r)\;
      \varphi_{i} (r)\;
      \displaystyle\frac{d\, V(r)}{dr}\;
      r^{2} dr, \\

  I_{\rm dip}\, (l_{f}) & = &
    \sqrt{\displaystyle\frac{1}{3}}\;
    \displaystyle\sum\limits_{\mu = \pm 1}
    \displaystyle\sum\limits_{m}
      h_{\mu}
    \displaystyle\int
        Y_{l_{f}m_{f}}^{*} ({\mathbf n}_{r}^{f})\;
        Y_{1,-\mu}(\mathbf{n}_{r})\; d\Omega.
\end{array}
\label{eq.2.7.1.7}$$ After use of formula (\[eq.2.5.1\]) for vectors $\mathbf{n}^{f}_{r}$ and $\mathbf{n}_{r}$ the angular integral $I_{\rm dip}\, (l_{f})$ is non-zero only at $$\begin{array}{lcl}
  m_{f} = -\mu = \pm 1, &
  l_{f} = 1
\end{array}
\label{eq.2.7.1.8}$$ and equals to $$\begin{array}{lcl}
  I_{\rm dip}\, (l_{f}=1) & = &
  \sqrt{\displaystyle\frac{1}{3}}\; \displaystyle\sum\limits_{\mu = \pm 1} h_{\mu} =
  \sqrt{\displaystyle\frac{1}{3}}\; \Bigl(h_{-1} + h_{+1}\Bigr) =
  \sqrt{\displaystyle\frac{1}{3}}\; \Bigl(-i\,\sqrt{2}\Bigr) =
  -i\; \sqrt{\displaystyle\frac{2}{3}}\; .
\end{array}
\label{eq.2.7.1.9}$$ Now the matrix element (\[eq.2.7.1.6\]) obtains a form: $$\begin{array}{lcl}
  p\,(k_{i}, k_{f}) =
  i\; \sqrt{\displaystyle\frac{2}{3}}\; \displaystyle\frac{J_{\rm dip}\, (1)}{w_{fi}}.
\end{array}
\label{eq.2.7.1.10}$$ Comparing selection rules (\[eq.2.7.1.8\]) in the dipole approximation with the selection rules (\[eq.2.5.3\]) in the multipole approach, one can see that they do not impose any restrictions on the emission of photons. *From eq. (\[eq.2.7.1.1\]) one can see that the wave function of photon in the dipole approximation has no any information about orientation of the vector $\mathbf{k}$ concerning vector $\mathbf{r}$ in any selected frame system. On the basis of such a fact it is logically to consider the emission of photons in the dipole approximation as isotropic.*

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