[CP2K-user] a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

[Fangyong Yan]

Dear CP2K developers,

I have a question regarding the translational invariant of Kohn-Sham 
orbital in the Gaussian plane-wave method. 

The Kohn-Sham orbital should be translational invariant, and if we express 
the orbital using Gaussian basis function, these Gaussian basis function 
needs also be translational invariant, in the paper "A hybrid Gaussian and 
plane wave density functional scheme", by LIPPERT, HUTTER and 
PARRINELLO, MOLECULAR PHYSICS, 1997, VOL. 92, NO. 3, 477-487. The Kohn-Sham 
orbital has been expanded by Gaussian basis function which includes the 
periodicity. (Eq. 4 and 5 in the paper).

However, in the recent paper of CP2K, "QUICKSTEP: Fast and accurate density 
functional calculations using a mixed Gaussian and plane waves 
approach", Joost VandeVondele, Matthias Krack , Fawzi Mohamed , Michele 
Parrinello , Thomas Chassaing , Jürg Hutter, Computer Physics 
Communications 167 (2005) 103–128, from Eq. 1, the definition of electron 
density, has been expanded based on Gaussian basis function, but these 
Gaussian basis function has not mentioned to take account of periodicity. 
But if the Gaussian basis function has not taken account the periodicity, 
the electron density is not translational invariant. 

Could you please help me with Eq. 1 regarding the periodicity of electron 
density in this recent paper of C2PK? 

Thank you very much!

Fangyong

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