text: In this Subsection we consider more conventional per- 
  turbations, magnetic interactions between the clusters, 
  Sk with j and k belong 
  e.g. the Heisenberg coupling Sj · 
  to diﬀerent tetrahedra. This has the advantage over the 
  previous phonon approach for not introducing additional 
  degrees of freedom. But it also has a signiﬁcant disad- 
  vantage: the perturbation does not commute with the 
  cluster Heisenberg Hamiltonian (2), so the cluster sin- 
  glet subspace will be mixed with other total spin states. 
  In this Subsection we will use the spin-chirality represen- 
  tation (6) for τ z. 
  
  will only list the results here. 
  The perturbation on x-links is given by 
  
  λy Hperturbation, y 
  
  Sk1 + sgn(Jy) 
  Sj4 + Sk3  (Sj3 
  Sk4) 
  
  Sj4)  (Sk3  Sk4)] 
  
  Pjk = 
  τ z 
  j,k = s 
  
  · 
  (Sj3 
  · 
  · 
  
  −  ·  − 
  
  · 
  Jy| · 
  The perturbation on z-links is 
  
  − | 
  
  Ps=±1 |  ih 
  Jcluster. 