Iterative diagonalization

Linear-scaling $\Rightarrow$ large systems $\Rightarrow$ large basis sets $\Rightarrow$ iterative diagonalization

Want to minimize the energy w.r.t. the wavefunctions $\{ \psi_i({\bf r}) \}$:

• Expand in a basis set: $\psi_i({\bf r}) = \displaystyle{\sum_{\mu} x_i^{\mu} \chi_{\mu}({\bf r})}$
• For a non-orthogonal basis set, solve the generalized eigenvalue equation:
  
  $$H_{\mu \nu} x_i^{\nu} - \varepsilon_i S_{\mu \nu} x_i^{\nu} = 0$$
  
  

• Need to distinguish contravariant and covariant quantities
• Correct gradient is therefore:
  
  $$g_i^{\mu} = S^{\mu \nu} H_{\nu \xi} x_i^{\xi} - \varepsilon_i x_i^{\mu}$$
  
  

Kinetic energy preconditioning

1mm

M. P. Teter, M. C. Payne and D. C. Allan, Phys. Rev. B 40 12255 (1989)
D. R. Bowler and M. J. Gillan, Comput. Phys. Commun. 112 103 (1998)