text: 4 
  
  an energy of interband transitions, which is roughly 2eV . 
  This would be consistent with Refs. 8,9. 
  just list the formulas that we used in our computations. 
  The conductivity σ(Ω) and the optical integral W (ωc) 
  are given by (see for example Ref. 35). 
  
  We begin with formulating our calculational basis in 
  the next section. Then we take up the four cases and 
  consider in each case the extent to which the Kubo sum is 
  satisﬁed up to the order of bandwidth and the functional 
  form and the sign of ∆W (ωc). The last section presents 
  our conclusions. 
  
  Π′′(Ω) 
  Ω 
  = − 
  
  ωc  ωc 
  
  Π′′(Ω) 
  Ω  π 
  2 
  W (ωc) =  σ′(Ω) dΩ = −  dΩ +  Π′(0) 
  
  0 
  Z 
  0+ 
  Z 
  II. OPTICAL INTEGRAL IN NORMAL AND 
  SUPERCONDUCTING STATES 
  (7b) 
  
  The generic formalism of the computation of the op- 
  tical conductivity and the optical integral has been dis- 
  cussed several times in the literature21–23,26,29 and we  where ‘X ′’ and ‘X ′′’ stand for real and imaginary parts 
  of X. We will restrict with T = 0. The polarization 
  operator Π(Ω) is (see Ref. 36) 