1.10. Example: The Schlapp Expression for ³Σ States

The calculation of rotational energy levels for a ³Σ state [11] proceeds in much the same fashion as for ²Π states. The basis set functions for the complete problem are represented by . Since we are dealing with a ³Σ state, Λ = 0 and S = 1. Since S = 1, Σ takes on the values + 1, 0, - 1. (footnote 6) Note that, even though the spin projection quantum number Σ will not be a good quantum number in the final ³Σ wave functions, it is perfectly acceptable, and indeed, from the point of view of the author, desirable, to use a basis set in which Σ is a good quantum number. Basis set functions for the problem thus have one of the following forms: | 0 1 1; 1 J M ⟩, | 0 1 0; 0 J M ⟩, or | 0 1 - 1; - 1 J M ⟩. For J = 0 only the second function exists, of course.

The matrix elements of the Hamiltonian in this basis set can again be represented by (1.16). Since nonvanishing matrix elements of are again diagonal in J and M, the matrix used to determine rotational energy levels for a ³Σ electronic state (Λ = 0, S = 1, Σ = ± 1, 0) has three rows and columns, labeled by the basis set functions given at the end of the preceeding paragraph.

Consider now the matrix elements of . If we employ the same reasoning used in obtaining the energies given in (1.17), we find that all three components of the nonrotating-molecule ³Σ state lie at the same energy. (There is no first order spin-orbit interaction AΛΣ in Σ electronic states because Λ = 0.) Nevertheless, states with different values of Ω do have different energies [12], except that pairs of states related to each other by a change in sign of both Λ and Σ are degenerate. From these considerations, we find that the matrix of is given by

(1.24)

The two states with Ω = ± 1 have been given the energy E and the state with Ω = 0 has been given the energy E - 2λ. Actually, in this problem, as in the ²Π problem, there is only one relevant energy separation. That separation is represented here by the quantity 2λ. (Experimental values for λ in the ³Σ ground state of several molecules are given in table VI of [12]).

Matrix elements of the rotational Hamiltonian (1.11) can be obtained from considerations identical to those for a ²Π state. The matrix of for J ≥ 1 is thus

(1.25)

where we have again used arguments (see chapter 2) to conclude that the matrix elements of B(L² - L_z²) in the basis set under consideration are all equal.

Unfortunately the sum of (1.24) and (1.25) does not immediately factor into any smaller diagonal blocks, so we must apparently find the roots of a cubic secular equation. However, there is one simplifying procedure which has not yet been employed. For reasons of symmetry, the size of the secular determinant for rotational energy levels can often be halved by working with the basis set functions 2^-½[ ± | -ΛS -Σ; -Ω J M ⟩] rather than with the individual functions themselves. In this connection we note that

(1.26)

Let us consider the matrices of and in a new basis set consisting of 2^-½[ | 0 1 1; 1 J M ⟩ + | 0 1 -1; -1 J M ⟩], | 0 1 0; 0 J M ⟩, and 2^-½[ | 0 1 1; 1 J M ⟩ - | 0 1 -1; -1 J M ⟩]. If the three rows and columns of the Hamiltonian matrix are labeled by these functions, we find that the matrix of is identical to that given in (1.24), but that the matrix of has the form

(1.27)

The sum of the matrices (1.24) and (1.27), i.e., + , now factors into a 2 × 2 diagonal block and a 1 × 1 diagonal block.

Before actually determining the rotational energy levels, we consider one further change in the Hamiltonian matrix. As mentioned above, the matrix of contains a single important adjustable parameter, corresponding to the single energy difference present in the nonrotating-molecule problem. On the other hand, the matrix elements of should contain more adjustable parameters than the one, i.e., B, which actually occurs in (1.25). Strictly speaking, for example, there should be three different values for B in (1.25). Because the internuclear distance will be slightly different for the states with Ω = ± 1 than it is for the state with Ω = 0, one value for B occurs in the matrix positions (1, 1) and (3, 3), one in the position (2, 2), and one in the off-diagonal positions (1, 2), (2, 1), (2, 3), and (3, 2). (The fact that differences in internuclear distance lead to three and only three values for B in (1.25) can be shown from symmetry considerations, as indicated in chapter 2.) In addition to the internuclear distance effect, all matrix elements of operators involving S, which were evaluated numerically from (1.13), using S = 1 and Σ = + 1, - 1, or 0, are slightly wrong, since S and Σ are not quite good quantum numbers. Since these matrix elements were evaluated numerically, they are somewhat difficult to find in (1.25), but the 2^½ occuring in the off-diagonal matrix elements, for example, is not exactly 2^½, because S and Σ, are not exactly good quantum numbers. Finally, has one value for the state with Ω = 0, and a slightly different value for the states with Ω = ± 1. Unfortunately, the extent of the various deviations mentioned above can only be quantitatively determined at the present time from experiment. For this reason it is common to introduce, in some way, additional parameters in (1.25), which are to be determined by fitting the experimental data. Since B is the only parameter occurring in (1.25), and since each matrix element contains B, it is possible to allow for all of the above-mentioned problems in a purely formal way by replacing the single parameter B by a set of B_eff's. It can be shown by symmetry arguments (see chapter 2) that a maximum of seven B_eff's could be used: three of which would occur in the positions (1, 1) and (3, 3); three of which would occur in the position (2, 2); and one of which would occur in the off-diagonal positions. [A single B_eff in the off-diagonal positions of (1.25) corresponds to a single B_eff in the off-diagonal positions of (1.27).]

We shall use in our calculations one value for B on the diagonal in (1.25) or (1.27), and one value for B, written as B - ½γ, off the diagonal. Such a decision leads ultimately to Schlapp's expression [11], and can be justified as follows. When 2λ in (1.24) is very small compared to electronic energies (less than 30 cm^-1, say), then the difference in equilibrium internuclear distance for the two multiplet components of a ³Σ state is expected to be very small, and only one coefficient for J(J + 1) is needed on the diagonal of (1.25). On the other hand, contamination of the ³Σ state by other electronic states through spin-orbit interaction will cause the matrix elements of the components of S to deviate perceptibly from those given in (1.13). On the diagonal, such deviations, as well as the differences in , can be allowed for by adjusting the parameter λ. Off the diagonal, such deviations can be allowed for by adjusting the parameter B.

In any case, the secular equation obtained from the sum of (1.24) and (1.27) modified as described above leads to the following energies for the three states of given J ≥ 1:

(1.28)

For J = 0 the energy is E - 2λ + B J(J + l) + 2B + . The last line of (1.28) can be rearranged to give Schlapp's expressions by expanding the square root as follows.

(1.29)

(The first approximate equality in (1.29) indicates the loss of terms in γ². The second approximate equality indicates a power series expansion of the radical. The third approximate equality represents the case (b) approximation B J >> | λ | for the small γ-dependent term.) Dropping the first two (J-independent) terms in (1.28) and making the substitutions J = N, J = N - 1, and J = N + 1 in the three energy expressions given in (1.28) and in (1.29), we obtain Schlapp's expressions

(1.30)

apart from the small J-independent term + ½γ. (N is assigned to the three levels of given J such that the level with N = J + 1 has the highest energy and the level with N = J - 1 has the lowest.

It is interesting to note that the parameter γ, which was introduced here to allow for some slight discrepancies in matrix elements of the spin operators, is generally introduced as a coupling parameter between the vectors N and S [11]. For ²Σ and ³Σ states, there is only one such parameter γ, and the two view points lead to identical results. For ⁴Σ states, however, the present approach leads in a natural way to the introduction of two parameters γ [13], whereas the vector coupling approach appears incorrectly still to require only one.