这是一份liverpool利物浦大学PHYS1042的成功案例

$$
\sum_{\lambda} F_{\lambda ‘ \lambda}^{j} C_{\lambda j}=\epsilon_{j} \sum_{\lambda} S_{\lambda^{\prime} \lambda} C_{\lambda j},
$$
where
$$
\begin{aligned}
F_{\lambda^{\prime} \lambda}^{j}=&\left\langle\chi_{\lambda^{\prime}}|h| \chi_{\lambda}\right\rangle+\sum_{\delta \kappa}\left[\gamma_{\delta_{k}}\left\langle\chi_{\lambda^{\prime}} \chi_{\delta}|g| \chi_{\lambda} \chi_{\chi^{\prime}}\right\rangle\right.\
&\left.-\gamma_{\delta \kappa}^{\text {exch }}\left\langle\chi_{\lambda^{\prime} \cdot} \chi_{\delta}|g| \chi_{\kappa} \chi_{\lambda}\right\rangle\right] .
\end{aligned}
$$
Here, $h$ is the one-electron part of the full Hamiltonian, $g$ is an electron-electron repulsion potential energy, and
$$
\begin{gathered}
\gamma_{\delta x}=\sum_{i}^{\prime} C_{\delta i} C_{\kappa i}, \
\gamma_{\delta \kappa}^{\text {exch }}=\sum_{i}^{n \prime} C_{\delta i} C_{\kappa i},
\end{gathered}
$$

PHYS104 COURSE NOTES ：

Show that
if $f(x)=x^{3}$, then $f^{\prime}(x)=3 x^{2}$.
Prove by induction that for each positive integer $n$, $f(x)=x^{n} \quad$ has derivative $\quad f^{\prime}(x)=n x^{n-1} .$
HINT:
$$
(x+h)^{k+1}-x^{k+1}=x(x+h)^{k}-x \cdot x^{k}+h(x+h)^{k} .
$$