.
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The Electron-Repulsion Integral and the-'. Independent-Particle Model for Helium
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Soo-Y. Lee National University of Singapore, Kent Ridge, Singapore 051 1, Republic of Singapore The total electronic energy for the helium-like atom, with Hamiltonian
A second method ( 3 , 4 )expands r;: in terms of the spherical harmonics. The orthogonality of the associated Legendre polynomials simplifies the algebra tremendously when the integrals are performed over the angular variahles. In both these methods. the final inteeral over the radial coordinates
in the independent-particle model, using the wavefunction W , 2 ) = 'bh(1) 4 , m
-- ( , - l l z ~ 3 / 2 e - z , , ) ( 1 1 - 1 1 2 ~ 3 / 2 e - Z ' 2 )
(2)
is given by (34
ranges in thk integrations over the radial coordinates, and this is discussed below. In this method, the volume element for the two-electron repulsion integral is written as r? s i n O ~ d O ~ d q ~ dsinwdwd~ r~r?~ d m
(4)
where the volume element of the second electron is expressed relative to that of the first electron, and the coordinates used are illustrated in Figure 1. The law of cosines yields The first integral in eqn. (3b) is straightforward to evaluate, and the result is simply twice the total energy -Z2/2 of the 1s state for a hvdrogen-like atom with nuclear charge Z. The . . second integral in eqn. (3b),known as the electron--repulsion integral, is more difficult to evaluate. There are various methods to evaluate the electron-repulsion integral. One way ( 1 , Z ) is to interpret the integral physically as the electrostatic energy of two overlapping, spherically symmetric charge clouds represented by the 1s orbitals.
Holding r l and r12 fixed and differentiating eqn. ( 5 ) yields
This allows us to replace the coordinate w with the coordinate r2 in eqn. (41, and the volume element becomes
Volume 60 Number 11 November 1983
935
The electron-repulsion integral is then given by
5 '52 where the repulsion term r;' has been eliminated. Integration over the angles 0 1 , PI, and x yields
v '1 =-=
5
2
12
Figure 2. S c h e m a t i c diagram of t h e t w o contributions It,l2 t o t h e repulsion integral. The long range effectof t h e repulsion r 2 m a k e s 12 11.
>
The limits of integration over the radial coordinates are often specified (5,6) to he where The integration is performed over rlz first, holding rl, r2 constant. This contradicts the condition by which we arrived a t the important eqn. (61, which requires that the integration over rz he performed first, keeping r l and r12 fixed. This contradiction can he a difficult point for the student. The limits of integration should correctly he given as
It is not immediately obvious that eqns. (9b) and (9c) will yield the same result. Proceeding in the same manner as Margenan and Murphy (5), we can write eqn. ( 9 4 as
for the case where r l > rlz, and
for the case where rl < r12. The two integrals I 1 and 1 2 are not equal, and this reflects the fact that the coordinate rlz is not equivalent to either of the coordinates rl or r2. The integration of eqns. (12) and (13) is straightforward, and yields 1, = 41 21216
(14)
and 12 =
11 2/21
(15)
Adding up gives 1=5218
(16)
which is idenrical t u what Maryenau and Murphy 151d)wintd wirh eon. (91,,. liut the derivariun ht.re satisties rht, ctmditions the use of eqn. (6). that An important consequence of the derivation we have given here is that Il is less than 1 2 . The situations that contribute to II and 1 2 are illustrated in Figure 2. The integral I 1 accounts largely for the case where the electrons are close together, whereas 1 2 accounts largely for the case where the electrons are far apart. The fact that I 1 is less than I 2 illustrates very well the importance of the long-range effect of the repulsion
rr;. Literature Clted
X Figwe 1 . Cmrdinate system far the helium-like atom. Spherical polar m d l n a t e s of one electron i s ( r , , f l , , q , ) , and of t h e s e c o n d (relative to t h e first) is (r32.W.x).
936
Journal of Chemical Education
(1) ~auling,L.. and Wilson, Jr., E. B.,"Int~oductiontoQuantum M~chanicr:'McGraw-Hill B m k Co.. New York, 1935. S e a a 5 and 8-7. (2) Kauunann. Walter."Quantum Chemistry. Anhtmdudio","AesdemiePress, New YWk, 1937, p p 285-287. (3) Eyring,H., Walter, L a n d Kimball. G. E.. "Qusntum Chemisizy.l'John Wiley & sons, New York. 1944. p. 103. (41 Anderson, J. M., '"InUoduetionto Quantum Chemistry: W. A. Be"jsmin.New York, 1969, pp. 162-163. (51 Margenau, H., and Murphy, 0 . M., "Th. Mathematies of Physics and Chemmtry:' D. Van Nostrand Co., Princeton, NJ, 1956, pp. 382383. (6) P h , Frank L., "Elementsry QusntumChmistrr' McCrsw-HillBwkCo., New York, 1968, pp. 180-181.
