Neutron Scattering

m
Eu
W
Figure 17 .7 : Eigenenergies of a tunnelling methyl groups . The potential V3=0 represents the
free rotor with quantum number J. Strong V3 approaches a harmonie oscillator model - quantum
number n - with equidistant librational modes (not yet reached at V3 = 257ncV ) .
For a purely 3-fold potential one obtains the (2M+1) dimensional Hamilton matrix
9 . . . + 23 0 0
23
0 0 0
0 . . .
4 + 0 0 0 0
0 . . .
0 1 + 23 0 0
23
0
0 H= . . .
0 0 0
V3
= 4 2 4
. . . 0
23
0 0 1 + 23 0 0
. . . 0 0
23
0 0 4+2 3 0
0 . . .
0 0
23
0 0 9 + 23
Such band matrices are easily diagonalised by standard programs . The resulting eigenenergies
represent librations split by the tunnel effect.
With increasing librational quantum number the tunnel splitting increases due to the increasing
overlap of wavefunctions in excited states . Fig .17 .7 shows the eigenenergies as a
fonction of increasing strength V3 of the hindering potential. One recognizes a hoge isotope
effect with deuteration (BD = ~2I) due a doubling of the scaled potential V' (17 .41) .
For zero potential the Hamilton matrix is already diagonal and the eigenvalues are those of
the free rotor J 2B .
17- 1 6