1. Computational Solid State Physics 計算物性学特論 第８回 8. Many-body effect II: Quantum Monte Carlo method

2. Quantum diffusion Monte Carlo method Diffusion Monte Carlo method to calculate the ground state Importance sampling method How to treat the Pauli principle: fixed node approximation

3. Schrödinger equation in atomic unit How to solve the Schrödinger equation for many electrons?

4. Imaginary Time The ground state wave function can be obtained in the limit of infinite time. Time-dependent Schrödinger equation

5. diffusionbranching Diffusion equation holds for Diffusion equation with branching process for the ground state wave function

6. Diffusion equation for particles Flux: Conservation of number of particles: D : diffusion constant, v d : drift velocity Diffusion equation diffusion flux drift flux

7. Rate equation R>0 : growth rate R<0: decay rate

8. Time-dependent Green ’ s function : Boundary Condition

9. Time evolution of wave function

10. Short time approximation

11. The transition probability from x to y can be simulated by random walk in 3N dimensions for N electron system. Green’s function of the classical diffusion equation

12. The branching process can be simulated by the creation or destruction of walkers with probability G B Green’s function of the rate equation

13. Importance sampling : Local energy : Quantum force : analytical trial fn. Diffusion Drift Branching Diffusion equation with branching process

14. Kinetic energy operator Drift term The transition probability from x to y can be simulated by biased random walk with quantum force F in 3N dimensions for N electron system. Biased diffusion Green ’ s function

15. Detailed balance condition To guarantee equilibrium Acceptance ratio of move of the walker from x to y

16. DMCImportance-sampled DMC suppression of branching process DMC and Importance-sampled DMC for the hydrogen atom Branching process:

17. Walker 1 Walker 2 Walker 3 Walker 4 Branching Biased diffusion Schematic of the Green’s function Monte Carlo calculation with importance sampling for 3 electrons

18. Evaluation of the ground state energy

19. How to remove the condition ? Fixed node approximation to treat wave functions with nodes Fixed phase approximation to treat complex wave functions

20. Fixed node approximation Wave function φ is assumed to have the same nodes with Ψ Ｄ. Importance sampling on condition

21. Pauli principle for n like-spin electrons Slater determinant Slater determinant has nodes.

22. Fixed phase approximation Wave function φ is assumed to have the same phase with Importance sampling on condition

23. Ground states of free electrons D.M.Ceperley, B.J.Alder: PRL 45(1980)566

24. Transition of the ground state of free electrons Unpolarized Fermi fluid Polarized Fermi fluid Wigner crystal n: concentration of free electrons

25. Problems 8 Calculate the ground state wave function of a hydrogen atom, using the diffusion Monte Carlo method. Consider how to calculate the ground state energy. Calculate the ground state of a hydrogen atom, using the diffusion Monte Carlo method with importance sampling method. Assume the trial function as follows. Derive the diffusion equation for in importance sampling method.