2123
CALCULATION OF INTERACTION ENERGY
Calculation of the Interaction Energy of One Molecule with Its Whole Surrounding. I.
Method and Application to Pure Nonpolar Compounds
by Marie-Joe6 Huron* Institut Francais d u PBtrole, 9bRueil-Malmaison, France
and Pierre Claverie Laboratoire de Chimie Quantique, Institut de Biologie Physico-Chimique, 75-Paris (be), France (Received January 14, 1972) Publication costs borne completely by The Journal of Physical Chemistry
A method of calculation of the dispersion and repulsion energies of a central molecule with its whole surrounding is proposed. The total energy is calculated as a sum of interaction potentials of atomic pairs with a correction of nonadditivity on the dispersion energy. The total energy is expressed as a function of energy integrals and “calibration constants” K,, . The energy integrals are calculated numerically outside a volume a that takes into account the form of the central molecule, as it is deduced from the van der Waals volume so as to reproduce the molecular volume. The constants K,, are defined as the ratio between energy calculated by discrete summation and energy calculated by integration for the simple case of packing of spheres of radius Ri around a sphere of radius R,. The method is applied to pure substances (hydrocarbons),and the calculated energies are compared with the vaporifiation energies. The agreement is very satisfactory.
I. Introduction The aim we pursue is to calculate interaction energies of molecules and to connect these calculated energies to the thermodynamic properties. Previously, we have used a wholly discrete mode1;l this one required long computation times since it was necessary to calculate several spatial configurations of the solvent molecules around the central molecule to obtain a partial statistical average. To obtain a more efficient method, we have tried a new model with integrations on the outside of the volume of the central molecule. I n the the central molecule is continuum models used so far,2,3a taken to be a sphere, and the outside solvent is a medium possessing volume characteristics (polarizability by unit of volume) which correspond to macroscopic experimental data. The agreement with the experiment essentially depends on the chosen values for the radius of the sphere. As we shall see, the effect of this choice is to take into consideration the nonuniform distribution of the attractive centers of the solvent immediately near the solute. We deviate from these models essentially by the two following points. (1) To take into account the form of the molecule, we do not replace it by a sphere, but by a volume which is deduced from the van der Waals volume so as to reproduce the molecular volume. ( 2 ) I n our model, the energy integrals are calibrated with regard to completely discrete models that are simple (packing of spheres of radius R j around a central sphere of radius R J . Therefore, it is a kind of interpolation between these discrete models rather than a calculation using macroscopic constants uniformly distributed in the volume attri-
buted to the solvent. It may be mentioned that Kihara and J h ~ have n ~ studied ~ the solubility of gases in liquids using a convex core model which also represents the real shape of the solute molecule in a more realistic way than the previous spherical approximations. The present model calculates dispersion energies and repulsion energies; consequently, it is convenient for nonpolar molecules in the cases where the electrostatic and induction energies are negligible. Later, we plan to extend the present model to polar molecules by using the dielectric constant of the solvent. The model is applied here to the calculation of the energies between identical molecules (pure substances), and calculated energies are compared with vaporization energies.
11. Description of the Used Model We consider a central molecule (solute molecule) surrounded by solvent molecules; we want to calculate the interaction energy of the solute molecule with its entire surrounding assumed infinite in volume. To calculate the energy, it is necessary to choose a potential law and to make a hypothesis on the distribution of solvent molecules around the solute molecule. The solute molecule and the solvent molecules are characterized by their geometry, the nature of their atoms, and by their molar volume. M.J. Huron and P. Claverie, Chem. Phys. Lett., 9, 194 (1971). (2) (a) B. Linder, J . Chem. Phys., 33, 666 (1960); (b) ibid., 35, 371 (1)
(1961). (3) (a) N. G. Bakhshiev, 0. P. Girin, and I. V. Piterskaya, Opt. Spectrosc., 24, 483 (1968); (b) T. Kihara and M . S. Jhon, Chem. Phys. Lett., 7, 559 (1970).
The Journal of Physical Chemistry, Vol. 76, N o . 16,1972
2124
MARIE-JOSB HURON AND PIERRE CLAVIDRIE
The solute-solvent interaction energy is calculated as a sum of interaction potentials of pairs of atoms;4 this is true for the dispersion energy calculated with the second order of perturbation. We also take into consideration (see section 114) the nonadditive contributions which appear a t the third order of perturbation.6-8 We use a potential function of the type of that used by Kitaygorod~ki~ for the dispersion energy and the repulsion energy. The sum of bond energies4 is not impossible but sets difficult problems which are not yet solved. 1. Expression of the Dispersion and Repulsion Energies. For nonpolar molecules, we can neglect electrostatic and induction energies, and ithe interaction energy between the atom i belonging to the solute molecule and the atom j belonging to the solvent molecule has the expression4
Rt and R j are, respectively, the van der Waals radii of atoms i and j; r a jis the distance between atoms i and j-
We call effective volume Vefifof the solvent (or of the solute) the molar volume of the solvent (or of the solute) divided by the Avogadro number. A solvent molecule contains 9zj atoms of type j; the volume dv of the solvent contains dv/Veffmolecules of solvent and nl dv/Veff atoms of type j to which is associated the energy
The interaction energy of the atom i of the solute with all the atoms of type j of the solvent is written
ne
The integration volume Qe, which is the outside volume of the volume Q of the solute molecule, will be specified in section 112; the “calibration constants” K t j will be defined in section 113. The interaction energy of the atom i of the solute with all the solvent is
A summation on all the atoms i of the solute gives us the total interaction energy of the solute with the solvent
The Journal of Physical Chemistry, Vol. 76, No. 16, 1072
De
I n formula 4, the summation on i is a summation on all the atoms i of the solute, but the summation on j is a summation on all the distinct species of atoms of the solvent; for instance, if the solute is the methane CH4, and the solvent the nitrobenzene CeHSN02, i will vary from 1 t o 5, j from 1 to 4, and the values of n, will be 6, 5, 1, 2. Finally, to calculate E , we have to calculate for each atom i of the solute the integrals .f.f.fDedv/r6 and .f.f.fneexp( -ar)dv, to determine calibration constants Ktj, and to choose the constants of the pair potential of formula 1. 2 . Calculation of the Integrals. It is important to notice that the integrals to calculate depend on the solute and on the species of the atoms of the solvent. For a given molar volume of the solute, we can calculate them, once for all, for every species of atoms; then it will be possible to calculate the energy for any solvent by combining these integrals according to formula 4. a . Transformation of Volume Integrals into Surface Integrals. Given the necessity to calculate numerically the integrals .f.f.fneF(r)dv, it is convenient to transform them into surface integrals. We explain in Appendix A how we can do this by proper use of Ostrogradski’s formula (eq 5-10). The volume integral is transformed into a surface integral by the relation
2 designates the boundary of the integration volume Qe. iZe(Qe) is the unit vector normal to 8, directed toward the outside of Qe, that is, toward the inside of Q. We have the relationiZ,(Qe) = -%,(Q). r is the distance from the atom i for which the integral is calculated to the current point. 3 is the unit vector F/r. We obtain, respectively, for the dispersion integral and for the repulsion integral
(4) M. J. Huron and P. Claverie, Chem. Phys. Lett., 4, 429 (1969); ibid., 11, 152 (1971). ( 5 ) B. M. Axilrod, J. Chem. Phys., 19, 719, 724 (1951). (6) T. Kihara, Advan. Chem. Phys., 1, 267 (1958). (7) N. R. Kestner and 0. Sinanogulu, J. Chem. Phus., 38, 1730 (1963). (8) H. ,Margenau and N. It. Kestner, “Theory of Intermolecular Forces, Pergamon Press, Elmsford, N. Y., 1969, Chapter 5. (9) A. I. Kitaygorodski, Tetrahedron, 14, 230 (1961); Proc. Acad. Sci. USSR,Phys. Chem. Sect., 137, 231 (1961).
CALCULATION OF INTERACTION ENERGY
Jj-ar [ar1 + (ar)2 + ?]?.Z,(Q)du 2 (ar) -
2-
2 125
(13)
2
We put a = C ~ / ( ~ R , R , ) ~ . ~ . We can also use Ostrogradski’s formula to calculate the volume $2 of the solute (see Appendix A, eq 14-16)
spheres. The integrals can be written as integrals on the whole sphere S iby introducing the characteristic function Ys,e of Ste; that is, the function defined by Ys,e(r) = 1 if the point r belongs to Sie, Ys,e(r) = 0 if the point r does not belong to Sie. We then have SSy(r)du
=
sSyg,e(r)g(r)du
(18)
SZ
St0
After the introduction of polar angles 0 and cp and of the variable u = -cos 0 and noting the radius of the sphere St, we obtain
SSg(r)du =SSys,.(r(e,cp))g(r(e,cp))rc2 sin ededcp = b. Choice of the Integration Surface. The integrasee SZ tion surface Z is selected in such a way as to take into account the form of the solute molecule. rtzs_:’ d u l z r y s ,e (r (u,cp g (r(u,cp))dcp (19) The volume $2 of the solute is deduced from the van der Waals volume of the solute in the following way. Equation 19 is calculated numerically by using a grid Each van der Waals radius Ri of the atom i of the solute of points [rk(k = 1, . . . , v)] uniformly distributed on molecule is multiplied by the same factor A, that is the sphere 8,; the integral 19 is then estimated by the such as the volume obtained by juxtaposition of sum spheres of radius h,R, and center i (i labels the atoms 1 ’ of the solute) is equal to the effective volume of the (20) Ys,e(rk)g(rd Vk=l solute. A factor A s is defined in the same way for the We have tried two types of grids: grids obtained as of the solute is not solvent. The volume $2 = Vsoluteeff the product of two uniform grids for the variables 0 homothetic of the van der Waals volume of the and (a and grids obtained by the Korobov’s methodlo solute, since interatomic distances remain unvarying. (see Appendix B). The comparison of the results has When the temperature T varies, the effective volume shown that Korobov’s grids give, for a same number of varies, and A, varies. This leads to a variation of the points of integration, a higher accuracy that is more energy E with the temperature, since molecules are at stable when the number of points increases; finally RC different distances according to the value of T . The used the Korobov’s grid with 610 points on each sphere variation of Vsolu+,eeff with T represents this effect of S,. For the molecules CO and CH4, we have calculated variation of E with T . the van der Waals area and volume exactly, and with Our factor X fulfills the same purpose as the “addi~ Korobov’s grid we have obtained an accuracy of tional thickness b” introduced by Kihara and J h ~ n , ~the 1/1000. namely to get for the solute molecule a convenient “effec3. Calculation of “Calibration Constants” K ijd and tive size” (taking into account the thermal expansion). KtJr. It is necessary to introduce the calibration conMore precisely, the additional radius b/2 relative to stants K i j to get, by the integration method, results the solute molecule corresponds t o our radius increase that are as close as possible to those given by a discrete R,(X - 1). sum over molecules packed around the solute molecule. The integration surface 2 is obtained by the juxtaConsider the case of a pure liquid made of spherical position of parts of surface of spheres (i,h,R,), the calcuatoms of radius a and calculate directly (without introlation of X, is made with the Raphson-Newton iteraduction of K i J d the ) integral of dispersion tion formula, the starting value being Xo = (Veff/
c
V v a n der Waals) ’/a
The calculation of the area of the constitutive parts of 2, the boundary of the volume $2 = Vsolutoeffcan be made by numerical integration. The van der Waals volume and the effective volume Veff = $2 = f(X,) are calculated with formula 17. c. Numerical Calculations of the Integrals. The integrals t o calculate have the form sszg(r)du where ~ ( 1 ’ ) is one of the integrands that appear in the right side of eq 12, 13, and 17. We call S i e the part of the sphere (i,X,Ri) that belongs to 2 , the rest of the sphere being inside other
ne
We can compare this result with the discrete sum which is known in this case” (10) N. M . Korobov, “Teoretiko-tchislovie metodi v priblixhenom analize” “Number-theoretical methods in applied analysis,” Gosudarstvennoe Izdatel’stvo, Fiziko-Matematitcheskey Literatury, Moscow, 1963 (in Russian), Chapter 111; (a) section 10, pp 146-148; (b) section 11, p 162. (11) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y., 1964, p 1040.
The Journal of Phvsical Chemistry, Vol. 76, No. 16,1972
212G
MARIE-JOS~ HURONAND PIERRE CLAVERIE m
71,/1‘?
2 %exp(-ai3rL) ~
= 14.454/do6
K , r C , ~ J ~ ~ e ~ ~ ( - a , 3=r CZ )du
2=1
ne
2=1
In formula 21 we have put, instead of Veff,its value (23) as n function of the distance to the nearest neighbors a,, = LX/(~R~R,)O.~ and (R,,o)2= 4RiR, do = ?a, and of the packing coefficient 1 / & ? . l z If in (21) we choosc R = a = d0/2 and R = do, we, nz is the number of atoms of type j situated at the disrespectively, obtain 47.39/do6 and 5.92/do6. None of tance r z of the central atom i. these two values of R gives satisfying results; this can When the solute is an atom i, the volume SZ is a be assigned to thc fact that the integral assumes an sphere of radius A&,; when the solvent is made of absolutely uniform distribution of the centers outside identical atoms j , the effective volume Vfeff of the the solute, while the discrete sum corresponds to a dissolvent is a sphere of radius A,R,. After integration tribution function that is not at all uniform. We and substitution of Vjerf, the constants K,, take the know that in the liquid the distribution of the moleform cules is not uniform at short distance; now it is the m distribution immediately near the central molecule which is important. The calibration constant K i j can be defined as the ratio between the energy calculated by discrete summation and the energy calculated by integration. This constant is analogous to the “factor f” introduced by Kihara and J h ~ in n their ~ ~ formula A, = fn A, and A, have the definition given in section II2b. V A B ( ~(SA ) + + sjavdpfor the interaction between the To calculate the summations of formulas 24 and 25, it solute molecule and the solvent. If A and B are taken is necessary to build a packing of spheres of radius to be spheres of equal radius a = d0/2, ( S A + p + s ) a v &R, around the central sphere of radius pX,R,. The becomes 4ap2, and x i t h U A B ( ~ )= l/p6 and n (the origin of p is the following: the sphere of radius X,R, ; ,is only used for the integration; to build the packing of number density of the solvent) = l/Veff, n I U A B ( ~(XA+p+Bjavdp ) reduces to our eq 21. Then spheres, we have to define packing radii Rip and RJp our K,, exactly corresponds to f. However, it must be for the types i and j . We fix them by requiring that emphasized that IGhara and Jhon use a single f, inthe packings of the spheres j alone (or i) correctly redependent from the solute, for the whole solute-solvent produce molecular volumes Tieff,which, respectively, potential (including both the dispersion and repulsion are (47/3) (XBRj)3and (4a/3) (X,RO3; for a packing of term), while we introduce specific K t j dand Ki; for the identical spheres, the distance do between nearest neighdispersion and repulsion term, and this for each kind of bors is connected to the effective volume by the relaand Kilr turn out to atoms pair ( i , j ) (and since the KzJd tion12 be different, they cannot be easily reduced to a single factor independent from the solute). Moreover, we define a method of calculation for the KiJdand Ki; depending only on the atom species i and We then obtain the packing radius RP j , while the f factor3b must be apparently empirically determined for every solvent. ‘L -I The constants K i j are directly calculable in the simplest case of packing of spheres. The main hypothesis that allows us to treat any molecule consists in assuming that the constants K,, We note which correspond to the packings of spheres can be used to calibrate in the general case; in a way, we take into consideration the distribution of the atoms of the solvent around the solute, avoiding the explicit conThe packing radius /3X,Rj is lower than the radius struction of a distribution function, for each type of A,R, of the effective sphere. When A, = A, = X (it is pairs of atoms and for each solute. For a pair ( i , j ) of the case for pure liquid), we can simplify the expression atoms, Lye obtain the equations defining K i , by equalof Ktld in an important way. Indeed, instead of ing calculated energy by integration, and calculated building a packing of spheres of radius PARj, around a energy by discrete summation sphere of radius PAR,, we can construct a packing of
.fzB
(12) E. A. Moelwyn-Hughes, “Physical Chemistry,” Pergamon Press, Elmsford, N.Y . , 1961, Chapter 13, p 551.
The Journal of Physical Chemistry, Vol. 76, No. 16,19’78
2127
CALCULATION OF INTERACTION ENERGY spheres of radius R5 around a sphere of radius R,; we pass from this packing t o the first one by a similarity of ratio PX; then, in the two packings, values of nzare the same, and we have r z = ,8Xrzowhere r I o is the same as r z for the packing of spheres (RJ; values of nz and rz0 are determined once for all, independently of the value of X that varies with T as the effective volume. K i f d takes the form .
K d j d= X6(R&)3
2
nl/(,8Xrlo)s =
1=1
where Rife = (4RiR,)0.5. The ratios (Ruo/rzO) only depend on the ratio of the radii R {/R,. I n the same way, K z j rtakes the form
m
C 2=1
(29)
Contrary to K i j d ,K,; depends on X, but the values of the n2and the most important exponentials can be calculated once for all. This way is practicable since the series quickly converges because of the rapid diminution of the exponentials; it is sufficient to take a small number of spheres (R,) to obtain the value of the series. We have written a program that allows us to generate a packing of spheres around any sphere and to obtain the values of the nz and r E oof eq 24, 25, 28, and 29. For the dispersion, we must go far enough to evaluate the series correctly; in practice, we have used between 90 and 150 spheres according to the cases, and we have replaced the missing part by an integral. We give in m
Table I the values of
C
z=
(Ri,0/rJ0)6. For K J , it is
1
sufficient to take into account the n1 spheres of the first shell all situated a t the distance rl0 = (R, R,) of the center i of the central atom and the n2 nearest spheres of the following shell. The last ones are not all rigorously at the same distance from the center i, but they are at distances sufficiently near so that we can take a single mean value r20. We then have
+
-
nL[exp(-ua,,rLO)]PA n1[exp(-aa,,rlo) IPx
+ nz[exp(-atyqO) 1''
(30)
4. Choice of the Constants of the Potential. Kitaygorodski has calculated his energies with the function of binary interactione
6
-e" exp(-artj/rij0) CY
z=1
Ri (central atom),
A
1.20 1.20 1.20 1.70 1.70 1.77 1.77
Rj,
A
1.20 1.70 1.77 1.20
1.70 1.20 1.77
5
(Rijo/Ti0)'
1=1
nl
14.454 6.971 6.643 18.216 14.454 20.629 14.454
12 6 6 14 12 18 12
ri"/Rij'
1.000 1.015 1.019 1.015 1.000 1.019 1.000
12%
raO/Rijo
4 3 3 3 4 2 4
1.453 1.301 1.347 1.110 1.453 1.187 1.453
where rrl0is the equilibrium distance which is deduced from the van der Waals radii by increasing those by about 1270.13 For a given value of r i j 0 , it is sufficient to fix a to determine el,. Kitaygorodski chose a! = 13 and obtained for the calculation of the sublimation energy of the crystal of methane too large a value:14 3.57 kcal/ mol, while the experimental value for the absolute zero is about 2.50815or 2.74 kcal/mol.14 Such an overvaluation of about 15% cannot be only explained by the nonadditivity of pair interaction energies, which would reduce the dispersion energy of about 7%.' To draw nearer to the experimental results, we have modified the value of a. We have calculated the energy of the CH4 crystal with the formula given in ref 16 for the means r-=6 a t d ex', using a = 13.80, rcco = 3.80 A, r m 0 = 2.68 A, and r C H o = 3.19 A. For a distance between carbon atoms of 4.16 i l l 5 we have obtained a total dispersion energy of -3.96 lical/mol, and a total repulsion energy of $1.18 kcal/ mol; after a reduction of 7y0on the dispersion energy (see later the nonadditivity effect) this gives a total energy of -2.66 kcal/mol. The same calculation for a distance of 4.11 Ale gives a total energy of -2.69 kcal/ mol (Ed after reduction = -4,19kcal/mol; E , = +1.50 kcal/mol). The value a = 13.80 seems to us satisfactory; formula 31 becomes eir = -0.1098
(;J + -
47,000 exp(-13.80r,,/r,,") (32)
m
1=1
m
Table I: Values of x ( R i j o / r l o ) 6n1, , n2, r I 0 / R i j 0 , and r2'/Rijo
ff
To go from (32) to (1) that contains van der Waals radii, we have to multiply the dispersion coefficient by ( Y C C ~ I ~and R Cthe ) ~ coefficient a by (2Rc/r~cO).~To evaluate thcse ratios, we have used YCCO = 3.80 ;1,9,14 (13) N. A. Ahmed, A. I. Kitaygorodski, and K. V . Mirskaya, Acta Crystallogr., Sect. B , 27, 867 (1971). (14) A. I. Kitaygorodski and K . V. Mirskaya, Sov. Phys. Crystallogr., 9, 137 (1964). (15) E.A. Mason and W. E. Rice, J . C h e w Phys., 22, 843 (1954). (16) A. I. Kitayyorodski and K. V. Mirskaya, Sou. Phys. CrystalZogr., 6,408 (1962). The JournaE of Physical Chemistry, Vol. 76, hTo.16,1072
2128
MARIE-JOS~ HURONAND PIERRECLAVERIE
Table I1 : Values of Calculated and Experimental Energies‘
Compound
Methane Ethane Propane Isobutane n-Butane Neopentane Isopentano n4entane n-Hexane Cyclohexane n-llecane n-Hexadecane n-Eicosane Benzene 4
T,OK
Molar volume, om8
h
111.66 184.52 231.08 298.15 261.42 298.15 272.65 298.15 282.65 298.15 298.15 301.00 298.15 309.22 298.15 341.49 298.15 353.87 298.15 447.27 298.15 559.94 298.15 617.0 298.15 353.25
37.8 54.7 75.9 89.48 97.7 105.48 96.6 101.43 119.68 123.31 117.38 117.96 116.10 118.38 131.6 140.9 108.74 116.99 195.90 235.82 294.08 396.15 359.83 513 9 89.40 96.04
1.397 1.382 1.424 1.533 1.456 1.513 1.442 1.474 1.492 1.511 1.471 1.477 1.455 1.467 1.434 1.489 1.388 1.447 1.409 1.555 1.385 1.652 1,376 1.704 1.332 1.376
I
Ecalad
Ed/2
Er/2
-2.111 -3.721 -4.361 -2.955 -4.843 -3.982 -5.159 -4.625 -5.023 -4.724 -5.542 -5.451 -6.006 -5.730 -7.508 -6.213 -8.181 -6.690 - 13.075 -7.912 -22.116 -9.334 -28.187 -9.921 -8.111 -6.799
0.199 0.379 0.318 0.094 0.542 0.144 0.323 0.228 0.207 0.168 0.271 0.255 0.339 0.294 0.490 0.268 0.738 0.389 1.012 0.200 2.032 0.107 2.761 0.074 1.365 0.823
E/2
-1.91 -3.34 -4.04 -2.86 -4.57 -3.84 -4.83 -4.40 -4.82 -4.56 -5.27 -5.20 -5.67 -5.44 -7.02 -5.94 -7.44 -6.30 -12.06 -7.71 -20.08 -9.23 -25.42 -9.85 -6.75 -5.97
Eexp
- I AEvexp
-1.73 -3.15 -4.03 -2.95 -4.57 -3.98 -4.81 -4.44 -4.85 -4.61 -5.34 -5.30 -5.72 -5.55 -6.95 -6.21 -7.30 -6.46 -11.68 -8.50 -18.79 -11.10 -23.5 -12.5 -7.50 -6.65
Eoalad/ A H yexP
1.955 3.517 4: 487 3.542 5 * 090 4.570 5.352 5.035 5.438 5.205 5.937 5.901 6.316 6.160 7.541 6.896 7.896 7.16 12,277 9.388 19.38 12.24 24.1 13.74 8.090 7.352
E~XR
1.10 1.06 1.00 0.97 1.00 0.96 1.00 0.99 0.99 0.99 0.99 0.98 0.99 0.98 1.01 0.96 1.02 0.97 1.03 0.91 1.06 0.83 1.08 0.79 0.91 0.90
In kcal/mol.
and Rc = 1.70 8 which is the van der Waals radius for saturated carbons given in ref 17. We then obtain etj =
-0.214(4R,RJ3/r,$
+
47,000 exp [ - 12.35rij/(4RtRj)OJ] (33)
It is this pair potential that we have used for the applications reported in section 111, taking the van der Waals radii proposed by Bondi’7 (except for RH i c aromatic molecules, see section 111): Rc = 1.70 A for saturated carbons; Rc = 1.77 8 for aromatic carbons, RH = 1.20 8 in all the cases. The potential eif of formula 33 is a true pair potential and not an effective pair potential. To take into consideration the nonadditivity of the dispersion energies, we have done a correction of 7% on tho total dispersion energy calculated with the hypothesis of additivity of binary interactions. A perturbation treatment to the third order would introduce forces between triplets; these forces have been evaluated to be 2-9% of the cohesive energy for crystals of rare gases5 and 2-11% for organic molecules.7vs For the interaction between two molecules of CH4 the reduction is 2lyO,7but for the total energy of a crystrtl or a liquid, the reduction of the sum of the dispersion energies is one-third of the previous The Journal of Physical Chemistry, V o l . 76, No. 16, 1072
percentage (ref 7, part F and G, rcf 8, p 164), that is 7%. We have assumed that the reduction of the total dispersion energy was 7% for all hydrocarbons.
111. Applications and Discussion The previously described model has been used for the study of pure substances. The chosen compounds are saturated hydrocarbons, these ones having negligible electrostatic energies because of the small charges carried by their atoms. The calculated energy of the central molecule with all its surrounding can be compared with twice the vaporization energy. We discuss in Appendix C the nature of the calculated energy. Table I1 contains the values of the calculated energies at the boiling point and a t 25”, and the values of the experimental vaporization enthalpies (ref 18 and 19, p (17) A. Bondi, “Physical Properties of Molecular Crystals, Liquids and Glasses,” Wiley, New York, N. Y . , 1968; A. Bondi, J. P h y s . Chem., 68, 441 (1964). (18) API, 44 Tables, “Selected xalues of Properties of Hydrocarbons and Related Compounds, American Petroleum Institute Research Project No. 44, Thermodynamics Research Center, Texas A&M University. (19) J. H. Hildebrand, J. M. Prausnitz, and R. L. Scott, “Regular and Related Solutions-The Solubility of Gases, Liquids and Solids,” Van Nostrand-Reinhold, Princeton, N. J., 1970.
CALCULATION OF INTERACTION ENERGY
217). Expcrimcntal vaporization cnergics are deduced from vaporization enthalpies by the relationlg IAEvexPI =
- RT
(34) R is the constant of perfect gases and T the absolute t cmpcraturc. RT starlds for A(PV) = PAV = P ( V g a 8 - Vliq). This replncemcnt is valid if V1iq
