I've seen a lot of articles, where people calculate free Gibbs energy of system using several methods in some sense undirect methods, but I've never seen it is being calculated using its definition:
$\Phi=U+pV-TS$
Imagine a big MD system at its equilibrium in a box with periodic boundary conditions. Now consider a smaller sub-box with walls transparent for particles (atoms), where are k particles right now. Suppose we know each particle's position *r\_i*, velocity *v\_i*, potential energy *\\pi\_i*, force acting on it *f\_i* and the sub-system's temperature *T*. And, mainly, we know per-atom enthropy *s\_i*. Can then the sub-system's free Gibbs energy be calculated as
\Phi_k=U_k+(pV)_k-T_kS_k;
U_k=\frac{m}{2}\sum{i=1}{k}v_i^2 + \sum_{i=1}{k}\pi_i
(pV)_k=kT_k-\frac{1}{3}\sum{i=1}^k r_i\cdot f_i
S_k=\sum_{i=1}^k s_i
Even though it may not look like that to you, the calculations are actually based on the definition.
Now, about your proposed formulae: while in coarse, semiclassical approximation the internal energy could sort of be calculated in the proposed one-particle manner, in reality it is quantum mechanical and you can no longer simply break it down like that.
The entropy is an even bigger fish here: it cannot be divided into "one-particle contributions" like that, it has to be calculated via a partition function or something equivalent according to the quasi ergodic hypothesis (mean value over time = ensemble mean value).
Once you start calculating a partition function (or an equivalent) for the entropy, you get the internal energy "on the fly", so it makes no sense to use coarse approximations here, anyway.
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