Comparison of two simple models for high frequency friction: Exponential vs. Gaussian wings

We describe new methods for ruling out unphysical forms for the high frequency friction $_{\omega \to \infty }^{\lim } \quad \beta (\omega )$ needed to compute vibrational energy relaxation times. These are based on the fluctuating force autocorrelation function$(faf)\mbox{ }C(t)=\langle \tilde {\Im }^2\rangle _0 ^{-1}\langle \tilde {\Im }(t)\tilde {\Im }\rangle _0 ,$which is proportional to the Fourier transform of $\beta (\omega )$. Here we compare two model faf's C$_{se}$(t) = sech (t/$\tau )$ and C$_{ga}$(t) = exp$\left[ {-\frac{1}{2}\left( {\frac{t}{\tau }} \right)^2} \right]\mbox{ }$. These give respective high frequency frictions which have incompatible exponential and Gaussian forms. We apply our procedures to eliminate C$_{se}$(t). We do this by showing from $\beta _{se}(\omega ) \quad \equiv $ $\frac{\langle \Im ^2\rangle _0 }{k_B T}\int_0^\infty {\cos \omega t} $C$_{se}$(t)dt that $_{\omega \to \infty }^{\lim } \beta _{se}(\omega )$ derives from the long time ``tail'' of C$_{se}$(t). We then note that C$_{se}$(t) is built only from short time quantities, rendering the form of this ``tail'' artifactual. Thus the exponential form of $_{\omega \to \infty }^{\lim } \beta (\omega )$, is also artifactual.