PX262 - J3 - molecules

approximation: neglecting the kinetic energy of the nuclei and solving the electronic problem
nuclei take the positions where the sum of electronic energy and nuclear coulomb energy is minimized, electrons bond the nuclei
types of bonding:
- covalent
- ionic
considering a covalent bond in H $_{2}$
the two protons are fixed at ${\vec{R}}_{A}$ and ${\vec{R}}_{B}$ , and the two electrons at ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$
the distance from $B$ to $A :$ $\vec{R} = {\vec{R}}_{A} - {\vec{R}}_{B}$
building a model of the electrons' ground state from hydrogen states which complies with the pauli exclusion principle

covalent bond

ϕ_{100, σ}^{A} ({\vec{r}}_{1}) = \frac{1}{\sqrt{π a_{0}^{3}}} \exp (- \frac{| {\vec{r}}_{1} - {\vec{R}}_{A} |}{a_{0}})

where, $σ =↑, ↓$

ϕ ({\vec{r}}_{1}, σ_{1}, {\vec{r}}_{2}, σ_{2}) = C (R) \underset{spatial}{\underset{⏟}{(ϕ_{100}^{A} ({\vec{r}}_{1}) ϕ_{100}^{B} ({\vec{r}}_{2}) + ϕ_{100}^{A} ({\vec{r}}_{2}) ϕ_{100}^{B} ({\vec{r}}_{1}))}} \times \underset{spin}{\underset{⏟}{(u_{1 ↑} u_{2 ↓} - u_{1 ↓} u_{2 ↑})}}

the charge is concentrated between the protons and each electron is more delocalized than in an isolated atom
$R_{0}$ is the bond length in the H $_{2}$ molecule $\sim 0.74$ Å

for a NaCl molecule:
Na : $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1}$
Cl : $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5}$
the molecule is well described by the Na $3 s^{1}$ valence electron transferring to the Cl $3 p$ shell so that both atoms have filled shell configurations
Na becomes positively charged, and Cl, negatively
this is called an ionic bond