The following exercises are concerned with determining the degeneracy of various electronic energy levels. a. Two low-lying but closely-spaced, excited energy levels of helium have the configuration \(1 s 2 p\). What are the values of \(L\) and \(S\) for each of these energy levels? For both levels, determine the multiplicity, the term symbol and degeneracy of each member of the multiplet, and the cumulative degeneracy (neglecting multiplet splitting). b. Two closely-spaced, excited energy levels of atomic oxygen have the configuration \(1 s^{2} 2 s^{2} 2 p^{3} 3 s\). What are the values of \(L\) and \(S\) for each energy level? For both levels, determine the multiplicity, the term symbol and degeneracy of each member of the multiplet, and the cumulative degeneracy. c. Consider the following molecular term symbols: \({ }^{3} \Sigma^{2} \Pi\). List the values of \(S, \Lambda\), the multiplicity, and the degeneracy for each term symbol. Provide the complete term-symbol classification (i.c., including multiplets) for each term component.

Term symbols are vital in quantum chemistry for describing the quantum states of electrons within atoms and molecules. They encapsulate both the electron configuration and the possible quantum states arising from spin-orbit coupling. Term symbols are given in the form \2S+1\L_J, where L represents the total orbital angular momentum quantum number, S indicates the total spin quantum number, and J is the total angular momentum quantum number, which combines both L and S via vector addition.

In the construction of a term symbol, L is denoted by a letter (S, P, D, F, etc.) corresponding to L values 0, 1, 2, 3, and so on, respectively. For instance, when dealing with helium's excited state configuration of 1s2p as per the exercise, there's one electron in the 2p orbital, which gives L a value of 1 represented by 'P'. Meanwhile, the spin S with one unpaired electron is \(\frac{1}{2}\) leading to a doublet state, indicated by the multiplicity portion of the term symbol: \2\. Therefore, the term symbol for helium's state is \2P\. This compact notation concisely conveys significant quantum mechanical information about the atom's energy states.

Quantum numbers are the 'addresses' that give the precise location of electrons within an atom. There are four main quantum numbers meant to describe every aspect of an electron's motion: n (principal quantum number), l (orbital angular momentum quantum number), m_l (magnetic quantum number), and s (spin quantum number).

The principal quantum number, n, determines the energy level or shell of an electron. For example, in the helium exercise, the electrons are in the n=1 and n=2 energy levels for the 1s and 2p orbitals, respectively. The orbital angular momentum quantum number, l, indicates the shape of the orbital; here, our configurations relate to s (0) and p (1) orbitals. The magnetic quantum number, m_l, specifies the orientation of the orbital in space, which for a p orbital is -1, 0, or 1. Finally, the spin quantum number helps determine the direction in which an electron spins, which can either be +\(\frac{1}{2}\) or -\(\frac{1}{2}\) as per fundamental principles of quantum mechanics.

Electron configuration is the distribution of electrons of an atom or molecule in atomic or molecular orbitals. For helium in the given exercise, the electron configuration is 1s2p, which indicates that there is a full 1s orbital and an electron in the 2p orbital. The electron configurations are particularly useful when predicting the chemical, electrical, and magnetic behavior of an atom.

In oxygen, which has more electrons, the configuration is a bit more complex: 1s² 2s² 2p³ 3s. Here, the 's' orbitals are fully occupied, and it's the three p electrons in the second shell that will mainly affect the chemistry of the atom, as these electrons are in the outermost shell and participate in bonding. Understanding the electron configuration allows us to approach the exercise on energy levels with confidence, as it directly influences the possible values of L and S, leading to term symbols and their related degeneracy.

Atomic multiplicity refers to the number of possible orientations of the total spin of electrons, which directly correlates with the number of closely spaced energy levels for a given electron configuration. It is calculated as 2S+1, where S is the total spin quantum number, and it indicates the number of spin states that an electron in a particular orbital can occupy.

Therefore, a multiplicity of 2, as observed in both helium and oxygen in our examples, signifies a doublet, implying two possible orientations. The multiplicity, as part of the term symbol, sheds light on the electron's spin behavior in the energy state of the atom. Higher multiplicity implies a greater number of unpaired electrons, leading to higher degeneracy as it multiplies the number of unique ways electrons can distribute among the available orbitals without violating the Pauli exclusion principle. When combined with the allowed values of m_l and m_s, the atomic multiplicity helps predict the degeneracy of specific energy levels, enhancing our understanding of the atom's energy spectrum.