^{1}, Michael V. Pak

^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

The interplay between electron-electron and electron-proton correlation is investigated within the framework of the nuclear-electronic orbital density functional theory (NEO-DFT) approach, which treats electrons and select protons quantum mechanically on the same level. Recently two electron-proton correlation functionals were developed from the electron-proton pair densities obtained from explicitly correlated wavefunctions. In these previous derivations, the kinetic energy contribution arising from electron-proton correlation was neglected. In this paper, an electron-proton correlation functional that includes this kinetic energy contribution is derived using the adiabatic connection formula in multicomponent DFT. The performance of the NEO-DFT approach using all three electron-proton correlation functionals in conjunction with three well-established electronic exchange-correlation functionals is assessed. NEO-DFT calculations with these electron-proton correlation functionals capture the increase in the hydrogen vibrational stretching frequencies arising from the inclusion of electron-electron correlation in model systems. Electron-proton and electron-electron correlation are found to be uncoupled and predominantly additive effects to the total energy for the model systems studied. Thus, electron-proton correlation functionals and electronic exchange-correlation functionals can be developed independently and subsequently combined together without re-parameterization.

We thank Chet Swalina and Chaehyuk Ko for helpful discussions. We also thank Benjamin Janesko for asking a question that motivated us to include the kinetic energy in the electron-proton functional. We gratefully acknowledge funding from AFOSR Grant FA9550-10-1-0081 and NSF Grant CHE-10-57875. A.S. thanks the Natural Sciences and Engineering Research Council of Canada for a PGS scholarship.

I. INTRODUCTION

II. THEORY

A. Multicomponent density functional theory

B. Electron-proton correlation functionals

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Electron correlation calculations
- 74.0
- Exchange correlation functionals
- 57.0
- Correlation functions
- 38.0
- Protons
- 28.0
- Density functional theory
- 19.0

## Tables

Electron-electron and electron-proton correlation energies for the [He-H-He]^{+} system at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set and a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent were used. The NEO-DFT calculations were performed using the EPC2 electron-proton correlation functional with three different electronic exchange-correlation functionals: B3LYP, BLYP, and PBE. The correlation energies *E* _{corr}(ee), *E* _{corr}(ep), and *E* _{corr}(ee, ep) are defined as the differences between the NEO-DFT(ee), NEO-DFT(ep), and NEO-DFT(ee,ep) energies, respectively, and the NEO-HF energy. The additivity error is defined as σ^{additivity} = *E* _{corr}(ee, ep) − *E* _{corr}(ee) − *E* _{corr}(ep). Energies are reported in atomic units.

Electron-electron and electron-proton correlation energies for the [He-H-He]^{+} system at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set and a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent were used. The NEO-DFT calculations were performed using the EPC2 electron-proton correlation functional with three different electronic exchange-correlation functionals: B3LYP, BLYP, and PBE. The correlation energies *E* _{corr}(ee), *E* _{corr}(ep), and *E* _{corr}(ee, ep) are defined as the differences between the NEO-DFT(ee), NEO-DFT(ep), and NEO-DFT(ee,ep) energies, respectively, and the NEO-HF energy. The additivity error is defined as σ^{additivity} = *E* _{corr}(ee, ep) − *E* _{corr}(ee) − *E* _{corr}(ep). Energies are reported in atomic units.

Vibrational frequencies in cm^{−1} corresponding to the hydrogen vibrational stretching motion calculated with the NEO-HF, NEO-DFT, and FGH methods for the [He-X-He]^{+} system with X = H, D, or T at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set was used for all calculations, and the NEO-HF and NEO-DFT calculations utilized a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The NEO-DFT calculations were performed using the EPC1, EPC2, and EPC2-KE electron-proton correlation functionals with the electronic exchange-correlation functional chosen to be the Hartree-Fock exchange. The FGH frequencies were obtained from the splitting between the relevant vibrational states, and the NEO frequencies were obtained from the variationally optimized exponent of the Gaussian nuclear basis function.

Vibrational frequencies in cm^{−1} corresponding to the hydrogen vibrational stretching motion calculated with the NEO-HF, NEO-DFT, and FGH methods for the [He-X-He]^{+} system with X = H, D, or T at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set was used for all calculations, and the NEO-HF and NEO-DFT calculations utilized a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The NEO-DFT calculations were performed using the EPC1, EPC2, and EPC2-KE electron-proton correlation functionals with the electronic exchange-correlation functional chosen to be the Hartree-Fock exchange. The FGH frequencies were obtained from the splitting between the relevant vibrational states, and the NEO frequencies were obtained from the variationally optimized exponent of the Gaussian nuclear basis function.

Vibrational frequencies in cm^{−1} corresponding to the hydrogen vibrational stretching motion calculated with the NEO-DFT and FGH methods for the [He-X-He]^{+} system with X = H, D, or T at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set was used for all calculations, and the NEO-DFT calculations utilized a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The NEO-DFT calculations were performed using the EPC1, EPC2, and EPC2-KE electron-proton correlation functionals with the electronic exchange-correlation functionals B3LYP, BLYP and PBE. The FGH frequencies were obtained from the splitting between the relevant vibrational states, and the NEO frequencies were obtained from the variationally optimized exponent of the Gaussian nuclear basis function.

Vibrational frequencies in cm^{−1} corresponding to the hydrogen vibrational stretching motion calculated with the NEO-DFT and FGH methods for the [He-X-He]^{+} system with X = H, D, or T at a fixed He−He distance of 1.955 Å. The cc-pVDZ electronic basis set was used for all calculations, and the NEO-DFT calculations utilized a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The NEO-DFT calculations were performed using the EPC1, EPC2, and EPC2-KE electron-proton correlation functionals with the electronic exchange-correlation functionals B3LYP, BLYP and PBE. The FGH frequencies were obtained from the splitting between the relevant vibrational states, and the NEO frequencies were obtained from the variationally optimized exponent of the Gaussian nuclear basis function.

Decomposition of the electron-proton correlation energy, *E* _{ epc }, into its kinetic and potential energy components, *T* _{ epc } and *V* _{ epc }, respectively, for the EPC2 and EPC2-KE functionals. These values were calculated for the [He-H-He]^{+} system at a fixed He−He distance of 1.955 Å with the cc-pVDZ electronic basis set and a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The results are given for Hartree-Fock exchange and the B3LYP, BLYP, and PBE electronic exchange-correlation functionals. Energies are reported in atomic units.

Decomposition of the electron-proton correlation energy, *E* _{ epc }, into its kinetic and potential energy components, *T* _{ epc } and *V* _{ epc }, respectively, for the EPC2 and EPC2-KE functionals. These values were calculated for the [He-H-He]^{+} system at a fixed He−He distance of 1.955 Å with the cc-pVDZ electronic basis set and a single *s*-type Gaussian nuclear basis function with a variationally optimized exponent. The results are given for Hartree-Fock exchange and the B3LYP, BLYP, and PBE electronic exchange-correlation functionals. Energies are reported in atomic units.

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