Desarrollo de (3.71) empleando la teoría de perturbaciones

$$E = E^{(0)} + E^{(1)} + E^{(2)} + \cdots$$ (3.79)

$$E^{(0)} = \langle \Psi^{0}\vert\hat{H}^{0}\vert\Psi^{0}\rangle$$ (3.80)

$$E^{(1)} = \langle \Psi^{0}\vert\hat{H}^{1}\vert\Psi^{0}\rangle = -\sum_{i}\langle \Psi^{0}\vert\hat{\mu}_{i}\vert\Psi^{0}\rangle F_{i}$$ (3.81)

$$E^{(2)} = \sum_{n}\left\{ \frac{\langle\Psi^{0}\vert\hat{H}^{1}\vert\Psi^{n... ...gle \langle\Psi^{n}\vert\hat{H}^{1}\vert\Psi^{0}\rangle} {E_{0}-E_{n}} \right\}$$

$$= \sum_{n}\sum_{i}\sum_{j}\left\{ \frac{\langle\Psi^{0}\vert\hat{\m... ...Psi^{n}\vert\hat{\mu}_{j}\vert\Psi^{0}\rangle} {E_{0}-E_{n}}\right\} F_{i}F_{j}$$ (3.82)

Comparando con las ecuaciones anteriores, tenemos que

$$\mu_{i}^{0} = \langle\Psi^{0}\vert\hat{\mu}_{i}\vert\Psi^{0}\rangle \\$$ (3.83)

$$\alpha_{ij} = -2\sum_{n}\left\{ \frac{\langle\Psi^{0}\vert\h... ...\vert\hat{\mu}_{j}\vert\Psi^{0}\rangle} {E_{0}-E_{n}} \right\}$$ (3.84)