## 4.3. Example: 1Π-1Σ+ Heterogeneous Perturbation

Since the two nonrotating-molecule wave functions of a 1Π state are characterized by Ω = ±1, while the nonrotating-molecule wave function of a 1Σ+ state is characterized by Ω = 0+, there can be no interaction between these states in the nonrotating molecule. A homogeneous perturbation of one of these states by the other is not possible.

However, the rotating-molecule operator of eq (1.10) has in general nonvanishing matrix elements between states with ΔΛ = ± 1 and ΔS = 0. Consequently, a heterogeneous perturbation of one of these states by the other is possible. This heterogeneous perturbation can be treated theoretically as follows.

Consider a 3 × 3 Hamiltonian matrix with rows and columns labeled by the three interacting wave functions  = |1 0 0; 1 J M⟩, | -1 0 0; -1 J M⟩ and | 0+ 0 0; 0 J M⟩. We find, as outlined below, that this matrix has the form

 (4.1)

where the upper left-hand 2 × 2 diagonal block represents the 1Π state by itself (arbitrarily placed at the energy origin in the nonrotating molecule), where the lower right-hand 1 × 1 diagonal block represents the 1Σ+ state by itself (located at energy EΣ in the nonrotating molecule), and where the off-diagonal elements represent the heterogeneous perturbation (Coriolis interaction) between the two states. Matrix elements for the 2 × 2 and 1 × 1 diagonal blocks were obtained as described in chapter  1. Off-diagonal matrix elements were obtained as follows.

The first line of terms in when is written as in eq (1.11), contains operators which give rise to nonvanishing matrix elements only if ΔΩ = 0. Consequently, the first line of (1.11) cannot be responsible for an interaction between the 1Π and the 1Σ+ states. The first and third terms of the second line of (1.11) give rise to nonvanishing matrix elements only if ΔΣ = ± 1. Consequently, these two terms can also not be responsible for the interaction under consideration here. The second term of the second line of (1.11) gives rise to nonvanishing matrix elements if ΔΛ = ± 1 and ΔS = ΔΣ = 0. This term thus does connect the 1Π and 1Σ+ states under consideration.

The second term of the second line of (1.11) contains four operators: J+J-L+L-. Matrix elements of J+ and J- can be obtained from expressions of the form (1.13) and (1.14), since both J and Ω are good quantum numbers in the basis set. Matrix elements of L+ and L- cannot be found from such expressions, since L is not a good quantum number. However, matrix elements of L± still obey the selection rule ΔΛ = ± 1, because Λ is a good quantum number. It can be shown by using symmetry arguments involving συ (see chap. 2), that

 (4.2)

where the + or - sign on the right-hand side of (4.2) allows for various possible sign choices and values of L in applying the transformation equation (2.11a) to the 1Π state. Neither sign choice in (4.2) leads to inconsistencies elsewhere in the calculation, since no matrix elements between the two components of the 1Π state are introduced. The relative phases of the electronic orbital wave functions |Λ = +1⟩ and |Λ = -1⟩ are fixed once and only once by the choice of sign in (4.2). The + sign was chosen in writing (4.1), where the quantity B⟨Π| L+ |Σ⟩, given by

 (4.3)

is considered to be an unknown adjustable (real) parameter. Energy levels can be obtained by diagonalizing (4.1).

Actually, (4.1) can be factored by symmetry into a 2 × 2 diagonal block and a 1 × 1 diagonal block, by using the functions 2-1/2[ | +1 0 0; +1 J M⟩ ± | -1 0 0; -1 J M⟩] as a basis set for the 1Π state, but this factorization will not be performed here.

Van Vleck [25] has introduced the phrase "pure precession" in connection with heterogeneous perturbations. From a quantum mechanical point of view, pure precession is said to occur whenever L is a good quantum number. If L is a good quantum number, it is possible to use (1.13) to evaluate the quantity ⟨Π|  L+  |Σ⟩ occurring in (4.1). Since this quantity is really a matrix element of the form ⟨Λ = + 1| L+  |Λ = 0⟩ we find

 (4.4)

The value [L(L + 1)]1/2 occurs when the Σ and Π states belong to the same L complex, i.e., when the Σ and Π states represent different projections of the same L along the internuclear axis. The value 0 occurs when the Σ and Π states belong to different L complexes.

It is interesting to note that the matrix (4.1) describes several apparently rather different situations. If the 1Σ+ state and the 1Π state are very nearly degenerate in the nonrotating molecule (EΣ ≅ 0), and if L is an approximately good quantum number equal to unity (L = 1), then the rotational energy levels which result from diagonalizing (4.1) correspond to those of a case (d) p-complex. (Pure case (d) occurs, from a quantum mechanical point of view, when L is a perfectly good quantum number, and when all 2L + 1 states corresponding to different projections of this L along the internuclear axis are exactly degenerate in the nonrotating molecule.) If the 1Σ+ state and the 1Π state are separated by a distance large compared to the distance between neighboring rotational levels (|EΣ| >> B J), then the rotational energy levels which result from diagonalizing (4.1) correspond to those of a normal 1Σ+ state and those of a normal 1Π state with some Λ-doubling. If the rotational energy levels of the 1Σ+ state cross those of the 1Π state for some value Jc of the rotational quantum number [i.e., if EΣ + BΣJc(Jc + 1) ≅ BΠJc(Jc + 1) for some Jc], and if the interaction between the 1Σ+ state and the 1Π state is very small ( | ⟨ Π | L+ | Σ ⟩ | << ), then the rotational energy levels resulting from diagonalizing (4.1) will correspond to those of a normal 1Σ+ state and those of a normal 1Π state, except for J values in the neighborhood of Jc. The rotational levels of the 1Σ+ state having J ~ Jc and the rotational levels of one component of the 1Π state having J ~ Jc will suffer small equal and opposite displacements, corresponding to what might be called the usual heterogeneous perturbation.