PX262 - J4 - interacting electrons and nuclei in matter

the total energy is the sum of the kinetic energies of electrons and nuclei, and the interactions between electrons and nuclei, all pairs of electrons, and all pairs of nuclei
even for simple approximations, TISE needs approximations and computational methods
TISE for electrons (neglecting the KE of the protons:)

{\hat{H}}_{w} ϕ_{λ} = E_{λ} ϕ_{λ}

anti-symmetry from the pauli exclusion principle:

ϕ_{λ} ({\vec{r}}_{1}, σ_{1}, {\vec{r}}_{2}, σ_{2}) = - ϕ_{λ} ({\vec{r}}_{2}, σ_{2}, {\vec{r}}_{1}, σ_{1})

a helium atom has 2 electrons, 1 nucleus:

\begin{aligned} \hat{H} & = (- \frac{ℏ^{2} \nabla_{1}^{2}}{2 m_{e}} - \frac{2 e^{2}}{4 π ε_{0} r_{1}}) + (- \frac{ℏ^{2} \nabla_{2}^{2}}{2 m_{e}} - \frac{2 e^{2}}{4 π ε_{0} r_{2}}) + \frac{e^{2}}{4 π ε_{0} | {\vec{r}}_{1} - r_{2} |} \\ \hat{H} & = {\hat{H}}_{1} ({\vec{r}}_{1}) + {\hat{H}}_{2} ({\vec{r}}_{2}) + {\hat{V}}_{e e} (| {\vec{r}}_{1} - {\vec{r}}_{2} |) \end{aligned}

if ${\vec{V}}_{e e}$ is ignored, the ground state wave function is:

ϕ_{g s} ({\vec{r}}_{1}, σ_{1}, {\vec{r}}_{2}, σ_{2}) = ϕ_{100} ({\vec{r}}_{1}) ϕ_{100} ({\vec{r}}_{2}) \times (↑↓)

the spin states:

(↑↓) = u_{1 ↑} u_{2 ↓} - u_{2 ↑} u_{1 ↓}

\begin{matrix} {\hat{H}}_{1} ({\vec{r}}_{1}) ϕ_{100} ({\vec{r}}_{1}) = - \frac{13.6}{1^{2}} 2^{2} ϕ_{100} ({\vec{r}}_{1}) \\ ∴ [{\hat{H}}_{1} ({\vec{r}}_{1}) + {\hat{H}}_{2} ({\vec{r}}_{2})] ϕ_{100} ({\vec{r}}_{1}) ϕ_{100} ({\vec{r}}_{2}) = - \frac{13.6}{1^{2}} 2^{2} \times 2 ϕ_{100} ({\vec{r}}_{1}) ϕ_{100} ({\vec{r}}_{2}) \\ E_{g s} = - 108.8 eV \end{matrix}

when ${\hat{V}}_{e e}$ is included, $E_{g s} = - 79.9$ eV
one electron screens the nuclear charge from the other