6.6: Solving for the energy

Considering the tight binding matrix of Equation (6.6.14), non trivial solutions for the weighting factors, \(c_{1}\) and \(c_{2}\) are obtained from

\[ \operatorname{det}\left|\begin{array}{cc}
\alpha_{1}-E & \beta_{12} \\
\beta_{21} & \alpha_{2}-E
\end{array}\right|=0 \nonumber \]

Let's assume that the hopping interactions are equal \(\beta_{12}=\beta_{21}=\beta\). We'll consider two cases for equal and different self energies.

Equal self energies \(\alpha_{1} = \alpha_{2} = \alpha\)

When \(\alpha_{1} = \alpha_{2} = \alpha\), the energy is

\[ E = \alpha \pm\beta \nonumber \]

Substituting the energy back into Equation (6.6.14) to obtain the coefficients \(c_{1}\) and \(c_{2}\) yields two normalized solutions:

\[ \varphi = \frac{\phi_{1}\pm\phi_{2}}{\sqrt{2}} \nonumber \]

These two orbitals can be defined by their parity: their symmetry if their position vectors are rotated. For example, we could exchange their coordinates. In this example the molecular orbital:

\[ \varphi = \frac{\phi_{1}+\phi_{2}}{\sqrt{2}} \nonumber \]

does not change sign under exchange of electrons. It is classified as having gerade symmetry, denoted by g, where gerade is German for even. In contrast, the other orbital:

\[ \varphi = \frac{\phi_{1}-\phi_{2}}{\sqrt{2}} \nonumber \]

does change sign under exchange of electrons. It is classified with ungerade symmetry, denoted by u, where ungerade is German for odd.

Figure \(\PageIndex{1}\): The probability density plotted for the two linear combinations of two frontier orbitals. Due to the increased electron density between the nuclei, the \(\phi_{1}+\phi_{2}\) has lower energy.

Since the molecular orbital:

\[ \varphi = \frac{\phi_{1}+\phi_{2}}{\sqrt{2}} \nonumber \]

has energy, \(E = \alpha+\beta\), below that of the self energy, \(\alpha\), of each atomic orbital, the molecule is stabilized in this configuration. This is known as a bonding orbital because it describes a stable chemical bond. Recall that \(\alpha\) and \(\beta\) are defined to be negative for an electron in a positively charge nuclear potential.

The other molecular orbital

\[ \varphi = \frac{\phi_{1}-\phi_{2}}{\sqrt{2}} \nonumber \]

has energy, \(E = \alpha-\beta\), greater than that of the self energy, \(\alpha\), of each atomic orbital. Thus, this configuration is not stable. It is known as an antibonding orbital.

Figure \(\PageIndex{2}\): Antibonding and bonding molecular potential energy curves. Note that the antibonding energy is typically substantially larger than shown.

Different self energies

If \(\left|\alpha_{1}-\alpha_{2}\right|\gg \beta\) and \(\alpha_{1}>\alpha_{2}\) then the solutions are:

\[ E = \alpha_{1} +\frac{\beta^{2}}{\alpha_{1}-\alpha_{2}},\ \ E=\alpha_{2} -\frac{\beta^{2}}{\alpha_{1}-\alpha_{2}} \nonumber \]

Figure \(\PageIndex{3}\): The strongest bonds are formed from atomic orbitals with similar energies. In these diagrams the constituent atomic orbitals are shown at left and right. The molecular orbitals are in the center.

Thus, the splitting increases with the similarity in energy of the participating atomic orbitals, i.e. the bonding orbital becomes more stable. This is a general attribute of the interaction between two quantum states. The more similar their initial energies, the stronger the interaction.