Wave Function Optimization and VMC Jeremy McMinis The University of Illinois July 26, 2012
Outline Motivation History of Wave Function Optimization Optimization in QMCPACK Multideterminant Example
Motivation: Variational Quantum Monte Carlo VMC Computes observables over a trial wave function Ô ψ 2 = dr ψ T (R) Ô ψ T (R) dr ψt (R) 2 The trial wave function is everything. It determines all observables! What are the errors in observables due to the trial wave function?
Motivation: Trial Wave Function Error VMC Energy: E v = ψ T Ĥ ψ T ψ T ψ T Trial Wave Function Error: VMC Energy Error: = ψ T = ψ 0 + δψ ĤψT ψ T ψ 2 T = E L ψ 2 T VMC Energy Variance: E = E L E 0 ψ 2 T = δψ Ĥ E 0 δψ ψ T ψ T = O[δψ 2 ] σ E = (E L E 0 ) 2 ψ 2 T = δψ (Ĥ E 0 ) 2 δψ ψ T ψ T = O[δψ 2 ]
Motivation: Trial Wave Function Error VMC Observables: O v = ψ T Ô ψ T ψ T ψ T VMC Observable Error: = ÔψT ψ T ψ 2 T = O L ψ 2 T O = O L O 0 ψ 2 T VMC Observable Variance: = ψ 0 2(Ô O 0) δψ ψ T ψ T = O[δψ] σ O = (O L O 0 ) 2 ψ 2 T = ψ 0 2(Ô O 0 ) 2 δψ ψ T ψ T Improve estimators and trial function! = O[1]
Motivation: Energy and Variance σ E = O[δψ 2 ], E = O[δψ 2 ] Kwon, Ceperley, Martin. Phys. Rev. B 48, 12037 (1993)
History Historical Tour of Wave Function Optimization By hand Variance Minimization: Conjugate Gradient Energy Minimization: Linear Method
History: First VMC Calculation McMillan. Phys. Rev. 138, A442 A451 (1965)
History: First Wave Function Optimization Lennard-Jones Potential V (r) = 4ɛ ( (σ/r) 12 (σ/r) 6) r = r ij = r i r j Wave function ψ T (R) = f (r ij ) i<j f (r) = exp( (a 1 /r) a 2 ) McMillan. Phys. Rev. 138, A442 A451 (1965)
History: Variance Minimization Umrigar, Wilson,Wilkins. Phys. Rev. Lett. 60, 1719 1722 (1988) http://apps.webofknowledge.com/full record.do?product=ua&qid=2&sid=2dioelij7c5jdplaai9
History: Variance Minimization Parameterized trial wave function, ψ T (p) = J (p j ) p kl D k D l, p = {p j, p kl } The Cost function, C(p): C(p, c) = c 1 (EL (p) E L (p) ) 2 ψ 2 T (p) +c 2 E L (p) E target ψ 2 T (p) +... Filippi, Umrigar. J. Chem. Phys. 105, 213 (1996) Minimize C(p) using anything (e.g. Levenberg-Marquardt) O[10 1 ] O[10 2 ] parameters
History: Variance Minimization Limits Complete basis set limit: σ and E L Minimization are the same. Finite Sample: Variance bound from below, Energy is not (EL (p) E L (p) ) 2 ψt 2 0 E L (p) E target ψ 2 T (p) 0 Simple example, H atom, 1 sample: ψ T (p, r) = exp( pr), E L (p) E 0 ψ 2 T (p) > r = ɛ E L (p, ɛ) E 0 = p2 2 + p 1 ɛ + 1 2 p opt = 1/ɛ E L (1/ɛ, ɛ) E 0 = 1 ɛ ( 1 2ɛ 1) + 1 2
History: Variance Minimization Problems Snajdr, Rothstein. J. Chem. Phys. 112, 4935 (2000)
History: Linear Method Umrigar, Toulouse, Filippi, Sorella, Hennig. Phys. Rev. Lett. 98, 110201 (2007)
Linear Method Using orthogonalized first order Taylor series expansion: ψ i (p 0, R) = p i ψ(p, R) p=p 0 ψ i (p 0, R) = ψ i (p 0, R) ψ i ψ ψ(p 0, R) ψ lin (p, R) = ψ(p 0, R) + p i ψ i (p 0, R) Build a generalized eigenvalue problem: H ij = ψ i ψ Ĥψ j ψ N p i=1 H p = E lin S p ψ 2, S ij = ψ i ψ Algorithm stabilized by H ii = H ii + (1 δ i0 ) exp λ ψ j ψ ψ 2
Linear Method For linear parameters (e.g. determinant coefficients) 1. Solve generalized eigenvalue problem: H p = E lin S p 2. New parameters are p = p 0 + p.
Linear Method For non-linear parameters (e.g. Jastrow, backflow, orbital, etc. coefficients) unknown normalization for wave function. ψ i (p 0, R) = p i N (p)ψ(p, R) p=p 0 ψ lin (p, R) = ψ(p 0, R) + N (p) p i ψ i (p 0, R) Once p is found, rescaling is admitted. p = p 0 + α p N p i=1 α set by ψ lin = ψ or line minimization.
QMCPACK Linear Algorithm C(c, p) = c E E L (p) + c σ (E L (p) E L (p) ) 2 While C new (c, p) C old (c, p) > ɛ Run VMC Fill H, S Generate {w i } Invert S H S 1 H While max( p i ) > p max H ii = H ii + (1 δ i0 )e λ Solve H p = E lin S p Quartic fit minimize, C(c, p + α p) Over {w i }, for α opt C new = C(c, p + α opt p)
Multideterminant Example Massive Multideterminant Expansion: ψ(p) = J (p j ) p kl D k D l, p = {p j, p kl } Number of CSF: 3461, Number of Determinants: 18427 Clark, Morales, JM, Kim, Scuseria. J. Chem. Phys. 135, 244105 (2011) Morales, JM, Clark, Kim, Scuseria. J. Chem. Theory Comput., 2012, 8 (7), pp 2181 2188
Multideterminant Example Morales, JM, Clark, Kim, Scuseria. J. Chem. Theory Comput., 2012, 8 (7), pp 2181 2188
Conclusions What we ve learned: How errors in the wave function effect observables Some history of wave function optimization Current state of the art: Linear method Optimization in QMCPACK and multideterminant expansions give excellent results
Beyond VMC Wave Function Optimization Problems: Fixed Node Error Must VMC and DMC energy minimum coincide? No! Usually if VMC energy is better, so is DMC. Methods: Overlap Maximization / Self Healing Energy minimization in DMC Direct nodal optimization Released Node