Molecule Types Symmetric Tops <Prev Next>

Symmetric Top Perturbation

Settings

Nucleus Index, starting from 1, of nuclear spin involved in perturbation; 0 (default) for those not involving a nuclear spin.
SymSelect Symmetry select
ScalePrev Scale factor with respect to preceding perturbation
JPower Twice power of N2 = N(N+1) (= J(J+1) for closed shell systems with S = 0).
zPower Power of Nz; If both zPower and pPower are non zero then the effective operator is either:
NJPower/2[Nz|zPower|, N+|pPower|+N-|pPower|]+
or NJPower/2[Nz|zPower|-1,[Nz, N+|pPower|+N-|pPower|]+]+ if zPower is odd and < -2
pPower Power of N±
lPower Select l dependence; valid values are 0 (l independent), 1 or -1. If lChange is 0, the matrix element is multiplied by l*lPower, otherwise the matrix element is multiplied by sign(l'-l)*lPower.
lChange Power of l+ or l-; < 0 implies opposite sign to change in K; valid values are -2..2
KSelect K the perturbation applies to
SPower Twice power of S2 = S(S+1)
SzPower Power of Sz. If this is combined with a rotational operator, the overall operator is:
[NJPower/2[Nz|zPower|, N+|pPower|+N-|pPower|]+,SzSzPower]+
SpPower Power of S±. If this is combined with a rotational operator, the overall operator is (if pPower and SpPower have the same sign):
[NJPower/2[Nz|zPower|,N+|pPower|]+, S+|SpPower|]+ + [NJPower/2[Nz|zPower|,N-|pPower|]+, S-|SpPower|]+
If pPower and SpPower have opposite signs:
[NJPower/2[Nz|zPower|,N+|pPower|]+, S-|SpPower|]+ + [NJPower/2[Nz|zPower|,N-|pPower|]+, S+|SpPower|]+
NSPower Power of N.S.

Currently only one of SzPower and SpPower can be non-zero.

Parameters

Value Size of perturbation.

Rotational Hamiltonian expressed as perturbations

The standard rotational Hamiltonian can be expressed entirely in terms of diagonal perturbations, as shown in the table below. This can be used as a basis for adding higher powers of the centrifugal distortion or other terms not provided as standard.


 Operator
 JPower
 zPower
 pPower
 lPower
 lChange
 Scale
Factor
 Notes
B N2 2
 0 0
 0 0 1
 Both terms required to replicate B[J(J+1) - K2]
Nz2 0
 2
 0
 0
 0
 -1
C Nz2 0 2 0 0
 0 1
DJ N4 ≡ N2(N+1)2 4 0 0 0 0 -1
DJK N2Nz2 ≡ N(N+1)K2 2 2
 0 0 0 -1
DK Nz4≡ K4 0 4
 0 0 0 -1
zeta
 Nzlz 0
 1
 0
 1
 0
 -2C
 In this form, parameter must be -2Cζ, not -2ζ
etaJ
 N2Nzlz 2
 1
 0
 1
 0
 1
etaK
 Nz3lz 0
 3
 0
 1
 0
 1
qplus
 N+2l+2+N-2l-2 0
 0
 2
 0
 2
 ½
qminus
 N+2l-2+N-2l+2 0
 0
 2
 0
 -2
 ½
r
 [N+l-2+N-l+2,Nz]+ 0
 1
 1
 0
 -2
 1
DqJ
 N2[N+2l+2+N-2l-2] 2
 0
 2
 0
 2
 ½
DqK
 [N+2l+2+N-2l-2,Nz2]+ 0
 2
 2
 0
 2
 ½
DrJ
 N2[N+l-2+N-l+2,Nz]+ 0
 1
 1
 0
 -2
 1
DrK
 [[N+l-2+N-l+2,Nz]+,Nz2]+ 0
 -3
 1
 0
 -2
 1
 zPower <= -3 flags use of two anticommutators
HJ N6 6 0 0 0 0 1
HJK N4Nz2 4 2
 0 0 0 1
HKJ N2Nz4 2 4
 0 0 0 1
HK Nz6 0 6
 0 0 0 1
LJ N8 8 0 0 0 0 1
LJJK N6Nz2 6 0 0 0 0 1
LJK N4Nz4 4 0 0 0 0 1
LKKJ N2Nz6 2 0 0 0 0 1
LK Nz8 0 0 0 0 0 1

Spin-Rotational Hamiltonian expressed as perturbations

The spin rotation terms can be expressed as perturbations.

 Operator
 JPower
 		zPower
 		pPower
 		lPower
 		lChange SzPower
 		SpPower
 		SPower
 NSPower
 Scale
Factor
ebb N+S- + S+N-+ N-S+ + S-N+
= 2(N+S- + N-S+)
= 8(NxSx+NySy)		 0
 0
 1
 0
 0
 0
 -1
 0
 0
 1/4
ecc
NzSz + SzNz
=2NzSz		 0
 1
 0
 0
 0
 1
 0
 0
 0
 ½
DsN N2N.S 2
 0
 0
 0
 0
 0
 0
 0
 1
 1
DsNK
 		[N2, NzSz]+ 2
 1
 0
 0
 0
 1
 0
 0
 0
 ½
DsKN
 		N.S Nz2 0
 2
 0
 0
 0
 1
 0
 0
 1
 1
DsK
 		[Nz3, Sz]+ = 2Nz3Sz
0
 3
 0
 0
 0
 1
 0
 0
 0
 ½
alpha
 Sz2 0
 0
 0
 0
 0
 2
 0
 0
 0
 3
S2 0
 0
 0
 0
 0
 0
 0
 2
 0
 -1
beta
 S+2-S-2 0
 0
 0
 0
 0
 0
 2
 0
 0
 ½
aeff
 		lzSz 0
 0
 0
 1
 0
 1
 0
 0
 0
 1