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We are then led to the set of equations (n # I ) : [V2 + ha - Uzz]Fz(r)= 0 + (1804 (18Ob) [V’ kE - unnIFn(r) = utn(r)Ft(r) Assuming further that U,,is spherically symmetric, Eq. (18Ob) is reduced to the form of Eq. (168), so that F, has the asymptotic The term with U,,, is probably as large, and when we omit it we rationalize that the method followed in the solution of Eqs. (180) is one of successive approximationsto the equationsderived when Us, is not ignored. ANALYTIC THEORY OF CHEMICAL REACTIONS 51 behavior given in Eq.
8 (177) Having displayed Eqs. (176) and (177) in full subscript language for completeness, before we carry on it would be well to suppress all the “ab” subscripts and abbreviate n’f and IZP as IZ’ and respectively. Equation (176) would then appear as follows: (V’ + k:)Fn(r) = 2 UnlnFn? n‘ (178) In Eq. (178) Born’s approximation amounts to taking all the terms in the series as zero with the exception of the one which overlaps with the initial-state function ylm,hereafter designated as y,. With the abbreviated notation this term is U,,F,, where, substituting k, for kzm, F, has the form F, = exp (ik,* r) (179) Despite the fact that the relative velocity is not large with respect to the internal motion, Born’s approximation is still fitting if all the matrix elements Un, are relatively small with U,,the largest of them.
Further details may be gleaned from consulting standard treatises. Suffice it to say here that Eq. (83) with the assumptions discussed above leads, for example, when n = 4, to the modified London formula shown below: E* = Q - [*(@ + 78 + +(a7 + 88 + *(By + a8 - - adfa - as)’]* (98) Here E* is either E or P depending on whether the integrals refer to H or V , and Q is identical with a sum of coulomb integrals between pairs of nuclei : - a? - 88)’ - By aB Q = Qus + Qyd + + Qpe + + QEY QBY Qua (99) * See Reference 62, where the undependable nature of these approximations is demonstrated conclusively.