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##### Discrete memory function of the VACF within the AR model

Starting from the memory function equation of the VACF (Eq. 4.72), the first step towards a numerical computation of the memory function consists in discretizing Eq. 4.72

| (4.79)

Eq. (4.79) is now subjected to a one-sided z-transform. Using that

| (4.80)

for any discrete function whose one-sided z-transform exists, one obtains from (4.79)

| (4.81)

using that . The one-sided z-transform of an arbitrary discrete function is defined as . Here it has been used that the one-sided z-transform of the discrete convolution integral is just the product . Inserting the definition of the one-sided z-transform for and , this equation can be rewritten as

| (4.82)

Note that the term proportional to z cancels out. The time dependent memory function is, in principle, obtained by comparing the coefficients of the series on the lhs and the rhs of Eq. (4.82). To construct a numerical method, we replace the series by polynomials of order N, where defines the time window for the memory function to be computed. After this first step a polynomial division is performed on the rhs of Eq. (4.82), and after a subsequent multiplication of both sides with one obtains the time dependent memory function, , by comparison of coefficients,

|  |  | (4.83)

Within nMOLDYN is replaced by the autocorrelation function calculated in the framework of the autoregressive model, , as in Eqs. 4.75 and 4.76. The coefficients are obtained by polynomial division and is a rest which does not contain information on the memory function within the time interval . The discrete memory function is therefore given by .

A remark concerning the discretization scheme (4.79) is in place here. The discrete convolution sum is effectively a first order approximation of the convolution integral. More sophisticated approximations could be used, but they would lead to less convenient expressions upon z-transformation. Correspondingly, we have chosen a first order approximation for the differentiation on the left-hand side of (4.72). In this way the first order (integro-)differential equation (4.72) is transformed into the first order difference equation (4.79).

However, this simple discretization scheme together with the use of the one-sided z-transform leads to a significant error in . It is clear from Eq. (4.72) that due to the symmetry of the autocorrelation function ( ), the derivative should vanish at However, in the discretized version it is approximated by a forward difference that is always negative. A higher-order calculation shows that the estimate for that results from the procedure described above should be doubled.

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