a probability distribution for which direct sampling is difficult. 1 1 The Metropolis–Hastings algorithm works by generating a sequence of sample values in such a way that, as more and more sample values are produced, the distribution of values more closely approximates the desired distribution, P ( x ) . These sample values are produced iteratively, with the distribution of the next sample being dependent only on the current one. Specifically, at each iteration, the algorithm picks a candidate for the next sample value based on the current one. Then, with some probability, the candidate is either accepted (in which case the candidate value is used in
3.1 Structured Bodies 85 The different versions of the quantum Monte Carlo method all share the common use of the technique to build accurate estimates to the multidimensional integrals that arise in the different formulations of the many-body problem. The quantum Monte Carlo methods allow for a direct treatment and description of complex many- body effects encoded in the wavefunction and offer numerically accurate solutions of the many-body problem. In principle, any physical system can be described by the many-body Schrödinger equation provided that the constituent particles are not moving at a speed comparable to that of light and that, therefore, relativistic effects can be neglected. There are, indeed, two versions of the Monte Carlo technique that have been applied to electronic structure problems: the variational (VMC) and the diffusion (DMC) one. In the QMC method we start from the consideration that the expectation value of themany-electron(say K )wavefunctionofthegroundstatereads(fortheequivalence between function and vector notations see Appendix A1): E 0 = 0 | H | 0 0 | 0 = ∗ 0 ( r ) H 0 ( r ) d r ∗ 0 ( r ) 0 ( r ) d r (3.1) in which r is the 3 K dimensional vector of electronic positions. The energy associated with the tentative function T is E T = ∗ T ( r ) H T ( r ) d r ∗ T ( r ) T ( r ) d r . According to the variational principle, E T is an upper limit to the (true) ground state energy E 0 . By reformulating the integral as follows: E T = | T ( r ) | 2 H T ( r ) T ( r ) d r | T ( r ) | 2 d r . (3.2) the VMC Monte Carlo method is then applied using the Metropolis–Hastings algo- rithm and a set of r values are generated in configuration space and at each of these points the energy (where H T ( r )/ T ( r ) is the “local energy”) is generated. For a sufficiently large sample of points, the average value is given by E V MC = 1 K K i = 1 H T ( r i ) T ( r i ) . (3.3) the next iteration) or rejected (in which case the candidate value is discarded, and current value is reused in the next iteration). The probability of acceptance is determined by comparing the values of the function f ( x ) of the current and candidate sample values with respect to the desired distribution P ( x ) .
86 3 Ab initio Electronic Structure for Few-Body Systems In the VMC method, therefore, it is crucial the choice of the tentative T ( r ) that determines the value of the observable computed by the simulation.
You've reached the end of your free preview.
Want to read all 219 pages?
- Fall '19
- dr. ahmed