hm…. yes, I am now studying theoretical chemistry, so, it is a bit different from nuclear physics. But since theoretical chemistry developed a lot of techniques, which were also used in nuclear physics, it is worth to study them. anyway~

In theoretical chemistry, for homonuclear diatomic molecule (such as H2, O2, N2, etc.) , the system should be the same under parity transfrom. There are two kind of symmetry that satisfy the partiy transform, which is even or odd function. In chemistry, they called the even function be gerade state, and odd function be the ungerade state. to see more.

Lets use H2 molecule to see how covalent and ionic bonding defined.

In H2 molecule, the two protons are located at A and B. An electron relative position to proton A and B are $r_A$ and $r_B$.

The spatial orbit can be even or odd for an single electron in ground state:

$\displaystyle \phi_g = \frac{1}{\sqrt{2}}(\psi(r_A) + \psi(r_B) )$

$\displaystyle \phi_u = \frac{1}{\sqrt{2}}(\psi(r_A) - \psi(r_B) )$

In chemistry, they use $\alpha$ and $\beta$ for an electron spin state, which can only be spin-up or spin-down. So, for two electrons H2 molecule, assume the orbit can only be s-orbit, the total spin can only be $S=0$ or $S=1$.

for spin-singlet $S=0$, the two electron wave functions or molecular orbitals must be symmetric, which are:

$\Phi_A = \chi_{00}(1,2) \phi_g(1) \phi_g(2)$

$\Phi_B = \chi_{00}(1,2) \phi_u(1) \phi_u(2)$

$\Phi_C = \chi_{00}(1,2) \frac{1}{\sqrt{2}} (\phi_g(1) \phi_u(2) + \phi_u(1) \phi_g(2))$

for spin-triplet $S=1$,

$\Phi_D = \chi_{1m_S}(1,2) \frac{1}{\sqrt{2}} (\phi_g(1) \phi_u(2) - \phi_u(1) \phi_g(2))$

We expand the $\phi_{g/u}$

$\displaystyle 2 \phi_g(1) \phi_g(2) = \begin{matrix} \psi(r_{A1})\psi(r_{B2}) + \psi(r_{B1})\psi(r_{A2}) + \\ \psi(r_{A1})\psi(r_{A2}) + \psi(r_{B1})\psi(r_{B2}) \end{matrix}$

The upper row is the covalent bonding, and the bottom row is the ionic bonding.

Let take a careful look on the covalent bonding

$\Phi_{cov} = \psi(r_{A1})\psi(r_{B2}) + \psi(r_{B1})\psi(r_{A2})$

The first term means the electron-1 is orbiting A and the electron-2 is orbiting B, the 2nd term is similar.

The electrons are being share among the nuclei.

The ionic bonding

$\Phi_{ion} = \psi(r_{A1})\psi(r_{A2}) + \psi(r_{B1})\psi(r_{B2})$

Both electrons are at the same nucleus at a time.

These agrees with our learning from high school chemistry class. yeah~

Note that, when minimizing the energy of H2 system, a single function $\Psi_A$ is not a good choice, it is better to include more molecular orbitals, say, $\Psi = a_A\Psi_A + a_B\Psi_B$. notice that the covalent and ionic component in $\Psi_B$ has different sign. After minimization, the ionic component is greatly suppressed.  In fact, the maximum component of ionic bond is only 20% when $R = 0.75 \textrm{\AA}$