The Theory of Rigid Rotors

In document Rotational and vibronic effects in molecular electronic spectra (Page 32-35)

1.6 THE ROTATIONAL ANALYSIS 1 Introduction

1.6.2 The Theory of Rigid Rotors

The energy of a rotating molecule can be attributed, to a good

f

approximation, to separate electronic, vibrational and rotational motions. The rotational energy levels are then approximately described by the eigenvalues of a rigid-rotor Hamiltonian. Any interactions between the motions may subsequently be treated as small perturbations coupling the rotational to the vibrational and electronic motion. The theory of the rotational energy levels of both rigid and nonrigid rotors is well known and is summarised in standard texts [e.g. Allen and Cross 1963; Townes and Schawlow 1955]. Only sufficient of the theory will be presented here to facilitate discussion in later sections.

Molecules are classified with reference to their principal moments of inertia, conventionally labelled with 1^< 1^ < I . From these, rotational constants A, B and C are defined in units of cm-1 by

A = h/87T2cI , and similar expressions for B and C. Symmetric top rotors a

have two moments of inertia of equal size, and are prolate if the unique axis is the A axis and oblate if it is C. Asymmetric rotors have all three rotational constants unequal, but are described in terms of the

t

symmetric rotor which they more closely resemble. The customary measure of the asymmetry of a rotor is

k = (2B- A - C)/(A~ C)

which has values running from -1 in the prolate limit to +1 in the

oblate limit. The energy levels of both types of rigid symmetric rotors may be obtained exactly and are described in terms of two quantum numbers J and K:

E(J,K ) - BJ(J-M) + ( A - B ) K 2

a a (prolate) (l.i)

E(J,Kc) = BJ(J + 1) - ( B - C ) K 2 (oblate) (1.2)

J quantizes the total angular momentum and the K Ts (=0, ±1, ±2, ..., ±J) quantize the component of J along the unique (figure) axis. Each level is doubly degenerate in K except for K = 0 . Selection rules for

transitions between symmetric top energy levels distinguish between (a) transitions in which the electric moment oscillates along the top axis (parallel bands) and (b) those in which it oscillates at right angles to the top axis (perpendicular bands), and are summarised below

AJ = 0, AJ - ±1 ±1, j AK = 0 AK = 0 parallel AJ = 0, ±1, AK = ±1 perpendicular

Asymmetric rotor wavefunctions may be defined in a basis of symmetric rotor wavefunctions. The energy matrix of the asymmetric top is still diagonal in J but loses the K degeneracy of the symmetric top

(asymmetry splitting). States are labelled by J (K ,K ), using the K

cl C »

levels of both the prolate and oblate top states with which they correlate in the symmetric top limits. The energy levels are then expressed in terms of the asymmetry parameter k by

E(JKa Kc) = >,(A + C)J(J + 1) +«,(A-C)E'(J 8«) . (1.3) The reduced rotor energies E' (J^ ;k) are the energies of an

l\c l y C

asymmetric rotor with inertial constants 1, k and -1 respectively. The

This matrix has a simple tricliagonal form since the only nonzero entries are of the form (K,K) and (K,K±2). This matrix may be immediately

decomposed into two submatrices E and 0. E involves only the elements with even K, and 0 contains only those with odd K. A further symmetry property allows the E and 0 matrices to be split further, yielding, in all, four submatrices which may be labelled according to the four

irreducible representations of the group D2. Fig. 1.6 shows the values of the energies obtained by diagonalising these submatrices for the case J = 40 over the full range of asymmetry k. The concept of asymmetry in the calculated levels may be best understood by examining the line AB of Fig. 1.6. The intersections of the curves with this line represent the energy levels of a prolate asymmetric top with k = -0.6. The lowest energy level for this top is degenerate (or very nearly so) in K , the

oblate top quantum number. The two levels may be correlated with the K = 0 and K = 1 levels of the prolate top. They are thus labelled

a a

40 (0,40) and 40 (1,40). This degeneracy remains as we progress along AB until we reach the levels in the vicinity of the 40 (10,30) and

40 (11,30) region. For the next few levels the splitting becomes more and more apparent. By the time the levels 40 (14,26) is reached it now completely split from its former degenerate partner 40 (15,26), which is to higher energy, and has now become degenerate with the level

40 (14,25). These two levels are thus degenerate in the prolate top

quantum number K . Thus the region above the diagonal from lower left to a

upper right is called the prolate top domain and that below is the oblate domain. The diagonal is the region of asymmetry.

Much of the subtlety in the spectra of, for example, a prolate asymmetric rotor occurs because of transitions between the lower levels of the top which are degenerate in the oblate top domain. No regular fine structure yet observed in a large molecule can be conclusively assigned to transitions between levels in the asymmetric region of this

f

curve.

The selection rules applying to J for an asymmetric rotor are the same as those for the symmetric rotor, since the energy matrices are still diagonal in J. Those for K are relaxed. They are restricted merely to even or odd changes. The transitions with A K = 0 , ± 1 are still, however, by far the most intense for asymmetric tops. The strict

B

In document Rotational and vibronic effects in molecular electronic spectra (Page 32-35)