Molecule Types Symmetric Tops

# Symmetric Top Perturbation

There is one of these for each perturbation parameter in each state. There are many possible perturbation types, as described below. To see the actual matrix elements used in any particular case, right click on the perturbation and select "Matrix Elements". The resulting expressions will be displayed in the log window, and should be understood as being multiplied by the Value parameter.

Note that if degenerate states are involved, careful selection of the lPower and lChange values will be required. For example, the common A1 - E Coriolis interaction requires pPower = 1, lPower = -1 and lChange = 1. If the combination of values are set wrong, the resulting matrix element will typically be zero, or give an error message about a symmetry mismatch.

## 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; the expressions and operators given here should be understood to be multiplied by this.

## 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. The scale factor indicates the factor the standard parameter must be multiplied by to use the given perturbation operator form.

 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 Hamiltonian expressed as perturbations

The following terms are also required if the electron spin is non-zero.
 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