   Menu Outline Exercise Questions Appendices Lectures Student Projects Books Sections Introduction Atoms Molecules Crystal Structure Crystal Diffraction Crystal Binding Photons Phonons Electrons Band Model Crystal Physics Semiconductors

PHY.K02UF Molecular and Solid State Physics

## Ionic bond potential

In an ionic bond one binding partner loses an electron to the other binding partner. For a large separation between the atoms, energy is required to transfer an electron from one atom to the other. For a small separation between the atoms, the energy is reduced by transferring an electron from one atom to the other. Consider a neutral Na atom separated by a long distance from a neutral Cl atom. We assign this configuration an energy of zero. An electron is then transferred from the Na atom to the Cl atom while they remain well separated. This will increase the energy by the ionization energy of Na, 5.14 eV, minus the electron affinity of Cl, 3.61 eV. The increase in energy is 1.53 eV. The ions are then slowly brought together and as they get closer, the energy of the system decreases like a Coulomb potential,

$$U_{\text{Coulomb}}=-\frac{e^2}{4\pi\epsilon_0R}+U_{\text{IE-EA}},$$

where $\epsilon_0$ is the permeativity constant and $U_{\text{IE-EA}}$ is the ionization energy minus the electron affinity. There is a critical separation $R_{\text{crit}}$ Where the energy of the charged ions is lower than the energy of the separated neutral atoms. This is the separation at which ionization occurs when neutral atoms are brought together. For NaCl,

$$R_{\text{crit}} = \frac{e^2}{4\pi\epsilon_0(1.53e)} = 9.41 \text{ Å}.$$

For separations less than the critical separation, ionic bonds can be approximated by a potential of the form,

$$U(R) = \lambda \exp\left(-\frac{R}{\rho}\right)-\frac{ae^2}{4\pi\epsilon_0R}+U_{\infty}.$$

Above the critical separation, the atoms are neutral an the bond potential is approximately zero. In the plot below, the red points are from a quantum mechanical calculation, the black line is the bond potential that was fit to the calculated energies, and the blue line is a Coulomb potential. The energies were calculated using Hartree-Fock aims without spin-polarization using the FHI-aims software package and the suggested "tight" settings. The bond strength of ionic bonds is typically a few eV.

NaCl
U(R) [eV]

$\lambda = 2199$ eV
$\rho = 0.301$ Å
$a= 1.1$
$U_{\infty}= 1.686$ eV

R [Å]