text: this term becomes 
  
  λ2 
  6Jcluster · 
  (λ2)/(32Jcluster) 
  
  − 
  
  = 
  −  ·  − 
  
  Another second order perturbation term r2λ2 
  
  PjkSk2 
  · 
  Hcluster k]−1(1 
  − 
  Pjk can be computed in the similar 
  τ x 
  j ). 
  Sj4)(1 
  (Sj3 
  − Pjk)[0 
  × 
  (Sj3 
  Sj4) 
  Pjk)Sk2 
  × 
  way and gives the result  Hcluster j − 
  (r2 λ2)/(32Jcluster)  (2 
  − 
  For one of the cross term 
  ·  − 
  
  Jz| 
  with λz = 4 
  Jcluster, r = sgn(Jz) is the sign of Jz. 
  The last term on the right-hand-side is to cancel the non- 
  trivial terms (r2 τ x 
  z/(32Jcluster) from the second 
  order perturbation of the ﬁrst term. Up to second order 
  perturbation this will produce 
  
  r λ2 
  PjkSj2 
  [0 
  − 
  − Pjk)Sk2 
  (1 
  
  Jzτ z 
  j τ z  k interactions. 
  
  × 
  × 
  − 
  (Sj3 
  Pjk  × 
  
  Finally we have been able to reduce the high order 
  interactions to at most three spin terms, the Hamiltonian 
  Hmagnetic is  Hcluster k]−1(1 
  − 