Robust All-Electron Optimization of Orbital-Free...

Robust All-Electron Optimization ofOrbital-Free Density-Functional Theory

Trygve Helgaker1, Tom Irons2, Matthew S. Ryley2, and Andrew M. Teale1,2

1Hylleraas Centre for Quantum Molecular Sciences,Department of Chemistry, University of Oslo, Norway

2School of Chemistry, University of Nottingham, University, UK

9th Molecular Quantum Mechanics Conference (MQM2019)

Kongresshaus Stadthalle Heidelberg,Heidelberg, Germany, 30 June – 5 July 2019

Trygve Helgaker (University of Oslo) Robust Optimization of Orbital-Free DFT MQM2019 1 / 22

Background

Orbital-free DFT

I In DFT, we calculate the N -electron ground-state energy in a given external potential v as

E(v,N) = infρ 7→N

(F (ρ) + (v|ρ)) , (v|ρ) =∫v(r)ρ(r) dr

I the universal density functional F (ρ) represents the intrinsic energy of the systemI modelling of F (ρ) is difficult – in particular, the kinetic energy

I In Kohn–Sham DFT, we calculate the kinetic energy exactly for a noninteracting reference system

F (ρ) = Ts(ρ) + J(ρ) + Exc(ρ)

I the noninteracting kinetic energy Ts(ρ) introduces orbitals, which are optimized self consistentlyI Kohn–Sham DFT has been enormously successful – the undisputed workhorse of quantum chemistry

I In orbital-free DFT, we stay closer to the origin of DFT, modelling the noninteracting kinetic energy explicitly

I circumventing orbitals, it offers greater simplicity and holds the promise of considerably lower costI ongoing research: Paul Madden, Emily Carter, Sam Trickey, Olga Lopez-Acevedo, Tomasz Wesolowski & others

I Two main challenges of orbital-free DFT:

I it is exceedingly difficult to model the kinetic energy accuratelyI it is difficult to optimize the density reliably and efficiently

I We present the first generally applicable all-electron molecular optimization method for orbital-free DFT


Background

Kinetic-energy functionals

I Simplest kinetic-energy functionals are those of Thomas and Fermi (1927) and of von Weizsacker (1935):

TTF(ρ) = CF

∫ρ5/3

(r)dr ← local density approximation TW(ρ) = 12

∫|∇ρ

1/2(r)|2 dr ← exact for two electrons

I many kinetic-energy functionals are generated as linear combinations of these

TTFλW(ρ) = TTF(ρ) + λTW(ρ) ← gradient-corrected Thomas–Fermi functional

I Baltin (5/9), YT65 (1/5), Lieb (0.1859), GE2 (1/9), OPT (0.697,0.599)

I However, GGA-type kinetic-energy functionals developed

I E00, P92, OL1

I Kinetic-energy errors for first-row atoms

I relative to the PBE noninteracting kinetic energyI very large errors – a difficult problem

I Note: errors on KS and OF densities differ widely

I errors reduce/increase with λ on KS/OF densitiesI GGAs are accurate only on KS densitiesI we can assess results with OF as well as KS densities


Background

Overview

I We will show that optimization of the OF-DFT energy is a minimax saddle-point problem

I we minimize over the density ρ and maximize over the chemical potential µ

E(v,N) = maxµ minρ(F (ρ) + (v − µ|ρ) + µN

)= minρ maxµ

(F (ρ) + (v − µ|ρ) + µN

)I We will demonstrate that a modification of Newton’s method converges robustly and efficiently

I typically in tens of iterationsI comparisons with other methods

I We will present calculations on atoms, diatomics and polyatomics

I all-electron calculations in Gaussian basis setsI comparison with Kohn–Sham noninteracting kinetic energiesI comparison on self-consistent orbital–free densities and Kohn–Sham densities


Ground-state energy and Legendre–Fenchel transformations

Ground-state energy

I Consider a molecule in an external potential v(r) (typically a nuclear attractive potential)

I to control the number of electrons in the system, we subtract from v(r) a uniform chemical potential µ

v(r)− µ = −∑I

ZI

|r−RI |− µ

I as we increase µ, more electrons are attracted to the system

I We are interested in the ground-state energy G(v, µ) of electrons in the potential v(r)− µ

I The Rayleigh–Ritz variation principle:

G(v, µ) = infΓ tr ΓH(v − µ)

I Γ is a grand-canonical ensemble density matrixI no constraints on the number of electrons

I Concavity of the energy G(v, µ) follows from

I the linearity of H(v − µ) in v − µI the Rayleigh–Ritz variation principle

I We will now Legendre–Fenchel (LF) transform G(v, µ) with respect to µ and v



1) LF transformation with respect to µ

I LF transformation of G(v, µ) with respect to µ gives the energy as a function of particle number N :

E(v,N) = supµ(G(v, µ) + µN

), G(v, µ) = infN

(E(v,N)− µN

),

I important: since G(v, µ) is concave in µ, it may be generated back again from E(v,N)I G(v, µ) and E(v,N) are dual functions: they contain the same information encoded in different ways

I Ground-state energy plotted against the dual variables µ and N for carbon and oxygen atoms

-10-20-30

-20

-40

-60

G(vC ,μ)

G(vO,μ)

2 4 6 8

-60

-40

-20

ℰ(vC,N)

ℰ(vO,N)



2) LF transformation with respect to v

I LF transformation of G(v, µ) with respect v gives the universal density functional F (ρ, µ):

F (ρ, µ) = supv (G(v, µ)− (v|ρ)) , G(v, µ) = infρ (F (ρ, µ) + (v|ρ))

I important: since G(v, µ) is concave in v, it may be generated back again from F (ρ, µ): they are dual functions

I Ground-state energy plotted against the dual variables v and ρ for a one-dimensional system

-2 -1 1 2

-2

-1

1

2

G(v,μ )

-2 -1 1 2

-2

-1

1

2

F(ρ,μ)

I The µ-dependent density functional F (ρ, µ) is related to F (ρ) as F (ρ, µ) = F (ρ)− µNρ.



Four-way correspondence

I We have established two transformations:

E(v,N) = supµ(G(v, µ) + µN

), G(v, µ) = infρ (F (ρ, µ) + (v|ρ))

I Combining these, we obtain the minimax problem for the generation of E(v,N) from F (ρ, µ):

E(v,N) = supµ infρ(F (ρ, µ) + (v|ρ) + µN

)= infρ supµ

(F (ρ, µ) + (v|ρ) + µN

)← saddle point

I Equality is shown using the theory of four-way correspondence of convex analysis (Rockafeller, 1968)

�(�� μ)

�(ρ� μ) ℰ(�� )

�(ρ� �)

�

�

μ

ρ


Ground-state energy by saddle-point optimization


I Optimization of the orbital-free DFT energy is minimax saddle-point problem:

E(v,N) = supµ infρ (F (ρ) + (v − µ|ρ) + µN)

I the exact functional is nondifferentiable (complicating the optimization) but approximate functionals are not...I also, the exact problem may be turned into a differentiable problem by Moreau–Yosida regularization at no loss of accuracy

I The square-root of the electron density is expanded in Gaussians:

ρc(r) = φ2c(r) ≥ 0, φc(r) =

∑i ciηp(r)

I with this parametrization, the OF-DFT optimization problem becomes

E(v,N) = maxµ minc L(c, µ), L(c, µ) = F (ρc) + (v − µ|ρc) + µN

I We are after the lowest first-order saddle point of the objective function L(ρ, µ)

I the Newton step is obtained by solving the following set of linear equations (noting that ∂2L/∂µ2 = 0):[∂2L∂c∂c

∂2L∂c∂µ

∂2L∂µ∂c

0

][∆c

∆µ

]= −

[∂L∂c∂L∂µ

]

I Newton’s method works well in the local region, giving fast, second-order convergence



Global region: trust-radius image minimization

I Assume no coupling between the mode to be maximized and those to be minimizedI saddle-point optimization is then equivalent to minimization after negating the mode to maximized

←− maximization becomes minimization along the image mode

I we may view this as minimizing an image function, obtained from the original function by negating one mode

I Trust-region image minimization (TRIM) was developed for transition-state optimizationsI Newton’s method on the image function with level shift for step-size controlI T. Helgaker, Chem. Phys. Lett. 182, 503 (1991)

I The convex–concave DFT objective function L(c, µ) and its convex–convex image function

I In the TRIM method applied to DFT, we solve the following level-shifted Newton step for the image:([∂2L∂c∂c −

∂2L∂c∂µ

− ∂2L∂µ∂c 0

]− ν

[1 0

0 1

])[∆c

∆µ

]= −

[∂L∂c

− ∂L∂µ

]← step size controlled by level-shift parameter ν



The CCH and LPS optimization methods

1 In the Cohen–Chan–Handy (CCH) method (2001), we apply Lagrange’s method to the constrained minimization

E(v,N) = minρc 7→N (F (ρc) + (v|ρ))

I the undetermined multiplier for the constraint ρc 7→ N becomes the chemical potential

L(ρc, µ) = F (ρc) + (v − µ|ρc) + µN,

[∂2L(ρc, µ)

∂c2

] [∆c

]= −

[∂L(ρc, µ)

∂c

]I outer iterations, where we bisect on µ until the desired number of particles N is reachedI inner iterations, where we minimize L(c, µ) with respect to c by Newton’s method

2 In the Levy–Perdew–Sahni (LPS) method (1984), we solve a Kohn–Sham one-electron eigenvalue problem

I if the functional contains TW, we have the one-electron Schodinger equation with chemical potential as an eigenvalue

F (ρ) = 12λ

∫|∇ρ

1/2(r)|2dr +G(ρ),

(− 1

2λ∇2

+ v(r) +δG(ρ)

δρ(r)

)ρ1/2

(r) = µ ρ1/2

(r)

I equivalent to a Kohn–Sham one-electron eigenvalue problem – quick to implementI any functional F (ρ) may be written in this way by writing G(ρ) = F (ρ)− λTW(ρ)I further developed by Lopez–Acevedo (2014) in the context of PAW methods by including λ

3 Conjugate-gradient method by Jiang and Yang (2004) and Carter (2008)


Atomic calculations

A simple example: He atom

I Basis of 21 even-tempered s-type Gaussians with exponents αn = 3n with −6 ≤ n ≤ 14

1.4× 10−3, . . . , 1/3, 1, 3, . . . , 4.8× 10

6

I PBE exchange–correlation functional and TFλvW kinetic-energy functionals (plot for TF + W functional)

Number of iterations:

LPSλ CCH TRIMvW 12 167 12TF + vW 237 169 17TF + 1/5 vW 2046 249 28TF + 1/9 vW 8539 220 20

I second-order convergence of TRIM

I an order of magnitude fewer steps than CCH

I LPSλ slowed down by damping for small λ


Atomic calculations

Convergence of µ and N in He atom

I Convergence of chemical potential µ and particle number N for helium atom with TW + vW functional

I the TRIM method converges µ and N to within 15 digits after a few iterationsI the LPSλ method maintains exact particle number in each iteration


Atomic calculations

Convergence for first-row atoms

I All first-row atoms converge in the same manner as the helium atom

Number of TF+1/5vW iterations

LPSλ CCH TRIMH 1627 221 54He 2046 249 28Li 2054 231 27Be 2018 228 19B 2028 260 24C 2026 289 17N 2017 176 15O 2003 164 14F 1995 271 15Ne 1996 217 14mean 1981 231 23

Comparison of optimization methods

LPSλ CCH TRIMimplementation KS based Newton Newton

two-step one-stepline-search trust-region

derivatives 1st 2nd 2ndlinear equations no yes yesdiagonalizaton yes no (yes)convergence linear linear quadratic# iterations 1000s 100s 10sparticle number exact optimized optimizedapplicability TFλvW general general


Atomic calculations

Thomas–Fermi atom

I The Thomas–Fermi universal density functional is given by

FTF(ρ) = CF

∫ρ5/3

(r)dr +1

2

∫∫ρ(r1)ρ(r2)

r12dr1dr2

I For a neutral atom, the Thomas–Fermi model gives the total energy

ETF(vZ , Z) = −0.7687Z7/3

I the density is unphysical, diverging as r−3/2 at the nucleus and decaying asymptotically as r−6

I The Thomas–Fermi atom is difficult to describe accurately in a Gaussian basis setI a 71s Gaussian basis with αn = 1.6n and −20 ≤ n ≤ 50 is in error by 2.2% for Z = 1 and 0.4% for Z = 10


Molecular calculations

Convergence for N2 molecule

I N2 molecule with E00 kinetic-energy functional and PBE exchange–correlation functional

I Large even-tempered primitive 18s7p4d3f Gaussian basis

−5 ≤ ns ≤ 12, −3 ≤ np ≤ 3, −2 ≤ nd ≤ 1, −2 ≤ nf ≤ 0

I Starting density: superposition of atomic densities (SAD)

I SAD guess is be essential for moleculesI LPSλ does not converge



Importance of atomic starting density

I Without the superposition of atomic densities (SAD), convergence would be very slow

I Comparison of SAD (blue) with a random starting guess (red) for N2 with E00+PBE functional in 18s7p4d3f basis



Hessian indexI In the optimization, the solution must have one negative eigenvalue (Hessian index 1)

I With a random starting density, Hessian index is typically largeI in atomic calculations with random starting density, Hessian index is rapidly reducedI in molecular calculations with random starting density, Hessian index may fluctuate strongly

I In molecular calculations with an atomic starting density, Hessian index is one or close to one in all iterationsI such calculations converge rapidly reach the local region

I N2 molecule – E00+PBE functional in the 18s7p4d3f basis



Potential-energy curves for N2

I A fundamental problem with orbital-free DFT is the description of chemical bindingI Teller nonbinding theorem: E. Teller, Rev. Mod. Phys. 34, 627 (1962)

I Potential-energy curves of N2 with γTTF + λTW

I increasing weight on the von Weizsacker functional increases binding

I The more general kinetic-energy GGAs also bind but are not necessarily in good agreement with experimentI E001 (Ernzerhof, 2000) and PBE2 (Karasiev, Trickey, and Harris, 2006)



Errors in kinetic energy of diatomic molecules

I The errors in kinetic-energy functionals relative to the PBE noninteracting kinetic energy

I different errors for self-consistent and post-Kohn–Sham energiesI remarkably smaller errors for the GGA kinetic energy functionals

I 45 diatomic molecules: AlBr, AlCl, AlF, As2 , BBr, BCl, BF, Br2 , Cl2 , ClBr, CO, CS, Cse, F2 , FBr, FCl, GaBr, GaCl, GaF, GeO, GeS, GeSe, K2 ,

KBr, KCl, KF, Li2 , LiBr, LiCl, LiF, LiK, LiNa, N2 , Na2 , NaBr, NaCl, NaF, NaK, NAs, NP, P2 , PAs, SiO, SiS, SiSe



Errors in kinetic energy of polyatomic molecules

I The errors in kinetic-energy functionals relative to the PBE noninteracting kinetic energy

I remarkably different errors for self-consistent and post-Kohn–Sham energies – often in opposite directions

Number of TRIM iterations:

mean min max medianBaltin 50 9 416 11YT65 32 12 96 18Lieb 37 13 192 15GE2 67 13 556 15OPT 26 11 115 11E00 46 13 731 18P92 50 11 351 19OL1 100 13 641 15

I 23 polyatomics: C2H4 , C2H4O, C3H4 , CH2O, CH3F, CH4 , FCCH, FCN, H2C2O, H2O, H2S, HCN, HCP, HFCO, HOF, LiF, LiH, N2O, NH3 , O3 , OCS, OF2 , SO2


Conclusions

Conclusions

I We have presented a robust and reliable all-electron molecular optimization method for orbital-free DFT

I requires typically tens of iterations although more difficult cases do occurI second-order local convergence give sharply optimized densities and chemical potentialsI requires the solution of linear Newton equations in each iterationI lowest Hessian eigenvalue needed – full diagonalization not needed

I All calculations performed with Quest developed in Andy Teale’s group

I Support:

I Centre for Advanced Study (CAS) at the Norwegian Academy of Science and LettersI Norwegian Research Council for Centres of Excellence CTCC & HylleraasI European Research Council


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