~ h e h t e g r aTransform l Approach TheElectron-Repulsion Integral and the'C;elium Atom Fernando R. Ornellas lnstituto de Estudos Avangados, CTA Caixa Postal 6044, SHo Jose dos Campos. SP, 12-and Paulo, Caixa Postal 20780, SHo Paulo, SP, 01498, Brasil When a student is introduced to the quantum mechanical m e t h d s lor treating thr helinm iltom, henr she isalwaysfaced with the prohlem of evaluating the elertron-repnlsion integral 1 I = SSd3rld3rz@(rl) &2) 1.
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4(?)= ~3/2r-112e-zr
Most standard texts usually evaluate this integral by one of the following methods: (1)an expansion of lrl - rd-' in terms of soherical harmonics (1.3): . . (2) . an exnlication of the coordinates of the second electron relativr to that of the first electron (.i n .. S1. areuments that this .. or (3) thc use of .ohvsical . integral represents the interaction electrost&c energy of two spherically symmetric electronic densities &I) and &rz) (6, 7). A discussion of the consistent way in which the integral given by eqn. (1) should he evaluated by the second method has recently been discussed by Lee ( 8 ) . In what follows we show that the integral I can still he evaluated bv another method, namely, t h e ~ o u r i etransform r method (F'I'MI (91,'l'he merits otthe YI'M are that it introduces the studmLs bot h too technique that is used to compure elertron-repulsion intcarals i n v o l \ . ~ norhltali ~ on different atoms and also to a t e c h q u e of evacuating integrals where the orbitals contain explicitly the interelectronic distance r;j (Hylleraas-type functions). F r a recent detailed presentation of the Fourier transform medod in the context of multi-center electron repulsion integrals, the reader is referred to the work of Partridge (10). An example of this same technique in the construction of correlated wavefunctions for the ground state of helium-like atoms is given by Bonham and Kohl (11). Still in the context of the helium atom we can also mention another merit of the FTM, namely, its applicability to the calculation of exchange integrals arising in such states as ls2s 'S and ls2s 3-. s
The basic step in the evaluation of eqn. (1) is to realize that Irl - rzl-I can be written as a Fourier transform
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The integrals over the coordinates rl and rz are equal and can he found in a table of integrals or he easily evaluated (choose the vector t parallel to the Z axis). They are given
(1)
where,
Substituting eqn. (2) into eqn. (1) gives
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Making use of eqn. (4) and carrying out the integration over the angular coordinates of the vector t, eqn. (3) reduces t o
The integral in eqn. (5) is again easily found in any table of integrals or directly evaluated by contour integration (46).Its vrlue is 5 ~ Z - ~ 2 - ' Substitution ~. of this value into eqn. (5) gives the well-known result
Finally, as mentioned in the introduction, this technique can also he extended to calculations related to excited states of helium. For example, the energy difference between the singlet state (ls2s IS) and the triplet state (1.92s 3S) is equal to 2 Klz, where Klz is the exchange integral (12). If the 1s and 2s functions are represented by hydrogen-like functions, a straightforward application of the procedure described here yields Klz = 1621729. Literature Cited (1) Eyrin8.H.. Walt%, J..and Kimball. G. E.. "Quantum Chemistry? Wilcy.New York, 1944.p. 103.
(2) Andemon,J. M., "The Introduditio" to Qvantvm Chcmiatry? W. A. Benjamin. NB. York, lW9,PP. 16W. (3) Karplus, M. and Porter R. N.,"Atoms and Molecules: An InMduetion for Stvdentr of Physical Chemistq? W. A. Benjamin,NewYork, 1910,~~. 173-7. (4) (a) Marsens". H.,and M w h y , E. M., "The Mathemstieof Phyaieand Chemistry," D. Van Nostrand Co., Princeton, NJ, 1956,pp. 382-3: (b)pp. 92-3. ( 5 ) Pilar, Frank L., "Elementary Quantum Chemistry? McCrsw-Hill. New York, 1968,
pp. 180-1.
(6) Paul@, h, and Wilson. Jr. E. B., "Idmductim fa Q i t t Meehaoie:'McCt-Ha, New York, 1935. Secs. 8-5 md 8-7. (7) Kauzmann. Waiter, "Quantum Chemistry. An Inttodunim," Academic Press. New York, 1957.p~.285-7. (8) Lee,Sw-Y., J. CHEM. EDuc, 60,935 (1983). (9) (a) Bonham, R. A . Poscher,J. L., andcar, H. L., J. Chem Phyr..40,3083(1964).(b) Bonhsm,R.A., J . M d Spect. 15,112(1965). (10) Partridge 111. Harry. Dissertation, Indiana University, Blwmington, IN, Dec 1978, 140 P. (11) Bonham,R A,, and Koh1.D. A,, J. Chom. Phys., 45,2471 (1966). (12) See re< (51, pp. 324-7.
