Zero Temperature QMC Outline of lecture. Variational Monte Carlo Diffusion Monte Carlo Fermion systems Coupled Electron-Ion Monte Carlo Applications Electron gas High pressure hydrogen. Notation. Individual coordinate of a particle r i All 3N coordinates R= (r 1 ,r 2 , …. r N )
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Quantum chemistry uses a product of single particle functions
With MC we can use any “computable” function.
Sample R from ||2 using MCMC.
Take average of local energy:
Optimize to get the best upper bound
Error in energy is 2nd order
Better wavefunction, lower variance! “Zero variance” principle. (non-classical)Variational Monte Carlo (VMC)
VMC calculation of ground state of liquid helium 4.
Applied MC techniques from classical liquid theory.
Ceperley, Chester and Kalos (1976) generalized to fermions.First Major QMC Calculation
Slater-Jastrow trial function.
kxTwist averaged boundary conditions
transitionPolarization of 3DEG
Simple trial function
Better trial function
Scaling (computational effort vs. size) is almost classical
Learn directly about what works in wavefunctions
No sign problem
Optimization is time consuming
Energy is insensitive to order parameter
Non-energetic properties are less accurate. O(1) vs. O(2) for energy.
Difficult to find out how accurate results are.
Favors simple states over more complicated states, e.g.
Solid over liquid
Polarized over unpolarizedSummary and problems with variational methods
What goes into the trial wave function comes out! “GIGO”
We need a more automatic method! Projector Monte Carlo
Requires interpretation of the wavefunction as a probability density.
But is it? Only in the boson ground state. Otherwise there are nodes. Come back to later.
Evolution = diffusion + drift + branching
make acceptance ratio>99% . Determines time step.
Commute through H
Ensemble evolves according to
Monte Carlo can add but not subtract
Negative walkersModel fermion problem: Particle in a box
Symmetric potential: V(r) =V(-r)
Antisymmetric state: f(r)=-f(-r)
Initial (trial) state
Final (exact) state
Sign of walkers fixed by initial position. They are allowed to diffuse freely.
f(r)= number of positive-negative walkers. Node is dynamically established by diffusion process. (cancellation of positive and negative walkers.)
EF and EBare Fermion, Bose energy (proportional to N)
With 0 <a < 4
It comes from a VMC simulation.
If we know the sign of the exact wavefunction (the nodes), we can solve the fermion problem with the fixed-node method.
Better trial function
RNSummary of T=0 methods:Variational(VMC), Fixed-node(FN), Released-node(RN)
-1 tail move
+1 head move
Tests on bcc hydrogen, N=16
Good results in a few slices p~20.
ionsCoupled electron ion MC
The Metropolis, Rosenbluth, Teller (1953) method:
a(SS*)= min [ 1 , exp ( - (E(S*)-E(S))/kBT ) ]
Given ergodicity, the distribution of the state, S, will be:
E(S)=energy of ionic arrangement “S”,
Only the difference in energy enters in the acceptance probability.
Average energy of Lennard-Jones liquid
< e > = E
< [e- E]2 > = 2
* OK because of central limit theorem for <
Path Integral MC for
T > EF/10
Diffusion MC T=0
Coupled-electron Ion MC
Path Integral MC with an effective potential
with the orbital from a rescaled LDA calculation.
“Reptile” spaceEnergy difference methods
H2 Bond-ordered phase
We find that the solid is stable below 500K.
Results for state.
approximate orbital on the real-space grid.
Energy from VMC (Jastrow-Slater)
Clear hysteresis in H-H2 transition.
An advantage of CEIMC: state.
Averages are almost free.
(M time slices to average over.)
a = min(1,exp[ - E(S)+E(S*) - v/2])