# What are the scientific justifications of the binding energy equation?

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)?

• You don't apply the basis set superposition error, but rather try to remove it. – Susi Lehtola Jun 18 '20 at 9:06
• @SusiLehtola of course! I added "correction". It is ok now? – Camps Jun 18 '20 at 10:25
• Yes, but it's also not specific to DFT but applies to any basis set calculation. – Susi Lehtola Jun 18 '20 at 11:14

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.

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.