ELECTRONIC CORRELATION IN RAREGASES
Sept. 20, 1956
4565
tions needed to calculate further integrals were also available there. Of the five integrals listed in Table IV, four were calculated by the methods given by Roothaan16 and Rudenberg.16 The notation used here is the same as that employed in these two papers. The [1SA 1SB 1 1SA 2pUA] integral was evaluated 1-I-I by a modification of Riidenberg’s18method. This is 3 1 1 - 1 1 I done by recognizing that if = 1SA2PU.4,it can be expressed in the form given by Rudenberg’s equaFrom here on this integral was calculated just tion (1.13). Now let f a refer to the 1 s A , f b to the like the two center exchange integrals, keeping the 2pUA orbital (or vice versa). Then equation changes made in mind, however. (1.13’) of Rudenberg’s paper becomes Most of the c€$(a,a) and By@) functions used were found in the tables of Kotani, Amemiya and 1 Simosel7 and in Hirschfelder and L i r ~ n e t t . ~Some d = ,jR ( f s Fb) B = 31 R ( i s i b ) more @ functions were calculated by a method given in Rudenberg’s paper. Some B:2(fl)funcand (1.13”) remains as is, while the formula for tions were calculated by the formula a({,7) will have to be determined anew, which is (20) BII”(P) = 1/53 B%(P) - Bou(B)) given here for la.42pUA in the same form as that used in Table I1 of Riidenberg’s paper (17) Kotani, Amemiya, and Simose, Proc. Phys.-Math. Soc., J a p a n ,
a
+
+
(;
(15) C. C . J . Roothaan, J . Chem. P h y s . , 19, 1445 (1951). (16) Klaus Riidenberg, i b i d . , 19, 1459 (1951).
[CONTRIBUTION FROM
20, Extra No. 1 (1938); 22, Extra No. 1 (1940).
BERKELEY4, CALIFORNIA
THE DEPARTMENT OF CHEMISTRY AND CHEMICAL ENGINEERING, UNIVERSITY O F CALIFORNIA, BERKELEY]
Electronic Correlation in Molecules.
11. The Rare Gases1
BY KENNETH S. P~TZER RECEIVED MAY31, 1956 The coefficient of the inverse sixth power London potential term derived from second virial coefficients and viscosities for the interaction of pairs of rare gas atoms is compared with approximate theoretical calculations. The usual theoretical formulas yield too small values probably because of the neglect of inner shell electrons. Empirical values for the effective number of polarizable electrons are obtained. These values may be used in connection with the Slater-Kirkwood formula to calculate approximate interaction energies.
We wish to select a formula, necessarily approximate, which may be used t o estimate that portion of the correlation energy which arises from nonoverlapping parts of the electron system. Our basis will lie in the theory of the attractive force between non-polar molecules. The rare gases are of particular interest since their atoms have closed shells and thus give examples with spherical symmetry. This theory was summarized in an excellent review by Margenau2 to which we shall make frequent reference. Two general methods have been used: the perturbation method by Eisenschitz and London3 and the variation method by Slater and Kirkwood.* In each case parallel calculations are made of the polarizability of the single atom or molecule and of the interaction energy between two such atoms or molecules. The sums of integrals which arise cannot be evaluated except in the simplest cases, but the same integrals appear in closely related sums in the (1) This research was assisted b y the American Petroleum Institute through Research Project 50. (2) H. Margenau, Reu Mod. Phys., 11, 1 (1939). (3) R. Eisenschitz and F. London, 2. Physik, 60, 491 (1930); F. London, Z. physik. Chem., B11, 222 (1930); Z. Physik, 65, 245
(1930). (1) J. c. Slater and J. G . Kirkwood, Phyr. Rev., 57, 682 (192.1).
two calculations. Thus one attempts to replace the sums of integrals in the interaction energy formula by the most nearly equivalent combination of the polarizability and related quantities. In the perturbation method the resulting formulas involve fi the oscillator strength which is the effective number of electrons participating in an optical transition, and Ei the excitation energy to the ith state. The formula for the polarizability a t frequency v is then (Margenaq2eq. c 8) a(.) =
e2ha m
fi
Ej2 - h2v2
The refractive indexes of many substances can be fitted to a more approximate formula involving a single f t and Et. Then the polarizability a t low frequency becomes
The more detailed formula for the interaction energy for a pair of atoms (or molecules) is E L = - - 3-
eM4
? ? EiEj(Ei + E j ) fifi
2 m2Re
-
(3)
