# How to calculate the energy of a single proton?

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An isolated proton will have energy given by the Schrodinger equation (just like any molecular system):

$$-\frac{\hbar^2}{2\mu}\nabla^2\psi + V\psi = E\psi$$

($\mu$ is reduced mass; $\hbar$ is $h/2\pi$; $V$ is potential energy; $\psi$ is the wavefunction)

The potential of this single-proton system is zero; the kinetic energy will depend on the momentum of the particle.

If you're using a generic quantum-chemistry software, the Born-Oppenheimer approximation is probably being applied, i.e. the proton ($\ce{H+}$) will be fixed in space with zero momentum. Thus its total energy will be zero, unless you deactivate BO-approximation or introduce another particle.

It's not typical to consider the energy of an isolated proton for obvious reasons (try finding one in nature!). Nonetheless it is calculable if the wavefunction is known.

Another way to consider its kinetic energy is by the classical equation $K = \frac{1}{2} mv^2$; if you consider an approximation of the lonely proton's speed as roughly that of atoms in liquid water, 1 angstrom per picosecond, you obtain ~ $5.03$ kJ/mol. This is similar to the thermal energy available at room temperature, $k_B T$, ~ 2.48 kJ/mol.

You're better off calculating energy of $\ce{H+}$ by getting energy for a molecule, and subtracting the energy of its deprotonated form. In that way you can evaluate proton affinity for a given molecule.

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### logical x 2

Updated on August 01, 2022

• logical x 2 over 1 year

Using e.g. the VWN5 functional, what is a simple way to calculate the energy of a single proton? Of course, it is not possible to perform a calculation of a proton in isolation, as it bears no electrons. So I first attempted to use the experimentally determined value (see e.g. M. Dewar and K. Dieter, J. Am. Chem. Soc. 1986, 108, 8075.), but this leads to highly erratic proton affinities.

Could it be that I need to determine the proton affinity for any given functional + basis set (e.g. VWN5, 6-31+G** in my case)? What would be an effective way to get a good rough estimate of the energy of a proton?

What "energy of single" proton? It can have any kinetic energy you can imagine.
• Ivan Neretin about 6 years
A good rough estimate of the energy of a proton is $0$.