Quantum Monte Carlo for Atoms, Molecules and Solids
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- The Short Time Approximation to the Greens Function
- The DMC walk
- The variational principle in DMC
DMCThe DMC method is a projector approach in which a stochastic imaginary-time evolution is used to improve a starting trial wave function. The governing equations can readily be obtained by multiplying the time-dependent Schrödinger Equation in imaginary time, Eq. (4), by a known approximate wave function T. One defines a new distribution given as the product of T and the unknown exact wave function, . The resultant equation has the form of a diffusion equation with drift, provided is positive definite. In general the exact and the approximate T will differ in various regions and their product will not be positive, therefore one imposes the fixed-node boundary condition which results in a positive distribution  . The wave function  is an approximation to Φ in that the nodes of are the same as ΨT.
There are two primary difficulties to be overcome to simulate the particles, i.e., the electrons in imaginary time. These are: how to create a stochastic process that generates the proper distribution, and how to obtain the appropriate Fermion ground state, not the lowest energy solution which would be the Boson ground state. The Short Time Approximation to the Green's FunctionIn order to create an ensemble of pseudo (imaginary time) particles with the distribution  we need a method similar to that in VMC that moves the particles of the system through a series of time steps. This aim can be achieved in a manner similar to VMC, by integrating the Schrödinger equation to obtain the function f at a time τ+δτ from the f at time τ,
The function  is the Green's function of the Hamiltonian, formally related by  . The exact Green's function is a solution to the Schrödinger equation, and is therefore unknown in general. However, we can approximate the Green's function as  which is exact in the limit that δτ →0 (since 𝒯 and 𝒱 do not commute). For the importance sampled equation the kinetic energy term is modified to include the drift term, so that expression for the short time approximate Green's function is
This is now composed of a gaussian probability distribution function with a mean that drifts with velocity  and spreads with time as  , and a rate term which grows or shrinks depending on the value of  relative to the fixed quantity  . Hence this Green's function has the expected behavior of a diffusion-like term multiplied by branching term.
The DMC walkThere are numerous methods to carry out the simulation (as well as other possible short-time approximations), but one of the simplest method is as follows: Create an initial ensemble of walkers,  , typically the result of a VMC simulation;
For each walker, each electron (with coordinates  ) will diffuse and drift one at a time for a time step δτ by,
where χ is a pseudo random number distributed according to the Gaussian part of  ;
If the electron crosses a node of the trial function the move is rejected and go back to (b) for the next electron. To insure detailed balance, accept the move of the electron with probability  where
Return to step (b), moving all the electrons in this walker. Compute the local energy and other quantities of interest. Calculate the multiplicity M (the branching probability) for this walker from the rate term in the Green’s function,
The quantity δτa is the effective time step of the move, taking into account the rejection of some moves. The effective time of the move is calculated from the mean-squared distance of move, which would be  without rejection. With rejection they move  and so one can compute,
The walker is then replicated or destroyed based on M. To do so create the integer  where ξ is a random number between zero and one, if  then make  copies, otherwise the multiplicity is zero and the walker is removed from the ensemble. (In some implementations the ratio  is kept as a weight for the walker, and in the Pure DMC method M is kept as a weight and the ensemble size is kept fixed.)
Weight the local energy and other quantities of interest by M. Return to step (b) for the remaining walkers. Compute the ensemble averages and continue to the next time step. At suitable intervals one can update the trial energy ET to regulate the size of the ensemble. It may also be necessary to “renormalize” the ensemble when it grows to large or too small. This can be done by randomly deleting or replicating walkers. Care must be taken as renormalization can cause an energy bias. At the end of the run the final energy and properties are computed and various statistical methods can be applied to determine the variance of the mean values. Repeat for several values of δτ . Because the simulation is only exact in the limit  it is necessary to either extrapolate to zero time or demonstrate that the bias is smaller than the statistical uncertainty of the result.
The variational principle in DMCThe DMC method is fundamentally a ground state method. No bounds exist similar to the variational principle or variance minimization for excited states. Nevertheless, excited states that are the lowest of a given symmetry are routinely addressed [refs.], and it has been shown that excited states of the same symmetry as a lower state can be computed with the fixed node method with excellent results [ref.]. Download 242 Kb. Share with your friends:
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