Wiener-Khinchin theorem for Fourier-Laplace transformation; new derivation, and application to molecular simulations

The Wiener-Khinchin theorem for Fourier-Laplace transformation (WKT-FLT) provides a general purpose method to calculate numerically the one-sided Fourier transformation of an arbitrary autocorrelation function (ACF) from molecular simulations. However, the existing derivation of the WKT-FLT equation includes some ambiguities. Moreover, the equation obtained always yields 2 artifacts in the data computed. In this work, we present a new derivation that eliminates these ambiguities and artifacts.
We define an ACF as a natural extension of the known ACF, and write its Fourier-Laplace transformation as the Fourier transformation of the product of the Heaviside unit step function and the ACF. As the result, we successfully obtain an equation for the WKT-FLT. Then, we discretized the WKT-FLT equation to compute the frequency-domain correlation function (FDCF) from numerical simulations. It is found that a correction term is needed in order to avoid over-counting and to satisfy the sum rule. With the correction term, an artifact previously observed in the real part of the FDCF disappears. On the other hand, another artifact still remains in the imaginary part of the FDCF. We will discuss the cause of this artifact.