1.9. Example: The Hill and Van Vleck Expression for 2Π States

The first example involves obtaining rotational energy expressions for a 2Π state of a diatomic molecule [10], which is well separated from other electronic states, and in which the spin-orbit splitting is small compared to such electronic separations. The rotational energy levels of such an electronic state correspond to Hund's case (a), Hund's case (b), or some intermediate case between these two.

The wave functions of the nonrotating molecule are represented by , where the quantum numbers Λ, S, Σ are all slightly bad. The quantum number Ω = Λ + Σ, however, is still good. The basis set functions for the complete (nonrotating plus rotating) problem are represented by . Since we are dealing with a 2Π state, Λ = ±  1 and S = ½. Since S = ½, Σ takes on the values ± ½. The quantum number M does not enter into the calculation of rotational energy levels for molecules unperturbed by the presence of external electric or magnetic fields. Hence, it will be ignored (even though the quantum number M will be carried along in the notation). The quantum number J will not be assigned a numerical value, since we are interested in the energy levels as a function of J. Basis set functions of interest for this problem thus have one of the following four forms:  | 1 ½ ½; J M ,  | 1 ½ - ½; ½ J M ,  | -1 ½ - ½; - J M , or  | -1 ½ ½; - ½ J M .

We must now set up the matrix of the Hamiltonian in the basis set given above. Let us represent a typical matrix element by the expression

 (1.16)

The complete Hamiltonian for a molecule in the absence of external fields is always diagonal in the quantum numbers J and M. Thus, for the problem under consideration, the matrix element given on the left of (1.16) vanishes unless J′ = J and M′ = M.

It can easily be seen, that if we restrict ourselves to matrix elements diagonal in J and M, and to the limited basis set appropriate for the rotational energy levels of a 2Π electronic state (Λ = ± 1, S = ½, Σ = ± ½), then the secular equation which must be solved to determine rotational energy levels is obtained from a Hamiltonian matrix having four rows and four columns. These rows and columns are labeled by the four basis set functions given just before eq (1.16). The elements of this 4 × 4 matrix are calculated as follows.

Consider first the matrix elements of . The quantum numbers of the basis set have been divided by a semicolon into two groups, reflecting the factorization of (1.12). Because of this factorization, matrix elements of must be diagonal in the three rotational quantum ΩJ M. Values for matrix elements of diagonal in these three quantum numbers can be obtained from eqs (1.7). If we label rows and columns of a 4 × 4 matrix by the four functions given just before eq (1.16), then we find that the matrix of has the form

 (1.17)

The parameter E represents the constant given in the third line of (1.7), and ± ½ A represents the spin splitting. We observe that only two distinct values for the energy occur, namely E + ½ A and E - ½ A. This is consistent with the general phenomenon that energy levels in the nonrotating molecule characterized by Ω ≠ 0 are doubly degenerate, the two degenerate functions having equal and opposite values of Ω (in this case Ω = ±  and Ω = ± ½). To be perfectly general we should allow the energy pattern to vary somewhat from that determined on the basis of simple spin-orbit interaction. However, in this case there is really only one relevant energy parameter in the nonrotating molecule, namely the distance between the Ω = ±  and Ω = ± ½ states. This one energy separation can be described completely by the single parameter A.

Matrix elements of the rotational Hamiltonian (1.11) in a basis set consisting of the four functions labeling the matrix (1.17) can be obtained as follows. The operators B(J2 - Jz2), B(S2 - Sz2) and -B(J+S- + J-S+) involve components of angular momenta for which the total magnitude quantum number and the z-axis projection quantum number characterize the basis set. Consequently, their matrix elements can be obtained immediately from (1.13). (footnote 4) Each of the operators + B(L+S- + L-S+) and - B(J+L- + J-L+) contains as a factor one of the quantities L± . The operators L±  have nonvanishing matrix elements only if the selection rule ΔΛ = ± 1 is satisfied. (This selection rule applies to L±  whenever Λ is a good quantum number, whether or not L is a good quantum number.) Since the limited basis set appropriate for the rotational energy levels of a 2Π state only contains functions characterized by Λ = ± 1, it is not possible to construct matrix elements satisfying the selection rules ΔΛ = ± 1 within this basis set. Thus, all matrix elements of L± , and hence of the two operators +B(L+S- + L-S+) and -B(J+L- + J-L+), vanish within this basis set. The remaining operator in (1.11), i.e., B(L2 - Lz2), is more difficult to deal with. From the statements in sect. 1.8, this operator has only diagonal matrix elements in the limited basis set under consideration. It can be shown (see chapter 2) that these four diagonal matrix elements are all equal. They are often represented by the symbol .

We now write down the matrix of analogous to the matrix of given in (1.17). When J ≥  it has the form

 (1.18)

The desired rotational energy levels are found by solving the secular equation obtained from the sum of the matrices for and . It can be seen that the sum of (1.17) and (1.18), i.e., + , contains two identical diagonal blocks. Individual diagonal blocks in a Hamiltonian matrix can always be diagonalized separately. Thus, we need only solve two secular equations of order two, rather than one secular equation of order four. Furthermore, these two secular equations are identical, so that the resultant energy levels occur in doubly degenerate pairs. The rotational energies obtained (twice each) from the secular equations are:

 (1.19)

(The algebraic operations necessary in obtaining (1.19) are simplified if one-half the trace is subtracted from each 2 × 2 matrix before diagonalizing it. This same quantity must then, of course, be added to the roots obtained from the secular equation.) Since the four basis set functions which label the rows and columns of (1.17) and (1.18) represent all the basis set functions for a given value of J, the two (doubly degenerate) energy levels given in (1.19) represent all the energy levels belonging to that value of J. The first two (J-independent) terms in (1.19) are often ignored in discussions of 2Π rotational energy levels.

When J = ½, the basis set functions labeling the first and third rows and columns of (1.17) and (1.18) do not exist. For this J value the Hamiltonian matrix factors into two identical 1 × 1 matrices, giving rise to a doubly degenerate energy level at E - ½ A + + B[J(J + 1) + ].

Limiting Hund's case (a) and case (b) behavior can be obtained by expanding the square roots in (1.19) as power series. Consider first case (a) behavior, which occurs, from a mathematical point of view, when | A | >> B J. If | A | >> B J, it is convenient to approximate the radical in (1.19) as follows.

 (1.20)

The approximation (1.20) to the radical must now be substituted into (1.19). Since we must both add and subtract (1.20), it is convenient at this time to replace | A | by A. The same two energy levels will be obtained by adding and subtracting | A | as by adding and subtracting A (though not necessarily in the same order). Keeping only terms through order B/A, and dropping the first two terms in (1.19), we obtain the two energy levels

 (1.21a) (1.21b)

which, apart from an extra + ½ B arising from , agree with the familiar [1] (p. 220) Hund's case (a) energy level expression B[J(J + l) - Ω2]. (footnote 5) If we retain also the terms of order B2/A2 in (1.20), we see that the coefficient of J(J + 1) in eqs (1.21a) and (1.21b), i.e., the effective B value, must be replaced by B(1 + B/A) and B(1 - B/A), respectively. This is also a well-known Hund's case (a) result [1] (p. 233).

Hund's case (b) occurs, from a mathematical point of view, when B J >> | A |. Pure Hund's case (b) occurs when A = 0. If A = 0, the last two terms of eqs (1.19) can be written

 (1.22a) (1.22b)

Both of these equations are of the form B[N(N + 1) - 1]. It is thus convenient to introduce an integer N, which is equal, to J + ½ for the higher energy level of given J, and equal to J - ½ for the lower energy level. Equations (1.22) then have the form of the familiar [1] (pp. 221-224) Hund's case (b) expression B[N(N + 1) - Λ2].

The significance of the quantum number N, which has arisen here in a somewhat formal way, can best be understood by examining the eigenfunctions of the matrix sum (1.17) + (1.18) when A = 0, i.e., the case (b) eigenfunctions. The two eigenfunctions obtained by diagonalizing the 2 ×  submatrix in the upper lefthand corner of (1.17) + (1.18) when A = 0 are given on the right-hand side of (1.23) as linear combinations of the basis set functions used to label the rows and columns of (1.17) and (1.18).

 (1.23)

It can be seen that the two basis set functions in a given linear combination are characterized by the same values of Λ, S, J, and M, but by different values of Σ and Ω. For this reason we say that Λ, S, J, and M are good quantum numbers in Hund's case (b), but that Σ and Ω are not.

Although the linear combinations of functions given on the right-hand side of (1.23) are not eigenfunctions of Sz and Jz, they do happen to be eigenfunctions of the operator (J - S)2, belonging to an eigenvalue 2 N(N + l). This can be demonstrated relatively easily. Since the functions are eigenfunctions of J2, Jz, S2, Sz, the expressions (1.13) and (1.14) can be used to determine the effect of (J - S)2 on these basis set functions and hence on any linear combination of them. Because the case (b) wave functions (1.23) belong to the eigenvalue 2 N(N + l) of the operator (J - S)2, we say that N is a good quantum number in case (b).

It is sometimes convenient to label the eigenfunctions represented by the linear combinations on the right-hand side of (1.23) by a single label of the form . These labels are found on the lefthand side of (1.23). The absence of a semicolon indicates that the functions cannot be factored into a nonrotating-molecule part and a rotating-molecule part.

Some authors also find it convenient to use functions of the form as basis set functions for diatomic molecule calculations. In this monograph we have chosen always to consider the nonrotating-molecule part of the problem separately from the rotating-molecule part. This decision requires us to use a basis set in which Ω and Σ, or perhaps just Ω, are defined, and in which N is not defined. Wave functions in which N is defined will therefore appear to arise here somewhat arbitrarily as linear combinations of the basis set functions, the linear combinations being obtained by diagonalizing appropriate Hamiltonian matrices.