (-2.826R)

The above semi-empirical potential was lower than the best BD theoretically derived potential over the entire range from zero to 0.8a (the difference increasing from about 1% to 17% in V

g

(R) and being about 40% in V(R) at 0.8a

Q

).°

Equation (6a) was noted [4] as being appreciably higher than the potential from the 1949

scattering data [10] for the range 1 to 2a . Very recent scattering experiments [18] for this range

yield a higher potential particularly at la , and are in better agreement with equation (6a). Some

of these points are apparent from the data in Table II which includes potential values derived

from equation (6a), the scattering data used as a guide in obtaining the latter equation, and also

the most recent scattering data [18] for small separations and some of the results of Phillipson's

elaborate theoretical potential calculation [19]. The last two sets of data indicate a sizeable

disagreement still existing between atom-atom scattering and quantum theory calculations near la Q

separation so that in fitting a semi-empirical function of equation (1) type a choice must be made

between fitting the theoretical or experimental data or a composite of the two.
3. SEMI-EMPIRICAL FUNCTIONS — Frost and Associates

Frost and Associates [20,21,22] proposed semi-empirical potential energy functions for diatomic

molecules and for the repulsion of inert gas atoms. The plan [20] was to cover the range from

the united atom to the separated atoms and to use the known properties of these two cases as

a guide in the choice of functions.

Initially [20] calculations were made for H and H in bound ground states, first with the simple two parameter potential

V(R) = (e /R - b) exp (-aR), (e is the electronic charge) (7)

and next with a more complicated one-parameter potential having the form of the energy for the H ion as obtained in a simple quantum mechanical calculation. Equation (7) was selected to

fix some theoretical considerations including:

a. V, the potential for nuclear motion is the sum of the nuclear repulsion and of V , the electronic energy.

b. The electronic energy V = V-e /R must remain finite as R vanishes, and should vary

as -e /R for large R.

c. V must be capable, with appropriate constants, of having a minimum.

With equation (7) two of the three spectroscopic constants, equilibrium separation, R , the disQ sociation energy, D , and the force constant, k , were used in various combinations to calculate

the third constant as well as other spectroscopic or related constants, i.e., higher derivatives

of the potential.

Subsequently equation (7) was modified [21] to apply to general diatomic molecules by re- placing e with c where _c was taken as Z Z e , the product of "effective" nuclear charges:

V(R) = (c/R - b) exp (-aR) (8)

The parameters .a, b and c were derived in terms of the spectroscopic constants D , R and k from the values at the minimum of the potential, its first derivative and its seconcf derivative

(equal to —k ). The relations for the parameters are: e

a = p/R , b = D (1+p) exp (p), c = D R p exp (p) (9) e e e e

where p = (1+k R

2/D r2 -1. e e e

These potential parameters were obtained for nine hydrides and ten homonuclear diatomic molecules;

in five instances the same element was common to the hydride and the homonuclear molecule.

The third and fourth derivatives of the potential of equation (8) were calculated and were compared

with values obtained from the experimental spectroscopic quantities, a and oi x , the vibrationrotation coupling and the anharmoniclty constants, respectively. The calculated results [21] were

of the right order of magnitude but were not better than those obtained with the Morse or some

other semi-empirical functions.

The constant c was studied, first to see if the same value of Z was obtained for the hydride of an element and its diatomic molecule; then a comparison was made with the effective

nuclear charge for the valence electrons as given by Slater's rules. For the latter comparison

there was correlation but for the heavier nuclei the correlation was poor.
In later work [22] semi-empirical potentials were proposed by Frost and Woodson for inert gas atoms and for diatomic molecules to include the l/R dispersion forces, and for the diatomic

ionic interaction if present. The inert gas potential suggested was

V = (c/R) exp (-aR) + b exp (-aR) - d [l/R

6 - F(aR)] (10)

where F(aR) is a function such that the bracketed term remains finite as R goes to zero. For diatomic molecules including the case of ionic interaction the first term in equation (10) was

replaced by

c'/R + [(c-c')/R] exp (-aR) (10a)

where _c' is the product of the ionic charges and c_ is as previously defined. It was then suggested that the interatomic potential in an alkali halide molecule could be represented as the sum of the interaction of the isoelectronic inert gas atom pair, i.e., for K + Br the pair Ar + Kr, plus

the Coulombic interaction of unit positive and negative point charges at the internuclear distance

R. For convenience the simple exponential-six potential

V(R) = A exp (-BR) - C/R6

of Whalley and Schneider 'including their parameters was adopted by Frost and Woodson to represent the "inert gas portion" of the alkali halide potential. Surprisingly close agreement

with experimental dissociation energies and equilibrium separations, usually within a few percent, was obtained for (Table III) a series of molecules from KC/^ (isoelectronic to Ar-Ar) through Csl

(isoelectronic to Xe-Xe). For mixed inert gas atom pairs the combining rules of Mason were

used. Probably a cancellation of neglected effects was responsible for the close agreement between

calculated and experimental data.

4. POTENTIAL FOR INERT GASES - Woolley

Wooley •* considered the behavior of the modified Buckingham exp-six potential which had

been used for a number of gases to fit second vlrial coefficient, transport and crystal data. This potential is a modification of the following:

V(R) = [e/(l-6/a)] £(e/a) exp [*(l-R/Rm

)] - (R/R) 6?. (U)
TABLE III (from Reference 22)

Comparison of Calculations with Observed Data for Some Alkali Halide Gas Molecules

Diatom R (calcd.

)

e R(obsd. ) a D (calcd.) D -(obsd.)

b

e

KGi 5.075 5.0392 0.1808 0.1814 KBr 5.3304 .1735

RbCl 5. 300

5.2664

.1739

.1763

Kl 5. 7596 .1622

CsCl 5. 555

5.4920

.1533

.1683

Rbl 6.0035 .1568

CsBr

5.846

5.8053

.1556

.1616 Csl 6. 354 6.2645 .1402 .1561 RbBr 5. 531 5. 5649 .1672 .1681

\ dRjR =

, = O; D

3. e e

= -V(R ) e

See A. Honig, M. Mandel, M. L. Stitch and C. H. Townes, Phys. Rev.

96. 629 (1954).

^Calculated from E. S. Rittner, J. Chem. Phys., .19, 1030 (1951).

where the potential has a minimum at R , - £ is V(R ) and (Z determines the steepness of the repulsive part of the potential. The potential of equation (11) has a spurious maximum at a small

separation, R . This has been corrected by arbitrarily defining the potential of equation (11) is infinite for TT less than R exp-six potential; max

With the latter adjustment one has the modified Buckingham

V(R) as in equation (11), R > R

V(R)

max

, R < R

(Ha) max

Woolley desired a more realistic potential in order to extend calculations to higher temperatures.

He sought an analytic function for the potential of inert gases that would extend to very small

separations and would include a term giving essentially an inverse sixth power attraction near

the potential minimum and beyond. At extremely close approach of a pair of inert gas atoms the interaction should be the Coulombic nuclear repulsion and with increasing separation, screening

of the nuclei by the electrons would cause a decay in repulsion more rapid than as l/R. A

very general type of function was proposed. An example of this function for which some calculations were made is (equation 10 of reference 2 5):
Y-5+M I 1+r l m' ' *

L '

v v m

/J (12)

/R)

6r

l-exP(-rR/Rm

) i 8")

1-exp (-y)

where /i = 8r/[exp (r) -1] and R is the equilibrium separation. m

The above is a sealed equation and the parameterT together with R introduce the influence of screening of the nuclear charges by the electron shells. In reference 25^T was related to the a

of the modified Buckingham equation, equation (11a), in several ways such as equating second

or third derivatives at the minimum for equation (11a) and for equation (12) with the factor (—

)

omitted.

o The eighth power correction factor defined f(R)/f(R )= (—) which multiplies the 'van der Waals

attraction type potential (R /R) in the second term in eq. 12 was made plausible as follows: The attractive term is proportional to the product of the effective polarizabilities of the interacting atoms; the effective polarizability when two atoms overlap is taken as that part contributed by the non-intersecting spherical electron cloud centered on each nucleus. From the estimate of the

effective polarizability a correction factor we define as _F, corresponding to f(R)/f(R ) and more

complicated than the latter, was obtained. The factor F_ was found to vary with the eighth power

of the distance R for very small separations; the factor f(R)/f(R ) has the same R dependence at small separations and in general similar behavior to F_.

The function equation (12) was graphed [25] for integral values of T from 11 to 15 for R/R

from about 0.1 to 1.4 and with the convenient linear ordinate log

]n [CV/i) + 2] to give coincidence

of curves at the minimum.

The curves (Figure 4) were compared with plots of repulsive potentials derived froni inert gas scattering data by Amdur and associates [26] (for the gases He through Xe) and Berry (for Ne and Ar)

.

The helium, neon and argon scattering data of Amdur and associates were fitted by T values

of about 11, 13, and 11, respectively; Berry's data for argon fitted the curve for V equal to 12. For krypton and xenon the potential from scattering deviated more from parallelism to the family

of curves of equation (12) than the scattering potentials for the other gases.

The behavior of the potential of equation (12) as the separation approached zero was examined [25] to see if it approached the Coulombic nuclear repulsion when the parameters T etc.,

chosen as above to fit the experimental data, were inserted. With the latter parameters the

coefficient [defined (Z Z ) ] of the l/R term of equation (12) for R approaching zero was within an order of magnitude ol 2?2p" for the inert gases. The values of (Z Z ) and Z Z for He, Ne,

Ar, Kr and Xe pairs were roughly, respectively: 10,4; 310, 100; 800, hfif §50, 1300; and 730, 2900.

5. EMPIRICAL POTENTIALS FITTED TO GAS OR GAS AND CRYSTAL DATA

5. 1 Mason and Rice

Mason and Rice [28], following a procedure of Corner [29], used experimental data from both the

crystalline and gaseous states to obtain parameters for the modified Buckingham exp-six potential

(equation 11a) for a series of non-polar molecules. 8 In their work [30] on hydrogen and helium which was based on experimental gas data only, they found that this exp-six potential was an improvement

over the Lennard-Jones (12-6) potential for reproducing second virial and viscosity coefficients.

7To reduce the scattering data to units of V/e and R/R , the values of e and R determined by

Mason and Rice (reference 28) were used.

8This work (reference 28) is referred to in reference 1, page 181, where the derived potential parameters are tabulated. We include a summary of reference 28, as the procedure for obtaining the parameters is of interest in discussing other researches.
In a subsequent paper [28], in which calculations were made on spherical (Ne, Ar, Kr, Xe,

CH ) and non-spherical (N , CO, CO ) molecules, a different approach was taken. The procedure

for obtaining the potential parameters was very briefly as follows. Crystal data, namely, the nearest neighbor distance or crystal lattice spacing r and the heat of sublimation at absolute zero &H{O a

) were used to obtain the values of the parameters R and £ corresponding to each one of a series of selected a values. With one of the latter pairs of values of R and e.the experimental

second virial and viscosity coefficients as functions of temperature were iitted to theoretical expressions for these coefficients to obtain a new value of a . If the a for the crystal and gas

properties agreed, all three parameters (a, R , £ ) were accepted as representative for the substance; otherwise the process was repeated for another pair of values of R and£ . m

The best values of the a s were in the range 12 to 14 for the spherical molecules.

For the spherical molecules the gas properties, second virial coefficient, viscosity, self and thermal diffusion coefficients were calculated for the Lennard-Jones (12-6) potential with the parameters R and £ /k derived by the Corner method from crystal or crystal and second virial data. The values obtained as a function of temperature were compared with corresponding exp-six

potential values and experimental data. No overall superiority was indicated for one type of empirical potential relative to the other.

It was noted [28] that for gases composed of non-spherical molecules the two parameter

Lennard-Jones (12-6) potential

V(R) = C [(RmA)12 - 2(Rm/R)

6

]

has not been too successful in fitting different properties with a single set of parameters, and

that the exp-six potential with three parameters might be more successful. However, from the analysis for this type molecule it was found necessary with the exp-six potential [eq. 11a] to choose different parameters to reproduce different properties. This behavior was considered an

indication that the assumptions of central intermolecular forces and elastic collisions are not adequate to describe the behavior of most real gases.

5. 2 Whalley and Schneider

A slightly different approach from that of Mason and Rice was taken by Whalley and Schneider [23],

The latter used their second virial coefficient measurements [31, 32] (for the range about 300 to 900 °K) on

argon, krypton and xenon to obtain the best parameters for the Lennard-Jones (12-6) and (9-6) potentials

and the modified Buckingham exp-six potential of equation (11a); these potentials were then used to derive crystal data.

9 See Appendix for more detailed outline of Corner's procedure.

The theoretical expressions (see reference 1, Chapters 3 and 8) for particular intermoleculer

potentials are usually developed in terms of the reduced temperature, T* = kT/e . For example,

the second virial coefficient in "reduced form" is B(T) = b B* (p, T*) where p is a parameter

characteristic of the potential (P is the a of the exp-six potential). b = (2/3 tt s where _s may be R or R depending on the potential form. For each type of potential ES* is calculated as a function of T* for a set of p values. For the modified Buckingham potential of equation

(11a) B* was tabulated by Mason and Rice [J. Chem. Phys. _22, 522 (1954)].

Actually, calculated results were given for CO and N which crystallize in their lowest temperature phases in the cubic clot.--packed lattice, the case for which the Corner relations had been

obtained at the time of the work done in reference 28. CO and O , for which the lowest temperature form was statecr^ as not cubic, were only examined in regard to fitting the second

virial and gas viscosity data with an exp-six potential and it was found that different parameters were needed for the two properties.

10
As far as fitting the second virial coefficients, the "goodness of fit", based upon the standard deviation of the difference of the experimental and calculated second virial coefficient, was

about the same for all three potentials. In the case of krypton and xenon the exp-six potential

gave equally good fits for a in the range 12 to 15; for argon, a good fit was obtained only with a

of 15.

TABLE IV

Potential Parameters and Crystal Data for Argon, Krypton and Xeno&

Argon

Potential e/k Rm

ro 4H(0°K)

°K oA

oA kcal/mole

L.J. (9-5) 89.6 4.08 3.93 1.718 ± 5 a

L.J. (12-6) 119.5 3.83 3. 76 2.033 ± 6 Exp-6, cC=15 13 1.5 3. 73 3. 74 2.128 ± 7 Experimental 3.81 1.998 +40

Krypton L.J. (9-6) 125. 1 3. 94 3. 78 2.400 i 6

L.J. (12-6) 166. 7 4.13 4.04 2.848 ± 8 Exp-6, ct=12 144.9 4. 35 4.2 1 2. 536 +10

d=15 183.6 3.96 3.90 2. 574 ± 8 Experimental 3.93 2. 733 ± 50 Xenon L.J. (9-6) 168.4 4.89 4.68 3. 236 ±23

L. f! (12-6) 225.3 4. 57 4.45 3. 850 +10

Exp-6, ct=12 196. 1 4.80 4.64 3. 397 ±18

cc=14 233. 3 4. 58 4.48 3.861 ±16

a=15 248.9 4. 50 4.41 4.043 +16

Experimental 4. 30 3.936

Note a: Value ± std. error. Std. errors for the other quantities are reported in reference 23.

Some of the results including the crystal data; namely nearest-neighbor distance, r , and heat

of sublimation AH (0°K) (obtained using Corner's [29] equations) are shown in Table IV. It should be noted (Table IV) that the potential parameters R and e varied considerably for the

different potentials.

It was concluded [23] that the crystal properties were moderately well predicted using any

of the potentials derived from the second virial data and that one potential could be chosen

which would fit either r or AH quite well but not both together; the potentials were considered

too simple to fit all three properties exactly [23],

It is of interest to compare the potential parameters of the present and previous sections. For A , Kr, and Xe, respectively, the LJ (12-6) parameters of Whalley and Schneider and of Mason

and Rice [28] (shown in parentheses) are: for R in A, 3.83 (3.87), 4.13 (4.04), 4.57 (4.46),

i.e., about 1 to 2 percent different; for e/k in ^t, 119.5 (119.3), 166.7 (159), 225.3 (231.2), or

11
about zero to 3 percent different. The exp-slx data is not so readily compared as the a's were

usually different and as is evident from Table I the other parameters are sensitive to a.

5. 3 Schamp and Associates

Experimental and theoretical work similar to that of Whalley and Schneider [23] (WS) was carried out on methane, an approximately spherical molecule, by Schamp etal. 33 The experimental temperature

range, 2 73° to 473 6 K was rather limited. Following the determination of the second and third virial coefficients, the WS procedure was used on the second virial data to obtain parameters for the LennardJones 12-6 and 9-6 potentials and the modified Buckingham exp-six potential (with a's from 12 to 15). With the standard deviation of the difference between calculated and experiment second

virial values as a criterion, and with due regard for experimental uncertainty, it was concluded

that no potential was superior to the others for representing the temperature dependence of the second virial. Similarly to the previous case (Table IV), the potential parameters varied considerably between potentials.

The various sets of potential parameters were used to obtain (from Corner's equations) the crystal data, namely the heat of sublimation, AH(0°K), and r , the nearest-neighbor distance

(measured between moleculur centers). The results are:

Potential r, in'

o A

AH(0°K), kcal/mole

L.J. (9-6) 4.41 2.12

L.J. (12-6) 4.22 2.51

Exp-6, a=l2 4.39 2.23

a=13 4.31 2.38

a=14 4.24 2. 52 a=15 4.18 2. 64

Experimental 4.16 2.51

The agreement between calculated and experimental values is best for the L.J. (12-6) potential

and the exp-six potential with a equal to 14 or 15.

The viscosity of methane was calculated for the various potentials listed above (except L. J. (9-6), for temperatures from 90 to 772 °K. From the average and the maximum deviations relative

to measured data, it was noted there was little difference between the potentials although the exp-six with a equal to 12 or 13 gave the poorest fits. For all potentials and at all temperatures

the calculated values were lower than those measured by an average of at least 3 percent.

5. 4 Konowalow and Hirschfelder

An analysis following Corner's procedure was made for the nonpolar gases Ne, Ar, Kr, Xe, CH4 and N by Konowalow and Hirschfelder [34] who assumed a Morse intermolecular potential:

where

V(R) = £ (x - 2x)

x = exp [-(c/R

q

) (R~Rm)]

(13)

As before, R is the separation at which V(R) is zero, S is the depth of the potential well and

R is the separation at the minimum. As a preliminary, calculations foi the above potential were made of the theoretical second virial coefficient [3 5]; lattice sums needed for using crystal

data to obtain potential parameters were also determined [34]. The experimental crystal data rQ

and AH were used for a series of c values to obtain pairs of values of e and R . With the

12
latter parameter values, theoretical second vlrlals were computed and the c selected which yielded

the best fit to the experimental virial coefficient at or near room temperature.

The parameters for the Morse potential were:

Ne 5.1 2. 775 3.152 43.99

Ar 5.0 3.386 3.855 144. 8 Kr 4.5 3. 510 4.038 182. 7 Xe 4.9 3.872 4.420 27£. 7 CH.

N.

4 4.9 3. 683 4.204 177.5

5.5 3. 579 4.030 134.4

28 The parameters R ande^k were compared [34] to similar parameters obtained by Mason and Rice

for the equation (II) exp-six and the Lennard-Jones (12-6) potentials. The separation at the mini- mum, R , agreed to within a percent for all three potentials. Except for neon, $/k was 15 J^ 2 percent greater than for the exp-six potential, for which in turn e/k was very slightly greater*

(by about one to three percent) than for^he L. I. (12-6) potential.

A test of the derived Morse potential was made by calculating the second virial coefficient

for temperature higher than room temperature (to about 800 °K) and also for Ne, Ar and N to lower than room temperature. The derived virial coefficients together with values obtained for the exp-six and the L.J. (12-6) potential, both sets based on the potential parameters of Mason

, and Rice , were all compared with experiment. Figure 5 shows these results for argon. Except

for krypton the agreement was considered good with the concordance of the first two slightly

| better than for the (12-6) potential. For krypton none of the three potentials fitted the experimental data closely over the temperature range. A good fit for krypton, however, was obtained

by using two sets of Morse parameters (e , c ; g , c , ) one for R<R the other for R> R ; the

potential, but not the first derivative, was required to be continuous at R.

5. 5 Extrapolations

In concluding this section of fitting empirical potentials to gas and crystal data, we refer briefly

to other work which suggests limitations of the method such as extrapolation of the potential to pair

separations outside the range for which the parameters were fitted and also application to other properties than those used for fitting. (Some evidence on the latter was mentioned for the data on Ar, Kr nd Xe by Whalley and Schneider).

12 Except for N and the L. J. of reference I.

(12-6) data for which the comparison values were from Table I-a

13

14,

In reference 28 the graph for krypton extends to less than 600 °K this partly explains the difference in conclusions in references 34 and 28; in the latter, it was noted that the L. J. (12-6) potential did not give a good fit for krypton although the exp-six did do so.

It should be noted that our manuscript, which was completed in the summer of 1963, contained

literature references only up to April 1963. Information that came to my attention (Dec. 1963)

work by Saxena and Gambhlr [S. C. Saxena and R. S. Gambhir, 6_{6), 577 (1963)] indicates that the low temperature second virial coefficient data for argon, krypton and methane are fitted much

better particularly at low temperatures by a Morse potential as compared to the Lennard-Jones

(12-6) or the modified Buckingham (exp-six) potential [m. B. (exp-six)] suggested combining

rules for computing the potential parameters for the Morse potential parameters for the Morse

potential between two unlike molecules from the known corresponding parameters of the like molecules. They find for the atom pairs, Ne-Ar, Kr-Ar and Ar-CH the overall agreement is reasonable and also somewhat better than that obtained with the L. J. (12-6) and m. B. (exp-six)

potentials.

13
As to the first point, Abrahamson [36] made theoretical calculations by the Thomas-FermiDirac statistical method of the repulsive interaction for like pairs of the inert gases, helium

through radon. A comparison of his results to semi-empirical, experimental and other theoretical

calculations was made (see Figures 6-12). He stressed the superiority of the modified Buckingham

exp-six potential over the Lennard-Jones (12-6) potential for representing repulsive interactions at very small separations, which we estimate as roughly l/2 R to 1/4 R .

Guggenheim and McGlashan [37] made an extensive study of the properties of argon, mostly

for the crystalline state, for a semi-empirical potential that they devised. This (GM) potential

had the following features; near the minimum, R= R , it was an anharmonic oscillator type with

cubic and quartic terms ,

V = -E + Ks

2 - As

3 + Bs4 where s = (R-R )/R ; m m

for distances R>1.4 R (i.e., greater than the nearest-neighbor distance r in the crystal, as

r is roughly equal to K ) a simple attractive term c./R was used with c taken from theoretical

calculation; between the first and second regions a graphical interpolation was introduced; for small distances R^d a hard sphere potential was employed; a graphical interpolation connected the point V(R=d)=0 with the first region - _d was adjusted to fit both high temperature second virial coefficient (gas) data and an average of the effective diameter obtained by inserting high temperature experimental viscosities into theoretical expressions for the viscosity of a gas of hard spheres.

The (GM) potential for argon atoms is shown in Figure 15.

The vibrational energy of the lattice was approximated by using the Einstein assumption of a single characteristic frequency v and first disregarding anharmonicity in the vibrations of the atoms about equilibrium. With this approximation and the above potential as a basis, expressions

for the face-centered cubic lattice were obtained for a) _v, b) the partition function, c) the molar

free energy, d) the total energy, e) the entropy and f) the equation of state — a), c), d) and f) all in terms of r and R and the other pair potential parameters. (Use was later made of the fact that a) and f) do not depend on the parameter E = -V(R ). To obtain a measure of the importance

of the quartic term calculations were made with B = O and B = A.

The procedure for getting the potential constants was roughly as follows. Values of _v were

obtained as a function of temperature from experimental entropy data by using equation (e). The

equation (a) and equation (f) (pv/RT - *••••) for zero pressure yield two equations, linear in K and —A and whose coefficients, the F's and G's.are 'mo simple functions of R and r :

from (a); v

2 R

° = KF. + AF + C.F, m 1 2 13

from (f); O = KG, + AG, + C,G, - (l/vR

2

) coth (h v/kT)

l 2 13 m

[KG

4

+ AG

fi + CjGg]

where T is the temperature and h and k. are the Planck and Boltzmann constants, respectively. For a chosen temperature, T = T (triple-point = 838°K), the pair of equations was solved repeatedly for series of trial values of R . The solutions for K and A were plotted as functions of R . The process was repeated for another suitable temperature, chosen as 1/2 T . The inter- section of the two K curves yielded a value of K and of R ; similarly the A curves yielded a value of A and of R . The values of R at the two intersection points needed to be the same.

This was found true when C. was varied over a modest range of its theoretical value.

With R , K and A known, the parameter _E was then found by assuming the experimental

enthalpy, H , to be equal to the total energy (reasonable at low pressure) and inserting H

in expression \d) for the total energy and solving for _E.

i

15 The two temperatures should differ widely and also the lesser must not be so low that the uncertainty

in the experimental entropy would be large and that the Einstein approximation would be a poor one. 14
Calculated values [37] of the temperature dependence (from 25° to T ) of the entropy, enthalpy and molar volume, and the pressure dependence of the molar volume for the crystalline

state, and also the second virial coefficient for the gas was obtained for the ,{GM) potential and

compared both with experiment and with similar data for a Lennard-Jones (12-6) pair potential.

The constants of the Jitter potential were fitted to crystal properties (lattice constant and heat of sublimation at 0°K).

For the second virial coefficient the two potentials were egually suitable for reproducing

experimental values. For the temperature dependence of entropy , total energy (enthalpy) and

molar volume (proportional to _r ) Guggenheim and McGlashan found their potential superior to

the L.J. (12-6) potential while for the pressure dependence of the molar volume the reverse was

true. Since Zucker actually obtained the parameters for L.J. (m-6) potentials with _m varied

between 10 and 14 and then chose that m value namely 12 which yielded the best fit to the experimental data for pressure dependence of molar volume, this superiority of the L.J. (12-6) is not unexpected.

The influence of anharmonicity in the vibrational motion of the atoms on the various calculated crystal properties for the (GM> potential (with B = A) was slight in contrast to the similar

effect of anharmonicity for the L.J. (12-6) potential. From an examination of the equations (e) and (f) it was observed that the effect of anharmonicity was much greater on the _pv product than

on entropy. However, it was found for the former quantity that, using accepted values of _E, K, A and C. the sign of the effect of anharmonicity depended on the magnitude of _B. The authors

believed [37] that they could not expect a reduction in their ignorance of B. by improvement in theory or experiment and concluded it was best to ignore the influence of anharmonicity of the vibrations

.

The lesser flexibility of the L.J. (12-6) compared to the [GMJ potential was emphasized by

Guggenheim and McGlashan who observed that the former potential is equivalent in the (iGM)

1

potential to fixing the constants K, A, B and C, all in terms of _E. Nevertheless the L.J. (12-6)

potential predicted all the crystal properties except entropy to within a few percent.

6. Delta Function Models

As a step in the direction of calculating the intermolecular potential by detailed quantum

theoretical methods, very approximate calculations have been made for simplified atomic and

molecular models. An example is the use of models in which delta function potentials [38, 39, 40]

are substitued for Coulomb potentials to represent the electron-nucleus interaction, together with

other simplifying assumptions. Normally the problem is treated as one-dimensional. Usually the case of the one electron hydrogenlike atom or two electron molecule is first solved and generalization to many-electron molecules made by some semi-empirical process based on physical

intuition.

The L.J. (12-6) parameters of Domb and Zucker [C. Domb and I. J. Zucker, Nature (London)

178 . 484 (1956) and I. J. Zucker, J. Chem. Phys. 2J5, 915 (1956) were used.

11 17 3 7 Since experimental entropy data were fitted to v, an exact fit would be expected for this property.

18 See previous footnote on page 14.

15
6. 1 Frost

Frost [38] solved the Schroedinger equation for the problem of a hydrogenlike atom with the

delta function potential taken as the limiting form of a square well potential of width w and

depth V as the well became indefinitely narrower and deeper while the "strength" £ = w V

remaine"CT constant. The correct atomic energy levels were obtained if the delta potential strength

_g was set equal to Ze /n (-e. is the electronic charge, _Ze the nuclear charge and n the principal

quantum number). This accurate result was attributed [38] to a cancellation of error from the use

of the delta function potential with a one-dimensional model.

For the molecule-ion H the solution [38] of a Schroedinger equation with a delta potential

centered at each nucleus was obtained. The results were less satisfactory; the ground state electronic energy being (over a range of nuclear separations) nearly half that for accurate calculations; also when the internuclear repulsion was added to the electronic energy to obtain a potential energy curve for internuclear motion, unsatisfactory values for the equilibrium separation R and dissociation energy D , occurred, m e

6. 2 Llppincott

Similarly to Frost, Llppincott [39] represented the interacting atoms by a one-dimensional

model in which atomic delta potentials, each a limiting form of a square well potential, are combined on bond formation to form a molecular delta potential; lnterelectronic repulsion was ignored.

To overcome the difficulty encountered on bond formation by Frost who used delta functions

centered at the nuclei, Llppincott allowed the delta potentials to shift or float symmetrically

from the nuclei. The arbitrary assumptions were made: "a) Bond formation can result from the coalescing of two shifted atomic delta functions of the correct separated atom energy into a molecular delta function in which the delta functions are located back on the nuclear positions

when the internuclear distance is equal to the equilibrium bond length. b) The interatomic attraction and internuclear repulsion may be implicity allowed for by the shifts of the delta functions

from the nuclear positions, with the result that the total energy of the system may be identified

with that obtained by solution of the Schroedinger equation for the model". Treating the molecularion H first, Llppincott found, on solving the Schroedinger equation with square potential wells

and then passing to the limit (delta potentials), a total energy

E = - (me

4

/2 J*2

) [1 + exp (-ca)]

2

, (14a)

where the variable c, roughly proportional to the delta potential strength, is given by

c = (me

2//) [1 + exp (-ca)]. (14b)

_a is the separation of atomic delta potentials, m, e and £' are the usual atomic constants. Equation

(14b) is solved for c by trial, but first a suitable relation between a and nuclear separation R

is required. A simple equation to fit assumption (b) and to yield a potential minimum was

assumed, namely

a = R[l - R /R + (R /R)

2

] (15) m m

where R is the nuclear separation at the energy minimum. From equation (15) we see that for stretched bonds a. is less than R and for compressed bonds a is greater than R; furthermore a is always greater than R except at equilibrium.

4 2

If the separated atom energy, E(R =«) = - me /2 K (here the energy of one hydrogen atom),

is subtracted from the total energy in equation (14a), one obtains the interaction potential V(R).

19 Floating wave functions, derived for Coulomb rather than delta potentials, were first used by

Gurnee and Magee [E. F. Gurnee and J. L. Magee, J. Chem. Phys. 18_, 142 (1950)] for two

bonding hydrogen atoms. Hurley [A. C. Hurley, Proc. Roy. Soc. A226 , 179 (1954)] also studied

the behavior of floating wave functions prior to Llppincott 1 s work.

16
For nuclear separations oi.interest the term exp (-2ca) in equation (14a) was small relative to exp

(-ca) and was neglected.

2/2 Bond properties such as c, R , dissociation enem»y D and force constant k = (d E/dR ) 39 t

""~ ""

m

""

s

j© ©cr were calculated for H and the results agreed well with experiment. The experimental value

of —c is defined in terms of other experimental bond properties as c = (1/2) k R /D . erne

The solution of the hydrogen molecule problem was considered similarly to the H example

and with the same one-dimensional potential. Spin and electron-electron repulsion were neglected

and the Schroedinger equation was separable for the two electrons. Thus equations like (14a) and

(14b) were obtained for each electron; equation (15) was again used to relate _a to R. Expressions

for energy _E and _c for this same problem treated as an equivalent one-electron problem followed

from the two-electron equations by introducing

c (H

2

) = (2) 1/2

c (H

2

+

) (16)

The bond properties obtained for H from the equivalent one-electron equations agreed with experi- ment but not as well as for H .

The equations for c and E for the one-electron equivalent of the H molecule were generalized to the homonuclear diatomic molecule M containing a total of _s electrons:

E = -s(me

4

/2K

2

) [1 + exp (-ca)]

2

(17a)

with

c = s 1/2(me

2/K2

) [1 + exp (-ca)] (17b)

and with equation (15) applied as before.

When equations (17) were applied to obtain c values for diatomic molecules of the elements

in the second row of the periodic table, Li through F, the agreement was not considered [39]

satisfactory for Li and F.

("39"! Further semi-empirical modifications were made to improve the agreement between predicted

and experimental bonding properties for a broad range of diatomic molecules. 2 * For example, noting that equation (17b) implicitly assumes the c value is proportional to the ionization potential of the hydrogen

atom and that.c is proportional to the delta potential strength, Lippincott modified equation (17b) for the diatomic molecule M by multiplying the right hand side by the factor I A . I and I are the

ionization potentials, respectively, of the element under consideration and of the hydrogen atom.

20This was also done in the energy equation for other results we quote later from reference 39.

21

It is interesting to observe that originally several workers [41,42] using regular quantum mechanical methods (Coulombic potentials, the three dimensional Schroedinger equation, etc.) found

that simple floating atomic orbitals of Is hydrogenic type yielded considerably improved binding

energies for the molecular systems H and H as compared to results for similar fixed orbitals. Then this problem was subsequently reexamined, [43,44,45] in some cases with a mixture of fixed and floating orbitals. The conclusions of the later authors are that the improvement is not as large as originally found afidf.that to introduce polarization the addition of 2 _p_ orbitals to the

Is type, both types fixed at the nuclei, effects as good or better improvement than floating wave

functions [44, 45] and with simpler calculations — with the floating orbitals 3-and 4-center

molecular integrals occur. It should be noted also that, in contrast to the delta potential model,

in the more rigorous treatment [41-45] the wave functions were not centered on the nucleus at the equilibrium separation. Instead the displacement of the electron orbitals from the nuclei and

also the orbital exponent, which governs the spread of the orbital, were varied to minimize the

total energy of the system at each of a series of internuclear separations.

17
6. 3 Mason and Vanderslice

Mason and Vanderslice [40] employed a delta model of an atom to explain the repulsive

interaction, first of a pair of hydrogen atoms then of a pair of inert gas atoms.

Upon considering Frost's work with such a model and the lack of quantitative agreement

when his results were applied to molecules, the former authors suggested reasons such as: a) neglect of interelectronic repulsion, b) uncertain basis for internuclear repulsion, and c) underestimation of overlap of each atomic orbital (AO) with the potential well of the other nucleus.

The elimination individually of the sources of error was presumed difficult and also would not preserve a simple model.

From consideration of a) Berlin's interpretation of the Hellman-Feynman electro static theo- rem , namely attraction is caused by accumulation of electronic charge between nuclei and

repulsion by removal of charge from this region, and b) Hurley's work [42,46] on the use of floating AO's to satisfy this theorem, Mason and Vanderslice proposed a delta function model in which the AO's "float" relative to the nuclei as the atoms approach. Since the delta potentials

determine the location of the centers of the AO's, the delta potentials also need to shift relative

to the nuclei. Thus this model is similar to Lippincott's; it was treated as one-dimensional.

The basic interaction calculated was that for two hydrogen atoms with parallel spins

( % state). If the procedure of Lippincott for a pair of bonding hydrogen atoms is followed, except

that each one-electron wave function is made anti-symmetric (a node midway between the nuclei), the solution of the Schroedinger equation yields a total energy:

E = -1/2 g

2

[1 - exp (-ca)]

2

= -1/2 g

2

[1 - 2 exp (-ca)], (18a)

in atomic units, and where

c = g [1 - exp (-ca)], (18b)

with a the separation between delta potentials and £ the delta potential strength. The interaction energy, the difference between total energy at some finite nuclear separation and that at infinite separation is

V (a) = g

2 exp (-ca), (19)

where as in equation (18a) the term in exp(-2ca) is discarded. £ is chosen to obtain the correct energy of two separated hydrogen atoms, that is (equation 18a) E (•) = 1/2 g must be one in atomic units, so jj is -/Fin atomic units.

To obtain the dependence of a on R it is assumed, following the Hellmann-Feynman theorem,

that the interaction being repulsive _a is greater than R but a. —* R as R —,>«>. Further for the united atom the Is atomic functions coalesce to a2 p function, an effect that is approximated by

taking a as 2a for R equal to 0. The relation adapted for a_, expressed in atomic units .a is — —

o

o

a = R + 2 exp (-R). (20)

See reference 1

23

It is stated in reference 40 that for the examples reported a three-dimensional treatment gave

identical final results to the one-dimensional method.

18
Mason and Vanderslice emphasize that in thermal collisions even at high temperature the

shift of the delta functions from the nuclei is relatively small so the sensitivity of the results in

this energy range to the particular dependence of a on R is not critical. We note in view of the nonlinear relation between a. and R, that V(R) through equation (19) is not strictly an exponential

in R.

The repulsive interaction of a pair of H atoms calculated from equations (18) - (20) for R between a. and 4a was compared with a ."most likely" curve composed of James-CoolidgePresent [5J and Hirschfelder-Linnett [6] data. The agreement was excellent for separations of 4a down to 2.4a. ; the delta potential data deviated by increasing amounts from the "most likely" value as —R was reduced below 2.4a , reaching a difference of 30 percent at 1.1a . ^o —

o

On considering the interaction of two inert gas atoms, it was assumed [40] that the total interaction energy is the sum of the average energies of each pair of electrons as for two hydrogen atoms. The interaction energy of two atoms each with n electrons is then from Equation (19)

V(a) = n g exp (-ca) (21)

Parameter _c is again obtained from equation ( 18b). The parameters jj and a need to be specified.

Lippincott's assumption was adopted [40], namely the delta potential strength ^j is proportional to

the ionization potential of the atom and hence to obtain the _g value for the atom pair A-A the _g_ value for hydrogen should be multiplied by the ratio of ionization potentials of atoms A and H,

gA=/F

"

IA

/lH

< 22)

For a., by analogy with equation (20) it is assumed that

a = R + 2 r exp (-R/r), in units of a , (23) o - 25

where r_ is the radius of the outermost electronic orbit of the isolated atom.

In the final form of the equations there are no adjustable parameters, the interaction potential is fixed when the ionization potential, the atomic number and the radius of the outermost

electronic orbit are introduced.

The interaction potentials were obtained for like atom pairs of the inert gases [40] and were compared (Figures 14-18) with experimental potentials derived from atom-atom scattering, or gas

property and crystal data, and in a few cases with detailed quantum theoretical calculations. A

few interaction potential values calculated from equations (18b) and (21)-(23) are shown in Table V

together with the comparison values. It was noted [40] that the agreement of the delta function

interaction potential values with experimentally derived values is as good or better than that of the detailed quantum theory calculations for helium, neon and argon, the only cases for which

such calculations were available.

The delta function model was also applied to the interaction of unlike atom pairs. Plausible

combining rules for this situation led to the following potential of interaction of an atom of type 1 with an atom of type 2:

V

12 = (n

i n2

)l/2

(g

l 9

2

) SXP [_(C

i ai + C

2

a

2 )/2] (24)

24

Reference 1, page 1062-63

250btained either from self-consistent-field calculations, or less accurately using screening

constants and equations derived for the average radius of hydrogen-like atomic orbitals. (see

reference 1, page 951 ff)

19
where n^ ^ and c, have the same meaning as n, _g, and _c in the preceding equations but refer to an atom of type J., a. is obtained from an equation like equation (23) withr in place of_r;

Cj is given by an equation like equation (18b) with c £ and a in place c , $7} a. A similar

procedure applies to a and c . Calculated values of V (R) were obtained with equation (24)

for H-He, He-Ar and Ne-Ar pairs and compared with potentials from atom-atom scattering data; agreement comparable to the homonuclear case resulted. Mason and Vanderslice point out [40] that because of the great oversimplification of the delta function model, its extension to other systems or the calculation of other properties must

be done with much caution.

7. A Semi-Classical Model of Atomic Interaction

An electrostatic model of the atom was proposed by Jefimenko [47] with features to permit

its application to pairwise interaction of both

TABLE V

Interaction Potential of Inert Gas Atom Pairs'

Detailed Gas and

R Delta Model Theoretical Calculation Scattering Crystal Data

a

o ev ev ev ev

Helium

1.0

2.0

19.8

2.02

26.9

b;

25.0

db 3.10. 3.27

d

9.0

C 16. e 3.3 e

2.5 0.58

f 0.90

Neon 3.M

1+.36 0.6U

0.077

2.15g 0.2l8g 0.88

h 0.075

1

Argon

3.0

k.O 5.0

6.69

0.33G

13.o\

2.05

3.90 J

0.257

1

, Krypton

>*.5 5.5 2.21

0.59

l.l*5m 0.U87m

Xenon

6.0 6.8 0.85

0.3U6

0.71°

0.262

n

* Footnotes on following page

a Delta potential from equations (18b), (21)-(2 3); other data from references below listed in reference

40 except values with notes d and _e.

b

S. Huzinaga, Pro. Theo. Phys. (Kyoto) XL, 512 (1957); 18., 139 (1957).

(Continued on next page)

20
(Footnotes to Table V) Reference 10.

d

? E. Phillipson, Phys. Rev. 125 , 1981 (1962).

'Reference 18.

Reference 11.

gW. E. Bleick and J. E. Mayer, J. Chem. Phys. 2, 252 (1934).

h

I. Amdur and E. A. Mason, Ibid. 23, 415 (1955).

ue fere nee 28.

^Amdur, Davenport and Kells, J. Chem. Phys. ,18_, 525 (1950).

k

M. Kunimune, Pro, Theo. Phys. (Kyoto) 5_, 412 (1950).

I. Amdur and E. A. Mason, J. Chem. Phys. 22., 670 (1954).

L

m

i. Amdur and E,. A. Mason, Ibid. 23, 2268 (1955).

n

l. Amdur and E. A. Mason, Ibid. 25, 624 (1956).

21
combining and non-combining atoms. The justification for an electrostatic view of the interatomic

forces was based on the Hellmann-Feynman Theorem which states essentially: In a molecule the force on an individual nucleus is that exerted electrostatically by the other nuclei and by the electrons where the latter are represented as a charge cloud whose density is given by the probability density determined quantum mechanically (as by solving the Schroedinger equation). Jefimenko

suggests it is plausible that approximate solutions to interatomic interactions can be obtained for electrostatic atom models that "are only partially compatible with the quantum mechanical requirements of the system under consideration".

The atom model is formed as follows: The electron charge cloud is a very thin spherical

shell of radius _a and total charge -Ne; the shell is centered about the nucleus which is a positive

charge of Ne units (not the actual charge Ze). N. is an effective charge factor; for hydrogen N is set equal to one, while for other elements (except inert gases) N is fixed by requiring that the ionization energy of the atom model with N electrons must equal the total experimental ionization

energy of the valence electrons of the element considered.

The energy of a one electron atom (the negative of the ionization energy), the sum pf the shell-nucleus attraction, -e /a, and the shell self—energy, + (1/2) (e /a), was obtained. If

a. is made equal to the Bohr radius, a = X /me , one obtains the correct energy, or negative

of ionization energy, of the hydrogen atom: WH H

= -e

4 m/2K

2

.

To obtain the ionization energy of an element X with effective charge factor N, e_ is replaced by

Ne in the equation for I to yield H

The effective charge factor is then

Ix

= (Ne) 4m/2/ (25)

Ne = (1^)1/4

(26)

A model of a homonuclear diatomic molecule is formed from two identical atomic models with inter- nuclear separation R. As the atoms interact each shell is assumed to retain its spherical shape

and remain rigidly attached to its nucleus. The electrostatic energy of the interacting systems

(shell -shell plus shell -nucleus plus shell -nucleus plus nucleus -nucleus ) was determined for the three cases (a) no shell interpenetration, lb) electron shells interpenetrate but nuclei remain

outside opposite shells, and (c) shells interpenetrate enough to enclose both nuclei (Figure 21) Q©

(a.) (h) (c)

Figure 21 - Semi-classical model of a homonuclear diatomic molecule. (after Jefimenko).

26Obviously this atomic energy has no minimum as a. is decreased.

22
The interaction energies obtainedl- 4_?J were

V(R) = ,R>2a (27a)

V(R) = -[(Ne) 2/4a 2 R] (2a-R) 2 ,a<R<2a (27b)

V(R) = [(Ne) 2 /4a 2 R] [8a 2 -(2a+R) 2

], R<a (27c)

These equations represent a potential having a well and with very approximately the character of a Morse potential except2 ' that the bottom of the well is a cusp (at R = a) instead of a bowl. This

deviation from reality can be removed by using an electron shell of finite thickness. The model thus

indicates an equilibrium separation Re equal to .a. The well depth V(R = a) equals the dissociation

energy Efe. From equations (26) and (27b) the latter energy is De

= (e 2 /4Re) (IXAh)1/z (28)

Values of De were calculated [42] from equation (28) for the homonuclear diatomic molecules of twenty elements by using experimental values of Rq and Ix- The substances included halogens;

alkalis; H2

; C2; O2 and the similar (Group VIA) divalently-bonded S2, Se 2, Te 2 ; N2 and the similar

(Group VA) trlvalently-bonded P 2 , As 2

, Sb2 , Bi2

. For the alkalis (except Li2 ) and H2 and O2 the agreement with measured dissociation energy values was better than 10 percent; for the halogens

the, calculated values were roughly 25 percent low; for many other substances it was noted [47] that the degree of agreement could not be judged because of uncertain experimental data. Considering

the simplicity of the model the agreement with experiment is good.

The potential curves in reduced form (V*(R*))= V/De with R* = R/Re

) for I2 and H2 were calculated

and compared with experiment. (Figures 22 and 23) For I2, except for the cusp shape at the bottom

of the potential well and for too shallow a slope for R* less than Re* the calculated and experimental

curves were fairly close. In the case of H2 the agreement was much poorer, the experimental

curve having a much wider potential well and a more gradual slope for R* greater than Rg*. It was

suggested [47] that for H-, the agreement with experiment could be improved by incorporating into

•the model some of the effects of the mutual perturbation of the charge distributions of the two atoms.

To obtain purely repulsive interactions between two atoms, such as two H atoms with parallel spins,

which quantum mechanically are related to spin orientation and the Pauli exclusion principle, it is necessary in the present model to introduce some concept equivalent to the exclusion principle.

This was done by postulating [47] a "penetrability" concept for electron shells: 1) two perfectly

penetrable shells can interpenetrate with no other interaction than the usual electrostatic one, as assumed in the previous derivation, 2) two perfectly impenetrable shells cannot penetrate at all.

The interaction energy of two atoms for case 2) will depend on the resistance the shells offer to shifting relative to their respective nuclei. In one extreme, infinite resistance to shifting leads

to the hard sphere model. The other extreme, no resistance to shifting, leads to an interatomic

repulsion, in reduced form

Vr(R*) = 2 [(2-R*) 2/(2 + R*)R*] (29)

27Other exceptions; equation (27) is not asymptotic to the R_-axis as R increases and as R* 0, V(R) varies l/R.

28As a matter of interest we calculated N for equation (26) to see how closely N compares with the actual number of valence electrons. The values of Ix/^H were taken from reference 47. The

calculated values of N are 0. 94 to 1. 06 for the halogens, 0. 73 to 0. 79 for the alkalis, 1. 2 to 1.4 for the divalent oxygen-like elements, and 1.4 to 1. 6 for the trivalent series N2

• • • Bi2

.

23
The true potential may be expected to lie between the above and the hard-sphere potential. V was

calculated [47] from equation (29) for H2 ( 3 £ state) and exhibited similar characteristics (Figure 20)

to a more accurate quantum mechanical curve 2 ^ but was lower and vanished at R = 2a while the accurate calculation vanishes at about 7. 5 a .

As a further test of the model the reduced attractive potential, [equation (27) in reduced form] was

used to calculate the reduce second virial coefficient. For low temperature, T*<1. 5, the model

potential results agreed with second virial values for the Lennard-Jones (12-6) potential; for higher

temperatures the agreement was poorer: (B*) .,, always negative approached zero at about T* = 10; B*L T kecame positive at T*^3. 5 and remained positive for increasing temperature. By arbitrarily

making the model potential become that of a hard sphere at the small separation corresponding to the zero of the equation (2 7) potential on the repulsive part of the curve, it was possible to reproduce the Lennard-Jones (12-6) B* vs T* curve fairly well. From experimental second virial data (for T*<1. 5) for the inert gases He through Xe, values were determined [47] of the effective charge factor N. They

were relatively small as may be expected in view of the weak interaction of the inert gases, and they

ranged from 0. 023 for He to 0. 16 for Xe.

While this model deviates greatly from the exact situation it does predict some features of atomic interactions.

The author wishes to thank Drs. Charles Beckett, Melville S. Green and H. W. Woolley of the National Bureau of Standards for their encouragement and helpful discussions during the preparation of this manuscript. He also wishes to thank the many persons who were so obliging in furnishing reprints

and unpublished data which was essential in carrying out this survey.

APPENDIX

Corner's method. The pair potential of equation (11a) is expressed as V = eV*(R*) that is a function of the reduced separation, R* = R/R,^. By performing a lattice sum the potential V (d, R*,£, a) for one

molecule in the crystal lattice displace d_ from equilibrium was obtained. From the latter potential

the vibrational frequency v is found for the one molecule with the others at rest. As an approximation

the average lattice vibrational frequency £ is taken to be V. The total energy of the lattice at 0°K,

UM(R*,£,a) is the sum of the potential energy Upe

= l/2NVjvj(0, R* • • •)» and the zero point energy

3/2Nhv. The condition for equilibrium lattice spacing (0°K) is that x = aU M/<?R* be zero.

The experimental heat of sublimation, AHexp(0°K) is obtained by adding to the observed heat

of sublimation the zero point energy, (9/8)NkT-Q, where Tj-, is the experimental Debye temperature.

The theoretical heat of sublimation at 0°K is the poten:ial energy Upe

of the crystal at equilibrium

spacing r . With4Hexp and r known, a value of a is selected. Then after setting U equal to 4H exp one obtains £ as a function of R* . Inverse interpolation of X shows the value of R* for which

X vanishes and the corresponding £ and Rm(rQis known) are found.

^Reference 1 page 1058; Hirschfelder-Linnett and James-Coolidge-Present data, loc. clt.

24
REFERENCES

1. Hlrschfelder, Curtiss and Bird, Molecular Theory of Gases and Liquids (John Wiley, New York 1954).

2. Y. P. Varshni, Rev. Mod. Phys. 29, 664 (1957).

3. Steele, Lippincott and Vanderslice, ibid. 34, 239(1962).

4. R. A. Buckingham, Trans. Far. Soc. j54, 453 (1958).

5. James, Coolidge, and Present, J. Chem. Phys. 4, 187 (1936).

6. J. O, Hlrschfelder and J. W. Linnett, ibid. 18, 130 (1950).

7. A. Dalgarno and N. Lynn, Proc. Phys. Soc. A69, 821(1956).

i 8. I. Amdur and E. A. Mason, J. Chem. Phys. 25_, 630 (1956).

9. Mason, Ross and Schatz, Ibid. 15, 626 (1956).

10. I. Amdur, Ibid. .17, 844(1949).

11. I. Amdur and A. L. Harkness, ibid. .22, 664(1954).

12. V. Griffing and J. F. Wehner, Ibid. .23, 1024 (1955).

13. M. Sakamoto and E. Ishiguro, Pro. Theor. Phys. (Kyoto) 15. 37(1956).

14. R. A. Buckingham and D. M. Duparc, Progress in International Research on Thermodynamic and

Transport Properties (Academic Press, New York 1962); see also J. Chem. Phys. 38., 275 (1963).

15. B. J. Ransil, J. Chem. Phys. 34, 2109 (1961).

16. W. Bingel, Ibid. JO, 1250 (1959).

17. Amdur, Jordan and Colgate, ibid. 34, 1525(1961).

18. I. Amdur and R. R. Bertrand, ibid. .36, 1078 (1962).

19. P. E. Phillipson, Phys. Rev. 125 , 1981 (1962).

20. A. A. Frost and B. Musulin, J. Chem. Phys. .22, 1017(1954).

21. Chen, GeHer and Frost, J. Phys. Chem. _61, 828(1957).

22. A. A. Frost and J. H. Woodson, J. Am. Chem. Soc. 80., 2615 (1958).

23. E. WhalleyandW. G. Schneider, J. Chem. Phys. 13, 1644(1955).

24. E. A. Mason, ibid. .23, 49 (1955).

25. H. Woolley, ibid. 31, 405 (I960).

26. I. Amdur and E. A. Mason, ibid. .25, 624 (1956) and preceding papers.

27. H. W. Berry, Phys. Rev. 94, 553 (1955).

25
28. E. A. Mason and W. E. Rice, J. Chem. Phys. 21, 843 (1954).

29. J. Corner, Trans. Far. Soc. 44, 914 (1948).

30. E. A. Mason and W. E. Rice, J. Chem. Phys. .22, 522 (1954).

31. Whalley, Lupien and Schneider, Can. J. Chem. _31, 722 (1953) and H, 633(1955).

32. E. Whalley and W. G. Schneider, Trans. Am. Soc. Mech. Engr. Tb, 1001(1954).

33. Schamp, Mason, Richardson and Altman, Phys. of Fluids, _1, 329(1958).

34. D. D. Konowalow and J. O. Hlrschfelder, Phys. of Fluids 4, 629(1961).

35. Konowalow, Taylor and Hlrschfelder, ibid. 4, 622(1961).

36. A. A. Abrahamson, Phys. Rev. 130 . 693 (1963).

37. E. A. Guggenheim and M. L. McGlashan, Pro. Roy. Soc. A255 , 456(1960).

38. A. A. Frost, J. Chem. Phys. 22, 1613 (1954); 25, 1150 (1956).

39. E. R. Lippincott, ibid. 26, 1978 (1957).

40. E. A. Mason and J. T. Vanderslice, ibid. .28, 432 (1958).

41. E. F. Gurnee and J. L. Magee, ibid. J8, 142(1950).

42. A. C. Hurley, Pro. Roy. Soc. A226 . 179(1954).

43. H. Shull and D. D. Ebbing, J. Chem. Phys. 28, 866(1958).

44. Geller, Frost and Lykos, Ibid. 36, 2693 (1962).

45. H. C. Bowen and J. W. Linnett, Mol. Phys. 4, 387 (1963).

46. A. C. Hurley, Pro. Roy. Soc. A226 , 170, 193 (1954).

47. O. Jefimenko, J. Chem. Phys. 37, 2125 (1962).

26
LIST OF FIGURES

4

Figure 1 - Variation of electronic interaction energy of two helium atoms with nuclear separation

(after Buckingham Reference 4).

Figure 2 - Variation of total interaction energy of two helium atoms with nuclear separation; theoretical

points as in Figure 1 (after Buckingham Reference 4).

Figure 3 - Potential energy curve for KC1: V and R in atomic units Full curve, Frost and Woodson

broken curve, Varshni. [Y. P. Varshni, Trans. Faraday Soc. , J33, 132(1958)]

Figure 4 - The potential function given by Equation (12) with integer v from 10 to 15. Lennard-Jones (6-12)

and (6-9) potentials and inert gas scattering data are also shown (after Woolley).

Figure 5 - Comparison with experiment of the second coefficients for argon calculated for the Morse,

Buckingham (exp 6), and the Lennard-Jones (12-6) potential models. The potential

parameters have been determined at least in part from crystal data.

Figure 6 - Repulsive interaction potentials for the He-He system. The first group of six references

relate to experimental work; the remainder to calculated results (after Abrahamson).

Figure 7 - Repulsive interaction potentials for the He-He system.

Figure 8 - Repulsive interaction potentials for the Ne-Ne system.

Figure 9 - Repulsive interaction potentials for the Ar-Ar system.

Figure 10 - Repulsive interaction potentials for the Kr-Kr system (after Abrahamson).

Figure 11 - Repulsive interaction potentials for the Xe-Xe system (after Abrahamson Reference 36).

Figure 12 - Repulsive interaction potential for the Xe-Xe system.

Figure 13 - Repulsive interaction potential for the Rn-Rn system (after Abrahamson).

Figure 14 - Repulsive interaction potentials for the Rn-Rn system (after Abrahamson).

Figure 15 - The interaction energy V plotted against the separation R of a pair of argon atoms for

each of the sets 1, 2, 3 and 4 of parameters given in Table 2. The parts of each curve,

calculated from formula (2-1) for R^^, and from formula (2-2) for R?-R 3, are shown

as full lines, and the parts drawn free-hand are shown as broken lines. The dotted

curve shown for comparison is the (6-12) interaction energy with the parameters of set 5

in Table 2.

27
Figure 16 - Energy of interaction of two helium atoms. The solid curve is calculated from the deltafunction model.

Figure 17 - Energy of interaction of two neon atoms. The solid curve is calculated from the deltafunction model.

Figure 18 - Energy of interaction of two argon atoms. The solid curve is calculated from the deltafunction model.

Figure 19 - Energy of interaction of two krypton atoms. The solid curve is calculated from the deltafunction model.

Figure 20 - Energy of interaction of two xenon atoms. The solid curve is calculated from the deltafunction model.

Figure 21 - Semi-classical model of a homonuclear diatomic molecule (after Jefimenko).

Figure 22 - Observed and calculated potential curves for H2.

Figure 23 - Observed and calculated potential curves for I 2.

Figure 24 - Comparison of the potential curve resulting from the semi-classical model with the curves

resulting from Lennard-Jones (5-12), (6-12), and (7-12) potentials.

28
o

i >

8

IO

/ Egn.(6)

/

L /

/

/

E(Be)-2E (He)

SCATTERING

R,a

Figure 1 - Variation of electronic interaction energy of two helium atoms with nuclear

separation4 (after Buckingham Reference 4)

a?
0.4

0.3

o

.5.

0.2

O.

EQN.(b)

AMDUR

,HARKNESS

M<

*

Figure 2 -

Variation of total interaction energy of two helium atoms with nuclear separation;

theoretical points as in Figure 1 (after Buckingham Reference 4)

30
°D/,a 'a

31
CVJ +

> 10

Figure 4 - The potential function given by Equation (12) with integer y from 10 to 15.

Lennard-Jone (6-12) and (6-9) potentials and inert gas scattering data are also shown (after Woolley).

32
20-

0-

-20

O

E

t6

o-40

CD

-60

-80

-100

ARGON

MORSE

(EXP-6)

200 400 600 800 1000

T,°K

Figure 5 - Comparison with experiment of the second coefficients for argon calculated

for the Morse, Buckingham (exp 6), and the Lennard-Jones (12-6) potential

models. The potential parameters have been determined at least in part from

crystal data.

33
10*

o

a

CM

a>

»»

>

— 1 1 He-He 1 i — — X BLAIS • BOHR —

A BUCKINGHAM GRIFFING SWEHNER —

+ CHAKRABORTI A HASHINO aHUZINAGA —

a LENNARD-JONES MOORE

1 — MASON & RICE MULLIKEN y

1 ~~ SCHNEIDER & YNTEMA T RANSIL —

L" __ 9 ROSEN

O ROSEN, MARGENAU, a PAGE

V SAKAMOTO a ISHIGURO

o SLATER

SLATER a KIRKWOOD

Figure 6 - "> u

Repulsive interaction potentials for the He-He system. The first group of

six references relate to experimental work; the remainder to calculated results

(after Abrahamson).

34
10 V

He -He =

Al AMDUR 1949

A2 AMDUR & HARKNESS 1954 =

A3 AMDUR, JORDAN, a COLGATE I96F

A4 AMDUR a BERTRAND 1962

D LENNARD-JONES (12-6)

O MOD. BUCKINGHAM (EXP-6) ~ A REPULSIVE PART OF (EXP-6) z PHILLIPSON

Figure 7 - Repulsive interaction potentials for the He-He system.

35
Figure 8 - Repulsive interaction potentials for the Ne-Ne system.

36
Figure 9 - Repulsive interaction potentials for the Ar-Ar system.

37
2 3

R,o

Figure 10 -

Repulsive interaction potentials for the Kr-Kr system (after Abrahamson).

38
I0

6

pr

I0

5 t

lor

O .^3 10" =- d

0)

10' =-

10

1.0

K>

rl

T

Xe-Xe

+ K0N0WAL0W a HIRSCHFELDER*.

MORSE

D MASON ft RICE: (12-6) __ O MASON a RICE:(EXP-6)

A REPULSIVE PART 0F(EXP-6) =

AMDUR SMASONIEXPTL. -

r BOHR

i i "M

Figure 11 - Repulsive interaction potentials for the Xe-Xe system (after Abrahamson

Reference 36).

39
10'

10'

10

o

o

M

<D

>

1.0

10

r\

"1 1

1 \—

Xe -Xe

+ K0N0WAL0W 8 HIRSCHFELDER;

MORSE

D MASON a RICEt (12-6)

O MASON a RICE» (EXP-6)

4 REPULSIVE PART 0F(EXP-6)

TF

+ %

+ \

+ ^AMOURS

VvMASON^EXPTL

+ \

BOHR

2 3 4 5 6 8

Figure 12 - Repulsive interaction potential for the Xe-Xe system.

40
\(f

Rn-Rn

a GREW 8 MUNDW2-6)-

OGREW 8 MUNDY:(EXP-6)

a REPULSIVE PART OF 3

(EXP-6)

Figure 13 - Repulsive interaction potential for the Rn-Rn system (after Abrahamson),

41
Rn-Rn

OGREW 8 MUNDY:(l2-6) -= o GREW a MUNDY:(EXP-6)

a REPULSIVE PART OF

(EXP-6)

I 2 3 4 5 6 7

R,a

Figure 14 - Repulsive interaction potentials for the Rn-Rn system (after Abrahamson).

42
o

>

KJ V : 1 1 1 __J

*, : • -<^rrr' *\ • " - -

."!"•"•"" "

'

50

v«-. V*. :#

00

\ \

y/>'

•

• s

y

/

50

t

V

t t

^-

Ri R2 R 3

1

1 1 1 I

R, &

Figure 15 - The Interaction energy V plotted against the separation R of a pair of argon

atoms for each of the sets 1, 2, 3 and 4 of parameters given in Table 2.

The parts of each curve, calculated from formula (2-1) for R <R<R , and

from formula (2-2) for R>R_, are shown as full lines, and the parts drawn

free-hand are shown as broken lines. The dotted curve shown for comparison is the (6-12) interaction energy with the parameters of set 5 in

Table 2.

43
.8

.6

o

a

CM

-

O R ROSEN

• HUZINAGA

EXPTL.

(A.H.)

-

~°rEXPTL.(M.R)

R,a

Figure 16 - Energy of interaction of two helium atoms. The solid curve is calculated

from the delta-function model.

44
o

o

<M

<1> >

r i

1

.08 o

-

BLEICK AND MAYER .

.06 -

-

\ .04 \ \ - \ \ \ \ \ \ \ \ \ \

\ ^— EXPTL.(A.M.) -

.02

\ \ \ \ \ \ \ \ \ \

X \ X N X^sxv

o

- EXPTL. (M.R.)^

I 1 i . .. 3 4 5

R,aQ

Figure 17 - Energy of interaction of two neon atoms. The solid curve is calculated from

the delta-function model.

45
.4

oo

(V) .2

>

-

"I-o

O KUNIMUNE

EXPTL. (A.D.K.)

EXPTL. (A.M.)

EXPTL. (M.R.)

3 4 5 6

R,Q

Figure 18 - Energy of interaction of two argon atoms. The solid curve is calculated

from the delta-function model.

46
.08

.06

o

o .04

CO

0)

.02

EXPTL.

(A.M.)

EXPTL. (W.S.)

-L 4 6

R, a

Figure 19 - Energy of interaction of two krypton atoms. The solid curve is calculated

from the delta-function model.

8

47
06

.04

o

o

CVJo> >

.02

EXPTL.

(A.M.)

E X P T L . ( W. S.

) 0GZZZ£ZSBssiasKmm*mm

8

R . a

o Figure 20 - Energy of interaction of two xenon atoms. The solid curve is calculated

from the delta-function model.

48
t

oc oMc

<D6

<D t—> io

•HoS6o

10

<DO3Co6s

(0

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too

J.e

CO

49
V

SEMI-CLASSICAL MODEL

HIRSCHFELDER etal

Su

( REPULSIVE STATE)

3 R

*

£

g

(NORMAL STATE)

Figure 22 - Observed and calculated potential curves for H.

50
v^

OBSERVE

CALCULATED

Figure 23 - Observed and calculated potential curves for I.

51
V

Figure 24 - Comparison of the potential curve resulting from the semi-classical

model with the curves resulting from Lennard-Jones (5-12), (6-12), and

(7-12) potentials.

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