# The Born Ion

One of the canonical examples for polar solvation is the Born ion: a nonpolarizable sphere with a single charge at its center surrounded by an aqueous medium. Consider the transfer of a non-polarizable ion between two dielectrics. In the initial state, the dielectric coefficient inside and outside the ion is $\epsilon_{\mathrm {in}}$, and in the final state, the dielectric coefficient inside the ion is $\epsilon_{\mathrm {in}}$ and the dielectric coefficient outside the ion is $\epsilon_{\mathrm {in}}$ In the absence of external ions, the polar solvation energy of this transfer for this system is given by

\[ \Delta_p G_{\mathrm{Born}}= \frac{q^2}{8\pi\epsilon_0 a}\left (\frac{1}{\epsilon_{\mathrm {out}}}-\frac{1}{\epsilon_{\mathrm {in}}}\right) \]

where q is the ion charge, a is the ion radius, and the two ε variables denote the two dielectric coefficients. This model assumes zero ionic strength.

Note that, in the case of transferring an ion from vacuum, or where $\epsilon_{\mathrm {in}} = 1$, the expression becomes

\[ \Delta_p G_{\mathrm{Born}}= \frac{q^2}{8\pi\epsilon_0 a}\left (\frac{1}{\epsilon_{\mathrm {out}}}-1\right) \]

For more information on the Born ion, see slides 24 and 25 of this presentation.

### Setting up and running the calculation

We can setup a PQR file for the Born ion for use with APBS with the contents:

We’re interested in performing two APBS calculations for the charging free energies in homogeneous and heterogeneous dielectric coefficients. We’ll assume the internal dielectric coefficient is 1 (e.g., a vacuum) and the external dielectric coefficient is 78.54 (e.g., water). for these settings, the polar Born ion solvation energy expression has the form

\[ \Delta_p G_{\mathrm{Born}} = -691.85 \biggl( \frac{z^2}{R} \biggr) \mathrm {kJ \, A/mol} \]

where z is the ion charge in electrons and R is the ion size in Å.

This solvation energy calculation can be setup in APBS with the following input file:

Running this example with a recent version of APBS should give an answer of -229.59 kJ/mol which is in good agreement with the -230.62 kJ/mol predicted by the analytic formula above.

**Note** that the Born example above can be easily generalized to other polar solvation energy calculations. For example, ions could be added to the solv ELEC, dielectric constants could be modified, surface definitions could be changed (in both ELEC sections!), or more complicated molecules could be examined. Many of the examples included with APBS (e.g., solv and ionize) also demonstrate solvation energy calculations.

**Note** that, as molecules get larger, it is important to examine the sensitivity of the calculated polar solvation energies with respect to grid spacings and dimensions.