Quantum Monte Carlo for excited state calculations

Transcription

1 Quantum Monte Carlo for excited state calculations Claudia Filippi MESA+ Institute for Nanotechnology, Universiteit Twente, The Netherlands Winter School in Theoretical Chemistry, Helsinki, Finland, Dec 2013

2 Outline Lecture 1: Continuum quantum Monte Carlo methods Variational Monte Carlo Jastrow-Slater wave functions Diffusion Monte Carlo Lecture 2: Quantum Monte Carlo and excited states Benchmark and validation: A headache FCI with random walks: A remarkable development Simple examples: Ethene and butadiene Assessment of conventional QMC From the gas phase to the protein environment Prospects and conclusions

3 Electronic structure methods Scaling Density functional theory methods N 3 Traditional post Hartree-Fock methods > N 6 Quantum Monte Carlo techniques N 4 Stochastic solution of Schrödinger equation Accurate calculations for medium-large systems Molecules of typically st/2nd-row atoms Relatively little experience with transition metals Solids (Si, C... Fe, MnO, FeO)

4 Electronic structure methods Scaling Density functional theory methods N 3 Traditional post Hartree-Fock methods > N 6 Quantum Monte Carlo techniques N 4 Stochastic solution of Schrödinger equation Accurate calculations for medium-large systems Molecules of typically st/2nd-row atoms Relatively little experience with transition metals Solids (Si, C... Fe, MnO, FeO) e.g. Geophysics: Bulk Fe with 96 atoms/cell (Alfé 2009)

5 Quantum Monte Carlo as a wave function based method Ψ(r 1,..., r N ) CASSCF, MRCI, CASPT2, CC... Can we construct an accurate but more compact Ψ? Explicit dependence on the inter-electronic distances r ij Use of non-orthogonal orbitals... Quantum Monte Carlo as alternative route

6 A different way of writing the expectation values Consider the expectation value of the Hamiltonian on Ψ E V = Ψ H Ψ Ψ Ψ = = dr HΨ(R) Ψ(R) dr Ψ (R)HΨ(R) dr Ψ (R)Ψ(R) Ψ(R) 2 dr Ψ(R) 2 E 0 = dr E L (R) ρ(r) = E L (R) ρ ρ(r) is a probability density and E L (R) = HΨ(R) Ψ(R) the local energy

7 Variational Monte Carlo: a random walk of the electrons Use Monte Carlo integration to compute expectation values Sample R from ρ(r) using Metropolis algorithm Average local energy E L (R) = HΨ(R) Ψ(R) to obtain E V as R E V = E L (R) ρ 1 M M E L (R i ) i=1 Random walk in 3N dimensions, R = (r 1,..., r N ) Just a trick to evaluate integrals in many dimensions

8 Is it really just a trick? Si 21 H 22 Number of electrons = 106 Number of dimensions = 318 Integral on a grid with 10 points/dimension points! MC is a powerful trick Freedom in form of the wave function Ψ

9 Are there any conditions on many-body Ψ to be used in VMC? Within VMC, we can use any computable wave function if Continuous, normalizable, proper symmetry Finite variance σ 2 = Ψ (H E V ) 2 Ψ Ψ Ψ = (E L (R) E V ) 2 ρ since the Monte Carlo error goes as err(e V ) σ M Zero variance principle: if Ψ Ψ 0, E L (R) does not fluctuate

10 Expectation values in variational Monte Carlo: The energy E V = Ψ H Ψ Ψ Ψ = E V = σ 2 = dr HΨ(R) Ψ(R) Ψ(R) 2 dr Ψ(R) 2 = dr E L (R) ρ(r) dr(e L (R) E V ) 2 ρ(r) dr E L (R) ρ(r) Estimate E V and σ from M independent samples as Ē V = 1 M σ 2 = 1 M M E L (R i ) i=1 M (E L (R i ) ĒV ) 2 i=1

11 Typical VMC run Example: Local energy and average energy of acetone (C 3 H 6 O) Energy (Hartree) σ VMC MC step E VMC = E L (R) ρ = ± Hartree ( steps) σ VMC = (E L (R) E VMC ) 2 ρ = 0.90 Hartree

12 Variational Monte Carlo and the generalized Metropolis algorithm How do we sample distribution function ρ(r) = Ψ(R) 2 dr Ψ(R) 2? Aim Obtain a set of {R 1, R 2,..., R M } distributed as ρ(r) Generate a Markov chain Start from arbitrary initial state R i Use stochastic transition matrix P(R f R i ) P(R f R i ) 0 P(R f R i ) = 1. R f as probability of making transition R i R f R Evolve the system by repeated application of P

13 How do we sample ρ(r)? To sample ρ, use P which satisfies stationarity condition : P(R f R i ) ρ(r i ) = ρ(r f ) R f i If we start with ρ, we continue to sample ρ Stationarity condition + stochastic property of P + ergodicity Any initial distribution will evolve to ρ

14 A more stringent condition In practice, we impose detailed balance condition P(R f R i ) ρ(r i ) = P(R i R f ) ρ(r f ) Stationarity condition can be obtained by summing over R i P(R f R i ) ρ(r i ) = P(R i R f ) ρ(r f ) = ρ(r f ) i i }{{} 1 Detailed balance is a sufficient but not necessary condition

15 How do we construct the transition matrix P in practice? Write transition matrix P as proposal T acceptance A P(R f R i ) = A(R f R i ) T (R f R i ) Let us rewrite the detailed balance condition P(R f R i ) ρ(r i ) = P(R i R f ) ρ(r f ) A(R f R i ) T (R f R i ) ρ(r i ) = A(R i R f ) T (R i R f ) ρ(r f ) A(R f R i ) A(R i R f ) = T (R i R f ) ρ(r f ) T (R f R i ) ρ(r i )

16 Choice of acceptance matrix A Original choice by Metropolis et al. maximizes the acceptance { A(R f R i ) = min 1, T (R } i R f ) ρ(r f ) T (R f R i ) ρ(r i ) Note: ρ(r) does not have to be normalized For complex Ψ we do not know the normalization! ρ(r) = Ψ(R) 2 Δ" Original Metropolis method { Symmetric T (R f R i ) = 1/ 3N A(R f R i ) = min 1, ρ(r } f) ρ(r i )

17 Better choices of proposal matrix T Sequential correlation M eff < M independent observations M eff = M T corr with T corr autocorrelation time of desired observable Aim is to achieve fast evolution of the system and reduce T corr Use freedom in choice of T : For example, use available trial Ψ T (R f R i ) = N exp [ (R f R i V(R i )τ) 2 ] 2τ with V(R i ) = Ψ(R i) Ψ(R i )

18 Acceptance and T corr for the total energy E V Example: All-electron Be atom with simple wave function Simple Metropolis T corr Ā Drift-diffusion transition τ T corr Ā Δ" 2 2τ

19 Typical VMC run Example: Local energy and average energy of acetone (C 3 H 6 O) Energy (Hartree) σ VMC MC step E VMC = E L (R) ρ = ± Hartree ( steps) σ VMC = (E L (R) E VMC ) 2 ρ = 0.90 Hartree

20 Variational Monte Carlo Freedom in choice of Ψ Monte Carlo integration allows the use of complex and accurate Ψ More compact representation of Ψ than in quantum chemistry Beyond c 0 D HF + c 1 D 1 + c 2 D too many determinants

21 Jastrow-Slater wave function Ψ(r 1,..., r N ) = J (r 1,..., r N ) k d k D k (r 1,..., r N ) J Jastrow correlation factor - Positive function of inter-particle distances - Explicit dependence on electron-electron distances - Takes care of divergences in potential ck D k Determinants of single-particle orbitals - Few and not millions of determinants as in quantum chemistry - Determines the nodal surface

22 Jastrow factor and divergences in the potential At interparticle coalescence points, the potential diverges as Z r iα 1 r ij for the electron-nucleus potential for the electron-electron potential Local energy HΨ Ψ = 1 2 i Ψ + V must be finite 2 Ψ i Kinetic energy must have opposite divergence to the potential V

23 Divergence in potential and Kato s cusp conditions Finite local energy as r ij 0 Ψ must satisfy: ˆΨ r ij rij =0 = µ ij q i q j Ψ(r ij = 0) where ˆΨ is a spherical average (we assumed Ψ(r ij = 0) 0) Electron-nucleus: µ = 1, q i = 1, q j = Z Ψ Ψ = Z r=0 Electron-electron: µ = 1 2, q i = 1, q j = 1 Ψ Ψ = 1/2 r=0

24 Cusp conditions and QMC wave functions Electron-electron cusps imposed through the Jastrow factor Example: Simple Jastrow factor J (r ij ) = i<j { exp b 0 r ij 1 + b r ij } with b 0 = 1 2 or b 0 = b 0 = 1 4 Imposes cusp conditions + keeps electrons apart 0 r ij Electron-nucleus cusps imposed through the determinantal part

25 The effect of the Jastrow factor Pair correlation function for electrons in the (110) plane of Si g (r, r ) with one electron is at the bond center Hood et al. Phys. Rev. Lett. 78, 3350 (1997)

26 Some comments on Jastrow factor More general Jastrow form with e-n, e-e and e-e-n terms exp {A(r iα )} exp {B(r ij )} exp {C(r iα, r jα, r ij )} α,i i<j α,i<j Polynomials of scaled variables, e.g. r = r/(1 + ar) J > 0 and becomes constant for large r i, r j and r ij Is this Jastrow factor adequate for multi-electron systems? Beyond e-e-n terms? Due to the exclusion principle, rare for 3 or more electrons to be close, since at least 2 electrons must have the same spin

27 Jastrow factor with e-e, e-e-n and e-e-e-n terms J E VMC E corr Li E HF VMC (%) σ VMC e-e (4) e-e-n (1) e-e-e-n (1) E exact Ne E HF e-e (2) e-e-n (1) e-e-e-n (3) E exact Huang, Umrigar, Nightingale, J. Chem. Phys. 107, 3007 (1997)

28 Dynamic and static correlation Ψ = Jastrow Determinants Two types of correlation Dynamic correlation Described by Jastrow factor Due to inter-electron repulsion Always present Static correlation Described by a linear combination of determinants Due to near-degeneracy of occupied and unoccupied orbitals Not always present

29 Construction of the wave function How do we obtain the parameters in the wave function? Ψ(r 1,..., r N ) = J k d k D k C 10 N 2 O 2 H 7 70 electrons and 21 atoms VTZ s-p basis + 1 p/d polarization Parameters in the Jastrow factor J ( 100) CI coefficients d k (< ) Linear coefficients in expansion of the orbitals (5540!)

30 Optimization of wave function by energy minimization The three most successful approaches Newton method (Umrigar and Filippi, 2005) E(α) E(α 0 ) + i E(α 0 ) α i E(α 0 ) α i j α i 2 α i α j i,j Linear method (Umrigar, Toulouse, Filippi, and Sorella, 2007) Ψ(α) Ψ(α 0 ) + i Ψ(α 0 ) α i α i Solution of H α = ES α in the basis of {Ψ(α 0 ), Ψ(α0 ) α i } Perturbative approach (Scemama and Filippi, 2006)

31 What is difficult about wave function optimization? Statistical error: Both a blessing and a curse! Ψ {αk } Energy and its derivatives wrt parameters {α k } E V = k E V = dr HΨ(R) Ψ(R) Ψ(R) 2 dr Ψ(R) 2 = E L Ψ 2 k Ψ Ψ E L + H kψ Ψ 2E V k Ψ = 2 Ψ (E L E V ) Ψ 2 k Ψ Ψ Ψ 2 The last expression is obtained using Hermiticity of H

32 Use gradient/hessian expressions with smaller fluctuations Two mathematically equivalent expressions of the energy gradient k E V = k Ψ Ψ E L + H kψ Ψ 2E V k Ψ k Ψ = 2 Ψ Ψ Ψ (E L E V ) 2 Ψ 2 Why using the last expression? δe Ψ =Ψ 0 Lower fluctuations 0 as Ψ Ψ 0 0 If you play similar tricks with the Hessian as with the gradient 5 orders of magnitude efficiency gain wrt using original Hessian C. Umrigar and C. Filippi, PRL 94, (2005)

33 Beyond VMC? Removing or reducing wave function bias? Projection Monte Carlo methods

34 Why going beyond VMC? Dependence of VMC from wave function Ψ D electron gas at a density r s =10 Energy (Ry) DMC JS VMC JS+BF VMC JS+3B+BF VMC JS+3B VMC JS DMC JS+3B+BF Variance ( r s (Ry/electron) 2 x ) Kwon, Ceperley, Martin, Phys. Rev. B 58, 6800 (1998)

35 Why going beyond VMC? Dependence on wave function: What goes in, comes out! Can we remove wave function bias? Projector Monte Carlo methods Construct an operator which inverts spectrum of H Use it to stochastically project the ground state of H Diffusion Monte Carlo exp[ τ(h E T )] Green s function Monte Carlo 1/(H E T ) Power Monte Carlo E T H

36 Diffusion Monte Carlo Consider initial guess Ψ (0) and repeatedly apply projection operator Ψ (n) = e τ(h ET) Ψ (n 1) Expand Ψ (0) on the eigenstates Ψ i with energies E i of H Ψ (n) = e nτ(h E T) Ψ (0) = i Ψ i Ψ (0) Ψ i e nτ(e i E T ) and obtain in the limit of n lim n Ψ(n) = Ψ 0 Ψ (0) Ψ 0 e nτ(e 0 E T ) If we choose E T E 0, we obtain lim n Ψ(n) = Ψ 0

37 How do we perform the projection? Rewrite projection equation in integral form Ψ(R, t + τ) = dr G(R, R, τ)ψ(r, t) where G(R, R, τ) = R e τ(h E T) R Can we sample the wave function? For the moment, assume we are dealing with bosons, so Ψ > 0 Can we interpret G(R, R, τ) as a transition probability? If yes, we can perform this integral by Monte Carlo integration

38 What can we say about the Green s function? G(R, R, τ) = R e τ(h E T) R G(R, R, τ) satisfies the imaginary-time Schrödinger equation (H E T )G(R, R 0, t) = G(R, R 0, t) t with G(R, R, 0) = δ(r R)

39 Can we interpret G(R, R, τ) as a transition probability? (1) H = T Imaginary-time Schrödinger equation is a diffusion equation G(R, R 0, t) = G(R, R 0, t) t The Green s function is given by a Gaussian G(R, R, τ) = (2πτ) 3N/2 exp [ (R R) 2 ] 2τ Positive and can be sampled

40 Can we interpret G(R, R, τ) as a transition probability? (2) H = V (V(R) E T )G(R, R 0, t) = G(R, R 0, t) t The Green s function is given by G(R, R, τ) = exp [ τ (V(R) E T )] δ(r R ), Positive but does not preserve the normalization It is a factor by which we multiply the distribution Ψ(R, t)

41 H = T + V and a combination of diffusion and branching Let us combine previous results G(R, R, τ) (2πτ) 3N/2 exp [ (R R) 2 ] exp [ τ (V(R) E T )] 2τ Diffusion + branching factor leading to survival/death/cloning Why? Trotter s theorem e (A+B)τ = e Aτ e Bτ + O(τ 2 ) Green s function in the short-time approximation to O(τ 2 )

42 Diffusion and branching in a harmonic potential V(x) Ψ(x) 0 Walkers proliferate/die in regions of lower/higher potential than E T

43 Time-step extrapolation DMC results must be extrapolated at short time-steps (τ 0) Example: Energy of Li 2 versus time-step τ

44 Problems with simple DMC algorithm G(R, R, τ) exp [ (R R) 2 ] exp [ τ (V(R) E T )] 2τ Potential can vary a lot, unbounded Exploding/dying population Solution Work with f (R, t) = Ψ T (R)Ψ(R, t) Evolution governed by drift-diffusion-branching Green s function G(R, R, τ) exp [ (R R τv(r)) 2 ] exp [ τ (E L (R) E T )] 2τ Local energy E L (R) is better behaved than the potential After many iterations, walkers distributed as Ψ T (R)Ψ 0 (R)

45 Electrons are fermions! We assumed that Ψ 0 > 0 and that we are dealing with bosons Fermions Ψ is antisymmetric and changes sign! Fermion Sign Problem All fermion QMC methods suffer from sign problems These sign problems look different but have the same flavour Arise when you treat something non-postive as probability density

46 The DMC Sign Problem How can we impose antisymmetry in simple DMC method? Idea Evolve separate positive and negative populations of walkers Simple 1D example: Antisymmetric wave function Ψ(x, τ = 0) Rewrite Ψ(x, τ = 0) as Ψ(x, τ=0) Ψ = Ψ + Ψ where Ψ + = 1 ( Ψ + Ψ) 2 Ψ(x, τ=0) Ψ(x, τ=0) Ψ = 1 2 ( Ψ Ψ) +

47 Particle in a box and the fermionic problem (1) The imaginary-time Schrödinger equation HΨ = Ψ t is linear, so solving it with the initial condition Ψ(x, t = 0) = Ψ + (x, t = 0) Ψ (x, t = 0) is equivalent to solving HΨ + = Ψ + t and HΨ = Ψ t separately and subtracting one solution from the other

48 Particle in a box and the fermionic problem (2) Since E s 0 < E a 0, both Ψ + and Ψ evolve to Ψ s 0 Ψ ± Antisymmetric component exponentially harder to extract Ψ + Ψ Ψ + + Ψ e E a 0 t e E s 0 t as t

49 The Fixed-Node Approximation Problem Small antisymmetric part swamped by random errors Solution Fix the nodes! (If you don t know them, guess them) impenetrable barrier

50 Fixed-node algorithm in simple DMC How do we impose that additional boundary condition? Annihilate walkers that bump into barrier (and into walls) This step enforces Ψ = 0 boundary conditions In each nodal pocket, evolution to ground state in pocket Numerically stable algorithm (no exponentially growing noise) Solution is exact if nodes are exact The computed energy is variational if nodes approximate Best solution consistent with the assumed nodes

51 For many electrons, what are the nodes? A complex beast Many-electron wave function Ψ(R) = Ψ(r 1, r 2,..., r N ) Node (dn 1)-dimensional surface where Ψ = 0 and across which Ψ changes sign A 2D slice through the 321-dimensional nodal surface of a gas of 161 spin-up electrons.

52 Use the nodes of trial Ψ T Fixed-node approximation Use the nodes of the best available trial Ψ T wave function Ψ(R)>0 R Ψ(R)=0 Find best solution with same nodes as trial wave function Ψ T Fixed-node solution exact if the nodes of trial Ψ T are exact

53 Have we solved all our problems? Results depend on the nodes of the trail wave function Ψ Diffusion Monte Carlo as a black-box approach? ɛ MAD for atomization energy of the G1 set DMC CCSD(T)/aug-cc-pVQZ HF orb Optimized orb CAS ɛ MAD kcal/mol Petruzielo, Toulouse, Umrigar, J. Chem. Phys. 136, (2012) With some effort on Ψ, we can do rather well

54 has yet to be fully tested. Here, the final FN-DMC results agree to within 0.25 kcal/mol with the best available estimates. These results show the potential of QMC for reliable estimation of Diffusion noncovalent molecular Monte interaction energies Carlo well below aschemical a black-box approach? accuracy. The calculations were performed on a diverse set of hydrogen and/or dispersion bound complexes for which reliable estimates of interaction energies already exist 8,39,40 Non-covalent and which were previouslyinteraction studied within QMC. energies for 6 compounds from S22 set 26,29,34,35 by au), in agreement with previous experience. 15,41 The considered test set consists of the dimers of ammonia, water, hydrogen fluoride, methane, ethene, and the ethene/ethyne complex (Figure 2). The larger considered complexes include DMC benzene/methane, with benzene/water, B3LYP/aug-cc-PVTZ and T-shape benzene dimer interaction orbitals energies of 3.33 versus ± 0.07 and 3.47 CCSD(T)/CBS ± 0.07, whereas (Figure 2). Figure 2. The set of molecules used in the present work (from top left, to bottom right): ammonia dimer, water dimer, hydrogen fluoride dimer, methane dimer, ethene dimer, and the complexes of ethene/ ethyne, benzene/methane, benzene/water, and benzene dimer T- shape. ADJUSTING THE QMC PROTOCOL The present methodology was developed via extensive testing and elimination of the biases that affect the final FN-DMC energies. Clearly, this has to be done in a step-by-step manner Orbital sets from other methods were mostly comparable; in the ammonia dimer complex, for instance, the HF nodes provide the same FN-DMC interaction energy as B3LYP ( 3.12 ± 0.07 vs 3.10 ± 0.06 kcal/mol) within the error bars, due to the FN error cancellation 26,28,29 (cf. Figure 1). Nevertheless, the total energies from B3LYP orbitals were found to be variationally lower than those from HF (in dimer Regarding the one-electron basis set, tests on the ammonia dimer confirm the crucial effect of augmentation functions (cf. ref 29). For the same system, TZV and QZV bases result in MAD = kcal/mol the aug-tzv and aug-qzv bases give 3.10 ± 0.06 and 3.13 ± 0.6 kcal/mol, so that the impact of augmentation is clearly visible and in accord with the reference value of 3.15 kcal/ mol. 40 On the other hand, the increase of basis set cardinality beyond the TZV level plays a smaller role than in the mainstream correlated wave function methods. In order to reduce the numerical cost of the calculations, effective core potentials (ECP) were employed for all elements (cf. Methods). Typically, this causes a mild dependence of the FN-DMC total energy on the Jastrow factor, 42,43 which cancels out in energy differences with an accuracy 1 kcal/mol. In our systems, elimination of this source of bias requires fully converged Jastrow factors including electron electron, electron nucleus, and electron electron nucleus terms so as to keep the target of 0.1 kcal/mol margin in energy differences. This is true except for the water dimer, where a standard Jastrow factor produces inaccurate energy difference ( 5.26 ± 0.09 kcal/mol, cf. Table 1), and a distinct Jastrow factor including unique parameter sets for nonequivalent atoms of the same type is required. 44 For the sake of completeness, we note that the model of ammonia dimer, taken from the S22 set, 39 is not a genuine hydrogen bonded case, where the same behavior would be expected, but a symmetrized transition structure that apparently does not require more parameters in the Jastrow factor. Note that a more economic variant of the correlation factor, with only electron electron and electron nucleus terms, doubles the average error on the considered test set, and therefore it would be inadequate for our purposes. 44 The parameters of the Jastrow factor were exhaustively optimized for each complex and its constituents separately, using a linear combination of energy and variance cost function. 45 We have found that for large complexes, 7 10 iterations of VMC optimization are sometimes necessary to reach the full convergence. The production protocol thus consists of (i) Slater Jastrow Dubecky et al., J. Chem. Theory Comput. 9, 4287 (2013) With practically no effort on Ψ, we can do rather well

55 Diffusion Monte Carlo as a black-box approach? Fragmentation energies of N 2 H 4, HNO 2, CH 3 OH, and CH 3 NH 2 MAD MAX (kcal/mol) DMC/cc-pVTZ 1 det DMC/cc-pVTZ J-LGVB CCSD(T)/cc-pVTZ CCSD(T)/cc-pVQZ CCSD(T)/cc-pV5Z Fracchia, Filippi, and Amovilli, J. Chem. Theory Comput. 8, 1943 (2012) Novel form of Ψ! +...

56 Alternatives to fixed-node DMC: Determinantal QMC Given single-particle basis, perform projection in determinant space Different way to deal with fermionic problem Determinantal QMC by Zhang and Krakauer Appears less plagued by fixed phase than DMC by FN Full-CI QMC by Alavi (see tomorrow) Start from Ψ CI = i c id i HΨ = Ψ t H ijc j = c i t

57 DMC in summary The fixed-node DMC method is Easy to do Stable Accurate enough for many applications in quantum chemistry... especially in large systems Not accurate enough for subtle correlation physics

58 Beauty of quantum Monte Carlo Highly parallelizable Ψ(r 1,..., r N ) Ensemble of walkers diffusing in 3N dimensions VMC Independent walkers Trival parallelization DMC Nearly independent walkers Few communications Easily take great advantage of parallel supercomputers! VOLUME 87, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 10 DECEMBER 2001 CPU Time (s) Si x H y Carbon { Number of electrons (a) MLW (b) Gaussian (c) Plane Wave (d) MLW Williamson, Hood, Grossman (2001) FIG. 2. CPU time on a 667 MHz EV67 alpha processor to move a configuration of electrons within DMC for SiH4, Si5H12, Si35H36, Si87H76, Si123H100, Si211H140, C20, C36, C60, C80, and C180. the Slater determinant allows us to significantly reduce the prefactor for this N 3 term. In larger systems where the determinant is increasingly sparse, it should be possible to reformulate the determinant update procedure to utilize this Up sparsity toand Siobtain 123 Ha better 100 size and scaling. C 180! Optical Emission of Silicon Nanocrystals Note, the discussion thus far involves the scaling of the computational cost of moving a single configuration of electrons. In practice, one calculates either (i) the total energy of the system, or (ii) the energy per atom, with a given statistical error. The statistical error, d, is related to the number of uncorrelated moves, M, by d s p M, where s 2 is the intrinsic variance of the system. Typically, the value of s 2 increases linearly with system size. Therefore, to calculate the total energy with a fixed d, the number of moves, M, must also increase linearly. When multiplied by our linear increase in the cost of each move, an N 2 size scaling is obtained. For quantities per atom, such as the binding energy of a bulk solid, s 2 still increases linearly with system size, but d is decreased by a

59 Human and computational cost of a typical QMC calculation Task Human time Computer time Choice of basis set, pseudo etc. 10% 5% DFT/HF/CI runs for Ψ setup 65% 10% Optimization of Ψ 20% 30% DMC calculation 5% 55%

60 Outlook on QMC Subjects of ongoing research Search for different forms of trial wave function Interatomic forces (see tomorrow) Let us attack transition metals! Alternatives to fixed-node DMC (see tomorrow)

61 Other applications of quantum Monte Carlo methods Electronic structure calculations Strongly correlated systems (Hubbard, t-j,...) Quantum spin systems (Ising, Heisenberg, XY,...) Liquid-solid helium, liquid-solid interface, droplets Atomic clusters Nuclear structure Lattice gauge theory Both zero (ground state) and finite temperature