Appendix D
Atomic and Molecular Term Symbols
D.1 Term Symbols for Multi-electron Atoms
The set of quantum numbers n, l, m l , s, m s associated with the set of quantum
mechanical operators {H,
l 2 , l z ,
s 2 , s z } apply to the individual electrons of a multielectron atom. In particular, the values taken by the orbital angular momentum
quantum number l (referred to as the ‘azimuthal’ quantum number in the old Bohr
theory of the hydrogen atom) are used to assign labels to the atomic orbitals in
which electrons are placed in the electron configuration model of atomic structure.
Each atomic orbital in this model is designated by the value of a principal quantum
number n, together with a letter that correlates with one of the allowed values for
the magnitude of the orbital angular momentum that is consistent with that principal
quantum number value. These letter designations are shown in Table D.1.
One of the main objectives of the quantum mechanical treatment of multielectron atoms is to determine both the energies of the various states that can be
occupied by such atoms and an appropriate way of identifying/labelling them. As
energies are the eigenvalues associated with the eigenstates of the multi-electron
Hamiltonian, and are obtained by solving the multi-electron Schrödinger equation,
and as the energy eigenfunctions will simultaneously be eigenfunctions of any
operator that commutes with the (appropriate) Hamiltonian, we may utilize the
eigenvalues associated with a set of such operators that also commute mutually with
one another to label both the eigenstates of the Hamiltonian and their corresponding
energy eigenvalues.
There are different levels at which the electrons of a multi-electron atom can
be considered. The scheme described in this appendix is based upon bringing in the
various terms making up the full Hamiltonian one at a time in the order in which they
contribute to the total energy eigenvalues of H, starting with the largest contribution
and proceeding stepwise to the smallest.
The simplest level is that of the so-called independent electron model, which
corresponds to the Hamiltonian having the structure H = H cf (the central-field
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
629
Atomic and Molecular Term Symbols
D.1 Term Symbols for Multi-electron Atoms
The set of quantum numbers n, l, m l , s, m s associated with the set of quantum
mechanical operators {H,
l 2 , l z ,
s 2 , s z } apply to the individual electrons of a multielectron atom. In particular, the values taken by the orbital angular momentum
quantum number l (referred to as the ‘azimuthal’ quantum number in the old Bohr
theory of the hydrogen atom) are used to assign labels to the atomic orbitals in
which electrons are placed in the electron configuration model of atomic structure.
Each atomic orbital in this model is designated by the value of a principal quantum
number n, together with a letter that correlates with one of the allowed values for
the magnitude of the orbital angular momentum that is consistent with that principal
quantum number value. These letter designations are shown in Table D.1.
One of the main objectives of the quantum mechanical treatment of multielectron atoms is to determine both the energies of the various states that can be
occupied by such atoms and an appropriate way of identifying/labelling them. As
energies are the eigenvalues associated with the eigenstates of the multi-electron
Hamiltonian, and are obtained by solving the multi-electron Schrödinger equation,
and as the energy eigenfunctions will simultaneously be eigenfunctions of any
operator that commutes with the (appropriate) Hamiltonian, we may utilize the
eigenvalues associated with a set of such operators that also commute mutually with
one another to label both the eigenstates of the Hamiltonian and their corresponding
energy eigenvalues.
There are different levels at which the electrons of a multi-electron atom can
be considered. The scheme described in this appendix is based upon bringing in the
various terms making up the full Hamiltonian one at a time in the order in which they
contribute to the total energy eigenvalues of H, starting with the largest contribution
and proceeding stepwise to the smallest.
The simplest level is that of the so-called independent electron model, which
corresponds to the Hamiltonian having the structure H = H cf (the central-field
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
629