What are the scientific justifications of the binding energy equation?

Question

There are many problems where we want to calculate the binding energy between two systems.

Normally, we have a system A, adsorbent (a surface, a nanotube, a protein, etc.), that interact with a system B, adsorbate (atoms, ions, molecules, etc.). The procedure is to calculate the energy of both systems separately $E_\text{A}$, and $E_\text{B}$ and to calculate the energy of the complex $E_{\text{AB}}$, then the binding energy can be calculated as:

$$ E_{\text{bind}} = E_{\text{AB}} - E_\text{A} + E_\text{B} \tag{1} $$

In case DFT calculations using basis set, we also need to apply the basis set superposition error (BSSE) correction (Molecular Physics, 1970, VOL. 19, NO. 4, 553-566).

The question is: what are the fundamentals of equation (1)?

Answer 1

The justification is simple and comes from a very fundamental law of thermodynamics: Internal energy is a complete differential form and is independent of intermediate states and only depends on start and final states: $$\Delta U = U_{2} - U_{1} $$

In your case: $U_{2} = E_{AB}$ and $U_{1} = E_{A} + E_{B}$ and if $\Delta U$ or binding energy $E_{\text{binding}}$ is positive, it means the start state is thermodynamically preferred but if $\Delta U$ is negative, it means the binding state is thermodynamically preferred. You can generalize it to more complex systems with more than two reacting systems or components.

Answer 2

In general, a system composed of $K$ interacting subsystems have a potential energy at a specific configuration of its parts. For instance, a system of $M$ nuclei and $N$ electrons can be separated into interacting subsystems with internal geometries, having $\{\mathbf{R}_A\}$ as nuclear positions for subsystem $A$ with $N_A$ electrons, $\{\mathbf{R}_B\}$ and $N_B$ for subsystem $B$, and so on. With all the remainder coordinates accounting for the distance between these subsystems, we can write:

$$ \mathbf{R}=\mathbf{R}_{\text{int}}+\mathbf{R}_{\text{ext}}=\sum_{A}^{K}\left[\sum_{a\in A}\left(\mathbf{R}_a+\sum_{b\in B}\mathbf{R}_{ba}\right)\right] $$

where $\mathbf{R}_a$ is the position of a nucleus $a$ within the subsystem $A$ (internal coordinates) and $\mathbf{R}_{ba}$ is the distance vector from a nucleus $b$ in the subsystem $B\neq A$ to the nucleus $a$.

According to this, we can write the total energy of the complex as one-system terms, two-system terms, and so on.

$$ E(\{\mathbf{R}\})=\sum_{A}^{K}E_A(\{\mathbf{R}_A\})+\frac{1}{2}\sum_{A}^{K}\sum_{B\neq A}^{K}E_{AB}(\{\mathbf{R}_B-\mathbf{R}_A\})+\dots $$

where the energy is adjusted to zero (subtracting the energy of the isolated subsystems from $E$) at infinite separation of the subsystems. Then, we can define a potential energy of interaction between the subsystems as

$$ \Delta E(\{\mathbf{R}\})=E(\{\mathbf{R}\})-\sum_{A}^{K}E_A(\mathbf{R}_A)=\frac{1}{2}\sum_{A}^{K}\sum_{B\neq A}^{K}E_{AB}(\{\mathbf{R}_B-\mathbf{R}_A\})+\dots $$

Now, this interaction energy can be calculated at any $\{\mathbf{R}\}$, however, most of the time we want to calculate the interaction energy at optimized geometries $\{\mathbf{R}^{\text{(opt)}}\}$:

$$ \Delta E=E(\{\mathbf{R}^{\text{(opt)}}\})-\sum_{A}^{K}E_A(\mathbf{R}^{\text{(opt)}}_A) $$

and this happen to be the binding energy with respect to a specific dissociation route from equilibrium.