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I know about s, p, d, and f-orbitals. However, my physical chemistry textbook does not mention what comes after f-orbitals. What are the next letters, and in what order are these higher orbitals filled?

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After s, p, d, and f, the next letters follow in alphabetical sequence: g, h, i, k, l, and so on. Note that j is omitted from the sequence: see Why is the letter J omitted in the spdf... sequence? for more information. Since g-subshells would have $l = 4$, the lowest g-subshell would be the $\mathrm{5g}$ subshell, bearing in mind $l$ must be strictly less than $n$.

According to the aufbau principle, orbitals are filled in increasing order of $(n+l)$. Amongst orbitals with similar $(n+l)$, those with smaller $n$ are filled first:

$$\begin{array}{ccc} \text{Orbital} & \hline (n+l) & n \\ \hline \mathrm{1s} & 1 & 1 \\ \hline \mathrm{2s} & 2 & 2 \\ \hline \mathrm{2p} & 3 & 2 \\ \mathrm{3s} & 3 & 3 \\ \hline \mathrm{3p} & 4 & 3 \\ \mathrm{4s} & 4 & 4 \\ \hline \mathrm{3d} & 5 & 3 \\ \hline \end{array}$$

Following this principle you would theoretically have the order

$$\mathrm{\underbrace{1s}_{\text{Period 1}} \to \underbrace{2s \to 2p}_{\text{Period 2}} \to \underbrace{3s \to 3p}_{\text{Period 3}} \to \underbrace{4s \to 3d \to 4p}_{\text{Period 4}} \to \underbrace{5s \to 4d \to 5p}_{\text{Period 5}} \to \underbrace{6s \to 4f \to 5d \to 6p}_{\text{Period 6}}} \\ \mathrm{\to \underbrace{7s \to 5f \to 6d \to 7p}_{\text{Period 7}} \to \underbrace{8s \to \mathbf{5g} \to 6f \to 7d \to 8p}_{\text{Period 8}} \to 9s \to \cdots} $$

However, at such large atomic numbers, the aufbau principle starts to fail. The subshells are much more closely spaced in energy and the exact electronic configurations are not so clear-cut: this is already evident within the d- and f-blocks where some anomalies are present.

On top of this, relativistic effects (which become significant at large atomic numbers) cause subshells to split into two distinct energy levels (except for the s-subshell). See also: Shape of the P1/2 Orbital.

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