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# Introduction to Diffusion Monte Carlo

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Introduction to Diffusion Monte Carlo

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### Introduction to Diffusion Monte Carlo

1. 1. Introduction to Quantum Monte Carlo Methods 2! Claudio Attaccalite http://attaccalite.com
2. 2. What we learned last time How to sample a given probability p(x) distribution with Metropolis Algorithm: Axx'=min1, px'T x, x' pxT x', x How to evaluate integrals in the form: Evaluate Quantum Mechanical Operators:  〈 f 〉=∫ f xpxdx= 1N Σ f xi where xi are distributed according to p(x) 〈 f  2 〉−〈 f 〉2  f=N 〈 A〉=∫ A dx ∫ x2 dx AL x= x  x =∫ AL xpxdx px= ∣x∣2 ∫∣x∣2dx
3. 3. Outline Path integral formulation of Quantum Mechanics Diffusion Monte Carlo One­Body density matrix and excitation energies
4. 4. Path Integral : classical action The path followed by the particle is the one that minimize: t b dt Lxt , ˙xt ;t  S=∫t a where S is the Classical Action and L is the Lagrangian Lxt  ,˙x t ;t=m2 ˙x t2−V xt  ;t  Only the extreme path contributes!!!!
5. 5. Path Integral: Quantum Mechanics In quantum mechanics non just the extreme path contributes to the probability amplitude K B, A= Σ[xt] over allpossiblepaths PB, A=∣K 2,1∣2 where [xt]=Aexp{iℏS[ xt]} Feynman's path integral formula B expiℏ K B, A=∫A S[B, A]Dxt
6. 6. From Path Integral to Schrödinger equation 1 X2 X1 X4 X5 X3 X... XM­1 XA XB M m2 SM=Σ j=1 x j−x j−1 −V x j2  2 It is possible to discretized the integral on the continuum into many intervals M slices of length =∣xi1−xi∣ ∞ K x2, t2 ; x1, t1 x1,t1  x2,t2=∫−∞ K B, A=lim ∞ ∫...∫expiℏ SM[2,1] dx1 A ... dxM−1 A  On each path the discretized classical action can be written as We want use this propagator in order to obtain the wave­function at time t2 in the position x2  xi , t= 1 ∞ expiℏ A∫−∞  , xi xi−1 , tdxi−1 Lxi−xi−1
7. 7. From Path Integral to Schrödinger equation 2 xi , t= 1 Substituting the discretized action ∞ expiℏ A∫−∞ m2 exp[−i  ℏ V xi ,t ] xi , tdxi−1 We call xi−1−xi= , then send  , to zero and compare left and right at the same order  A=2 i ℏ  m 1/2 −ℏ i ∂ ∂t = −ℏ2 2m ∂2 ∂ x2V x ,t  At the order 0 we get the normalization constant At the order 1 we get the Schroedinger equation!
8. 8. Cafe Moment ∞ dx0K x ,t ,x0,0 x0,0 x ,t =∫−∞ What we have: ­> I=∫ f x1,. .., xNpx1,. .., xNdx1. ..dxN What we want: ­>
9. 9. Imaginary Time Evolution We want to solve the Schrödinger equation in imaginary time: ℏ ∂ =it ∂  = ℏ 2m ∂2 ∂ x2 [V x−Er ] The formal solution is: x , =exp[−H −ER ℏ ] x0 ,0 ∞ cnnxe− If we expand in a eigenfunction of H: x , =Σ n=0 En−ER ℏ if ER > E0 if ER < E0 if ER = E0 limt ∞   =∞ limt ∞  =o limt ∞  =0 Tree Possibility:
10. 10. From Path Integral to DMC: 1 ∞ dx0K x , , x0,0 x0,0  x ,=∫−∞ Using Feynman path integral the imaginary time evolution can be rewritten as lim N∞ ∞ ...∫−∞ ∫−∞ ∞  m 2ℏ   N/2 exp{−  N [ m ℏ Σj =1 2  xi−x j−12V xi−En ]} K x, ,x0 ,0 is equal to and as usual we rewrite this integral as K x, ,x0 ,0=lim N ∞ ∞ Π j=1 ∫−∞ N−1 dx j Σ N Wxn×Pxn , xn1 x0,0 n=1
11. 11. From Path Integral to DMC: 2 K x, ,x0 ,0=lim N ∞ ∞ Π j=1 ∫−∞ N−1 dx j Σ N Wxn×Pxn , xn−1 x0,0 n=1 2 ℏ exp[−mxn−xn−12 Pxn ,xn−1= m 2 ℏ  ] Wxn=exp[−[V xn −ER ]  2ℏ  ] A Gaussian probability distribution A Weight Function N Pxn , xn−1 Px0, x1,... xn , xN= x0,0Π i=1 N WxN f x1,... xn , xN=Π i=1 If we define: I=∫ f x1,. .., xN Px1,. .., xN dx1. ..dxN we have
12. 12. The Algorithm We want generate the probability distribution N Pxn , xn−1 Px0, x1,... xn , xN= x0,0Π i=1 and sample N WxN f x1,... xn , xN=Π i=1 Generate points distributed on (x0,0) x1 is generate from x0 sampling P(xn,xn­1 ) (a Gaussian) the weight function is evaluated W(x1) x ,∞=0 X
13. 13. An example H and H2 Convergence of the Energy H molecule versus  H atom wave­function and energy
14. 14. Application to Silicon: one body density matrix r , r '=Σi , j i , ji r j r ' i r  LDA local orbitals The matrix elements are calculated as: i , j=N∫∗iri jr ' r ' ,r 2,.... , r N r1,. .., r N ∣r1,. .., r N2∣dr 'dr 1. ..dr N
15. 15. Results on Silicon Max difference between ii QMC and LDA is 0.00625 Max off­diagonal element 0.0014(1)
16. 16. Results on Silicon: 2 QMC one­body­density 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%
17. 17. Reference SISSA Lectures on Numerical methods for strongly correlated electrons 4th draft S. Sorella G. E. Santoro and F. Becca (2008) Introduction to Diffusion Monte Carlo Method I. Kostin, B. Faber and K. Schulten, physics/9702023v1 (1995) Quantum Monte Carlo calculations of the one­body density matrix and excitation energies of silicon P. R. C. Kent et al. Phys. Rev. B 57 15293 (1998) FreeScience.info­> Quantum Monte Carlo http://www.freescience.info/books.php?id=35
18. 18. From Path Integral to Schrödinger equation: 1+1/2 xi , t= 1 Substituting the discretized action ∞ expiℏ A∫−∞  exp[−i mxi−xi−12 ℏ V xi ,t]xi−1 , tdxi−1 We call xi−1−xi= and send to zero ∂t xi , t= 1 A∫−∞ xi , t ∂ ∞ expim2 ℏ  [1−i ℏ V xi ,t...] [ xi ,t  ∂ ∂ xi  xi ,t 12 2  xi , t]dxi−1 2 ∂2 ∂ xi ∞ exp[imℏ2 1 A∫−∞ 2ℏ  ]d=1 and  ∂ ∂t =−i ℏ V − ℏ  2m ∂2 ∂ x2