By N. H. March, M. P. Tosi (auth.)
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Extra info for Atomic Dynamics in Liquids
3 Effective interatomic potential and the Percus-Yevick theory The crippling limitation of the above approximation is that it assumes an interaction with a Fourier transform. For liquids, in general, the interaction has almost a hard core, and we cannot Fourier transform it. The Percus-Yevick (1958) method is an attempt to produce an effective potential which will replace the Fourier components J)(k) above. The idea behind it is to use the fJk as collective coordinates. For them to afford an accurate approximation we must be able to express the Hamiltonian in terms of them and the corresponding momenta, and then to treat them as if they were 'almost' independent.
It will be instructive to introduce this formalism through a study of the simplest timedependent correlation function, which is the velocity autocorrelation function for a particle in a classical fluid, before proceeding to the discussion of S(k, w). 1) where v(t) is the velocity of a specified particle at time t. One reason for the importance of this function is that it is related to the mean square distance, u(t) say, travelled by a particle over a time interval t. 2) ¢(t) is especially suitable for treating the theory of the Brownian motion of a colloidal particle in a liquid, but it can also be used to describe the irregular zig-zag motion of a particle of a fluid under the impacts of the other particles.
74) and (2. 64). 78) This is in excellent agreement (within about 10%) of the empirical value of a. If we had used the Born-Green asymptotic form, we would have been quite wrong. 64) for argon and enables the equation of state to be calculated directly from the parameters in the force law (Woodhead-Galloway, Gaskell and March, 1968). Actually Longuet-Higgins and Widom used the results of machine calculations for Phs(P, T ). If we are content with slightly less accuracy, we could use the Percus-Yevick result.