Molecule Types Symmetric Tops <Prev Next>

Symmetric Top State

This allows setting rotational constants and other values specific to each state in the molecule.


Colour Colour - set to "None" to take value from elsewhere as explained in Determining Colours and J ranges.
RveSelect Only include given rovibronic symmetry - for normal use set to 'all'
S Electron Spin
Symmetry Symmetry
Kmin Minimum |K| to use; "all" (default) for no lower limit.
Kmax Maximum |K| to use; "all" (default) for no upper limit.


The Hamiltonian used is (for prolate tops read A for C):

+ BN(N+1) + (C-B)K2
+ (-2Cζ + ηJN(N+1) + ηKK2) lK
- DJN2(N+1)2 - DJKN(N+1)K2 - DKK4
+ HJN3(N+1)3 + HJKN2(N+1)2K2 + HKJN(N+1)K4 + HKK6
+ LJN4(N+1)4 + LJJKN3(N+1)3K2 + LJKN2(N+1)2K4 + LKKJN(N+1)K6 + LKK8
+ ½εbb(N+S-+N-S+) +  εccNzSz
+ DsNN2N.S + ½DsNK(N2NzSz+NzSzN2) + DsKNN.SNz2
+ α(3Sz2-S2) + ½β(S+2+S-2)
+ aefflSz

with the following off-diagonal matrix elements, which are responsible for l-doubling::

<N, K+2, l+1|H|N, K, l-1> = ½ (q+ + DqJN(N+1) + DqK(K2+(K+2)2)) [ (N(N+1) - K(K-1)) (N(N+1) - K(K+1)) ]½
<N, K+2, l-1|H|N, K, l+1> = ½ q- [ (N(N+1) - K(K-1)) (N(N+1) - K(K+1)) ]½
<N, K+1, l-1|H|N, K, l+1> = (r + DrJN(N+1) + DrK(K2+(K+1)2)) (2K+1) (N(N+1) - K(K+1))½

A note on the sign convention for these constants is appropriate. This is discussed in a G. J.Cartwright and I. M. Mills, J. Molec. Spectrosc.34, 415 (1970), though that they define their constants, q(+) and q(-),  in a way to factor out the expected vibrational dependence, specifically:

q+ = ½ρ[(vt+1)2 - lt2]½q(+)
- = ½ρ[(vt+1)2 - lt2]½q(-)

The constant ρ defines the sign convention, and Cartwright and Mills suggest a value of -1 should be used for this. The sign convention can also be defined by looking at the spitting of the l = K = ±1 levels. Note that equation (5) of this paper is slightly unclear; the magnitude of the spitting between these otherwise degenerate levels is q+N(N+1) with the PGOPHER definition of q rather then the Cartwright and Mills definition. The PGOPHER definition means a positive value of q+ puts the even J, K=1 A1 rotational level below the A2 level, whereas a positive value of the Cartwright and Mills constant, q(+), gives the opposite order.

l doubling is also observed in degenerate electronic states of symmetric top molecules; while the physical origin is different, the J and K dependence of the matrix elements is the same so PGOPHER can be used for such cases also. The classic example is the B state of NH3 (see for example M. N. R. Ashfold, R. N. Dixon, N. Little, R. J. Stickland and C. M. Western, J. Chem. Phys., 89, 1754 (1988)) with the published q values being simply -q+ as used by PGOPHER.

Origin State Origin.
Width Linewidth (rotation independent) for state; see Width and Lifetime effects
B Rotational constants perpendicular to symmetry axis.
C A or C - the rotational constant about symmetry axis.
DJ J2(J+1)2 Quartic Centrifugal Distortion
DJK J(J+1)K2 Quartic Centrifugal Distortion
DK K4 Quartic Centrifugal Distortion
zeta=ζ Coriolis coupling constant.
etaJ=ηJ J(J+1) dependence of Coriolis coupling constant.
etaK=ηK K2 dependence of Coriolis coupling constant.
qplus=q+ l doubling constant.
qminus=q- l doubling constant.
r l doubling constant.
DqJ Centrifugal distortion of qplus l doubling constant.
DqK Centrifugal distortion of qplus l doubling constant.
DrJ Centrifugal distortion of r l doubling constant.
DrK Centrifugal distortion of r l doubling constant.
HJ J3(J+1)3 Sextic Centrifugal Distortion
HJK J2(J+1)2K2 Sextic Centrifugal Distortion
HKJ J(J+1)K4 Sextic Centrifugal Distortion
HK K6 Sextic Centrifugal Distortion
LJ J4(J+1)4 Octic Centrifugal Distortion
LJJK J3(J+1)3K2 Octic Centrifugal Distortion
LJK J2(J+1)2K4 Octic Centrifugal Distortion
LKKJ J(J+1)K6 Octic Centrifugal Distortion
LK K8 Octic Centrifugal Distortion
ebb xx/yy Spin-Rotation interaction
ecc zz Spin-Rotation interaction
DsN quartic spin rotation parameter
DsNK quartic spin rotation parameter
DsKN quartic spin rotation parameter
DsK quartic spin rotation parameter
alpha Diagonal spin-spin coupling constant (=D/3)
beta Off-Diagonal spin-spin coupling constant (=E)
aeff Effective spin-orbit parameter
wK multiplier of <K2> for linewidth; see Width and Lifetime effects
wJ multiplier of <J(J+1)-K2> for linewidth; see Width and Lifetime effects
After 7.1.323 the definition of the spin-spin parameters alpha and beta changed to be consistent with the literature. Values in older files are converted on loading by multiplying alpha by sqrt(5) and beta by sqrt(6).