When can you express the Helmholtz energy as a sum of terms?

Question

In both excess Gibbs/Helmholtz energy models the total Gibbs/Helmholtz energy is the sum of multiple contributions. For example, in the SAFT equation of state the total Helmholtz energy is

$$ A = A^\text{KE} + A^\text{HS} + A^\text{disp} + A^\text{chain} + A^\text{assoc}\,. $$ Here $\text{KE}$, $\text{HS}$, $\text{disp}$, $\text{chain}$, and $\text{assoc}$ denote the kinetic energy, hard-sphere, dispersion, chain, and association contributions to the Helmholtz energy.

I am struggling to understand when you can write the Helmholtz energy as the sum of multiple terms. Presumably each of the Helmholtz energy contributions corresponds to a term in the Hamiltonian

$H = \text{KE}(\mathbf{p}_1,\cdots, \mathbf{p}_N) + U^\text{HS}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{disp}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{chain}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{assoc}(\mathbf{q}_1,\cdots, \mathbf{q}_N) \,.$

Here $H$ is the Hamiltonian, $\mathbf{p}_i$ is the momentum of molecule $i$ and $\mathbf{q}_i$ is the position of molecule $i$. $U^\text{x}$ indicates the total pairwise energy of type $\text{x}$ between $N$ molecules. $\text{KE}$ is the total kinetic energy.

Is it an approximation or can you show from the Canonical partition function (see below) that each Helmholtz energy corresponds to a particular term in the Hamiltonian?

Note: The partition function for a continuous and classical system with N molecules is $$ \mathcal{Z} = \frac{1}{h^{3N}N!} \int e^{-\beta H(p_1,\cdots,p_N,q_1,\cdots,q_N)}\text{d}^3\mathbf{p}_1\cdots\text{d}^3\mathbf{p}_N\text{d}^3\mathbf{q}_1\cdots\text{d}^3\mathbf{q}_N. $$ Here $h$ is Planck's constant and $\beta$ is the inverse temperature.

The partition function is related to the Helmholtz energy through $A=-\frac{1}{\beta}\ln \mathcal{Z}$.

Answer 1

The way you wrote the Hamiltonian, $$H = \text{KE}(\mathbf{p}_1,\cdots, \mathbf{p}_N) + U^\text{HS}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{disp}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{chain}(\mathbf{q}_1,\cdots, \mathbf{q}_N)+ U^\text{assoc}(\mathbf{q}_1,\cdots, \mathbf{q}_N) ,$$ would only allow expressing the Helmholtz free energy as a sum of an ideal gas contribution (originating from the kinetic energy term) and a configurational contribution from all the potential energy terms depending on the same degrees of freedom.

However, in some cases, such additive decomposition of the free energy is possible because each term of the Helmholtz free energy comes from an integral over different configurational degrees of freedom. In such a case, the sum of contributions to the Hamiltonian depending on different variables implies the factorization of the canonical partition function and consequently a sum of different contributions to the free energy.

There are also cases where different terms depend on the same degrees of freedom, but one of them dominates. In such cases, perturbative approaches or a variational approach like that based on Gibbs-Bogolioubov inequality can justify, as an approximation, the additive decomposition of the free energy.