J. Chem. Phys. 151, 124102 (2019); https://doi.org/10.1063/1.5119124 151, 124102

© 2019 Author(s).

Multicomponent density functional theory:Including the density gradient in theelectron-proton correlation functional forhydrogen and deuteriumCite as: J. Chem. Phys. 151, 124102 (2019); https://doi.org/10.1063/1.5119124Submitted: 08 July 2019 . Accepted: 29 August 2019 . Published Online: 23 September 2019

Zhen Tao , Yang Yang , and Sharon Hammes-Schiffer

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Multicomponent density functional theory:Including the density gradient in theelectron-proton correlation functionalfor hydrogen and deuterium

Cite as: J. Chem. Phys. 151, 124102 (2019); doi: 10.1063/1.5119124Submitted: 8 July 2019 • Accepted: 29 August 2019 •Published Online: 23 September 2019

Zhen Tao, Yang Yang, and Sharon Hammes-Schiffera)

AFFILIATIONSDepartment of Chemistry, Yale University, 225 Prospect Street, New Haven, Connecticut 06520, USA

a)E-mail: sharon.hammes-schiffer@yale.edu

ABSTRACTMulticomponent density functional theory (DFT) has many practical advantages for incorporating nuclear quantum effects into quan-tum chemistry calculations. Within the nuclear-electronic orbital (NEO) framework, specified nuclei, typically protons, are treated quan-tum mechanically on the same level as the electrons. Previously, electron-proton correlation functionals based on the local densityapproximation (LDA), denoted epc17 and epc18, were developed and shown to provide more accurate proton densities and ener-gies compared to the neglect of electron-proton correlation, but a quantitatively accurate description of both densities and energiessimultaneously has remained elusive. Herein, an electron-proton correlation functional that depends on the electron and proton den-sity gradients, as well as the densities, is derived and implemented. Compared to the LDA functionals, the resulting generalized gra-dient approximation functional, denoted epc19, is able to simultaneously provide accurate proton densities and energies, as well asreproduce the impact of nuclear quantum effects on optimized geometries. In addition, without further parameterization, the NEO-DFT/epc19 method provides accurate densities and energies for deuterium as well as hydrogen. These results demonstrate that theform of the epc19 functional is able to capture the essential aspects of electron-proton correlation and highlights the importance ofincluding gradient terms. This approach will enable the exploration of nuclear quantum effects and isotope effects in a wide range ofsystems.

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I. INTRODUCTION

Most quantum chemistry methods such as ab initio wave func-tion theory and density functional theory (DFT) treat nuclei asfixed point charges and invoke the Born-Oppenheimer separationbetween the motions of the nuclei and the electrons. Such calcu-lations cannot directly capture nuclear quantum effects, includingzero-point energy, nuclear delocalization, and vibrational excita-tion energies, which are important for calculating properties suchas vibrationally averaged geometries, geometric isotope effects, andelectron-proton vibronic states in proton coupled electron-transfersystems.1,2 Many approaches have been developed to describethese nuclear quantum effects to varying degrees. For example,

zero-point energy and vibrational frequencies are commonly cal-culated by diagonalizing a Hessian matrix, which is an efficientapproach but does not include anharmonic effects. A variety of per-turbative and grid-based methods have been developed to includeanharmonic effects in vibrational frequency calculations.3–5 How-ever, typically these methods do not include nuclear quantumeffects during the optimization of geometries, the generation ofreaction paths, or the propagation of reaction dynamics. Pathintegral molecular dynamics methods6,7 incorporate nuclear quan-tum effects but require a significant amount of conformationalsampling.

An alternative approach is the nuclear-electronic orbital (NEO)method,8 which treats electrons and select nuclei, typically protons,

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quantum mechanically on the same level. In the NEO method, acombined nuclear-electronic Schrödinger equation is solved withmolecular orbital techniques. At least two classical nuclei are neededto eliminate difficulties with the translational and rotational degreesof freedom. The NEO method has the advantage of includingthe zero-point energy and nuclear delocalization of the quan-tum protons during the self-consistent field (SCF) procedure. BothNEO wave function methods8,9 and NEO density functional theory(NEO-DFT) methods10,11 have been developed. A practical advan-tage of NEO-DFT is that it has the formal scaling of conven-tional electronic DFT, which is O(N3) or O(N4), depending on thetype of electronic functional, and density fitting methods12 mayimprove the scaling. Other multicomponent DFT methods havealso been explored,13–18 but the main obstacle has been the lack ofelectron-proton correlation functionals that provide even qualita-tively accurate proton densities, energies, and vibrational frequen-cies. Electron-proton correlation is highly significant because ofthe attractive interaction between an electron and a proton, andthe neglect or inadequate description of this correlation leads tounphysical behavior due to severely overlocalized proton densities.19

In previous work, a number of important mathematical proper-ties of the exact universal multicomponent density functional werederived.20

Recently, a series of electron-proton correlation functionalswere developed in the framework of NEO-DFT and have beenshown to provide accurate proton densities,21 energies,22 geome-tries,22 and vibrational frequencies,23 again demonstrating theimportance of electron-proton correlation. These epc17 and epc18functionals21,24 were based on the Colle-Salvetti formalism,25 whichproduced the popular Lee-Yang-Parr (LYP) electronic correlationfunctional,26 with several essential differences required for describ-ing electron-proton correlation. The epc17 and epc18 functionalscould be parameterized to provide highly accurate proton densitiesor energies but not both simultaneously, although the epc17-2 andepc18-2 functionals achieved a reasonable compromise. In contrastto the LYP electronic correlation functional, the epc17 and epc18functionals are based on the local density approximation (LDA)and thus depend only on the electron and proton densities. Herein,we derive an electron-proton correlation functional, denoted epc19,that also depends on the gradients of the electron and proton den-sities to further enhance the accuracy. Due to its dependence on thegradient terms, this functional belongs to the set of generalized gra-dient approximation (GGA) functionals and is analogous to the LYPelectronic correlation functional.

This paper is organized as follows: Section II outlines theformalism underlying the epc19 functional and the parameteriza-tion procedure, with the full details provided in the supplementarymaterial. Section III presents the performance of the epc19 func-tional for computing proton densities and energies, proton affini-ties, and geometry optimizations for both hydrogen and deuterium.A summary and discussion of future directions are provided inSec. IV.

II. THEORYMulticomponent DFT, in which more than one type of par-

ticle is treated quantum mechanically, has been formulated as anextension of the Hohenberg-Kohn theorems.11,15,16,18,20,27 Within

the NEO-DFT framework, the total energy can be expressed as afunctional of the electron and proton densities, ρe and ρ p, respec-tively,

E[ρe, ρ p] = (Ts[ρe] + Ts[ρ p

]) + (Veext[ρ

e] + Vp

ext[ρp])

+ (Jee[ρe] + Jpp[ρ p] + Jep[ρe, ρ p

])

+ (Eexc[ρ

e] + Ep

xc[ρp]) + Eepc

[ρe, ρ p]. (1)

The quantities in each parenthesis are as follows: the noninter-acting kinetic energy of the electrons and protons; the interac-tion of the external potential with the electrons and protons;the electron-electron, proton-proton, and electron-proton mean-field Coulomb interactions; the exchange-correlation energies ofthe electrons and protons; and the electron-proton correlationenergy. The Kohn-Sham formalism can be used in conjunctionwith Eq. (1) to determine the electron and proton Kohn-Shamorbitals and densities, as well as the total energy, for a given clas-sical nuclear configuration. Within this framework, existing elec-tronic exchange-correlation functionals can be used.24 Because thequantum nuclei are sufficiently localized in molecular systems, theproton exchange-correlation functional can be approximated by thediagonal Hartree-Fock exchange terms to eliminate self-interactionerror.28

The development of the epc19 functional starts with the sameanalog of the Colle-Salvetti25 wavefunction ansatz as used in thedevelopment of the epc17 and epc18 functionals,

Ψ(xe, xp) = ΨFCIe (xe)Ψ

FCIp (xp) ∏

i∈Ne ,I∈Np

[1 − φ(ri,e, rI,p)], (2)

where xe and xp represent the combined spatial and spin coordi-nates of all electrons and protons, respectively, and ri,e and rI,p rep-resent the spatial coordinates of individual electrons and protons,respectively. The protons only adopt high spin configurations inseparate orbitals to avoid unphysically large Coulomb repulsion, asis reasonable in molecular systems because each proton is spatiallylocalized.29 In Eq. (2), ΨFCI

e and ΨFCIp are the electron and proton

wavefunctions from the full configuration interaction (FCI) treat-ment of electron-electron exchange-correlation and proton-protonexchange-correlation, respectively, each of which includes only amean-field Coulomb interaction between electrons and protons.Electron-proton correlation is included through the correlation fac-tor φ(ri,e, ri,p), which has the form

φ(ri,e, ri,p) = exp[−β2(R)r2

][1 − ξ(R)(1 − r)], (3)

where r = ∣r∣ = |ri,e−rI,p| and R is the center of mass betweenthe ith electron and the Ith proton. The electron-proton correla-tion energy is defined to be the difference between the exact totalenergy associated with Eq. (1) and the sum of the FCI energies cor-responding to the electron and proton FCI wavefunctions with thedouble counting of the mean-field electron-proton Coulomb energyremoved.

The form of ξ(R) in Eq. (3) can be determined as a func-tion of the inverse correlation length β(R), as shown previously.21

The inverse correlation length β(R) is assumed to be inversely pro-portional to the electron-proton Wigner-Seitz radius rs,ep, whichis defined to be the geometric mean of the electron and protonWigner-Seitz radii from a uniform electron or proton gas model:

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rs,ep = (rs,ers,p)1/2. Using the expressions for the individualWigner-Seitz radii, rs,e(R) ∝ [ρe(R)]−1/3 and rs,p(R)∝ [ρp(R)]−1/3, the inverse correlation length is expressed asβ(R)∝ [ρe(R)]1/6[ρp(R)]1/6. This expression was used to developthe epc17 functionals.21 The alternative assumption that β(R) isproportional to the arithmetic mean of the inverse electron and pro-ton Wigner-Seitz radii was used to develop the epc18 functionals,24

which perform similarly to the epc17 functionals. Note that the

inverse of β(R) can be interpreted as the length scale over whichelectron-proton correlation exerts a significant contribution. Thus,in contrast to prior efforts at developing electron-proton correla-tion functionals,30,31 the inverse correlation length should depend onboth the electron and proton densities, although the specific form issomewhat flexible.

Following additional approximations analogous to thoseinvoked in the Colle-Salvetti formulation, the electron-proton cor-relation energy is expressed as

Eepc≈ −4π∫ dRPFCI

2,ep(R,0)∫ dre−2β2(R)r2

[1 − ξ(R)(1 − r)]2r + 8π∫ dRPFCI2,ep(R,0)∫ dre−β

2(R)r2

× [1 − ξ(R)(1 − r)]r − 2π/3∫ dR(∇2rP

FCI2,ep(R,r))∣

r=0 ∫dre−2β2

(R)r2

[1 − ξ(R)(1 − r)]2r3

+ 4π/3∫ dR(∇2rP

FCI2,ep(R,r))∣

r=0 ∫dre−β

2(R)r2

[1 − ξ(R)(1 − r)]r3, (4)

where PFCI2,ep(R, 0) = ρFCIe (R)ρFCIp (R) ≈ ρe(R)ρp(R). In the epc17 and epc18 functionals, the last two terms involving the second-order

derivatives of the pair density were neglected, producing LDA functionals that depend only on the electron and proton densities. Inthe present work, we include all four of the terms in Eq. (4). After integration, we approximate the GGA functional with the followingexpression:

Eepc[ρe, ρp,∇ρe,∇ρp] = −∫ dR

ρe(R)ρp(R)a − b[ρe(R)]1/2[ρp(R)]1/2 + cρe(R)ρp(R)

⎧⎪⎪⎨⎪⎪⎩

1 − d⎛

⎝

[ρe(R)ρp(R)]−1/3

(1 + mp)2

× [m2p∇

2ρe(R)ρe(R)

− 2mp∇ρe(R) ⋅ ∇ρp(R)

ρe(R)ρp(R)+∇

2ρp(R)ρp(R)

])exp⎡⎢⎢⎢⎢⎣

−k[ρe(R)ρp(R)]1/6

⎤⎥⎥⎥⎥⎦

⎫⎪⎪⎬⎪⎪⎭

, (5)

where mp is the proton mass, and the second derivative terms canbe converted to gradient terms through integration by parts. Thisfunctional has five parameters: a, b, c, d, and k. Note that settingd = 0 recovers the functional form of epc17.

The GGA functional depends on the proton mass, in contrastto the LDA functionals developed previously. The explicit depen-dence of the epc19 functional on the proton mass suggests that acommon set of parameters could potentially describe both hydrogenand deuterium. However, the validity of some of the approxima-tions underlying these functionals may be different for hydrogenand deuterium and therefore could require the use of differentparameters for describing hydrogen vs deuterium. Nevertheless,for simplicity, we use the same set of parameters to describeboth isotopes with the epc19 functional. Although the energiesand densities for deuterium are of similar accuracy as those forhydrogen, more quantitatively accurate results could potentiallybe obtained for deuterium with reparameterization of the epc19functional.

Another potential complication with deuterium is that it isa boson rather than a fermion. For molecular systems, however,the nuclear orbitals are sufficiently localized, with only one deu-terium per orbital in the lowest-energy state, that the energyexpressions derived for hydrogen are also applicable to deuterium.

Moreover, these bosonic effects impact the deuterium-deuteriumexchange interactions but do not impact the validity of theepc19 functional. In this paper, these issues are not directlyrelevant because the molecules studied contain only a singledeuterium.

III. RESULTSWe determined the epc19 parameters for hydrogen by mini-

mizing the errors in the energies and the root-mean-square devi-ations (RMSDs) of the three-dimensional proton densities forHCN and FHF−. The reference proton vibrational ground stateenergies and proton densities were obtained from Fourier gridHamiltonian (FGH)32,33 calculations corresponding to the hydro-gen moving on a three-dimensional potential energy surface.This surface was generated by moving the proton along a gridwith 32 grid points per dimension and performing conventionalDFT/B3LYP/def2-QZVP26,34,35 single-point energy calculations ateach grid point. The resulting proton vibrational ground state ener-gies and proton densities are considered to be numerically accuratefor these types of electronically adiabatic systems and serve as thebenchmark. For the NEO-DFT calculations, the proton and all elec-trons were treated quantum mechanically. The geometries of HCN

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TABLE I. RMSD of three-dimensional proton densities for FHF− and HCN anddeuterated counterparts.a,b

(a.u.) no-epc epc17-1 epc17-2 epc19

FHF− 0.814 0.121 0.190 0.118HCN 0.735 0.170 0.331 0.222FDF− 1.13 0.128 0.185 0.048DCN 1.00 0.142 0.452 0.296

aThe RMSD was computed as the square root of the sum of the squares of the differencesbetween the NEO-DFT and FGH proton density at each grid point, summing over theentire three-dimensional grid and dividing by the number of grid points.bThe cubic grid for FHF− and FDF− spans the range from −0.5610 Å to 0.5984 Å, bothalong and perpendicular to the F–F axis, where the F atoms are located at±1.1507 Å. Thecubic grid for HCN and DCN spans the range from 0.3245 Å to 1.8652 Å along the C–Naxis and from −0.7455 Å to 0.7952 Å perpendicular to this axis, where the carbon andnitrogen atoms are positioned at 0.0 Å and −1.1463 Å, respectively.

and FHF− were optimized at the conventional DFT/B3LYP levelwith the def2-TZVPD36 basis set, and the heavy atoms were fixedat these geometries for both the NEO calculations and the FGHcalculations. All of the NEO calculations were performed with theB3LYP electronic functional and the def2-QZVP35 electronic basisset, with an even-tempered 8s8p8d proton basis set.21,37 During theparameterization, the basis functions associated with the quantumproton were centered at the equilibrium position of the hydrogenobtained through conventional DFT geometry optimization. After

TABLE II. Energy errora for FHF− and HCN and deuterated counterparts.

(eV) no-epc epc17-1 epc17-2 epc19

FHF− 0.66 −0.79 −0.12 −0.08HCN 0.85 −0.75 0.09 0.11FDF− 0.49 −0.82 −0.15 −0.09DCN 0.63 −0.78 0.03 0.08

aThe energy error is defined as the difference between the NEO-DFT energy and theenergy of the ground proton vibrational state obtained from the FGH calculation. Here,the absolute energies can be compared because the lowest-energy point on the gridpotential energy surface corresponds to the conventional electronic DFT energy at theequilibrium geometry used for the NEO-DFT calculations.

selecting the parameters for the epc19 functional, the calculationswere performed again while variationally optimizing the basis func-tion center positions to produce the energies and proton densitiespresented herein, with the exception of the proton affinities, whichwere computed with the basis function centers at the equilibriumhydrogen positions optimized with conventional DFT. All of thesecalculations were performed with an in-house developer version ofGAMESS.38

In addition to minimizing the errors in the energies and pro-ton densities, we examined the shapes of the proton densities toavoid unphysical features and confirmed that the linear equilibriumgeometries for HCN and FHF− were retained. The epc19 functional

FIG. 1. On-axis (upper panels) and off-axis (lower panels) proton densities for HCN (left panels) and FHF− (right panels) calculated with the FGH grid-based method as thereference (solid black line), NEO-DFT/no-epc (dashed red line), NEO-DFT/epc17-2 (dotted blue line), and NEO-DFT/epc19 (dashed-dotted magenta line). The on-axis protondensity is the slice along the line connecting the heavy atoms, and the off-axis proton density is the slice perpendicular to this line and passing through the maximum densitygrid point, which is located at 2.02 a.u. for HCN and 0.000 a.u. for FHF−. The C and N atoms are located at 0.000 a.u. and −2.1662 a.u., respectively, and the F atoms arelocated at ±2.1746 a.u. The basis function centers associated with the H were optimized variationally for each functional.

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parameters a, b, c, d, and k determined from this procedure are1.9, 1.3, 8.1, 1600, and 8.2, respectively. During the parameteriza-tion procedure, we found that smaller values of k tended to producebent equilibrium geometries for HCN and/or FHF−. Given this rela-tively large value of k, which dampens the overall contribution fromthe gradient-dependent term in Eq. (5), a relatively large value ofd was required to obtain accurate proton energies and densities.Although these parameters were determined by fitting only to datafor hydrogen, this same set of parameters was used to describe deu-terium as well. The parameter space is sufficiently extensive thatmultiple parameter sets could produce similar or even better results,but these parameters provide the desired level of accuracy for thepresent work.

The proton densities and energies computed with the vari-ous electron-proton correlation functionals for HCN and FHF−

are compared to each other and to the grid reference in Tables Iand II, respectively. Table I provides the root-mean square devi-ations (RMSDs) for the NEO-DFT proton densities with respectto the grid-based reference densities. The epc19 functional pro-duces proton densities reasonably comparable to those obtainedwith the epc17-1 functional, which was previously parameterizedsolely for accurate proton densities.21 The proton densities cal-culated with various electron-proton correlation functionals aredepicted in Fig. 1, with one-dimensional slices corresponding tothe proton density along the axis connecting the heavy atoms (onaxis) or the proton density perpendicular to this axis (off axis).The neglect of electron-proton correlation, denoted NEO-DFT/no-epc, produces highly overlocalized proton densities, as also indi-cated by the large RMSDs for the NEO-DFT/no-epc proton densitiesgiven in Table I. The proton densities calculated with the epc17-1,21

epc17-2, and epc19 functionals are significantly more delocalizedand agree much better with the grid-based reference, with slightvariations in the shapes. Note that the proton densities computedwith the epc17-1 functional are presented in Ref. 21 but are notshown in Fig. 1 for clarity. Table II provides the difference betweenthe NEO-DFT energies and the grid-based reference energies. Theepc19 functional produces energies for HCN and FHF− compara-ble to those obtained with the epc17-2 functional, which has beenwidely used within the NEO-DFT framework because of its accurateenergies. In contrast, the energies provided by the epc17-1 functionaland without electron-proton correlation (i.e., no-epc) are physicallyunreasonable.

To further test the epc19 functional, we computed the protonaffinities for a set of 23 molecules. The proton affinity is definedas the negative enthalpy corresponding to species A gaining a pro-ton to produce species AH+. Within the NEO-DFT framework, theproton affinity is defined to be the difference between the conven-tional DFT energy of A and the NEO-DFT energy of AH+, withthe addition of 5/2 RT to account for the conversion from energyto enthalpy and translational energy changes.22 NEO calculationsinherently include the vibrational energy associated with the quan-tum protons, and the change in the vibrational energy associatedwith the classical nuclei is assumed to be negligible upon protona-tion.9 All of the proton affinity calculations were performed with theB3LYP electronic functional and the def2-QZVP35 electronic basisset. The NEO calculations were performed with an even-tempered10s10p10d proton basis set22,37 centered at the equilibrium posi-tion of the proton obtained through a conventional DFT geometry

optimization. Table III illustrates that the epc19 and epc17-2 func-tionals achieve the same level of accuracy for computing this set ofproton affinities with a mean unsigned error (MUE) of 0.06 eV. Notethat this MUE is within the experimental uncertainty of ∼0.09 eV, asreported by Hunter and Lias.39

A significant advantage of the NEO approach over conven-tional electronic structure methods is that nuclear quantum effectssuch as proton delocalization and zero-point energy can be includedin geometry optimizations. To investigate the accuracy of the epc19functional for geometry optimizations, we calculated the equi-librium F–F bond length for FHF− with the NEO-DFT methodusing the epc19 functional and compared it to the equilibrium

TABLE III. Proton affinities computed with NEO-DFT using epc19 and epc17-2electron-proton correlation functionals.

(eV) Experiment39–43 epc19 epc17-2

AminesNH3 8.85 8.90 8.89CH3NH2 9.32 9.36 9.37CH3CH2NH2 9.45 9.51 9.52CH3CH2CH2NH2 9.51 9.57 9.58(CH3)2NH2 9.63 9.65 9.67(CH3)3NH2 9.84 9.82 9.85

MUE 0.04 0.05

InorganicsCN− 15.31 15.18 15.21HS− 15.31 15.23 15.27NOO− 14.75 14.85 14.84

MUE 0.10 0.07

CarboxylatesHCOO− 14.97 14.97 14.95CH3COO− 15.11 15.15 15.12CH3CH2COO− 15.07 15.12 15.10CH3CH2CH2COO− 15.03 15.11 15.08CH3CH2CH2CH2COO− 15.01 15.10 15.08CH3COCOO− 14.46 14.46 14.43CH2FCOO− 14.71 14.69 14.66CHF2COO− 14.32 14.38 14.35CF3COO− 13.99 14.10 14.07CH2ClCOO− 14.58 14.55 14.53CH2ClCH2COO− 14.78 14.67 14.64

MUE 0.05 0.05

AromaticsC6H5O− 15.24 15.18 15.17C6H5COO− 14.75 14.54 14.53C6H5NH2 9.15 9.15 9.16

MUE 0.09 0.10

Overall MUE 0.06 0.06

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TABLE IV. Equilibrium fluorine-fluorine distance in Å calculated for FHF− and FDF−.

Standard DFT Reference NEO-DFT/epc19 NEO-DFT/epc17-2

FHF− 2.2978 2.3185 2.3302 2.3206FDF− 2.2978 2.3130 2.3294 2.3185Difference 0.0000 0.0055 0.0008 0.0021

bond length obtained with conventional electronic DFT, the grid-based reference, and NEO-DFT/epc17-2. The grid-based referenceequilibrium F–F distance was determined using the FGH methodat the B3LYP/def2-QZVP level of theory to compute the energyfor a range of F–F distances with a step size of 0.0005 Å inthe region near the minimum. The NEO-DFT equilibrium F–Fdistances were obtained with a step size of 0.0001 Å using thesame electronic functional and basis set with the 10s10p10d pro-ton basis set. As shown previously,22 conventional DFT geometryoptimization underestimates the equilibrium F–F distance. Table IVillustrates that both NEO-DFT/epc17-2 and NEO-DFT/epc19 cor-rectly describe the increase in the equilibrium F–F distance by∼0.02–0.03 Å when the nuclear quantum effects of the proton areincluded.

Tables I and II also provide the errors in the proton densi-ties and energies for DCN and FDF−, where hydrogen has beenreplaced by deuterium. Without parameterization for deuterium,

the NEO-DFT/epc19 method produces similar errors in the ener-gies and densities for deuterium as for hydrogen. Analogously,even though the epc17-2 functional was parameterized for hydro-gen, the NEO-DFT/epc17-2 method produces similar errors in theenergies and densities for both hydrogen and deuterium. However,the epc19 functional continues to provide significantly more accu-rate nuclear densities than does the epc17-2 functional, as illus-trated by Fig. 2. The qualitative increase in the equilibrium F–Fdistance for FDF− compared to conventional DFT is reproducedby both the epc17-2 and epc19 functionals, although they overes-timate the equilibrium distance by 0.0055 Å and 0.0164 Å, respec-tively. Moreover, both functionals qualitatively reproduce the geo-metric isotope effect, where the equilibrium F–F distance is smallerfor FDF− than for FHF−, although the quantitative differences areunderestimated. These results suggest that reparameterizing for deu-terium may be important for capturing subtle quantitative geometricisotope effects.

FIG. 2. On-axis (upper panels) and off-axis (lower panels) proton densities for DCN (left panels) and FDF− (right panels) calculated with the FGH grid-based methodas the reference (solid black line), NEO-DFT/no-epc (dashed red line), NEO-DFT/epc17-2 (dotted blue line), and NEO-DFT/epc19 (dashed-dotted magenta line). Thepositions of the heavy nuclei and the axes associated with the density slices are the same as those used in Fig. 1. The basis function centers associated with theD were optimized variationally for each functional. Note that the small shoulder in the DCN on-axis density could be eliminated by parameterization specifically fordeuterium.

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IV. CONCLUSIONIn this paper, we derived and parameterized the GGA ver-

sion of the epc17 functional within the NEO-DFT frameworkby including the gradient terms with respect to both the elec-tron and proton densities. The resulting epc19 functional pro-vides comparable energies to the epc17-2 functional and signif-icantly more accurate proton densities, achieving the objectiveof producing accurate energies and densities simultaneously. Inaddition, the epc19 functional achieves a similar level of accu-racy as the epc17-2 functional for the proton affinities associatedwith a set of 23 molecules and reproduces the increased equilib-rium F–F distance for FHF− upon inclusion of nuclear quantumeffects. These combined results demonstrate an overall improve-ment over the previous LDA-type electron-proton correlation func-tionals when the gradient terms are included. However, we empha-size that the parameterization depends on the weighting of thesevarious properties, and a different weighting could improve anyone of these properties, possibly at the expense of another prop-erty. The objective of this paper is to provide the form of a GGAfunctional that may be parameterized to optimize the properties ofinterest.

These functionals were also applied to deuterated systemsto enable the investigation of hydrogen/deuterium isotope effects.Without further parameterization, the epc17-2 and epc19 func-tionals each produced energies and densities for deuterium thatare of similar accuracy as those computed for hydrogen withthe same functional. As observed for hydrogen, the nuclear den-sities for deuterium are significantly more accurate when com-puted with the epc19 functional than with the epc17-2 func-tional. Moreover, both functionals qualitatively reproduced thegeometric isotope effect for FHF− vs FDF−, where the equilib-rium F–F distance is greater for hydrogen than for deuterium,although quantitative accuracy may require separate parameters fordeuterium.

This work illustrates that the form of the epc19 functionalis able to capture the essential aspects of electron-proton cor-relation. However, more systematic parameterization procedureswith larger datasets will be necessary to develop more quanti-tatively accurate functionals for both hydrogen and deuterium.Future studies will focus on applying the NEO-DFT methodto systems for which nuclear quantum effects such as zero-point energy and nuclear delocalization are important. The useof this method to investigate geometric isotope effects, as wellas equilibrium and kinetic isotope effects, is another promisingdirection.

SUPPLEMENTARY MATERIAL

See the supplementary material for detailed formalism under-lying the epc19 functional.

ACKNOWLEDGMENTSThe authors thank Tanner Culpitt, Patrick Schneider,

Dr. Fabijan Pavoševic, and Benjamin Rousseau for helpful discus-sions. This work was supported by the National Science Foundation(Grant No. CHE-1762018).

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