Computational Physics

12.2 The variational Monte Carlo method 383 A particular diffusion equation which we shall encounter later in this chapter is ∂ρ ...

384 Quantum Monte Carlo methods explicitly here, because we need to find the Green’s function for a more complicated type of dif ...

12.2 The variational Monte Carlo method 385 We again separate the exponent into two terms, one containingˆxand the otherpˆ,at th ...

386 Quantum Monte Carlo methods 12.2.5 The Fokker–Planck equation approach to VMC The VMC method described inSections 12.2.1and1 ...

12.3 Diffusion Monte Carlo 387 accepted with probability min(1,qRR′), where qRR′= ωR′Rρ(R′) ωRR′ρ(R) . (12.52) Note that the fra ...

388 Quantum Monte Carlo methods For largeτthe ground state energyEGdominates in the sum by a factor exp[−τ(E 1 − EG)]; therefore ...

12.3 Diffusion Monte Carlo 389 and that after the last branching step their actual number isM, then we adjustETas ET=E 0 +αln ( ...

390 Quantum Monte Carlo methods 0 0.5 1 1.5 2 2.5 3 3.5 4
r Figure 12.1. Ground state wave function (timesr^2 ) for the three- ...

12.3 Diffusion Monte Carlo 391 Next we analyse the helium atom using the diffusion Monte Carlo method. This is less successful. ...

392 Quantum Monte Carlo methods The FP-diffusion term will be used to diffuse the walkers, whereas the ‘potential’ EL(R)−ETis us ...

12.3 Diffusion Monte Carlo 393 Evaluateq=exp{− τ[ELocal(R′)+ELocal(R)]/ 2 −ET}; Eliminate the walker or create new ones atR′, de ...

394 Quantum Monte Carlo methods 12.3.4 Problems with fermion calculations We have described how the simulation of a diffusion pr ...

12.3 Diffusion Monte Carlo 395 we assume the spins of theNfermions to be given, then the nodes form (3N− 1)-dimensional hypersur ...

396 Quantum Monte Carlo methods Now perform two independent DMC calculations, one withφ−and the other withφ+as a starting distri ...

12.3 Diffusion Monte Carlo 397 (=0) +(=0) −(=0) −() () +() (a) (b) (c) (d) Figure 12.2. Evolution of the distributio ...

398 Quantum Monte Carlo methods 12.4 Path-integral Monte Carlo In Chapter 11 we saw that the partition function of a classical l ...

12.4 Path-integral Monte Carlo 399 Nspinless particles with coordinatesRi, the partition function can be written as ∫ dR 0 〈R 0 ...

400 Quantum Monte Carlo methods Figure 12.3. Classical system described by the path integral of the two elec- trons in the heliu ...

12.4 Path-integral Monte Carlo 401  R Figure 12.4. The path integral for a one-dimensional system. The vertical axes areR-axes ...

402 Quantum Monte Carlo methods for largeM. The error is then given by [11, 12] α^2 M ∑ m>m′ |[Hm,Hm′]|e−α ∑ m|Hm|, (12.74) w ...