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 Outline

I Lecture 1: Continuum quantum Monte Carlo methods

− Variational Monte Carlo − Jastrow-Slater wave functions − Diffusion Monte Carlo

I 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 e.g. Geophysics: Bulk Fe with 96 atoms/cell (Alf´e2009)

Electronic structure methods Scaling − Density functional theory methods N3

− Traditional post Hartree-Fock methods > N6

− Quantum Monte Carlo techniques N4

Stochastic solution of Schr¨odingerequation Accurate calculations for medium-large systems → Molecules of typically 10-50 1st/2nd-row atoms → Relatively little experience with transition metals → Solids (Si, C ... Fe, MnO, FeO) Electronic structure methods Scaling − Density functional theory methods N3

− Traditional post Hartree-Fock methods > N6

− Quantum Monte Carlo techniques N4

Stochastic solution of Schr¨odingerequation Accurate calculations for medium-large systems → Molecules of typically 10-50 1st/2nd-row atoms → Relatively little experience with transition metals → Solids (Si, C ... Fe, MnO, FeO) e.g. Geophysics: Bulk Fe with 96 atoms/cell (Alf´e2009) Quantum Monte Carlo as a wave function based method

Ψ(r1,..., rN ) → CASSCF, MRCI, CASPT2, CC ...

Can we construct an accurate but more compact Ψ?

Explicit dependence on the inter-electronic distances rij Use of non-orthogonal orbitals ...

→ Quantum Monte Carlo as alternative route A different way of writing the expectation values

Consider the expectation value of the Hamiltonian on Ψ

hΨ|H|Ψi R dR Ψ∗(R)HΨ(R) E = = ≥ E V hΨ|Ψi R dR Ψ∗(R)Ψ(R) 0

Z HΨ(R) |Ψ(R)|2 = dR Ψ(R) R dR|Ψ(R)|2

Z ←− = dR EL(R) ρ(R)= hEL(R)iρ

HΨ(R) ρ(R) is a probability density and E (R) = the local energy L Ψ(R) 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 HΨ(R) . Average local energy E (R) = to obtain E as L Ψ(R) V

M 1 X E = hE (R)i ≈ E (R ) V L ρ M L i i=1 R

Random walk in 3N dimensions, R = (r1,..., rN )

Just a trick to evaluate integrals in many dimensions Is it really “just” a trick?

Si21H22

Number of electrons 4 × 21 + 22 = 106 Number of dimensions 3 × 106 = 318

Integral on a grid with 10 points/dimension → 10318 points!

MC is a powerful trick ⇒ Freedom in form of the wave function Ψ 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

hΨ|(H − E )2|Ψi σ2 = V = h(E (R) − E )2i hΨ|Ψi L V ρ

σ since the Monte Carlo error goes as err(EV ) ∼ √ M

Zero variance principle: if Ψ → Ψ0, EL(R) does not fluctuate Expectation values in variational Monte Carlo: The energy

hΨ|H|Ψi Z HΨ(R) |Ψ(R)|2 Z E = = dR = dR E (R) ρ(R) V hΨ|Ψi Ψ(R) R dR|Ψ(R)|2 L Z EV = dR EL(R) ρ(R) Z 2 2 σ = dR(EL(R) − EV ) ρ(R)

Estimate EV and σ from M independent samples as

M 1 X E¯ = E (R ) V M L i i=1 M 1 X σ¯2 = (E (R ) − E¯ )2 M L i V i=1 Typical VMC run

Example: Local energy and average energy of acetone (C3H6O) -34

-35

-36 σ VMC

-37 Energy (Hartree) -38

-39 0 500 1000 1500 2000 MC step

EVMC = hEL(R)iρ = −36.542 ± 0.001 Hartree (40×20000 steps)

2 σVMC = h(EL(R) − EVMC) iρ = 0.90 Hartree Variational Monte Carlo and the generalized Metropolis algorithm

|Ψ(R)|2 How do we sample distribution function ρ(R) = ? R dR|Ψ(R)|2

Aim → Obtain a set of {R1, R2,..., RM } distributed as ρ(R)

Generate a Markov chain

. Start from arbitrary initial state Ri

. Use stochastic transition matrix P(Rf |Ri) X P(Rf |Ri) ≥ 0 P(Rf |Ri) = 1.

Rf R as probability of making transition Ri → Rf

. Evolve the system by repeated application of P How do we sample ρ(R)?

To sample ρ, use P which satisfies stationarity condition :

X P(Rf |Ri) ρ(Ri) = ρ(Rf ) ∀ Rf 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 A more stringent condition

In practice, we impose detailed balance condition

P(Rf |Ri) ρ(Ri) = P(Ri|Rf ) ρ(Rf )

Stationarity condition can be obtained by summing over Ri X X P(Rf |Ri) ρ(Ri) = P(Ri|Rf ) ρ(Rf ) = ρ(Rf ) i i | {z } 1

Detailed balance is a sufficient but not necessary condition How do we construct the transition matrix P in practice?

Write transition matrix P as proposal T × acceptance A

P(Rf |Ri) = A(Rf |Ri) T (Rf |Ri)

Let us rewrite the detailed balance condition

P(Rf |Ri) ρ(Ri) = P(Ri|Rf ) ρ(Rf )

A(Rf |Ri) T (Rf |Ri) ρ(Ri) = A(Ri|Rf ) T (Ri|Rf ) ρ(Rf )

A(R |R ) T (R |R ) ρ(R ) ⇒ f i = i f f A(Ri|Rf ) T (Rf |Ri) ρ(Ri) Choice of acceptance matrix A

Original choice by Metropolis et al. maximizes the acceptance

  T (Ri|Rf ) ρ(Rf ) A(Rf |Ri) = min 1, T (Rf |Ri) ρ(Ri)

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

  3N ρ(Rf ) Symmetric T (Rf |Ri) = 1/∆ ⇒ A(Rf |Ri) = min 1, ρ(Ri) Better choices of proposal matrix T

Sequential correlation ⇒ Meff < M independent observations

M Meff = Tcorr with Tcorr autocorrelation time of desired observable

Aim is to achieve fast evolution of the system and reduce Tcorr

Use freedom in choice of T : For example, use available trial Ψ

 2  (Rf − Ri − V(Ri)τ) ∇Ψ(Ri) T (Rf |Ri) = N exp − with V(Ri) = 2τ Ψ(Ri) Acceptance and Tcorr for the total energy EV

Example: All-electron Be atom with simple wave function

Simple Metropolis Δ" ∆ Tcorr A¯

1.00 41 0.17 0.75 21 0.28 0.50 17 0.46 0.20 45 0.75

Drift-diffusion transition

τ Tcorr A¯ 0.100 13 0.42 0.050 7 0.66 √ 0.020 8 0.87 2 2τ ↔ ∆ 0.010 14 0.94 Typical VMC run

Example: Local energy and average energy of acetone (C3H6O) -34

-35

-36 σ VMC

-37 Energy (Hartree) -38

-39 0 500 1000 1500 2000 MC step

EVMC = hEL(R)iρ = −36.542 ± 0.001 Hartree (40×20000 steps)

2 σVMC = h(EL(R) − EVMC) iρ = 0.90 Hartree 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 c0DHF + c1D1 + c2D2 + ... too many determinants Jastrow-Slater wave function

X Ψ(r1,..., rN ) = J (r1,..., rN ) dk Dk (r1,..., rN ) k

J −→ Jastrow correlation factor

- Positive function of inter-particle distances - Explicit dependence on electron-electron distances - Takes care of divergences in potential X ck Dk −→ Determinants of single-particle orbitals

- Few and not millions of determinants as in quantum chemistry - Determines the nodal surface Jastrow factor and divergences in the potential

At interparticle coalescence points, the potential diverges as Z − for the electron-nucleus potential riα 1 for the electron-electron potential rij

HΨ 1 X ∇2Ψ Local energy = − i + V must be finite Ψ 2 Ψ i

⇒ Kinetic energy must have opposite divergence to the potential V Divergence in potential and Kato’s cusp conditions

Finite local energy as rij → 0 ⇒ Ψ must satisfy:

∂Ψˆ

= µij qi qj Ψ(rij = 0) ∂rij rij =0 where Ψˆ is a spherical average (we assumed Ψ(rij = 0) 6= 0) 0 Ψ • Electron-nucleus: µ = 1, qi = 1, qj = −Z ⇒ = −Z Ψ r=0 0 1 Ψ • Electron-electron: µ = , qi = 1, qj = 1 ⇒ = 1/2 2 Ψ r=0 Cusp conditions and QMC wave functions

. Electron-electron cusps imposed through the Jastrow factor

Example: Simple Jastrow factor   Y rij J (rij ) = exp b0 1 + b rij i

1 1 with b↑↓ = or b↑↑ = b↓↓ = 0 2 0 0 4

Imposes cusp conditions + keeps electrons apart rij 0

. Electron-nucleus cusps imposed through the determinantal part The effect of the Jastrow factor

Pair correlation function for ↑↓ electrons in the (110) plane of Si

0 g↑↓(r, r ) with one electron is at the bond center

Hood et al. Phys. Rev. Lett. 78, 3350 (1997) Some comments on Jastrow factor

More general Jastrow form with e-n, e-e and e-e-n terms

Y Y Y exp {A(riα)} exp {B(rij )} exp {C(riα, rjα, rij )} α,i i

. Polynomials of scaled variables, e.g.r ¯ = r/(1 + ar)

J > 0 and becomes constant for large ri , rj and rij

. 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 Jastrow factor with e-e, e-e-n and e-e-e-n terms

corr J EVMC EVMC (%) σVMC Li EHF -7.43273 0 e-e -7.47427(4) 91.6 0.240 + e-e-n -7.47788(1) 99.6 0.037 + e-e-e-n -7.47797(1) 99.8 0.028

Eexact -7.47806 100 0

Ne EHF -128.5471 0 e-e -128.713(2) 42.5 1.90 + e-e-n -128.9008(1) 90.6 0.90 + e-e-e-n -128.9029(3) 91.1 0.88

Eexact -128.9376 100 0

Huang, Umrigar, Nightingale, J. Chem. Phys. 107, 3007 (1997) 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 Construction of the wave function

How do we obtain the parameters in the wave function? X Ψ(r1,..., rN ) = J dk Dk k

− C10N2O2H7 70 electrons and 21 atoms VTZ s-p basis + 1 p/d polarization

. Parameters in the Jastrow factor J (≈ 100)

. CI coefficients dk (< 10 − 100) . Linear coefficients in expansion of the orbitals (5540 !) Optimization of wave function by energy minimization

The three most successful approaches

• Newton method (Umrigar and Filippi, 2005)

0 2 0 0 X ∂E(α ) 1 X ∂ E(α ) E(α) ≈ E(α ) + ∆αi + ∆αi ∆j ∂αi 2 ∂αi ∂αj i i,j

• Linear method (Umrigar, Toulouse, Filippi, and Sorella, 2007)

0 0 X ∂Ψ(α ) Ψ(α) ≈ Ψ(α ) + ∆αi ∂αi i ∂Ψ(α0) Solution of H ∆α = ES ∆α in the basis of {Ψ(α0), } ∂αi • Perturbative approach (Scemama and Filippi, 2006) What is difficult about wave function optimization?

Statistical error: Both a blessing and a curse!

Ψ{αk } → Energy and its derivatives wrt parameters {αk }

Z HΨ(R) |Ψ(R)|2 E = dR = hE i 2 V Ψ(R) R dR|Ψ(R)|2 L Ψ

  ∂k Ψ H∂k Ψ ∂k Ψ ∂k EV = EL + − 2EV Ψ Ψ Ψ Ψ2   ∂k Ψ =2 (EL − EV ) Ψ Ψ2

The last expression is obtained using Hermiticity of H Use gradient/Hessian expressions with smaller fluctuations

Two mathematically equivalent expressions of the energy gradient

    ∂k Ψ H∂k Ψ ∂k Ψ ∂k Ψ ∂k EV = EL + − 2EV =2 (EL − EV ) Ψ Ψ Ψ Ψ2 Ψ Ψ2

Why using the last expression? δE Ψ = Ψ0

0 Lower fluctuations → 0 as Ψ → Ψ0

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, 150201 (2005) Beyond VMC?

Removing or reducing wave function bias?

⇒ Projection Monte Carlo methods Why going beyond VMC?

Dependence of VMC from wave function Ψ

-0.1070

3D electron gas at a density rs=10 VMC JS -0.1075 VMC JS+3B

-0.1080 VMC JS+BF Energy (Ry) DMC JS VMC JS+3B+BF -0.1085

DMC JS+3B+BF -0.1090 0 0.02 0.04 0.06 0.08 4 2 x Variance ( rs (Ry/electron) )

Kwon, Ceperley, Martin, Phys. Rev. B 58, 6800 (1998) 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 − ET)]

Green’s function Monte Carlo1 /(H − ET)

Power Monte Carlo ET − H Diffusion Monte Carlo

Consider initial guess Ψ(0) and repeatedly apply projection operator

Ψ(n) = e−τ(H−ET)Ψ(n−1)

(0) Expand Ψ on the eigenstates Ψi with energies Ei of H

(n) −nτ(H−ET) (0) X (0) −nτ(Ei −ET) Ψ = e Ψ = Ψi hΨ |Ψi ie i and obtain in the limit of n → ∞

(n) (0) −nτ(E0−ET) lim Ψ = Ψ0hΨ |Ψ0ie n→∞

(n) If we choose ET ≈ E0, we obtain lim Ψ = Ψ0 n→∞ How do we perform the projection?

Rewrite projection equation in integral form

Z Ψ(R0, t + τ) = dR G(R0, R, τ)Ψ(R, t) where G(R0, R, τ) = hR0|e−τ(H−ET)|Ri

. Can we sample the wave function? For the moment, assume we are dealing with bosons , so Ψ > 0

. Can we interpret G(R0, R, τ) as a transition probability? If yes, we can perform this integral by Monte Carlo integration What can we say about the Green’s function?

G(R0, R, τ) = hR0|e−τ(H−ET)|Ri

G(R0, R, τ) satisfies the imaginary-time Schr¨odingerequation

∂G(R, R , t) (H − E )G(R, R , t) = − 0 T 0 ∂t with G(R0, R, 0) = δ(R0 − R) Can we interpret G(R0, R, τ) as a transition probability? (1)

H = T

Imaginary-time Schr¨odinger equation is a diffusion equation

1 ∂G(R, R , t) − ∇2G(R, R , t) = − 0 2 0 ∂t The Green’s function is given by a Gaussian

 (R0 − R)2  G(R0, R, τ) = (2πτ)−3N/2 exp − 2τ

Positive and can be sampled Can we interpret G(R0, R, τ) as a transition probability? (2)

H = V

∂G(R, R , t) (V(R) − E )G(R, R , t) = − 0 , T 0 ∂t The Green’s function is given by

0 0 G(R , R, τ) = exp [−τ (V(R) − ET)] δ(R − R )

Positive but does not preserve the normalization

It is a factor by which we multiply the distribution Ψ(R, t) H = T + V and a combination of diffusion and branching

Let us combine previous results

 (R0 − R)2  G(R0, R, τ) ≈ (2πτ)−3N/2 exp − exp [−τ (V(R) − E )] 2τ T

Diffusion + branching factor leading to survival/death/cloning

Why? Trotter’s theorem → e(A+B)τ = eAτ eBτ + O(τ 2)

→ Green’s function in the short-time approximation to O(τ 2) Diffusion and branching in a harmonic potential

V(x)

Ψ 0(x)

Walkers proliferate/die in regions of lower/higher potential than ET Time-step extrapolation

DMC results must be extrapolated at short time-steps (τ → 0)

Example: Energy of Li2 versus time-step τ Problems with simple DMC algorithm

 (R0 − R)2  G(R0, R, τ) ≈ exp − exp [−τ (V(R) − E )] 2τ T Potential can vary a lot, unbounded → Exploding/dying population

Solution → Work with f (R, t) = ΨT(R)Ψ(R, t)

Evolution governed by drift-diffusion-branching Green’s function

 (R0 − R − τV(R))2  G˜(R0, R, τ) ≈ exp − exp [−τ (E (R) − E )] 2τ L T

Local energy EL(R) is better behaved than the potential

→ After many iterations, walkers distributed as ΨT(R)Ψ0(R) Electrons are fermions!

We assumed that Ψ0 > 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 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 1 Ψ+ = (|Ψ| + Ψ) 2 Ψ(x,τ=0 ) Ψ(x,τ=0 ) 1 + − Ψ = (|Ψ| − Ψ) − 2 Particle in a box and the fermionic problem (1)

The imaginary-time Schr¨odingerequation

∂Ψ HΨ = − ∂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

∂Ψ ∂Ψ HΨ = − + and HΨ = − − + ∂t − ∂t separately and subtracting one solution from the other Particle in a box and the fermionic problem (2)

s a s . Since E0 < E0 , both Ψ+ and Ψ− evolve to Ψ0

Ψ± −→

. Antisymmetric component exponentially harder to extract

−E at |Ψ+ − Ψ−| e 0 ∝ −E s t as t → ∞ |Ψ+ + Ψ−| e 0 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 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 For many electrons, what are the nodes? A complex beast

Many-electron wave function Ψ(R) = Ψ(r1, r2,..., rN )

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. Use the nodes of trial ΨT → Fixed-node approximation

Use the nodes of the best available trial ΨT wave function

Ψ(R)=0 Ψ (R)>0 R

Find best solution with same nodes as trial wave function ΨT

Fixed-node solution exact if the nodes of trial ΨT are exact 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?

MAD for atomization energy of the G1 set

DMC CCSD(T)/aug-cc-pVQZ HF orb Optimized orb CAS MAD 3.1 2.1 1.2 2.8 kcal/mol

Petruzielo, Toulouse, Umrigar, J. Chem. Phys. 136, 124116 (2012)

With “some” effort on Ψ, we can do rather well Journal of Chemical Theory and Computation Letter

that FN-DMC with single-determinant trial functions is able to and a positive definite Jastrow term12 explicitly describing the approach the CCSD(T)/CBS reference to within 0.1 kcal/mol interparticle correlations. Remarkably, we have found that (one standard deviation errors are reported) for small single-reference wave functions filled with B3LYP/aug-TZV complexes. In addition, the identified easy-to-use protocol is orbitals reach the desired accuracy criterion for the whole test tested on larger complexes, where the reliability of CCSD(T) set; consequently, multiple determinants were not considered. has yet to be fully tested. Here, the final FN-DMC results agree Orbital sets from other methods were mostly comparable; in to within 0.25 kcal/mol with the best available estimates. These the ammonia dimer complex, for instance, the HF nodes results show the potential of QMC for reliable estimation of provide the same FN-DMC interaction energy as B3LYP Diffusionnoncovalent molecular Monte interaction energies Carlo well below as chemical a black-box(−3.12 ± 0.07 vs approach?−3.10 ± 0.06 kcal/mol) within the error bars, accuracy. due to the FN error cancellation26,28,29 (cf. Figure 1). The calculations were performed on a diverse set of Nevertheless, the total energies from B3LYP orbitals were hydrogen and/or dispersion bound complexes for which 8,39,40 found to be variationally lower than those from HF (in dimer reliable estimates of interaction energies already exist by ∼0.001 au), in agreement with previous experience.15,41 and which were previously studied within QMC.26,29,34,35 The Non-covalent interaction energiesRegarding for the 6 one-electron compounds basis set, tests on from the ammonia S22 set considered test set consists of the dimers of ammonia, water, dimer confirm the crucial effect of augmentation functions (cf. hydrogen fluoride, methane, ethene, and the ethene/ethyne ref 29). For the same system, TZV and QZV bases result in complex (Figure 2). The larger considered complexes include ± ± DMCbenzene/methane, with benzene/water, B3LYP/aug-cc-PVTZ and T-shape benzene dimer interaction orbitals energies of −3.33 versus0.07 and − CCSD(T)/CBS3.47 0.07, whereas ± (Figure 2). 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, elec- tron−nucleus,∆ andMAD electron−=electron 0−.nucleus058 terms kcal/mol 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 Figure 2. The set of molecules used in the present work (from top left, factor. Note that a more economic variant of the correlation to bottom right): ammonia dimer, water dimer, hydrogen fluoride factor, with only electron−electron and electron−nucleus dimer, methane dimer, ethene dimer, and the complexes of ethene/ terms, doubles the average error on the considered test set, Dubeckyethyne, benzene/methane,et al., benzene/water, J. Chem. and benzene Theory dimer T- Comput.and therefore it would9, be 4287 inadequate (2013) for our purposes.44 The shape. 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 ■ ADJUSTING THE QMC PROTOCOL found that for large complexes, 7−10 iterations of VMC WithThe present “practically methodology was developed no” via extensiveeffort testing on Ψ,optimization we are can sometimes donecessary rather to reach well the full and elimination of the biases that affect the final FN-DMC convergence. energies. Clearly, this has to be done in a step-by-step manner The production protocol thus consists of (i) Slater−Jastrow since several sets of parameters enter the multistage refinement trial wave functions of B3LYP/aug-TZV quality, (ii) a strategy16,21 on the way to the final FN-DMC results. The converged VMC optimization of the Jastrow factor with sequence of the steps includes (i) the construction of the trial electron−electron, electron−nucleus, and electron−electron− wave function, (ii) its VMC optimization, and (iii) FN-DMC nucleus terms, and (iii) a FN-DMC ground-state projection 43 production calculation. The tasks i and ii involve optimizations using the T-moves scheme and a time step of 0.005 au. Note which affect the final interaction energies obtained in iii as the that the VMC reoptimization of orbitals has not been explored, differences of the statistically independent total energies. although it could be tested in the future as well. The error bars We employ trial wave functions of the Slater-Jastrow were converged to at least ∼0.1 kcal/mol in the projection time type,10,11 in general, a product of the sum of determinants of several thousands of atomic units.

4289 dx.doi.org/10.1021/ct4006739 | J. Chem. Theory Comput. 2013, 9, 4287−4292 Diffusion Monte Carlo as a black-box approach?

Fragmentation energies of N2H4, HNO2, CH3OH, and CH3NH2

MAD MAX (kcal/mol) DMC/cc-pVTZ 1 det 1.9 2.7 DMC/cc-pVTZ J-LGVB 0.4 1.0 CCSD(T)/cc-pVTZ 2.8 3.5 CCSD(T)/cc-pVQZ 1.1 1.6 CCSD(T)/cc-pV5Z 0.6 1.0

Fracchia, Filippi, and Amovilli, J. Chem. Theory Comput. 8, 1943 (2012)

Novel form of Ψ!

... +

→ 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) P Start from ΨCI = i ci Di ∂Ψ ∂c HΨ = − → H c = − i ∂t ij j ∂t DMC in summary

The fixed-node DMC method is

I Easy to do

I Stable

I Accurate enough for many applications in quantum chemistry ... especially in large systems

I Not accurate enough for subtle correlation physics Beauty of quantum Monte Carlo → Highly parallelizable

Ψ(r1,..., rN ) → 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

30 the Slater determinant allows us to significantly reduce the 3 (a) MLW prefactor for this N term. In larger systems where the 25 SixHy (b) Gaussian determinant is increasingly sparse, it should be possible { (c) Plane Wave to reformulate the determinant update procedure to utilize 20 Carbon (d) MLW Optical EmisthissUpion osparsityf Silic toon Nandan Siocryobtain123stals Ha better100sizeandscaling. C180 ! A R T I C L E S s) ( Note, the discussion thus far involves the scaling of the

ime 15 T computational cost of moving a single configuration of

U electrons. In practice, one calculates either (i) the total CP 10 energy of the system, or (ii) the energy per atom, with a given statistical error. The statistical error, d, is related 5 to the number of uncorrelated moves, M, by d ෇ s͞pM, where s2 is the intrinsic variance of the system. Typi- 0 2 0 200 400 600 800 1000 cally, the value of s increases linearly with system size. Therefore, to calculate the total energy with a fixed d, the Number of electrons number of moves, M, must also increase linearly. When FIG. 2. CPU time on a 667 MHz EV67 alpha processor to multiplied by our linear increase in the cost of each move, Williamson,move a configuration Hood,of electrons within GrossmanDMC for SiH4 (2001), Si5H12, an N2 size scaling is obtained. For quantities per atom, Si H , Si H , Si H , Si H , C , C , C , C , and 35 36 87 76 123 100 211 140 20 36 60 80 such as the binding energy of a bulk solid, s2 still in- C180. creases linearly with system size, but d is decreased by a functions in a plane wave basis scales as approximately factor of N, and, hence, the number of required moves, M, N3. The exact scaling is determined by the volume of the actually decreases linearly with system size. Therefore the supercell chosen for each system. The computational cost cost of calculating energies per atom is now independent of the Gaussian basis also scales as N3, but with a smaller of system size. prefactor as the number of basis functions per atom is much To illustrate the range of systems that can now feasibly smaller. The calculations using the truncated MLW func- be studied within QMC using truncated MLW functions tions demonstrate that the CPU time required to move a in the Slater determinant, we have calculated total ener- single configuration of electrons scales approximately lin- gies of a series of carbon fullerenes. In Fig. 3 we plot the early with the number of electrons. The deviations from binding energy per atom of C20, C36, C60, C80, and C180 linearity in the hydrogenated silicon cluster MLW curves fullerenes. Line (a) shows the binding energies calculated are mainly due to differing ratios of hydrogen to silicon using LDA; line (b) shows the binding energy calculated atoms; for the carbon fullerenes, they are due to differ- within fixed node DMC. The LDA calculations were per- ent strain in the clusters requiring slightly different cutoff formed at the G point of the Brillouin zone, using a cutoff radii for the MLW functions. For the systems we com- of 40 Ry. The same pseudopotentials were used in the pared (SiH4 and Si5H12), the DMC energies for all three LDA and DMC calculations. Six points were used in the basis sets agreed within 0.001 hartree per atom statistical QMC angular integration for the nonlocal pseudopotential. error bars. Once the cost of evaluating the Slater determinant has been reduced to linear scaling, it is interesting to ask how large a system one can study before other parts of the al- 8.5 gorithm will begin to dominate and the linear scaling will 8.0 be lost. For the Si211H140 system (984 electrons), approxi- mately 10% of the calculation involves the remaining parts 2 3 7.5 (a) of the algorithm that scale as N and N . With relatively Expt. bulk minor algorithmic improvements, the cost of these terms per Atom (eV) value Figure 2. Thegy two left columns show charge density isosurfaces of the HOMO and LUMO orbitals of 1 nm clusters with different surface structures and could be dramatically reduced, extending the linear regimedifferent passivants.7.0 The silicon atoms are gray, the hydrogen atoms are white, and the oxygen atoms are red. The isosurfaces are chosen at 50% of the to several thousand electrons. In particular, we envisagemaximum amplitude. The right column shows vectors proportional to the displacement of each atom during the Stokes shift (see text). The displacements have been magnified by (b)10 for clarity. LDA (i) the electron-ion interaction could be rewritten with the 6.5 sum over ions precomputed so the local part scales linearly. DMC The nonlocal contribution already scales linearly duestrutoctures andBinding Ener rapidly decreases to less than 0.1 Å as the cluster surface, for example, Si29H34O (Figure 2c), show smaller rms size increases to6.0 2 nm. In these clusters, Figure 2a shows that displacements than do the completely hydrogenated clusters. the cutoff in the range of the interaction. (ii) The electron- ∼ 20 36 60 80 180 ∞ electron interaction could be rewritten to scale linearlythebychange in charge density resulting from the HOMO to In these reconstructed clusters, the charge density change LUMO excitation is distributed relativNeulmybeevre onf lCy athtormous ghout the associated with the HOMO-LUMO excitation is localized at writing it as a sum of short- and long-ranged pieces [1,7], cluster and hence all atoms experience forces of approximately the surface, and therefore the surface atoms experience the or using Greengard’s multipole expansion [20]. (iii) To up- FIG. 3. Binding energy per atom (a) within LDA, and 3 equal magnitude.(b) withinThisDMCequalityof carbonof forcesfullerenes.is confirmedDMCbystatisticalthe greatesterror force. Similar to hydrogenated clusters, as the size of date the Slater determinant, we adopt the N scalingsimilarpro- magnitudebars are ofsmallerthe vectorthandisplacementsthe symbols.plottedFor incomparison,the thewereconstructed clusters increases, the change in charge density cedure based on storing the inverse of the transpose ofrightthecolumnhaveofaddedFigure0.182a. TheseeV zerodisplacementspoint energyshowto thatthe thebulk graphitedue to the excitation of a single electron from the HOMO to matrix from Ref. [21]. Our introduction of sparsityshapeinto of ehydrogenatedxperimental bindingclustersenerchangesgy [22].from spherical to LUMO is distributed over a larger area, and hence the force on elliptical upon excitation of one electron. As the size of the each individual surface atom again decreases with size; this is 246406-3 hydrogenated clusters increases, the relative change in charge 246406-3confirmed by the decreasing rms displacements of the larger density around each atom due to the excitation of a single reconstructed clusters. electron decreases inversely proportionally to the number of In the clusters with oxygen double bonded to the surface, atoms, and hence the rms displacements also decrease (see the for example, Si35H35O (Figure 2d), the rms displacement is first five rows of Table 1). slightly larger. However, in this case, considering the rms The clusters with reconstructed surfaces, for example, Si29H24 displacement is somewhat misleading as almost all of the atomic (Figure 2b), and those with Si-O-Si bridged oxygen on the relaxation is concentrated on the double bonded oxygen atom.

J. AM. CHEM. SOC. 9 VOL. 125, NO. 9, 2003 2789 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% 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) Other applications of quantum Monte Carlo methods

I Electronic structure calculations

I Strongly correlated systems (Hubbard, t-J, ...)

I Quantum spin systems (Ising, Heisenberg, XY, ...)

I Liquid-solid helium, liquid-solid interface, droplets

I Atomic clusters

I Nuclear structure

I Lattice gauge theory

Both zero (ground state) and finite temperature