Forces in QMC

From QMC

Discussion during CECAM QMC Workshop 2007

Contents 1 Introduction 2 L. Mitas 3 S. Moroni 4 A. Badinski 4.1 Pulay nodal terms in accurate DMC forces

[edit] Introduction

Problems in calculating forces in QMC:

• Time step behavior appears to worsen when calculating forces
• Force is more sensitive to trial function error than energy
• Should one calulate finite differences in or derivatives of the energy?

[edit] L. Mitas

• Calculate QMC energies during an MD run
  • Tested on silane optical gap as molecule vibrates
  • QMC parallels LDA-DFT values
• Continuous DMC - drag the wave function along with the ions during DMC run
• Finite difference, correlated sampling, Umrigar-Filippi modifications to Green function
• ~50X more expensive than energy calculation (one needs to do many short DMC runs for correlated sampling)
• Calculated H₂ vibration with no input from DFT

[edit] S. Moroni

• Derivatives of the fixed-node energy easier with path integral MC
• drift-diffusion is partly ignored by finite difference method
• nodal action:
• estimated nodal distance:
• better estmate:
• space-warp transformation applied to nodal distance
• tested on Li₂ nodes from HF with STO-3G basis
  • calculated force consistent with E from interpolation

[edit] A. Badinski

[edit] Pulay nodal terms in accurate DMC forces

• two methods to calculate forces:
  • mixed - use trial wave function
  • pure - use DMC wave function
• first-order terms in derivative of total energy with respect to nucleus coordinate
  • mixed: Hellmann-Feynman term, volume term (Reynolds approximation), nodal term
  • pure : Hellman-Feynman term, nodal term
• test on GeH
  • start close to equilibrium geometry
  • no e-e interaction (nodal terms arise from kinetic energy)
  • use single det. trial wf. w/ 4 (different in quality) basis sets
  • local pp. (to avoid infinite variance)
  • calculate partial rather than total derivs.
  • use extrapolation and future walking to calculate HFT
  • also calculate E & grad. from pot. E curves
  • mixed est.
    • significant basis set dependence
    • adding V term improves mixed HFT
    • nodal term much smaller than V term
  • pure est.
    • error is order of magnitude smaller than mixed
    • HFT from future walking not better than extrapolation
    • adding N term to HFT(extrapol.) "overshoots"
    • adding N term to HFT(future) gives exact within error
    • remaining errors due to Reynolds approx. (neglecting higher derivative terms) and evaluation of derivative of Psi trial