Example: Rotational Intensity Distribution in a

Step (*i*). Rotational energy levels and rotational wave functions for
each of the two vibrational-electronic states involved in a given optical
transition can be determined as described in
chapter 1. For molecules in the
absence of external fields, it is thus necessary to diagonalize matrices having
rows and columns labeled by a set of functions characterized by the same value
of *J* and the same value of *M* (*J* and *M* are good
quantum numbers in free space), but by different values of the other quantum
numbers of the basis set (see sect. 1.9 and
sect. 1.10). Furthermore, if numerical
calculations are to be performed, it is necessary to consider a number of
different matrices, corresponding to different numerical values for the quantum
number *J*. The quantum number *M* need not be assigned a numerical
value, since matrix elements of the Hamiltonian operator for a molecule in free
space are independent of *M* [7] (p. 49).

For the ^{4}Δ - ^{6}Σ^{+} example
considered here, we must diagonalize two sets of Hamiltonian matrices. One set
is of dimension 8 × 8, with rows and columns labeled by functions
of the ^{4}Δ upper state, the functions being
characterized by a fixed numerical value of *J*, a fixed algebraic value
of *M*, and by Λ = ±2,
Σ = ± ,
±. It is convenient to represent an
individual matrix from this set of 8 × 8 matrices by the symbol
*H _{u}*(

It is often desirable to perform a factorization in intensity calculations like that used in writing (1-27) from (1-25), since this allows parity selection rules to be taken into account at once. For simplicity of presentation, such a factorization is not performed below.

The diagonalization of a given upper state Hamiltonian matrix
*H _{u}*(

(3.35) |

are diagonal if the columns of *U* contain the eigenvectors of
*H _{u}* and the columns of

The matrices *U*, *U*^{-1}, *L*, and
*L*^{-1}, obtained from step (*i*), will be used in
step (*iii*).

Step (*ii*). The determination and systematic handling of basis set matrix
elements of the dipole moment operator for a
^{4}Δ - ^{6}Σ^{+}*µ _{Z}* satisfy the selection
rule Δ

We see from eq (3.12) that a matrix
element of *µ _{Z}* can be represented as the sum of three
terms.

(3.36) |

The first factor in each of the three terms in (3.36) represents a quantity
which can only be calculated from a rather complete knowledge of the electronic
wave functions. These quantities are analogous to the parameters
*µ*_{||} and *µ*_{⊥} introduced in
sect. 3.4 and
sect. 3.5. Following the selection rules
of eq (3.3), and making use of various
symmetry arguments (chap. 2), we find that the
following independent intensity parameters *µ _{i}* must be
considered for a

(3.37) |

The quantities *µ*_{1} through *µ*_{6} correspond to
perpendicular transition moments; the quantities *µ*_{7} through
*µ*_{9} correspond to parallel transition moments. The
relationships between matrix elements in eqs (3.37) correspond to using
*L* = 2 and the lower sign choice in
eq (2.11a). It can be shown (see
sect. 2.11 and
sect. 3.5) that for this choice of phases, the
parameters *µ*_{1} through *µ*_{9} are all real,
though not necessarily positive. (If, for example, a value of
*L* = 2 and the upper sign choice had been used in (2.11a), then
the parameters *µ*_{1} through *µ*_{9} would all have
been pure imaginary.)

The second factor in each of the three terms in (3.36) contains quantum mechanical matrix elements (in the basis set ) of elements of the direction cosine matrix. These matrix elements can be obtained from table 6.

The submatrices *µ _{b}*(

It will become apparent in the description of
step (*iii*) below that it is desirable to
define a matrix *µ _{b}*(

(3.38) |

It is thus possible to construct a simplified table 6, given here as
table 7, to be used in calculating matrix elements of elements of the
direction cosine matrix, *when these matrix elements are to be used in
intensity calculations for unpolarized light and for molecules in the absence
external fields.*

Nonvanishing matrix elements , where | |||

Factor | J′ = J + 1 |
J′ = J |
J′ = J - 1 |
---|---|---|---|

f(J′; J) · h(_{Z}J′; J)^{ } |
[12 (J + 1)]^{-1/2} |
J (J + 1)/(2 J+1)]^{-1/2} |
[12 J]^{-1/2} |

g(_{z}J′, Ω; J, Ω)^{ }g(_{x}J′,
Ω ± 1; J, Ω)ig
(_{y}J&prime, Ω ± 1; J, Ω) |
2[(J + Ω + 1) (J - Ω + 1)]^{1/2}J
± Ω ± 1)
(J ± Ω + 2)]^{1/2} |
2Ω^{ }[( J Ω)
J ± Ω + 1)]^{1/2} |
2[(J + Ω) (J - Ω]^{1/2}J Ω)
(J
Ω - 1)]^{1/2} |

As an example of the use of table 7, we consider
*µ _{b}*(

(3.39) |

if the rows of this matrix are labeled by the ^{4}Δ functions
characterized by
Λ = + 2,
Σ = + ,
+ ,
- ,
- and
Λ = - 2,
Σ = + ,
+ ,
- ,
- , respectively, and
if the columns are labeled by the ^{6}Σ^{+} functions
characterized by
Λ = 0^{+},
Σ = +,
+ ,
+ ,
- ,
- ,
- , respectively.

The quantities *c _{i}* are taken from table 7 to be

(3.40) |

It may occasionally be convenient to rewrite (3.39) as the sum of nine
matrices, each matrix depending on only one of the intensity parameters
*µ _{i}*.

Step (*iii*). We must now transform from matrix elements of
*µ _{Z}* between

(3.41) |

The matrix *µ _{b}*(

To determine the intensity *I*(α*J*′,
β*J*″) of a transition between a rotational level
α*J*′ of the ^{4}Δ state and a rotational level
β*J*″ of the ^{6}Σ^{+} state, it is
necessary to square the absolute value of the matrix element of
*µ _{Z}* between pairs of final wave functions
|α

(3.42) |

It is at this point that the usefulness of the matrix
*µ*(*J*′; *J*″) becomes apparent. As mentioned in
the discussion of step (ii), the *M* dependence
of all elements of a given (fixed *J*′, *J*Prime;, *M*)
matrix *µ _{b}*(

(3.43) |

i.e., if table 7 is used to calculate direction cosine
matrix elements, then the summation over *M* in intensity calculations is
taken care of automatically.

From the discussion of this section, it is apparent that intensity expressions
obtained by squaring the appropriate element of *µ*(*J*′;
*J*″) will in general be expressed for
^{4}Δ - ^{6}Σ^{+}
transition as functions of the nine unknown intensity parameters
*µ _{i}*. The values of these parameters must be determined by
fitting the calculated results to observed intensity data. For any particular
transition, many of the