Final answer:
The allowed spectroscopic notations based on the rules of quantum numbers are 1s¹ and 4s². Notations such as 1d³, 3p⁷, and 6h²⁰ are not allowed as they either involve orbitals that do not exist at the specified energy level or exceed the maximum number of electrons that can be accommodated in a subshell.
Step-by-step explanation:
The spectroscopic notations correspond to the different ways atomic orbitals are represented, referring to the electron configurations of atoms. The notation consists of a number (the principal quantum number n), a letter (indicating the type of orbital: s, p, d, f, g, h, etc.), and a superscripted number (showing the number of electrons in that orbital).
The rules that govern these notations are based on the quantum numbers and the possible subshell designations for each principal energy level. Here we assess whether the given notations are allowed:
- 1s¹: This is allowed. The 1st principal energy level has only an 's' subshell, and it can hold up to 2 electrons.
- 1d³: Not allowed. The 'd' subshell does not exist for n=1.
- 4s²: This is allowed. 's' orbital can exist for n=4 and can hold up to 2 electrons.
- 3p⁷: Not allowed. A 'p' orbital can hold a maximum of 6 electrons only.
- 6h²⁰: Not allowed. 'h' orbitals do not exist for practical purposes in observable chemistry, and even if they did, the superscript exceeds the maximum number for a subshell, which is determined by the equation 2(2l+1), where l is the azimuthal quantum number for h, which would be 5, thus allowing a maximum of 22 electrons.
Therefore, the only allowed spectroscopic notations from the options given are 1s¹ and 4s².