A theoretical study is performed for the excitation of a single atom localized in the center of twisted light modes. Here we present the explicit dependence of excitation rates on critical parameters, such as the polarization of light, its orbital angular momentum projection, and the orientation of its propagation axis with respect to the atomic quantization axis. The effect of a spatial spread of the atom is also considered in detail. The expressions for transition rates obtained in this work can be used for any atom of arbitrary electronic configuration. For definiteness we apply them to the specific case of $^{2}S_{1/2} (F=0) rightarrow; ^{2}F_{7/2} (F=3, M=0)$ electric octupole (E3) transition in $^{171}$Yb$^{+}$ ion. Our analytical and numerical results are suitable for the analysis and planning of future experiments on the excitation of electric-dipole-forbidden transitions by twisted light modes in optical atomic clocks.
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