### 3.6. Intensity Calculations When Closed-Form Expressions Cannot Be Obtained. Example: Rotational Intensity Distribution in a 4Δ - 6Σ+ Transition

The calculation of rotational intensity distributions when some of the pertinent expressions cannot be written down in closed form can conveniently be divided into three parts: (i) determination of wave functions for rotational levels of the upper and lower electronic states of the transition, expressed in terms of basis set functions for the upper and lower states, respectively; (ii) determination of matrix elements of the dipole moment operator between functions in the upper state basis set and functions in the lower state basis set; (iii) formation of linear combinations of basis set dipole moment matrix elements to obtain dipole moment matrix elements between upper state final wave functions and lower state final wave functions. Each of these three steps must be performed numerically, presumably by a modern electronic computer. It is convenient in what follows to indicate the numerical steps in matrix notation.

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 4Δ - 6Σ+ 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 4Δ 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 Hu(J′), where J′ is the J-value characterizing this particular upper state Hamiltonian matrix. The other set of matrices is of dimension 6 × 6, with rows and columns labeled by the functions of the 6Σ+ lower state, the functions being characterized by a fixed numerical value of J, a fixed algebraic value of M, and by Λ = 0, Σ = ±, ± , ± . It is convenient to represent an individual matrix from this set of 6 × 6 matrices by the symbol Hl(J″), where J″ is the J-value characterizing this particular lower state Hamiltonian matrix.

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 Hu(J′) will be accomplished by a particular transformation matrix U(J′); the diagonalization of a given lower state Hamiltonian matrix Hl(J″) will be accomplished by a particular transformation matrix L(J″). As is well known [17] (pp. 26-28), the product matrices

 (3.35)

are diagonal if the columns of U contain the eigenvectors of Hu and the columns of L contain the eigenvectors of Hl.

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 4Δ - 6Σ+ transition can be carried out as follows. The most general matrix element of the dipole moment operator in the basis set under consideration has the form . However, nonvanishing matrix elements of µZ satisfy the selection rule ΔM = 0; it is thus convenient to set M′ = M″ = M in this general matrix element. Furthermore, the Hamiltonian operator for a molecule in free space only mixes together basis set functions characterized by the same value of J and the same value of M (step (i)). Consequently, the transformation from matrix elements of the dipole moment operator between basis set functions to matrix elements of the dipole moment operator between final wave functions (step (iii)) never requires adding together basis set matrix elements characterized by different values of J′, different values of J″, or different values of M. It is thus convenient to group the basis set matrix elements into submatrices µb(J′, J″; M) characterized by fixed values of J′, J″, and M. For the present example of a 4Δ - 6Σ+ transition, these submatrices are of dimension 8 × 6; their rows are labeled by the eight basis set functions of fixed J′ and M belonging to the upper 4Δ state; their columns are labeled by the six basis set functions of fixed J″ and M belonging to the lower 6Σ+ state.

We see from eq (3.12) that a matrix element of µZ 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 µi must be considered for a 4Δ - 6Σ+ transition.

 (3.37)

The quantities µ1 through µ6 correspond to perpendicular transition moments; the quantities µ7 through µ9 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 µ1 through µ9 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 µ1 through µ9 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 µb(J′, J″; M) can thus be evaluated numerically when numerical values for J′, J″; M and for the µi of (3.37) have been chosen.

It will become apparent in the description of step (iii) below that it is desirable to define a matrix µb(J′; J″), which has the same dimensions as µb(J′, J″; M), but which is independent of the quantum number M. It can be seen from table 6 and from (3.36) that the M-dependence of each element of a given (i.e., fixed J′, J″, M) matrix µb(J′, J″; M) is the same. In fact, this M-dependence is given by the quantity hZ(J′, M; J″, M) in table 6. The M-independent matrix µb(J′; J″) is now defined just like the M-dependent matrix µb(J′, J″; M), except that the M-dependent quantity hZ(J′, M; J″, M) given in the fourth row of table 6 is replaced by the M-independent quantity hZ(J′; J″), where

 (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.

### TABLE 7. A simplified table 6, for use in intensity calculations for unpolarized light and for molecules in the absence of external fields.(Image of Table 7)

Nonvanishing matrix elements , where s = x, y, or z, can be replaced by the product of three factors: f(J′; J) · gs(J′, Ω′; J, Ω) · hZ(J′; J&Pime;). The product f · hZ of the first and third factors is given in row one of this table; the second factor gs is chosen from rows two and three.
Factor J′ = J + 1 J′ = J J′ = J - 1
f(J′; J) · hZ(J′; J)  [12 (J + 1)]-1/2 [12 J (J + 1)/(2 J+1)]-1/2 [12 J]-1/2
gz(J′, Ω; J, Ω)  gx(J′, Ω ± 1; J, Ω) or
igy (J&prime, Ω ± 1; J, Ω)
2[(J + Ω + 1) (J - Ω + 1)]1/2
[(J ± Ω ± 1) (J ± Ω + 2)]1/2

[(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 µb(JJ) for the 4Δ - 6Σ+ transition under discussion in this section. Making use of (3.36), (3.37), and table 7, we find that µb(JJ) has the form

 (3.39)

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

The quantities ci 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 µi.

Step (iii). We must now transform from matrix elements of µZ between basis set functions to matrix elements of µZ between final wave functions. This transformation can be carried out as follows [5] (pp. 208-210):

 (3.41)

The matrix µb(J′, J″; M) has been defined in step (ii). An element lying at the intersection of a given row and column of this matrix represents the matrix element of µZ between the 4Δ basis set function labeling the row and the 6Σ+ basis set function labeling the column. The two transformation matrices U(J) and L (J) are defined in the text associated with (3.35). The matrix µ(J′, J″; M) contains matrix elements of µZ between final wave functions. An element lying at the intersection of a given row and column of this matrix represents the matrix element of µZ between the 4Δ final wave function labeling the corresponding row of U-1 (the corresponding column of U) and the 6Σ+ final wave function labeling the corresponding column of L. For bookkeeping purposes it is convenient to represent the element of (J′, J″; M) lying at the αth row and βth column as µ(J′, J″; M).

To determine the intensity IJ′, βJ″) of a transition between a rotational level αJ′ of the 4Δ state and a rotational level βJ″ of the 6Σ+ state, it is necessary to square the absolute value of the matrix element of µZ between pairs of final wave functions |αJ′ M′⟩ and |βJ″ M″⟩, and then to sum over all M′ and M″. Since nonvanishing matrix elements of µZ are diagonal in M, this double sum reduces immediately to a single sum. In the notation of this section we can write.

 (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′, JPrime;, M) matrix µb(J′, J″; M) is the same, and is given by the quantity hZ(J′, M; J″, M) of table 6. It follows from (3.41) that the M dependence of all elements of µ(J′, J″; M) is also given by hZ(J″, M; J″; J″) of (3.38) and therefore the matrix µ(J′; J″) have been defined in just such a way that

 (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 4Δ - 6Σ+ transition as functions of the nine unknown intensity parameters µi. The values of these parameters must be determined by fitting the calculated results to observed intensity data. For any particular transition, many of the µi may be negligibly small. For example, Klynning [21], who used essentially the procedure described in this section, obtained good qualitative agreement between the calculated and the observed rotational intensity distribution in a 4Σ - 2Π transition in SnH, even though he considered as nonzero only two of the five possible intensity parameters for such a transition.