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 E_Σ 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 σ_υ (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 (E_Σ ≅ 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 (|E_Σ| >> 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 J_c of the rotational quantum number [i.e., if E_Σ + B_ΣJ_c(J_c + 1) ≅ B_ΠJ_c(J_c + 1) for some J_c], 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 J_c. The rotational levels of the ¹Σ⁺ state having J ~ J_c and the rotational levels of one component of the ¹Π state having J ~ J_c will suffer small equal and opposite displacements, corresponding to what might be called the usual heterogeneous perturbation.