Computational Physics - Department of Physics

14.5 Variational Monte Carlo for atoms 469 1 R ∂ ∂r (r^2 ∂R ∂r )+ 2 mrke^2 ̄h^2 + 2 mr^2 ̄h^2 E=Cr, (14.14) whereCrandCφare cons ...

470 14 Quantum Monte Carlo Methods − ̄h^2 r^2 2 m ( ∂ ∂r(r 2 ∂R(r) ∂r ) ) − ke^2 r R(r)+ ̄h^2 l(l+ 1 ) 2 mr^2 R(r) =E R(r). Intr ...

14.5 Variational Monte Carlo for atoms 471 Hydrogen-like atomic radial functions l\n 1 2 3 0 exp(−Zr) ( 2 −r)exp(−Zr/ 2 ) ( 27 − ...

472 14 Quantum Monte Carlo Methods The ground state of the hydrogen atom has the energyλ=− 1 / 2 , orE=− 13. 6 eV. The exact wav ...

14.5 Variational Monte Carlo for atoms 473 depends only on the dimensionless radiusρ. It is the solution of a one-dimensional di ...

474 14 Quantum Monte Carlo Methods the exact result, and the variance is zero, as it should. The reason is that we have used the ...

14.5 Variational Monte Carlo for atoms 475 monic oscillator are some of the few cases where we can use a trial wave function pro ...

476 14 Quantum Monte Carlo Methods V(r 1 ,r 2 ) =− 2 ke^2 r 1 − 2 ke^2 r 2 + ke^2 r 12 , with the electrons separated at a dista ...

14.5 Variational Monte Carlo for atoms 477 The energy becomes then ( ̂h 1 ψa(r 1 ) ) ψb(r 2 )+ ( ̂h 2 ψb(r 2 ) ) ψa(r 1 ) = (Ea+ ...

478 14 Quantum Monte Carlo Methods whereRT(r 1 )is the radial part of the wave function for electron 1. We have also used that t ...

14.5 Variational Monte Carlo for atoms 479 wherer 1 , 2 are dimensionless radii andαis a variational parameter which is to be in ...

480 14 Quantum Monte Carlo Methods where the inter-particle separationri jfor simplicity is written asr. In additions=ri+riand t ...

14.5 Variational Monte Carlo for atoms 481 Initialize: Setrold,α andΨT−α(rold) Suggest a move Compute acceptance ratioR Generate ...

482 14 Quantum Monte Carlo Methods // the step length and its squared inverse for the second derivative defineh 0.001 defineh2 1 ...

14.5 Variational Monte Carlo for atoms 483 The implementation of the brute force Metropolis algorithmis shown in the next functi ...

484 14 Quantum Monte Carlo Methods // update energies energy += delta_e; energy2 += delta_e*delta_e; } } // end of loop over MC ...

14.5 Variational Monte Carlo for atoms 485 r_single_particle; doubler_plus,r_minus; // allocate matrices which contain the posit ...

486 14 Quantum Monte Carlo Methods The ground state isE=− 2. 85 in atomic units orE=− 77. 5 eV. The experimental value is− 78 ...

14.5 Variational Monte Carlo for atoms 487 Secondly, when using the simple symmetric form ofω(rold,rnew), one has to keep in min ...

488 14 Quantum Monte Carlo Methods ∂f ∂t =∑ i D∂ ∂xi ( ∂ ∂xi −Fi(F) ) f (14.26) wherefis the continuous distribution of walkers. ...