Download Angular Momentum Theory for Diatomic Molecules by Brain Judd (Auth.) PDF
By Brain Judd (Auth.)
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Sample text
30) is a n example of a Fredholm equation of t h e second kind [ 3 8 ] . T h e general m e t h o d for solving such a n equation is t o e x p a n d b o t h kernel a n d function in some series of orthonormal polynomials, thereby obtaining a series of equations for t h e coefficients in t h e expansion of t h e function. T h e n a t u r a l choice for our polynomials is t h e four-dimen sional spherical harmonics. If we specialize t o t h e hydrogen a t o m , we can set R = 0, corresponding t o a charge Ze a t t h e origin.
1). Freed [ 3 4 ] has described in detail the procedures to follow when the method of reversed angular momentum is adopted. It is not necessary to take Van Vleck's extreme position. Carrington et al. 8) can be corrected by setting V = V> λ / = — λ,", V = , V i for then λ" satisfies the commutation relations of an ordinary angular momentum vector. Alternatively, all these adjustments can be avoided b y accepting Eq. 7), and this is the course we shall follow here. 2). 3 REDUCED MATRIX ELEMENTS An important matrix element for eigenstates of the rigid rotator is (JMN I Dp™ I J'M'N').
I t is n o t difficult t o show t h a t 2 ΚΛοο(Ω) = 1 2 ( 2 7 T ) - / C n - l ( C O S X) . Furthermore, r 2(rc-l) \ W (nOO, n'00 | n " 0 0 ) = (00, 00 | 00) η" i(n = (η'ΥηηΟ^Δίη - #/ 0 - 1) i(n' - 1) 0 - 1) h(n" 1) 0 1, n' - 1, n " - 1), where Δ = 1 if t h e t r i a d (n — 1, n' — 1, η " — 1) satisfies t h e t r i a n g u l a r condition a n d if, in addition, η + n' + n" is o d d : otherwise Δ = 0. 7. So E q . 31) becomes 2 C w- i ( c o s x ) C n ' - i ( c o s x ) = Σ f m {2τ η"/nn ) f(nn'n") n" X A(n - 1, n' - 1, n " - l ) C n - - i ( c o s X) .