Noura Dawass

5.2. M ETHODS 5 95 g βγ ( r ) = n βγ ( r ) N γ c ( r ) (5.12) Note that RDFs are symmetric, i.e. g i j ( r ) = g ji ( r ). The expressions above are used to find RDFs of the pseudo-binary mixture ( α and ω ), resulting from com- bining the identity of β and γ into ω . For this new system, we need to find ex- pressions for g ωω ( r ) and g ωα ( r ). To derive an expression for g ωω ( r ), we start with the general RDF expression for similar molecules (Eq. (5.2) ) g ωω ( r ) = n ωω ( r ) N ω c ( r ) (5.13) The local number of molecules n ωω ( r ) is composed of different contributions: n ββ ( r ), n βγ ( r ), n γβ ( r ), and n γγ ( r ). The probability that the central molecule is of type β or of type γ is N β / N ω and N γ / N ω , respectively. N ω is the total number of indistinguishable molecules. In this case, N ω = N β + N γ . Substituting these terms in Eq. (5.13) yields g ωω ( r ) = ³ N β N ω n ββ ( r ) + N γ N ω n γγ ( r ) + N γ N ω n γβ ( r ) + N β N ω n βγ ( r ) ´ N ω c ( r ) = ¡ N β n ββ ( r ) + N γ n γγ ( r ) + N γ n γβ ( r ) + N β n βγ ( r ) ¢ N 2 ω c ( r ) (5.14) Multiplying and diving the nominator by N β N γ N β N γ , yields g ωω ( r ) = µ N 2 β N γ N β N γ n ββ ( r ) + N 2 γ N β N β N γ n γγ ( r ) + N 2 γ N β N β N γ n γβ ( r ) + N 2 β N γ N β N γ n βγ ( r ) ¶ N 2 ω c ( r ) (5.15) Using Eqs. (5.5) , (5.6) , (5.11) and (5.12) results in g ωω ( r ) = µ N 2 β N γ N γ g ββ ( r ) + N 2 γ N β N β g γγ ( r ) + N 2 γ N β N γ g γβ ( r ) + N 2 β N γ N β g βγ ( r ) ¶ N 2 ω (5.16) The functions g βγ ( r ) and g γβ ( r ) are equal, and Eq. (5.16) can be further simpli- fied to g ωω ( r ) = N 2 β g ββ ( r ) + N 2 γ g γγ ( r ) + 2 N β N γ g βγ ( r ) N 2 ω (5.17)