Molecule Types Symmetric Tops | <Prev Next> |

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*):

Origin

+ BN(N+1) + (C-B)K^{2}

+ (-2Cζ + η_{J}*N*(*N*+1)
+ η_{K}K^{2}) lK^{
}- D_{J}N^{2}(N+1)^{2} - D_{JK}N(N+1)K^{2} - D_{K}K^{4}

+ H_{J}N^{3}(N+1)^{3} + H_{JK}N^{2}(N+1)^{2}K^{2} + H_{KJ}N(N+1)K^{4}
+ H_{K}K^{6}

+ L_{J}N^{4}(N+1)^{4} + L_{JJK}N^{3}(N+1)^{3}K^{2} + L_{JK}N^{2}(N+1)^{2}K^{4} + L_{KKJ}N(N+1)K^{6}
+ L_{K}K^{8}

+ ½ε_{bb}(N_{+}S_{-}+N_{-}S_{+}) +
ε_{cc}N_{z}S_{z}

+*D*^{s}_{N}N^{2}N.S + ½*D*^{s}_{NK}(N^{2}N_{z}S_{z}+N_{z}S_{z}N^{2}) +* **D*^{s}_{KN}N.**S**N_{z}^{2}

+ α(3*S*_{z}^{2}-**S**^{2})
+ ½β(*S*_{+}^{2}+*S*_{-}^{2})

+*a*_{eff}lS_{z}

+ BN(N+1) + (C-B)K

+ (-2Cζ + η

+ H

+ L

+ ½ε

+

+ α(3

+

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_{+}
+ D_{qJ}N(N+1) + D_{qK}(K^{2}+(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 + D_{rJ}N(N+1) + D_{rK}(K^{2}+(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[(v_{t}+1)^{2}-l_{t}^{2}]^{½}q^{(+)}

q_{-}= ½ρ[(v_{t}+1)^{2}-l_{t}^{2}]^{½}q^{(-)}

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 A_{1} rotational level below the A_{2}
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 NH_{3}
(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 | J^{2}(J+1)^{2}
Quartic Centrifugal Distortion |

DJK | J(J+1)K^{2} Quartic
Centrifugal Distortion |

DK | K^{4}
Quartic Centrifugal Distortion |

zeta=ζ | Coriolis coupling constant. |

etaJ=η_{J} |
J(J+1) dependence of Coriolis coupling constant. |

etaK=η_{K} |
_{}K^{2}
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 |
J^{3}(J+1)^{3} Sextic
Centrifugal Distortion |

HJK |
J^{2}(J+1)^{2}K^{2} Sextic
Centrifugal Distortion |

HKJ |
J(J+1)K^{4} Sextic
Centrifugal Distortion |

HK |
K^{6}
Sextic Centrifugal Distortion |

LJ |
J^{4}(J+1)^{4} Octic
Centrifugal Distortion |

LJJK |
J^{3}(J+1)^{3}K^{2} Octic
Centrifugal Distortion |

LJK |
J^{2}(J+1)^{2}K^{4} Octic
Centrifugal Distortion |

LKKJ |
J(J+1)K^{6} Octic
Centrifugal Distortion |

LK |
K^{8}
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 <K^{2}>
for
linewidth;
see
Width and Lifetime effects |

wJ |
multiplier of <J(J+1)-K^{2}> for linewidth; see Width and Lifetime effects |