Optimization of quantum Monte Carlo (QMC) wave functions by energy minimization Julien Toulouse, Cyrus Umrigar, Roland Assaraf Cornell Theory Center, Cornell University, Ithaca, New York, USA. Laboratoire de Chimie Théorique, Université Pierre et Marie Curie, Paris, France. Email: toulouse@tc.cornell.edu Web page: www.lct.jussieu.fr/pagesperso/toulouse/ February 2007
Introduction To fully benefit of the considerable flexibility in the form of the many-body wave functions used in QMC, it is crucial to be able to efficiently optimize their parameters. So far, variance minimization in correlated sampling [1] has been the most frequently used optimization method because it is far more efficient than straightforward energy minimization. Recently, clever energy minimization techniques have however started to appear [2-8] in order to overcome the limitations of variance minimization. We present here the development and the application of a new, very robust and efficient energy minimization method [9-10] to simultaneously optimize all parameters in QMC wave functions.
Parametrized many-body QMC wave functions We use standard Jastrow-Slater wave functions : N CSF Ψ(p) = J(α) c i C i (λ) where J(α) is a Jastrow factor and C i (λ) is a configuration state function (CSF), itself consisting of a symmetry-adapted linear combination of Slater determinants. i=1 The Slater determinants are constructed from orbitals expanded in a (localized) one-electron basis : φ µ (r) = N basis ν=1 λ µν χ ν (r) The parameters p to be optimized are the Jastrow parameters α, the CSF coefficients c and the orbital coefficients λ. (For the orbitals, in practice, we optimize a set of non-redundant orbital rotation parameters.)
Optimization of QMC wave functions Optimization is important both in VMC and in DMC for reducing the systematic bias due to the trial wave function reducing the statistical uncertainty How to optimize? Until recently : minimization of the variance of the local energy σ 2 = E L (R) 2 E L (R) 2 OK for Jastrow parameters but does not work well for the CSF and orbital parameters Today, better : minimization of the total energy E L (R) the total energy is a better criterion works well for all parameters
Linear energy minimization method in VMC [9,10] Expansion of the wave function around p 0 to first order in p : Ψ [1] (p) = Ψ 0 + i p i Ψ i where Ψ 0 = Ψ(p 0 ) and Ψ i = Ψ(p 0 )/ p i. Normalization of wave function chosen so that the derivatives Ψ i are orthogonal to Ψ 0. Minimization of the energy = generalized eigenvalue equation : min p Ψ [1] (p) Ĥ Ψ[1] (p) Ψ [1] (p) Ψ [1] (p) where H ij = Ψ i Ĥ Ψ j and S ij = Ψ i Ψ j. = H p = E S p Use a non-symmetric Monte Carlo estimator of the Hamiltonian matrix H ij reducing the fluctuations on p according to Nightingale s strong zero-variance principle [4]. Stabilization : H ij H ij + a δ ij (1 δ i0 ) where a 0.
Simultaneous optimization of all parameters in VMC Optimization of 24 Jastrow parameters, 49 CSF parameters and 64 orbital parameters for the C 2 molecule : -75.4-75.855 Energy (Hartree) -75.5-75.6-75.7-75.8 Energy (Hartree) -75.86-75.865-75.87-75.875-75.88 2 3 4 5 6 Iterations -75.9 0 1 2 3 4 5 6 Iterations = The energy converges with an accuracy of 10 3 Hartree in about 4 iterations
Systematic improvement in quantum Monte Carlo For C 2 molecule : VMC and DMC total energies for a series of fully optimized Jastrow-Slater wave functions : -75.8-75.82 Energy (Hartree) -75.84-75.86-75.88-75.9 VMC DMC CCSD(T)/cc-pVQZ -75.92 Exact -75.94 J*SD J*CAS(8,5) J*CAS(8,7) J*CAS(8,8) J*RAS(8,26) Wave function = Systematic improvement in VMC and DMC!
Well depths of second-row homonuclear diatomic molecules VMC and DMC errors in well depths for fully optimized single-determinant (SD) and multi-determinant (CAS) Jastrow-Slater wave functions : Error in well depth (ev) 0-0.5-1 DMC J SD DMC J CAS VMC J CAS -1.5 VMC J SD MCSCF CAS Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Molecules = Near chemical accuracy in DMC with Jastrow CAS
Illustration : QMC calculations of intracule densities Intracule density (system-average spherical-average pair density) I(u) = dωu drψ(r) 2 δ(r ij u) 4π i<j Usual histogram estimator with large statistical uncertainties : I(u) i<j dωu 4π drψ(r) 2 1 [u u/2, u+ u/2](r ij ) u 3 In this work : improved estimator with much smaller statistical uncertainties : I(u) = 1 dωu drψ(r) 2 r j Ψ(R) r ij u 2π 4π Ψ(R) r ij u 3 i<j (+ refinements)
Correlation hole of the C 2 molecule VMC calculations of 4πu 2 [I(u) I HF (u)] using a series of Jastrow-Slater wave functions : 4 π u 2 [ I (u) - I HF (u) ] (a.u.) 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2 Jastrow HF Jastrow SD Jastrow CAS(8,5) Jastrow CAS(8,7) Jastrow CAS(8,8) 0 1 2 3 4 5 u (a.u.) = Systematic improvement of the correlation hole
Conclusions Summary Efficient wave function optimization methods by energy minimization in VMC are now available. Achievement of systematic improvement in VMC and DMC as the number of variational parameters increases thanks to proper optimization. Perspectives Optimization of the exponents of the basis functions. Direct minimization of the DMC energy. Geometry optimization.
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