A PARTIAL DIFFERENTIAL EQUATION FOR THE RKR INTEGRAL.
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Date
1989
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Ohio State University
Abstract
Imposing the requirement the RKR effective potentials derived from different values of J should differ only by the nuclear rotational energy $\not h^{2}J(J+1)/2\mu R^{2}$, results in a nonlinear partial differential equation relating derivatives of the RKR f and g integrals, from which we can derive restrictions on the permitted forms of rovibrational energy expressions or relations between vibrational and rotational Dunham coefficients. In the RKR approach to deriving diatomic potential energy curve from term energies we construct the integral $S(Y,\xi)=\sqrt{2}\int^{\eta(U)}_{0}[U-E(\eta,\xi)]^{1/2}d_{\eta}.$ where $\xi=\not h^{2}J(J+1)/\mu$ and $\eta =\not h(v+1/2)/\sqrt{\mu}$. The inner and outer turning points are calculated from $R^{-}_{+}=[f/g+f^{2}]^{1/2}\pm f$ where $f(U,\xi)=\partial S/\partial U$ and $g(U,\xi)=-2\partial S/\partial \xi$. The Born-Oppenheimer approximation requires that RKR effective potentials, $U_{\xi}(R)$, derived for various values of J (or $\xi$) should all correspond to the same rotationless potential, $V_{0}(R) = U_{\xi}(R) = \xi/(2R^{2})$. By substituting the turning points into this last relation, and differentiating with respect to U and $\xi$, we derive a partial differential for the RKR integral, $\left(\frac{\partial S}{\partial U}\right)^{2} \frac{\partial^{2}S}{\partial \xi^{2}}+\left[ -2\left(\frac{\partial S}{\partial \xi}\right)\left(\frac{\partial S}{\partial U}\right)+8\left(\frac{\partial S}{\partial \xi}\right)^{2}\left(\frac{\partial S}{\partial U}\right)^{2}\right]\frac{\partial^{2}S}{\partial \xi \partial U}+\left(\frac{\partial S}{\partial \xi}\right)^{2}\frac{\partial^{2}S}{\partial U^{2}}=0.$ While this equation is formidable to solve directly, we can derive required relations between Dunham coefficients by formal power series manipulation. Furthermore, some simple energy expressions such as $E(vJ) = T(J) + \omega(J)(v+1/2)$ [harmonic oscillator] and $E(n, \ell) = -1/2(n-\delta_{\ell})^{2}$ [Rydberg atom] can be manipulated analytically, giving interesting new results.
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Author Institution: Molecular Physics Laboratory, SRI International