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I mean I know they’re $\ce{d_{-2}}$, $\ce{d_{-1}}$, $\ce{d_{0}}$, $\ce{d_{1}}$, $\ce{d_{2}}$ but how do these numbers relate to each one of $\ce{d}$ orbitals $\ce{d_{z_2}}$, $\ce{d_{xz}}$, $\ce{d_{yz}}$, $\ce{d_{xy}}$, $\ce{d_{x_2 - y_2}}$)? I couldn't find a definitive answer anywhere.

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    $\begingroup$ They don't. $\,$ $\endgroup$ – Ivan Neretin Sep 15 '18 at 17:21
  • $\begingroup$ Katie, $\ce{d_{z_2}}$, $\ce{d_{xz}}$, $\ce{d_{yz}}$, $\ce{d_{xy}}$, $\ce{d_{x_2 - y_2}}$ are linear combinations of $\ce{d_{-2}}$, $\ce{d_{-1}}$, $\ce{d_{0}}$, $\ce{d_{1}}$, $\ce{d_{2}}$. See around page 338 of Quantum Chemistry (MCQUARRIE, D., 2nd ed.). $\endgroup$ – Felipe S. S. Schneider Sep 18 '18 at 13:45
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On p. 238 of my copy of McQuarrie (1983; his Table 6-6), he lists the following for the hydrogenlike atomic wave functions (expressed as real functions), describing these as "commonly used": \begin{align*} n=3, l = 2, m=0: & \ \psi_{3d_{z^2}} \ ... \\ n=3, l = 2, m=\pm 1: & \ \psi_{3d_{xz }} \ ... \\ : & \ \psi_{3d_{yz }} \ ... \\ n=3, l = 2, m=\pm 2: & \ \psi_{3d_{x^2-y^2 }} \ ... \\ : & \ \psi_{3d_{xy }} \ ... \\ \end{align*}

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