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Introduction to Diffusion Monte Carlo

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- 1. Introduction to Quantum Monte Carlo Methods 2! Claudio Attaccalite http://attaccalite.com
- 2. What we learned last time How to sample a given probability p(x) distribution with Metropolis Algorithm: Axx'=min1, px'T x, x' pxT x', x How to evaluate integrals in the form: Evaluate Quantum Mechanical Operators: 〈 f 〉=∫ f xpxdx= 1N Σ f xi where xi are distributed according to p(x) 〈 f 2 〉−〈 f 〉2 f=N 〈 A〉=∫ A dx ∫ x2 dx AL x= x x =∫ AL xpxdx px= ∣x∣2 ∫∣x∣2dx
- 3. Outline Path integral formulation of Quantum Mechanics Diffusion Monte Carlo OneBody density matrix and excitation energies
- 4. Path Integral : classical action The path followed by the particle is the one that minimize: t b dt Lxt , ˙xt ;t S=∫t a where S is the Classical Action and L is the Lagrangian Lxt ,˙x t ;t=m2 ˙x t2−V xt ;t Only the extreme path contributes!!!!
- 5. Path Integral: Quantum Mechanics In quantum mechanics non just the extreme path contributes to the probability amplitude K B, A= Σ[xt] over allpossiblepaths PB, A=∣K 2,1∣2 where [xt]=Aexp{iℏS[ xt]} Feynman's path integral formula B expiℏ K B, A=∫A S[B, A]Dxt
- 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 j2 2 It is possible to discretized the integral on the continuum into many intervals M slices of length =∣xi1−xi∣ ∞ K x2, t2 ; x1, t1 x1,t1 x2,t2=∫−∞ K B, A=lim ∞ ∫...∫expiℏ 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 ∞ expiℏ A∫−∞ , xi xi−1 , tdxi−1 Lxi−xi−1
- 7. From Path Integral to Schrödinger equation 2 xi , t= 1 Substituting the discretized action ∞ expiℏ A∫−∞ m2 exp[−i ℏ V xi ,t ] xi , tdxi−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 ∂ x2V x ,t At the order 0 we get the normalization constant At the order 1 we get the Schroedinger equation!
- 8. Cafe Moment ∞ dx0K x ,t ,x0,0 x0,0 x ,t =∫−∞ What we have: > I=∫ f x1,. .., xNpx1,. .., xNdx1. ..dxN What we want: >
- 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 ∞ cnnxe− 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. 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 2ℏ N/2 exp{− N [ m ℏ Σj =1 2 xi−x j−12V 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 Wxn×Pxn , xn1 x0,0 n=1
- 11. From Path Integral to DMC: 2 K x, ,x0 ,0=lim N ∞ ∞ Π j=1 ∫−∞ N−1 dx j Σ N Wxn×Pxn , xn−1 x0,0 n=1 2 ℏ exp[−mxn−xn−12 Pxn ,xn−1= m 2 ℏ ] Wxn=exp[−[V xn −ER ] 2ℏ ] A Gaussian probability distribution A Weight Function N Pxn , xn−1 Px0, x1,... xn , xN= x0,0Π i=1 N WxN f x1,... xn , xN=Π i=1 If we define: I=∫ f x1,. .., xN Px1,. .., xN dx1. ..dxN we have
- 12. The Algorithm We want generate the probability distribution N Pxn , xn−1 Px0, x1,... xn , xN= x0,0Π i=1 and sample N WxN 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. An example H and H2 Convergence of the Energy H molecule versus H atom wavefunction and energy
- 14. Application to Silicon: one body density matrix r , r '=Σi , j i , ji r j r ' i r LDA local orbitals The matrix elements are calculated as: i , j=N∫∗iri jr ' r ' ,r 2,.... , r N r1,. .., r N ∣r1,. .., r N2∣dr 'dr 1. ..dr N
- 15. Results on Silicon Max difference between ii QMC and LDA is 0.00625 Max offdiagonal element 0.0014(1)
- 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. 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. From Path Integral to Schrödinger equation: 1+1/2 xi , t= 1 Substituting the discretized action ∞ expiℏ A∫−∞ exp[−i mxi−xi−12 ℏ V xi ,t]xi−1 , tdxi−1 We call xi−1−xi= and send to zero ∂t xi , t= 1 A∫−∞ xi , t ∂ ∞ expim2 ℏ [1−i ℏ V xi ,t...] [ xi ,t ∂ ∂ xi xi ,t 12 2 xi , t]dxi−1 2 ∂2 ∂ xi ∞ exp[imℏ2 1 A∫−∞ 2ℏ ]d=1 and ∂ ∂t =−i ℏ V − ℏ 2m ∂2 ∂ x2

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