# Asymmetric Top Perturbation

There will be 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.

## Settings

 Nucleus Index, starting from 1, of the 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 N(N+1) aPower Power of Na bPower Power of Nb cPower Power of Nc pPower Power of N±. KSelect K that the perturbation applies to; set to "all" (default) to apply to all K. Currently requires that K= K' = Kselect SaPower Power of Sa. SbPower Power of Sb. ScPower Power of Sc. SpPower Power of S± - operator is S+SpPower + S-SpPower SPower Twice power of S2 = S(S+1) NSPower Power of N.S.

Currently the N± and S± operators can only be combined with the a/b/c style operators for the axis corresponding to the z axis so, for example, in the Ir representation N± can be combined with Na or Sa but not Nb or Sb.

The effective operator for rotational perturbations (i.e. where Nucleus = 0) when pPower is zero is:
NJPower [NaaPower, [NbbPower, NccPower]+]+
where:
[An, Bm]+ = AnBm + AnBm
provided that the special cases with n or m zero are taken as:
[An, B0]+ = An
[A0, Bm]+ = Bm
See AllowComplex and PhaseAdjust for options in handling imaginary operators. Note that versions before 7.1.138 multiplied perturbation matrix elements by a phase factor to ensure that all the matrix elements were real; the factor was 1 except where the overall operator involved an odd power of Ny, when it was i.

When pPower is non zero the effective operator is:
NJPower [NzzPower, (N+pPower + N-pPower)]+
with the same definitions as above. z will correspond to a, b or c depending on the representation.

The electron spin operators are similar, with the general form being:
or just Sp or N.SNSPowerSSPower/2 if only one is present. Currently the N.SNSPowerSSPower/2 operator is the only spin operator that can be combined with another spin operator.

If both rotation and spin operators are present, the most general form is:

NJPower/2NzzPower(N+pPower + N-pPower) Sp N.SNSPowerSSPower/2 + SSPower/2N.SNSPower Sp(N+pPower + N-pPower)NzzPowerNJPower/2

or

NJPower/2NzzPower[NxxPower, NyyPower]+ Sp N.SNSPowerSSPower/2 + SSPower/2N.SNSPower Sp [NxxPower, NyyPower]+ NzzPowerNJPower/2
As an example, consider the operator bPower = 1, cPower = 1, which PGOPHER displays as bc. This notionally corresponds to the operator NbNc + NcNb. In the Ir representation this is equivalent to NxNy + NyNx. Expressed in terms of raising and lowering operators this is -i(N+2- N-2)/2 so the matrix elements are:
<N,K-2|bc|N,K> = i sqrt((N*(N+1)+(-K+1)*(K-2))*(N*(N+1)-K*(K-1)))/2
<N,K+2|bc|N,K> = -i sqrt((N*(N+1)+(-K-1)*(K+2))*(N*(N+1)-K*(K+1)))/2
Note that, for this matrix element, or any other matrix element that is imaginary, allowing the matrix elements to be complex is required to ensure that the overall Hamiltonian matrix is Hermitian. If AllowComplex is false then the matrix elements used are simply the imaginary matrix elements multiplied by -i, which can result in a Hamiltonian matrix that is not symmetric. In this case <N,K-2|bc|N,K> =  sqrt(x) and <N,K+2|bc|N,K> = -sqrt(x), but these two matrix elements should have the same sign.

## 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. zPower is not an actual setting; it corresponds to aPower, bPower or cPower depending on the representation used. The scale factor indicates the factor the standard parameter must be multiplied by to use the given perturbation operator form.

 Operator Reduction JPower aPower bPower cPower pPower zPower Scale Factor A Na2 A/S 0 2 0 0 0 0 1 B Nb2 A/S 0 0 2 0 0 0 1 C Nc2 A/S 0 0 0 2 0 0 1 BDelta N+2 + N-2 A/S 0 0 0 0 2 0 ¼ DJ N4 ≡ N2(N+1)2 A/S 4 0 0 0 0 0 -1 DJK N2Nz2 ≡ N(N+1)K2 A/S 2 0 0 0 0 2 -1 DK Nz4≡ K4 A/S 0 0 0 0 0 4 -1 deltaJ N2(N+2 + N-2) A 2 0 0 0 2 0 -1 N2(N+2 + N-2) S 2 0 0 0 2 0 1 deltaK [Nz2, N+2 + N-2]+ A 0 0 0 0 2 2 -½ N+4 + N-4 S 0 0 0 0 4 0 1 HJ N6 A/S 6 0 0 0 0 0 1 HJK N4Nz2 A/S 4 0 0 0 0 2 1 HKJ N2Nz4 A/S 2 0 0 0 0 4 1 HK Nz6 A/S 0 0 0 0 0 6 1 phiJ N4(N+2 + N-2) A/S 4 0 0 0 2 0 1 phiJK N2[Nz2, N+2 + N-2]+ A 2 0 0 0 2 2 ½ N2(N+4 + N-4) S 2 0 0 0 4 0 1 phiK [Nz4, N+2 + N-2]+ A 0 0 0 0 2 4 ½ N+6 + N-6 S 0 0 0 0 6 0 1 LJ N8 A/S 8 0 0 0 0 0 ¼ LJJK N6Nz2 A/S 6 0 0 0 0 2 1 LJK N4Nz4 A/S 4 0 0 0 0 4 1 LKKJ N2Nz6 A/S 2 0 0 0 0 6 1 LK Nz8 A/S 0 0 0 0 0 8 1 llJ N6(N+2 + N-2) A/S 6 0 0 0 2 0 1 llJK N4[Nz2, N+2 + N-2]+ A 4 0 0 0 2 2 ½ N4(N+4 + N-4) S 4 0 0 0 4 0 1 llKJ N2[Nz4, N+2 + N-2]+ A 2 0 0 0 2 4 ½ N2(N+6 + N-6) S 2 0 0 0 6 0 1 llK [Nz6, N+2 + N-2]+ A 0 0 0 0 2 6 ½ N+8 + N-8 S 0 0 0 0 8 0 1

## Spin-Rotational Hamiltonian expressed as perturbations

The spin rotation terms can be expressed as perturbations. Note that terms such as eabbar need two perturbation items to reproduce the corresponding term in Hamiltonian.
 Operator aPower bPower cPower SaPower SbPower ScPower Scale Factor Notes eaa NaSa + SaNa =2NaSa 1 0 0 1 0 0 ½ = 2NaSa = 2SaNa as Na and Sa commute ebb NbSb + SbNb =2NbSb 0 1 0 0 1 0 ½ = 2NbSb = 2SbNb as Nb and Sb commute ecc NbSb + SbNb =2NbSb 0 0 1 0 0 1 ½ = 2NcSc = 2ScNc as Nc and Sc commute eabbar NaSb + SbNa 1 0 0 0 1 0 ½ Two perturbations required NbSa + SaNb 0 1 0 1 0 0 ½ eacbar NaSc + ScNa 1 0 0 0 0 1 ½ Two perturbations required NcSa + SaNc 0 0 1 1 0 0 ½ ebcbar NbSc + ScNb 0 1 0 0 0 1 ½ Two perturbations required NcSb + SbNc 0 0 1 0 1 0 ½
The quartic spin rotation terms are expressed in terms of the effective constants zPower and SzPower, where the z axis corresponds to a, b or c depending on the representation used.
 Operator Reduction JPower pPower zPower SpPower SzPower NSPower Scale Factor DsN N2N.S A/S 2 0 0 0 0 1 1 DsNK [N2, NzSz]+ A/S 2 0 1 0 1 0 ½ DsKN N.S Nz2 A/S 0 0 2 0 0 1 1 DsK [Nz3, Sz] = 2Nz3Sz A/S 0 0 3 0 1 0 ½ ds1 N.S(N+2 + N-2) A/S 0 2 0 0 0 1 1 ds2 [N+2 + N-2, NzSz]+ A 0 2 1 0 1 0 ½ [N+3, S+]+ + [N-3, S-]+ = 2(N+3S++N-3S-) S 0 3 0 1 0 0 ½
Note cancelling factors of 2 and ½ in DsK.and the S form for ds2. They arise because the operator forms are derived allowing for the fact that the components of N and S do not, in general, commute, though in this case they do.

## Spin-Spin Hamiltonian expressed as perturbations

 Operator SxPower SyPower SzPower SpPower Scale Factor alpha Sx2 2 0 0 0 -1 3Sz2-S2 = 2Sz2- Sx2-Sy2 Sy2 0 2 0 0 -1 Sz2 0 0 2 0 2 beta S+2 + S-2 = 2(Sx2-Sy2) 0 0 0 2 ½
Note that three terms are required to replicate  alpha. The assignment of x, y and z is determined by the Representation.