Equações

A função de ondas é:

$\vec{\psi} = \mathcal{\vec{R}} \cdot \mathcal{\vec{Y}}$

(I)

onde temos a função de ondas radial:

$\mathcal{\vec{R}} = f_{R}(Z,n,l,a0,\vec{r}) = \xi(e^{-\frac{\vec{\rho}}{2}})(\vec{\rho})^{l}f_{lag}(\beta,\gamma,\vec{\rho})$

(II)

em que:

$\vec{\rho} = f_{\rho}(\alpha\vec{r}) = \alpha\vec{r}$

(III)

$\alpha = f_{\alpha}(Z,n,a_0) = \frac{2Z}{n \cdot a_0}$

(IV)

$\beta = f_{\beta}(n,l) = (n-l-1)$

(V)

$\gamma = f_{\gamma}(l) = (2l+1)$

(VI)

$\lambda = f_{\lambda}(\beta,\gamma) = (\beta + \gamma)$

(VII)

$ \xi = f_{\xi}(n,\alpha,\beta,\lambda) = \sqrt{\frac{\alpha^{3}\beta!}{2n(n+1)!}} \cdot \frac{1}{\lambda!}$

(VIII)

e a função de ondas angular:

$\mathcal{\vec{Y}} = f_{Y}(l,|m|,\vec{\theta},\vec{\phi})$

$\sqrt{2}(-1)^{m}(-1)^{\|m\|}\sqrt{\frac{2l+1}{4\\pi}\frac{(1-\|m\|)!}{(1+\|m\|)!}}\ \text{imag}(\vec{\sigma})$	∀m<0
$ (-1)^{m} \sqrt{\frac{2l+1}{4\pi} \frac{(1-m)!}{(1+m)!}} \ \vec{\sigma}$	∀m=0
$ \sqrt{2}(-1)^{m}(-1)^{\|m\|}\sqrt{\\frac{2l+1}{4\pi}\frac{(1-m)!}{(1+\|m\|)!}}\ \text{real}(\vec{\sigma})$	∀m>0

(IX)

em que:

$\vec{\sigma} = f_{\vec{\sigma}}(l,m,\vec{\theta},\vec{\phi}) = e^{i|m|\vec{\\phi}} \cdot f_{\text{leg}}(l,|m|,\cos(\vec{\theta}))$

(X)