Noura Dawass

3 44 F INITE -S IZE E FFECTS 3.3.2. I NVERSE - N FINITE - SIZE CORRECTION A simple method to correct for the finite-size effect observed in radial distribu- tion functions was presented by Krüger et al. [74] , where the difference between g ( r ) and g ∞ ( r ) is expanded in a Taylor series in 1/ N [125] : g N 1 αβ ( r ) = g ∞ αβ ( r ) + c ( r ) N 1 + O Ã 1 N 2 1 , r ! (3.5) Here, g N 1 αβ ( r ) is the radial distribution function for a closed system with N 1 molecules, g ∞ αβ ( r ) is the RDF corrected for the finite-size effect, and c ( r ) is a func- tion that describes deviation from an open system. The function c ( r ) is usually not known, but can be estimated using two systems with different sizes at the same thermodynamic state (same density/pressure, temperature, and composi- tion). The term O ( 1 N 2 1 , r ) is the error associated with the truncation in Eq. (3.5) , which is a function of the number of molecules used as well as r . From Eq. (3.5) , the corrected g ∞ αβ ( r ) can be expressed as: g ∞ αβ ( r ) = N 1 g N 1 αβ ( r ) − N 2 g N 2 αβ ( r ) N 1 − N 2 (3.6) where the subscripts 1 and 2 refer to two systems with different number of molecules, but with same density/pressure, temperature, and composition. This method of correcting for the finite-size effect is straightforward, but it re- quires two different set of simulations, with different box sizes and number of molecules. The box sizes should not be too different and the resulting g ∞ αβ ( r ) can only be extended to the size of the smallest system. Another shortcoming of this method is related to the numerical accuracy arising from subtracting two num- bers of the same magnitude, both in numerator and denominator. The resulting numerical instabilities are increased when using two system sizes that are very close to each other [126] . 3.3.3. C ORTES -H UERTO ET AL . CORRECTION Another RDF correction is proposed by Cortes-Huerto et al. [83] . These au- thors define KBIs from finite systems in terms of fluctuations of the number of molecules as in the work of Krüger et al [74] . Their study considers the fluctua- tions inside a cubic subvolume (as oppose to spherical subvolumes, but the use of a different subvolume geometry should not affect the values of the KBIs at the thermodynamic limit G ∞ αβ [122] . The KBIs are also defined in terms of integrals over the RDF of the system (Eq. (1.18) ). To compute KBIs using finite volumes, these authors modify the L.H.S of Eq. (1.18) to include finite effects of RDFs and