text: Again consider two clusters j and k. For simplicity 
  of notations deﬁne a projection operator 
  PjPk, 
  where 
  Pj,k is projection into the singlet subspace of clus- 
  τ z 
  j,k = 
  ter j and k, respectively, 
  Pj,k = 
  s 
  . For a given perturbation λ Hperturbation with small 
  | 
  parameter λ (in factor λ/Jcluster is the expansion param- 
  eter), lowest two orders of the perturbation series are 
  
  =λy[Sj1 
  Jy| 
  with λy = 
  4 
  p  | 
  
  λz Hperturbation, z 
  
  Pjk + λ2 
  Hcluster k]−1(1 
  
  = λz[Sj2 
  Jz| 
  with λz = 4 
  
  (Sk3 
  
  Sk4) + sgn(Jz) 
  Sk4). 
  
  Sk2  (Sj3  Sj4)]  λ 
  
  PjkHperturbation 
  Hcluster j − 
  [0 
  × 
  
  PjkHperturbation(1 
  
  − Pjk) 
  − Pjk)Hperturbation 
  · 
  (Sj3  × 
  Sj4 + Sk3 
  ·  ·  × 
  
  Pjk 
  (15) 
  − | 
  
  · 
  Jz| · 
  
  · 
  − 
  
  Jcluster. 
  p| 
  The entire Hamiltonian Hmagnetic reads explicitly as, 