Tuning the range separation parameter in periodic systems
Wenfei Li, Vojtech Vlcek, Helen Eisenberg, Eran Rabani, Roi Baer, Daniel Neuhauser
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Tuning the range separation parameter in periodic systems
Wenfei Li, Vojtech Vlcek, a) Helen Eisenberg, Eran Rabani,
3, 4
Roi Baer, and Daniel Neuhauser Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90095,USA Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem,Jerusalem 91904, Israel Department of Chemistry, University of California, and Materials Sciences Division,Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University,Tel Aviv 69978, Israel
Kohn-Sham DFT with optimally tuned range-separated hybrid (RSH) functionals provides accurate andnonempirical fundamental gaps for a wide variety of finite-size systems. The standard tuning procedure relieson calculation of total energies of charged systems and thus cannot be applied to periodic solids. Here,we develop a framework for tuning the range separation parameter that can be used for periodic and openboundary conditions. The basic idea is to choose the range parameter that results in a stationary point wherethe fundamental gap obtained by RSH matches the gap obtained from a G W over RSH calculation. Theproposed framework is therefore analogous to eigenvalue self-consistent GW (scGW). We assess the methodfor various solids and obtain very good agreement with scGW results.Due to its high accuracy and low computational cost,Kohn-Sham DFT (KS-DFT) is one of the most prevalenttools for probing the electronic structure of both molecu-lar and periodic systems. However, KS-DFT with localand semilocal (LDA and GGA) functionals often severelyunderestimate the fundamental band gap.
Several al-ternative frameworks have been developed to tackle theproblem. One is the GW approximation, where thequasiparticle excitation energies are obtained by solvinga Dyson equation. In practice, calculations are typicallycarried out with the G W approximation, where the selfenergy is applied as a perturbative correction to KS-DFTorbital energies. For many systems this approach pro-vides quasiparticle gaps that are in good agreement withexperimental band gaps. Another route is the generalized Kohn-Sham DFT(GKS-DFT) method where instead of a local exchange-correlation potential, the effective Hamiltonian is non-local. DFT with hybrid functionals, in either the originalfractional exchange (e.g., B3LYP ) or range-separatedflavor, are all part of the GKS framework.
In hybrid functionals, certain fraction of Fock exchangeis incorporated, and this choice can be justified by con-sidering two facts. First, in semilocal approximations,due to the existence of self interaction the exchange-correlation functional does not have the correct asymp-totic form, while in Hartree-Fock theory the one-particleself interaction is eliminated through the balance betweenthe Hartree and Fock exchange terms. Therefore, inclu-sion of Fock exchange helps achieving the desired asymp-totic behavior of exchange-correlation functionals.
A second, related, aspect of the self-interactionproblem is that for the exact exchange-correlationfunctional the total energy curve, as a function of a) Current address: Department of Chemistry and Biochemistry,University of California, Santa Barbara, California 93106, USA particle number, should be composed of line segmentsjoining the energies at integer electron numbers.
However, DFT with local and semilocal approximationsis convex at fractional charge, while Hartree-Fock isconcave. Therefore, by incorporating Fock exchange,hybrid functionals provide a way of enforcing piecewiselinearity. In fact, studies have shown that optimallytuned range separated hybrid (OT-RSH) functionalsproduce total energy curves that are almost piecewiselinear.
In OT-RSH the Coulomb interaction between electronsis separated into short-range and long-range parts: r = − erf( γr ) r + erf( γr ) r . The short range part is thenapproximated using local or semilocal approximations,which preseve the cancellation of errors between the ex-change and the correlation functional. In long-range ex-change functionals, the long-range part is calculated withFock exchange to offset the self-interaction error and en-force the correct long-range asymptotice behavior of thefunctional. The range-separation parameter γ is chosento maintain a balance between the long-range and short-range exchange, and γ − is an effective screening length.With this partition, the overall exchange-correlationenergy becomes E XC = E C + E lF X + E sX , where E lF X , E sX are the long range Fock exchange and short-rangelocal/semilocal exchange, and the superscripts "l" and"s" refer to long-range and short-range respectively. Theaction of the exchange-correlation part of the Hamilto-nian is then: ˆ V XC ψ ( r ) = ˆ K l ψ ( r ) + [ v C ( r ) + v sX ( r )] ψ ( r )= − ˆ dr ′ u l ( | r − r ′ | ) ρ ( r, r ′ ) ψ ( r ′ ) + v sXC ( r ) ψ ( r ) (1)where ρ ( r, r ′ ) is the density matrix of the system, u l ( r ) = erf( γ | r | ) | r | is the long-range part of the Coulomb interaction,while v sXC is the short range exchange-correlation poten-tial.The one unknown is then the range-separation param-eter. For finite sized systems, a self-consistent optimaltuning procedure chooses γ to ensure that Koopman’stheorem is obeyed , i.e., to minimize the targetfunction: J ( γ ) = | ǫ γH + IP γ | , (2)where ǫ γH is the HOMO energy of the electron sys-tem, and the ionization potential is given by IP γ ≡ [ E γ ( N ) − E γ ( N − δ )] /δ , where N is the number of elec-trons in the neutral system, and δ is a small fractionalcharge. Optimally tuned RSH (OT-RSH) functionalshave been applied to study various molecular systemsand nanocrystals, yielding band gaps in good agree-ments with GW and/or experimental results. However, this procedure is not applicable to periodicsolids, where total energy calculations of charged systemsare problematic.There are several ways of obtaining γ for solids. Formolecular solids, satisfactory results can be obtained with γ tuned for isolated molecules. For solids in general,various attempts were made to connect γ with the opticaldielectric constant ǫ ∞ . Here we devise instead an approach for systematicallytuning the range-separation parameter for periodic sys-tems solely based on first principle calculations. The ideais to perform RSH calculations as well as G W calcula-tions with RSH as starting points. Two sets of band gapswill then be obtained, each being functions of the param-eter γ . The optimal γ is determined such that the twogaps agree with each other.We emphasize that this proposed techinque is notjust a modified G W , but is closely related to the self-consistent GW (scGW) method. In previous work, wehave shown that perhaps the simplest self-consistent GWmethod is ev-sc GW with scissors operator; namely, self-consistently, updating the G operator through a scissorsshift of the occupied vs. virtuals states; this amounts torepeatedly writing: ˆΣ( t ) ∝ ˆΣ( t ) e − i ∆ θ ( t ) t ,where we introduce the time domain self-energy ˆΣ( t ) , ∆ is the difference between the quasiparticle band gap andthe DFT band gap, and θ ( t ) is the Heaveside step func-tion. We demonstrated that such a self-cosistent proce-dure can open up the fundamental band gap and improvethe accuracy of calculated gaps over one-shot G W . The current approach for finding γ gives ∆ = 0 , andthus amounts to finding a stationary point for this self-consistent procedure.In the following discussions, we will refer to thistuning procedure as OT-GW/RSH. OT-GW/RSH can be implemented with any conven-tional GW code, such as VASP. Since many of theOT-GW/RSH applications are envisioned to eventuallytake place for large (potentially disordered) systems, wehave also studied here the use of the method with ourrecent linear-scaling stochastic GW (sGW) approach,which has been succesfully applied to systems of , electrons and more. To carry out sGW calculationswith RSH, we also applied here a stochastic method forapplying and propagating long-range Fock exchange.
In sGW, the quasiparticle energy in G W formulationis obtained as first-order perturbation to the Kohn-Shamorbital energies: ǫ QP = ǫ KS + h φ F | ˆΣ P ( ǫ QP ) + ˆ K − ˆ V XC | φ F i (3)where φ F ( r ) is typically the HOMO or LUMO, ˆΣ P isthe dynamical polarization self energy, ˆ K is the full Fockexchange operator, and ˆ V XC is the exchange-correlationpart of the GKS-Hamiltonian as in equation (1). In sGW,we calculate the matrix elements of ˆΣ P in the time do-main. Detailed accounts of the sGW method are foundin previous works. One important issue is that the Generalized Kohn-Sham Hamiltonian ˆ h contains the long-range ex-change potential, which depends on the density matrix: ρ ( r, r ′ ) = P i,occ φ i ( r ) φ ∗ i ( r ′ ) . For large systems, the num-ber of occupied orbitals is large, making the applica-tion of long-range exchange computationally demand-ing. To solve this problem, we implemented stochas-tic long-range Fock exchange in the sGW code, as donerecently. Detailed explanation of the method can befound in references.
In short, we use for the pur-pose of stochastic exchange a total number of N ζ stochas-tic orbitals, each being a linear combination of occupiedstates that are obtained by a low-band-pass flter of awhite-noise function, | ζ i = q Θ( µ − ˆ h ) | ζ i . Here, we in-troduced the chemical potential µ , the white noise func-tion is chosen as ζ ( r ) ∝ ± , and the application of Θ operator is carried out by a Chebyshev expansion. Thedensity matrix is then approximated by: ρ ( r, r ′ ) ≈ [ ζ ( r ) ζ ∗ ( r ′ )] . (4)Further, the long-range Coulomb potential is approx-imated as u l ( | r − r ′ | ) = [ χ ( r ) χ ∗ ( r ′ )] , where χ ( r ) is con-structed by Fourier transforming a stochastic combina-tion of the square root of the Fourier components of thelong-range potential, p ˜ u l ( k ) . These two random repre-sentations make the long range exchange operator a sumof separable terms. Then, the action of the long-rangeexchange operator becomes:Figure 1: Fundamental band gaps (in eV) as functionsof γ for a LiF × × supercell. ˆ K l ψ = − [ ζ ( r ) χ ( r ) ˆ dr ′ ζ ∗ ( r ′ ) χ ∗ ( r ′ ) ψ ( r ′ )] (5) = − N ζ X ζ ζ ( r ) χ ( r ) ˆ dr ′ ζ ∗ ( r ′ ) χ ∗ ( r ′ ) ψ ( r ′ ) (6)We note that a part of the GW calculation, the actionof the short-time e − i ˆ K l dt is required; a one-term Taylorexpansion is used, in conjunction with re-normalizationof the orbitals after the short-time propagator is ap-plied. Finally, in calculating the final quaiparticle energyaccording to equation (3), we note that the difference ˆ K − ˆ V XC involves the term ˆ K s = ˆ K − ˆ K l . In our code,this term is calculated in a similar manner to ˆ K l , but nowusing the Fourier transofrm of the short range potential. I. RESULTS AND DISCUSSIONS
We assessed the tuning procedure for several solids us-ing VASP. For two of the systems, we additionally per-formed sGW calculations. As an illustration of the tun-ing procedure, we plot the fundamental gaps calculatedfrom sGW/RSH and RSH, as functions of the range-separation parameter γ for a LiF × × supercell.It is evident from Fig. 1 that the RSH DFT results aremore sensitive to γ than the GW gaps. This is expected,as γ affects the GW gap only indirectly by changing theDFT starting point. A summary of the calculated fun-damental band gaps are given in Table 1.With the exception of Si, results obtained using OT-GW/RSH are in good agreement with that of self-consistent GW (scGW). This is consistent with our no-tion that OT-GW/RSH is equivalent to finding a sta-tionary point in scGW. We also note that both OT-GW/RSH and scGW tend to over-estimate the experi-mental value. In fact, this over-estimation has been re-ported in the literature, and this performance has beenascribed to an under-estimation of the dielectric screen-ing in random-phase approximation (RPA) adopted in OT-GW/RSH scGW Exp.VASP sGWZnO 3.7 3.8 a a Si 1.24 1.24 1.41 a a LiF 15.9 15.8 15.9 a a CdO 0.99 0.98 b b Table I: Fundamental band gaps with OT-GW/RSHusing VASP and sGW, in eV. Results fromself-consistent GW (scGW) as well as experiments(Exp.) are also reported. References: a) b) γ (VASP) γ (sGW) γ (Fitted) ǫ ∞ ZnO 0.12 0.09 3.14 a Si 0.029 0.029 0.019 11.68 b LiF 0.286 0.297 0.2 1.92 c CdO 0.1 0.15 2.3 d Table II: Optimally tuned γ in Bohr − , obtained fromOT-GW/RSH with VASP and sGW. We also reportvalues fitted from dielectric constant using the empiricalformula from the work of Baer et al. References:a) b) c) d) GW calculations.
Possible ways to fix the situationwere proposed, and this will be explored in our futurework.As for the present work, we emphasize that OT-GW/RSH provides a way of tuning the range-separationparameter for periodic systems, and it produces good re-sults. To see this, the fitted optimal γ are reported inTable 2. For reference, we compare the optimal γ ob-tained from OT-GW/RSH to that calculated using theempirical formula of Baer et al. The two methods giveresults that are overall consistent though not identical.This indicates that OT-GW/RSH is an effective methodfor obtaining the range-separation parameter in periodicsystems from first principles only.
ACKNOWLEDGEMENTS
The authors acknowledge support from the Center forComputational Study of Excited State Phenomena in En-ergy Materials (C2SEPEM) at the Lawrence BerkeleyNational Laboratory, which is funded by the U.S. De-partment of Energy, Office of Science, Basic Energy Sci-ences, Materials Sciences and Engineering Division underContract No. DE-AC02-05CH11231 as part of the Com-putational Materials Sciences Program. Computationalresources were supplied through the XSEDE allocationTG-CHE170058. In addition, R.B. gratefully acknowl-edge the support from the US−Israel Binational ScienceFoundation (BSF) under Grant No. 2018368.
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