## 1.6. Rotating-Molecule Hamiltonian

The Hamiltonian operator corresponding to the rotational part of the problem in
diatomic molecules [3] (pp. 6-16) must now be
examined in some detail. for any molecule can be written as a sum of three products, each
product consisting of a rotational constant for the molecule and the square of
one of the three components of the rotational angular momentum of the molecule
[6] (pp. 273-284). Since there can be no
rotational angular momentum of a diatomic molecule about its internuclear axis
(the *z* axis), the third component of the rotational angular momentum is
zero, and hence absent from . for diatomic molecules is thus written as

where the *rotational* angular momentum is expressed in the second line
of (1.10) as the *total* angular momentum (**J**) *minus* the
electronic *orbital* and *spin* angular momenta (**L** and
**S**). For the purposes of calculation, is often written in the form

where
*J*_{±} = *J*_{x} ±
*iJ*_{y}, *L*_{±} =
*L*_{x} ± *iL*_{y}, and
*S*_{±} = *S*_{x} ±
*iS*_{y}.

The extremely thorough student will find that the total angular momentum
operator **J** for linear molecules is somewhat peculiar, since its
molecule-fixed components do not obey angular-momentum-type commutation
relations. Furthermore, the operator for linear molecules does not have precisely the simple
form indicated in (1.10). Nevertheless, it can be shown
that correct results are obtained by ignoring the peculiarities associated with
**J** in linear molecules and by treating **J** like the angular momentum
operator defined for nonlinear molecules in [6].

The complete Hamiltonian
describing the nonrotational and rotational parts of the problem is of course
given by (1.1). It is the matrix of the
Hamiltonian (1.1) which will ultimately be diagonalized to obtain molecular
energy levels and molecular wave functions.

Loosely speaking, one can see in eq (1.10) the
origin of the various entries in column 2 of
table 2. Forgetting for a moment the
absence of *R*_{z} in (1.10), and remembering that the eigenvalue
associated with the sum of the squares of the components of an angular momentum
operator has the form ^{2}
*X*(*X* + 1), we note that: if the operators
**L** and **S** in the rotational Hamiltonian (1.10) can both be ignored,
then one might expect rotational energies to be given by
*B J*(*J* + 1), since *J* is the quantum number
associated with **J**^{2}; if the operator **L** in (1.10) can be
ignored, but the operator **S** cannot be, then one might expect rotational
energies to be given by *B N*(*N* + 1), since *N*
is the quantum number associated with
(**J** - **S**)^{2}; if neither **L** nor **S**
in (1.10) can be ignored, then one might expect rotational energies to be given
by *B R*(*R* + 1), since *R* is the quantum
number associated with (**J** - **L** -
**S**)^{2}.

Since the operators **L** and **S** in (1.10)
affect the course of the rotational energy levels only through the four cross
terms *J*_{x}L_{x}, *J*_{y}L_{y},
*J*_{x}S_{x}, *J*_{y}S_{y}, and since
the selection rules for nonvanishing matrix elements of *L*_{x},
*L*_{y}, and *S*_{x}, *S*_{y} are
ΔΛ and ΔΣ = ± 1, respectively, we see
that the effects of **L** and/or **S** in (1.10) can be ignored when the
separation between states of the nonrotating molecule satisfying the selection
rules ΔΛ and/or ΔΣ = ± 1 is large
compared to *B J*.