# Significance of single point energy when calculating interaction energies

I am currently investigating about the interaction behavior of a few atoms in certain conditions.

• Is it possible to use the concept of single point energy to represent the atomic interaction energies or I have to go other way around?
• What is the basic difference between potential energy and single point energy?
• The very language you use is a bit strange, so I'm not sure we are on the same page. How do you actually investigate "the interaction behavior of few atoms"? – Wildcat Jun 7 '15 at 18:22
• @Wildcat sorry for that. I'm basically using DFT/mp2 to obtain the interaction energy(potential energy or single point energy) between atomic pairs like (O...H) or (N...H) or (F...H). – diffracteD Jun 9 '15 at 3:25
• What you can obtain using electronic structure calculations is the single point energy. If you do a number of calculations for few different nuclear configurations you than single point energies will define the potential energy for nuclear motion. But it is not the interaction energy. For details, see my updated answer. – Wildcat Jun 9 '15 at 8:30

Single point energy arises in the framework of the Born–Oppenheimer approximation and corresponds to just one point on the potential energy surface. Physically it is the total energy of the molecular system with its nuclei beeing fixed (or clamped) at some particular locations in space. In other words, it is total energy of the molecular system within the so-called clamped nuclei approximation.

Mathematically, if you develop the Born–Oppenheimer approximation step-by-step you can easily see that single point energy it is the sum of the electronic energy and nuclear repulsion potential energy, $$U = E_{\mathrm{e}} + V_{\mathrm{nn}} \, ,$$ where the electronic energy $E_{\mathrm{e}}$ is the solution of the electronic Schrödinger equation, $$\hat{H}_{\mathrm{e}} \psi_{\mathrm{e}}(\vec{r}_{\mathrm{e}}) = E_{\mathrm{e}} \psi_{\mathrm{e}}(\vec{r}_{\mathrm{e}}) \, .$$ The fact that at this point we use the symbol $U$ which (alongside with $V$) is usually used for potential energy to mean the single point energy is justified a little later. Namely, when we introduce the Born-Oppenheimer approximation which give rise to the nuclear Schrödinger equation, $$\Big( \hat{T}_{\mathrm{n}} + U(\vec{r}_{\mathrm{n}}) \Big) \psi_{\mathrm{n}}(\vec{r}_{\mathrm{n}}) = E \psi_{\mathrm{n}}(\vec{r}_{\mathrm{n}}) \, ,$$ it is easy to recognize that the values of the single point energy $U$ for all possible nuclear configurations define the potential energy for nuclear motion. So, it is in this sense that the single point energy is related to the potential energy.

Update: it became clear that OP misunderstood the notion of the single point energy $U$. Indeed, once we do few single point calculations for different nuclear configurations, the resulting $U(\vec{r}_{\mathrm{n}})$ is the potential energy for nuclear motion. However, it is not the interaction energy between some fragments, though, the interaction energy contributes to it. So if one wants to obtain the interaction energy one has to decompose the potential energy $U(\vec{r}_{\mathrm{n}})$ into its parts.

There different ways to perform the energy decomposition, just to name a few without any particular order:

• SAPT (Symmetry-Adapted Perturbation Theory) a separate program (few of them, to be more precise) which can be interfaced with different quantum chemistry codes.
• NEDA (Natural Energy Decomposition Analysis) which is available as a part of NBO package.
• Morokuma decomposition already available in some quantum chemistry codes (GAMESS-US, for instance).
• LMO-EDA (Localised Molecular Orbital Energy Decomposition Analysis) also available in some quantum chemistry codes (GAMESS-US, for instance).
• Ok, so what I'm getting is single point energy from DFT/mp2. If I get these single point energies for multiple atomic-configurations(as possible) in a particular distance-range then am I to get the potential energy surface (applicable to those atomic configurations and limited to that distance-range) ?! – diffracteD Jun 9 '15 at 8:44
• Yes, you need to scan the PES (as we call it) at least in the direction of one coordinate, say, by elongating the bond. You will end up with the curve which is basically a slice of the PES. But interaction energy is just a part of the potential energy, so you have to decompose the potential energy than. – Wildcat Jun 9 '15 at 8:54
• can you refer some literature for me on this matter. Thank you. – diffracteD Jun 10 '15 at 3:34

Single point energy is a point on the potential energy surface.
Suppose you have a single atom which has both potential energy (Due to interaction between electrons and protons and since there is no other atom or external force, all the potential interactions come from interaction between it's own electrons and protons.) and kinetic energy due to electronic motion. So we can write a hamiltonian (PE+KE) here. Here if you want to optimize this geometry you just need to optimize electonic hamiltonian because there is only one nucleus. It can be called single point energy as well as optimized energy. Now for two H atom, each have KE and also PE due to previously discussed interaction plus interaction between electrons and nucleus of each other. For example equlibrium H-H distance is 0.745 Angstrom (B3LYP theory). So if you put two H in a distance of 0.75 or greater, they will attract each other and if you put them in a distance less than equilibrium distance they will repel each other. And eventually they will come to equilibrium distance. So your potential energy curve will look like morse potential curve. This is optimization and the lowest point you find is optimized distance. Now, you want to find the force or you just want to know the energy when they are 0.8 Angstrom apart. What you can do is you can freeze their nucleus at those position and solve the schrodinger equation. Here since there is no nuclear degree of freedom, it will only optimize electronic hamiltonian to remove any kind of electronic overlap or to make the orbital orthogonal. This is called the single point energy. If you find all the single point energy for all distance, you can plot the potential energy surface and you can also find the lowest point on the PES, which is optimized geometry or you can find the maximum energy, which would be transition energy.

• thanks for elaborate explanation. But how can I be able to explain the greater pot. energy of [C-C-C-O H-N-C-C] compared to [C-O H-N]. While all that matters as per my interest is the pot. energy between O and H atoms. – diffracteD Jun 8 '15 at 4:21
• Your question is not clear. Can you please make it clear? I think you are having problem with the term used in computational chemistry because your question was also ambiguous. – Osman Mamun Jun 8 '15 at 19:49
• I need to get the potential energy between O and H atom(given a particular cartesian geometry) using DFT/mp2. Its all going fine. But if I add some adjacent atoms too with this calculation (viz. C-C-C-O....H-N-C-C) then the potential energy value of interaction between O and H is getting higher. May be this is because dispersion issue. But how can I be able to explain it more scientifically .!? – diffracteD Jun 9 '15 at 3:20