Michael V. Pak

Inclusion of explicit electron-proton correlation in the nuclear-electronic orbital approach using Gaussian-type geminal functions

The nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach for including electron-proton correlation in mixed nuclear-electronic wavefunctions is presented. A general ansatz for the nuclear-electronic wavefunction that includes explicit dependence on the nuclear-electronic distances with Gaussian-type geminal functions is proposed. Based on this ansatz, expressions are derived for the total energy and the electronic and nuclear Fock operators for multielectron systems. The explicit electron-proton correlation is incorporated directly into the self-consistent-field procedure for optimizing the nuclear-electronic wavefunction. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically, and only electron-proton correlation is treated explicitly. Electron-electron correlation can be included by combining the NEO-XCHF approach with perturbation theory, density functional theory, and multiconfigurational methods. Previous nuclear-electronic orbital-based methods produce nuclear densities that are too localized, resulting in abnormally high stretching frequencies and inaccuracies in other properties relying on these densities. The application of the NEO-XCHF approach to the [He-H-He](+) model system illustrates that this approach includes the significant electron-proton correlation, thereby leading to an accurate description of the nuclear density. The agreement between the proton densities obtained with the NEO-XCHF and grid-based methods validates the underlying theory and the implementation of the NEO-XCHF method.

Investigation of isotope effects with the nuclear-electronic orbital approach

This paper addresses fundamental issues that arise in the application of the nuclear-electronic orbital (NEO) approach to systems with equivalent quantum nuclei. Our analysis illustrates that Hartree-Fock nuclear wave functions do not provide physically reasonable descriptions of systems comprised of equivalent low-spin fermions or equivalent bosons. The physical basis for this breakdown is that the ionic terms dominate due to the localized nature of the nuclear orbitals. Multi-configurational wave functions that include only covalent terms provide physically reasonable descriptions of these types of systems. The application of the NEO approach to a variety of chemical systems is presented to elucidate the isotope effects on the geometries and electronic wave functions. Deuteration of hydrogen halides, water, ammonia, and hydronium ion decreases the bond length and the magnitude of negative partial atomic charge on the heavy atom. These results are consistent with experimental spectroscopic data. Deuteration at the beta position for formate anion and a series of amines increases the magnitude of negative partial atomic charge on the protonation site for the unprotonated species. This observation is consistent with the experimentally observed increase in basicity upon deuteration at the beta position for carboxylic acids and amines.

Nuclear-electronic orbital nonorthogonal configuration interaction approach

The nuclear-electronic orbital nonorthogonal configuration interaction (NEO-NOCI) approach is presented. In this framework, the hydrogen nuclei are treated quantum mechanically on the same level as the electrons, and a mixed nuclear-electronic time-independent Schrodinger equation is solved with molecular orbital techniques. For hydrogen transfer systems, the transferring hydrogen is represented by two basis function centers to allow delocalization of the nuclear wave function. In the two-state NEO-NOCI approach, the ground and excited state delocalized nuclear-electronic wave functions are expressed as linear combinations of two nonorthogonal localized nuclear-electronic wave functions obtained at the NEO-Hartree-Fock level. The advantages of the NEO-NOCI approach are the removal of the adiabatic separation between the electrons and the quantum nuclei, the computational efficiency, the potential for systematic improvement by enhancing the basis sets and number of configurations, and the applicability to a broad range of chemical systems. The tunneling splitting is determined by the energy difference between the two delocalized vibronic states. The hydrogen tunneling splittings calculated with the NEO-NOCI approach for the [He-H-He]+ model system with a range of fixed He-He distances are in excellent agreement with NEO-full CI and Fourier grid calculations. These benchmarking calculations indicate that NEO-NOCI is a promising approach for the calculation of delocalized, bilobal hydrogen wave functions and the corresponding hydrogen tunneling splittings.

Derivation of an Electron-Proton Correlation Functional for Multicomponent Density Functional Theory within the Nuclear-Electronic Orbital Approach.

Multicomponent density functional theory enables the quantum mechanical treatment of electrons and selected hydrogen nuclei. An electron-proton correlation functional is derived from the electron-proton pair density associated with a recently proposed ansatz for the explicitly correlated nuclear-electronic wave function. This ansatz allows the retention of all terms in the pair density, and the resulting functional is expected to scale properly and to be computationally efficient. Applications to model systems illustrate that it provides accurate nuclear densities.

Multicomponent density functional theory study of the interplay between electron-electron and electron-proton correlation

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.

Properties of the exact universal functional in multicomponent density functional theory

Multicomponent density functional theory has been developed to treat systems with more than one type of quantum particle, such as electrons and nuclei, in an external potential. The existence of the exact universal multicomponent density functional in terms of the one-particle densities for each type of quantum particle has been proven. In the present paper, a number of important mathematical properties of the exact universal multicomponent density functional are derived. The expression relating the electron-proton pair density to the one-particle densities leads to an inequality for the potential energy component of the electron-proton correlation functional under well-defined conditions. General inequalities for the kinetic energy correlation functionals and the total electron-proton correlation functional are also derived. The coordinate scaling analysis leads to mathematical inequalities describing the effect of scaled densities on the kinetic, potential, and total energy functionals. The adiabatic connection formula defines the exact electron-proton functional in terms of an adiabatic scaling parameter that smoothly connects the noninteracting system with the fully interacting system. The virial expression provides the relation between the exact kinetic and potential energy functionals for the ground state densities of multicomponent systems. These mathematical relationships provide insight into the fundamental properties of the exact universal multicomponent density functional and serve as a guide for the development of approximate electron-proton density functionals.

Quantum treatment of protons with the reduced explicitly correlated Hartree-Fock approach

The nuclear-electronic orbital (NEO) approach treats select nuclei quantum mechanically on the same level as the electrons and includes nonadiabatic effects between the electrons and the quantum nuclei. The practical implementation of this approach is challenging due to the significance of electron-nucleus dynamical correlation. Herein, we present a general extension of the previously developed reduced NEO explicitly correlated Hartree-Fock (RXCHF) approach, in which only select electronic orbitals are explicitly correlated to each quantum nuclear orbital via Gaussian-type geminal functions. Approximations of the electronic exchange between the geminal-coupled electronic orbitals and the other electronic orbitals are also explored. This general approach enables computationally tractable yet accurate calculations on molecular systems with quantum protons. The RXCHF method is applied to the hydrogen cyanide (HCN) and FHF(-) systems, where the proton and all electrons are treated quantum mechanically. For the HCN system, only the two electronic orbitals associated with the CH covalent bond are geminal-coupled to the proton orbital. For the FHF(-) system, only the four electronic orbitals associated with the two FH covalent bonds are geminal-coupled to the proton orbital. For both systems, the RXCHF method produces qualitatively accurate nuclear densities, in contrast to mean field-based NEO approaches. The development and implementation of the RXCHF method provide the framework to perform calculations on systems such as proton-coupled electron transfer reactions, where electron-proton nonadiabatic effects are important.

Reduced explicitly correlated Hartree-Fock approach within the nuclear-electronic orbital framework: Theoretical formulation

The nuclear-electronic orbital (NEO) method treats electrons and select nuclei quantum mechanically on the same level to extend beyond the Born-Oppenheimer approximation. Electron-nucleus dynamical correlation has been found to be highly significant due to the attractive Coulomb interaction. The explicitly correlated Hartree-Fock (NEO-XCHF) approach includes explicit electron-nucleus correlation with Gaussian-type geminal functions during the variational optimization of the nuclear-electronic wavefunction. Although accurate for small model systems, the NEO-XCHF method is computationally impractical for larger chemical systems. In this paper, we develop the reduced explicitly correlated Hartree-Fock approach, denoted NEO-RXCHF, where only select electronic orbitals are explicitly correlated to the nuclear orbitals. By explicitly correlating only the relevant electronic orbitals to the nuclear orbitals, the NEO-RXCHF approach avoids problems that can arise when all electronic orbitals are explicitly correlated to the nuclear orbitals in the same manner. We examine three different NEO-RXCHF methods that differ in the treatment of the exchange between the geminal-coupled electronic orbitals and the other electronic orbitals: NEO-RXCHF-fe is fully antisymmetric with respect to exchange of all electronic coordinates and includes all electronic exchange terms; NEO-RXCHF-ne neglects the exchange between the geminal-coupled electronic orbitals and the other electronic orbitals; and NEO-RXCHF-ae includes approximate exchange terms between the geminal-coupled electronic orbitals and the other electronic orbitals. The latter two NEO-RXCHF methods offer substantial computational savings over the NEO-XCHF approach. The NEO-RXCHF approach is applicable to a wide range of chemical systems that exhibit non-Born-Oppenheimer effects between electrons and nuclei, as well as positron-containing molecular systems.

Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+Ĝ), where Ĝ is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+Ĝ) rather than (1+Ĝ). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.

Modeling positrons in molecular electronic structure calculations with the nuclear-electronic orbital method.

The nuclear-electronic orbital (NEO) method was modified and extended to positron systems for studying mixed positronic-electronic wavefunctions, replacing the mass of the proton with the mass of the positron. Within the modified NEO framework, the NEO-HF (Hartree-Fock) method provides the energy corresponding to the single-configuration mixed positronic-electronic wavefunction, minimized with respect to the molecular orbitals expressed as linear combinations of Gaussian basis functions. The electron-electron and electron-positron correlation can be treated in the NEO framework with second-order perturbation theory (NEO-MP2) or multiconfigurational methods such as the full configuration interaction (NEO-FCI) and complete active space self-consistent-field (NEO-CASSCF) methods. In addition to implementing these methods for positronic systems, strategies for calculating electron-positron annihilation rates using NEO-HF, NEO-MP2, and NEO-FCI wavefunctions were also developed. To apply the NEO method to the positronium hydride (PsH) system, positronic and electronic basis sets were optimized at the NEO-FCI level and used to compute NEO-MP2 and NEO-FCI energies and annihilation rates. The effects of basis set size on NEO-MP2 and NEO-FCI correlation energies and annihilation rates were compared. Even-tempered electronic and positronic basis sets were also optimized for the e+LiH molecule at the NEO-MP2 level and used to compute the equilibrium bond length and vibrational energy.