Introduction to Quantum Monte Carlo Methods 2! Claudio Attaccalite

What we learned last time 1) How to sample a given probability p(x) distribution with Metropolis Algorithm: 2) How to evaluate integrals in the form: 3) Evaluate Quantum Mechanical Operators: where x i are distributed according to p(x)

Outline ● Path integral formulation of Quantum Mechanics ● Diffusion Monte Carlo ● One-Body density matrix and excitation energies

Path Integral : classical action where S is the Classical Action and L is the Lagrangian The path followed by the particle is the one that minimize: Only the extreme path contributes!!!!

Path Integral: Quantum Mechanics In quantum mechanics non just the extreme path contributes to the probability amplitude where Feynman's path integral formula

From Path Integral to Schrödinger equation 1 XAXA XBXB X1X1 X2X2 X3X3 X4X4 X5X5 X... X M- 1 It is possible to discretized the integral on the continuum into many intervals M slices of length On each path the discretized classical action can be written as We want use this propagator in order to obtain the wave-function at time t 2 in the position x 2

From Path Integral to Schrödinger equation 2 We call, then send Substituting the discretized action and compare left and right at the same order t to zero At the order 0 we get the normalization constant At the order 1 we get the Schroedinger equation!

Cafe Moment What we want: -> What we have: ->

Imaginary Time Evolution We want to solve the Schrödinger equation in imaginary time: The formal solution is: If we expand in a eigenfunction of H: ● if E R > E 0 ● if E R < E 0 ● if E R = E 0 Tree Possibility:

From Path Integral to DMC: 1 Using Feynman path integral the imaginary time evolution can be rewritten as is equal to and as usual we rewrite this integral as

From Path Integral to DMC: 2 A Gaussian probability distribution A Weight Function If we define: we have

The Algorithm We want generate the probability distribution and sample Generate points distributed on Y(x 0,0) x 1 is generate from x 0 sampling P(x n,x n-1 ) (a Gaussian) the weight function is evaluated W(x 1 ) X

An example H and H 2 Convergence of the Energy H molecule versus t H atom wave-function and energy

Application to Silicon: one body density matrix LDA local orbitals The matrix elements are calculated as:

Results on Silicon Max difference between r ii QMC and LDA is Max off-diagonal element (1)

Results on Silicon: 2 QMC one-body-density matrix on the 110 plane where r is fixed on the center of the bonding Difference between QMC and LDA for r=r' is 1.7%

Reference ● SISSA Lectures on Numerical methods for strongly correlated electrons 4 th draft S. Sorella G. E. Santoro and F. Becca (2008) ● Introduction to Diffusion Monte Carlo Method I. Kostin, B. Faber and K. Schulten, physics/ v1 (1995) ● Quantum Monte Carlo calculations of the one-body density matrix and excitation energies of silicon P. R. C. Kent et al. Phys. Rev. B (1998) ● FreeScience.info-> Quantum Monte Carlo

From Path Integral to Schrödinger equation: 1+1/2 We call and send t to zero: Substituting the discretized action and