Finite-size errors in QMC

From QMC

Discussion during CECAM QMC Workshop 2007

Contents 1 Introduction 2 R. Needs 2.1 Two approaches for removing Coulomb finite-size effects 2.2 Equivalence of the two approaches 2.3 Conclusions 3 D. Ceperley 3.1 Condensed version of PRL 97, 076404 (2006) 4 R. Gaudoin

[edit] Introduction

• Without twist averaging, what is the best way to get finite k-pt correction from DFT or HF?
• Pros and cons of MPC vs S(k)
• Is the quadratic coefficient at k=0?
• Residual correction 1/N?
• Kinetic energy correction form Jastrow.
• Any additional problems for charged defects.

[edit] R. Needs

[edit] Two approaches for removing Coulomb finite-size effects

• MPC: PRB 53, 1814 (1996); PRB 55 R4851 (1997)
  • Interaction of electrons with their exchange-correlation holes should be exactly 1/r, but Ewald interaction converges only slowly to 1/4
  • Use Ewald interaction to evaluate Hartree energy
  • Use 1/r interaction within minimum image convention to evaluate interaction of each electron with its exchange-correlation hole.
• Chiesa correction (think in reciprocal space): PRL 97, 076404 (2006)
  • Leading-order error due to omission of G=0

[edit] Equivalence of the two approaches

• MPC is perfect if exchange-correlation hole fits into simulation cell; addition the Chiesa correction to the Ewalk energy is an approximation to the MPC.
• If exchange-correlation hole does not fit into simulation cell, MPC is not perfect; Chiesa correction can still make use of the known form of the structure factor at small k.
• The two approaches give very similar results in practice.

[edit] Conclusions

• Leading-order Chiesa kinetic-energy correction works well at low densities but not high density.
• Real-space method for k-point sampling errors?

[edit] D. Ceperley

[edit] Condensed version of PRL 97, 076404 (2006)

• Coulomb correction:
• Kinetic-energy correction:
• Key region is at and others
• Make an interpolation, e.g.,

[edit] R. Gaudoin

Modeling the spherically averaged structure factor