Nuclear-electronic Orbital Framework Research Articles

Vibrational Spectra of Highly Anharmonic Water Clusters: Molecular Dynamics and Harmonic Analysis Revisited with Constrained Nuclear-Electronic Orbital Methods.

Vibrational spectroscopy is widely used to gain insights into structural and dynamic properties of chemical, biological, and materials systems. Thus, an efficient and accurate method to simulate vibrational spectra is desired. In this paper, we justify and employ a microcanonical molecular simulation scheme to calculate the vibrational spectra of three challenging water clusters: the neutral water dimer (H4O2), the protonated water trimer (H7O3+), and the protonated water tetramer (H9O4+). We find that with the accurate description of quantum nuclear delocalization effects through the constrained nuclear-electronic orbital framework, including vibrational mode coupling effects through molecular dynamics simulations can additionally improve the vibrational spectrum calculations. In contrast, without the quantum nuclear delocalization picture, conventional ab initio molecular dynamics may even lead to less accurate results than harmonic analysis.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals.

Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.

Assessing Implicit and Explicit Polarizable Solvation Models for Nuclear-Electronic Orbital Systems: Quantum Proton Polarization and Solvation Energetics.

Accurate simulations of many chemical processes require the inclusion of both nuclear quantum effects and a solvent environment. The nuclear-electronic orbital (NEO) approach, which treats electrons and select nuclei quantum mechanically on the same level, combined with a polarizable continuum model (PCM) for the solvent environment, addresses this challenge in a computationally practical manner. In this work, the NEO-PCM approach is extended beyond the IEF-PCM (integral equation formalism PCM) and C-PCM (conductor PCM) approaches to the SS(V)PE (surface and simulation of volume polarization for electrostatics) and ddCOSMO (domain decomposed conductor-like screening model) approaches. IEF-PCM, SS(V)PE, C-PCM, and ddCOSMO all exhibit similar solvation energies as well as comparable nuclear polarization within the NEO framework. The calculations show that the nuclear density does not leak out of the molecular cavity because it is much more localized than the electronic density. Finally, the polarization of quantized protons is analyzed in both continuum solvent and explicit solvent environments described by the polarizable MB-pol model, illustrating the impact of specific hydrogen-bonding interactions captured only by explicit solvation. These calculations highlight the relationship among solvation formalism, nuclear polarization, and energetics.

A quantum computing implementation of nuclearelectronic orbital (NEO) theory: Toward an exact pre-Born-Oppenheimer formulation of molecular quantum systems.

Nuclear quantum phenomena beyond the Born-Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron-nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron-nuclear problem on near-term quantum computers, based upon the Nuclear-Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10-6 hartree and entanglement entropy to within 10-4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.

Electronic Born-Oppenheimer approximation in nuclear-electronic orbital dynamics.

Within the nuclear-electronic orbital (NEO) framework, the real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics. Herein, the electronic Born-Oppenheimer (BO) approximation within the NEO framework is presented. In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of-magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting. For the intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.

Triple electron-electron-proton excitations and second-order approximations in nuclear-electronic orbital coupled cluster methods.

The accurate description of nuclear quantum effects, such as zero-point energy, is important for modeling a wide range of chemical and biological processes. Within the nuclear-electronic orbital (NEO) approach, such effects are incorporated in a computationally efficient way by treating electrons and select nuclei, typically protons, quantum mechanically with molecular orbital techniques. Herein, we implement and test a NEO coupled cluster method that explicitly includes the triple electron-electron-proton excitations, where two electrons and one proton are excited simultaneously, using automatic differentiation Our calculations show that this NEO-CCSDTeep method provides highly accurate proton densities and proton affinities, outperforming any previously studied NEO method. These examples highlight the importance of the triple electron-electron-proton excitations for an accurate description of nuclear quantum effects. Additionally, we also implement and test the second-order approximate coupled cluster with singles and doubles (NEO-CC2) method as well as its scaled-opposite-spin (SOS) versions. The NEO-SOS'-CC2 method, which scales the electron-proton correlation energy as well as the opposite-spin and same-spin components of the electron-electron correlation energy, achieves nearly the same accuracy as the NEO-CCSDTeep method for the properties studied. Because of its low computational cost, this method will enable a wide range of chemical and photochemical applications for large molecular systems. This work sets the stage for a variety of developments and applications within the NEO framework.

Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear-Electronic Orbital Method.

In multicomponent quantum chemistry, more than one type of particle is treated quantum mechanically with either density functional theory or wave function based methods. In particular, the nuclear-electronic orbital (NEO) approach treats specified nuclei, typically hydrogen nuclei, on the same level as the electrons. This approach enables the incorporation of nuclear quantum effects, such as nuclear delocalization, anharmonicity, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects directly into quantum chemistry calculations. Such effects impact optimized geometries, molecular vibrational frequencies, reaction paths, isotope effects, and dynamical simulations. Multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) achieve an optimal balance between computational efficiency and accuracy for computing ground and excited state properties, respectively. Multicomponent wave function based methods, such as the coupled cluster singles and doubles (NEO-CCSD) method for ground states and the equation-of-motion counterpart (NEO-EOM-CCSD) for excited states, attain similar accuracy without requiring any parametrization and can be systematically improved but are more computationally expensive. Variants of the orbital-optimized perturbation theory (NEO-OOMP2) method achieve nearly the accuracy of NEO-CCSD for ground states with significantly lower computational cost. Additional approaches for computing excited electronic, vibrational, and vibronic states include the delta self-consistent field (NEO-ΔSCF), complete active space SCF (NEO-CASSCF), and nonorthogonal configuration interaction methods. Multireference methods are particularly important for describing hydrogen tunneling processes. Other types of multicomponent systems, such as those containing electrons and positrons, have also been studied within the NEO framework. The NEO approach allows the incorporation of nuclear quantum effects and non-Born-Oppenheimer effects for specified nuclei into quantum chemistry calculations in an accessible and computationally efficient manner.

Molecular Vibrational Frequencies with Multiple Quantum Protons within the Nuclear-Electronic Orbital Framework.

The nuclear-electronic orbital (NEO) approach treats all electrons and specified nuclei, typically protons, on the same quantum mechanical level. Proton vibrational excitations can be calculated using multicomponent time-dependent density functional theory (NEO-TDDFT) for fixed classical nuclei. Recently the NEO-DFT(V) approach was developed to enable the calculation of molecular vibrational frequencies for modes composed of both classical and quantum nuclei. This approach uses input from NEO-TDDFT to construct an extended NEO Hessian that depends on the expectation values of the quantum protons as well as the classical nuclear coordinates. Herein strategies are devised for extending these approaches to molecules with multiple quantum protons in a self-contained, effective, and computationally practical manner. The NEO-TDDFT method is shown to describe vibrational excitations corresponding to collective nuclear motions, such as linear combinations of proton vibrational excitations. The NEO-DFT(V) approach is shown to incorporate the most significant anharmonic effects in the molecular vibrations, particularly for the hydrogen stretching modes. These theoretical strategies pave the way for a wide range of multicomponent quantum chemistry applications.

Multicomponent coupled cluster singles and doubles and Brueckner doubles methods: Proton densities and energies.

The nuclear-electronic orbital (NEO) framework enables computationally practical coupled cluster calculations of multicomponent molecular systems, in which all electrons and specified nuclei, typically protons, are treated quantum mechanically. In addition to energies, computing accurate proton densities is essential for the calculation of reliable molecular properties, including vibrationally averaged geometries and vibrational frequencies. Herein, the Lagrangian formalism for the multicomponent coupled cluster with single and double excitations (NEO-CCSD) method is derived and implemented. The multicomponent coupled cluster with double excitations method using optimized Brueckner orbitals, denoted as NEO-BCCD, is also developed. Both of these methods are used to compute the proton densities for two molecular systems. The results illustrate that orbital relaxation effects, which can be included either indirectly with the NEO-CCSD method or directly with the NEO-BCCD method, are critical for computing even qualitatively accurate proton densities. Both methods are also able to provide accurate proton affinities and vibrationally averaged optimized geometries. This Lagrangian formalism will enable the calculation of other properties such as analytical nuclear gradients and Hessians with NEO coupled cluster methods. Moreover, the accuracy of these methods may be improved systematically by the inclusion of higher-order excitations. Thus, this work provides the foundation for a wide range of future methodological developments and applications within the NEO framework.

Diagonal Born-Oppenheimer Corrections within the Nuclear-Electronic Orbital Framework.

The nuclear-electronic orbital (NEO) method treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons. This approach invokes the Born-Oppenheimer separation between the quantum and classical nuclei, as well as the conventional separation between the electrons and classical nuclei. To test the validity of this additional adiabatic approximation, herein the diagonal Born-Oppenheimer correction (DBOC) within the NEO framework is derived, analyzed, and calculated numerically for a set of eight molecules. Inclusion of the NEO DBOC is found to change the equilibrium bond lengths by only ∼10-4 Å and the heavy atom vibrational stretching frequencies by ∼1-2 cm-1 per quantum proton bonded to an atom participating in the vibrational mode. These results imply that the DBOC does not significantly impact molecular properties computed with the NEO approach, although it can be included when necessary. Understanding the physical characteristics and quantitative contributions of the DBOC has broad implications for applications of multicomponent density functional theory and wave function methods.

Molecular Vibrational Frequencies within the Nuclear-Electronic Orbital Framework.

A significant challenge for multicomponent quantum chemistry methods is the calculation of vibrational frequencies for comparison to experiment. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically key protons, quantum mechanically. The Born-Oppenheimer separation between the quantum and classical nuclei prevents the direct calculation of vibrational frequencies corresponding to modes composed of both types of nuclei. Herein an effective strategy for calculating the vibrational frequencies of the entire molecule within the NEO framework is devised and implemented. This strategy requires diagonalization of an extended NEO Hessian that depends on the expectation values of the quantum nuclei as well as the coordinates of the classical nuclei and is constructed with input from multicomponent time-dependent density functional theory (NEO-TDDFT). Application of this NEO-DFT(V) approach to molecular systems illustrates that it accurately incorporates the most significant anharmonic effects. This general theoretical formulation opens up a broad spectrum of new directions for multicomponent quantum chemistry.

Multicomponent Coupled Cluster Singles and Doubles Theory within the Nuclear-Electronic Orbital Framework.

The nuclear-electronic orbital (NEO) method treats all electrons and specified nuclei, typically protons, quantum mechanically on the same level with molecular orbital techniques. This approach directly includes nuclear delocalization, anharmonicity, and zero point energy contributions of the quantum nuclei in the self-consistent-field procedure for solving the time-independent Schrödinger equation. Herein the multicomponent wave function based methods configuration interaction singles and doubles (CISD) and coupled cluster singles and doubles (CCSD) are implemented within the NEO framework and are applied to molecular systems. In contrast to the NEO-HF (Hartree-Fock) and NEO-CISD methods, which produce proton densities that are much too localized, the NEO-CCSD method produces accurate proton densities in reasonable agreement with a grid-based reference. Moreover, the NEO-CCSD method also predicts accurate proton affinities in agreement with experimental measurements for a set of 12 molecules. An advantage of the NEO-CCSD method is its ability to include nuclear quantum effects, such as proton delocalization and zero point energy, during geometry optimizations and nuclear dynamics simulations. The NEO-CCSD method is a promising, parameter free approach for including nuclear quantum effects in high-level electronic structure calculations of molecular systems.

Multicomponent Time-Dependent Density Functional Theory: Proton and Electron Excitation Energies.

The quantum mechanical treatment of both electrons and protons in the calculation of excited state properties is critical for describing nonadiabatic processes such as photoinduced proton-coupled electron transfer. Multicomponent density functional theory enables the consistent quantum mechanical treatment of more than one type of particle and has been implemented previously for studying ground state molecular properties within the nuclear-electronic orbital (NEO) framework, where all electrons and specified protons are treated quantum mechanically. To enable the study of excited state molecular properties, herein the linear response multicomponent time-dependent density functional theory (TDDFT) is derived and implemented within the NEO framework. Initial applications to FHF- and HCN illustrate that NEO-TDDFT provides accurate proton and electron excitation energies within a single calculation. As its computational cost is similar to that of conventional electronic TDDFT, the NEO-TDDFT approach is promising for diverse applications, particularly nonadiabatic proton transfer reactions, which may exhibit mixed electron-proton vibronic excitations.

Calculation of Positron Binding Energies and Electron-Positron Annihilation Rates for Atomic Systems with the Reduced Explicitly Correlated Hartree-Fock Method in the Nuclear-Electronic Orbital Framework.

Although the binding of a positron to a neutral atom has not been directly observed experimentally, high-level theoretical methods have predicted that a positron will bind to a neutral atom. In the present study, the binding energies of a positron to lithium, sodium, beryllium, and magnesium, as well as the electron-positron annihilation rates for these systems, are calculated using the reduced explicitly correlated Hartree-Fock (RXCHF) method within the nuclear-electronic orbital (NEO) framework. Due to the lack of explicit electron-positron correlation, NEO Hartree-Fock and full configuration interaction calculations with reasonable electronic and positronic basis sets do not predict positron binding to any of these atoms. In contrast, the RXCHF calculations predict positron binding energies and electron-positron annihilation rates in qualitative agreement with previous highly accurate but computationally expensive stochastic variational method calculations. These results illustrate that the RXCHF method can successfully describe the binding of a positron to a neutral species with no dipole moment. Moreover, the RXCHF method will be computationally tractable for calculating positron binding to molecular systems. The RXCHF approach offers a balance of accuracy and computational tractability for studying these types of positronic systems.

Multicomponent density functional theory embedding formulation.

Multicomponent density functional theory (DFT) methods have been developed to treat two types of particles, such as electrons and nuclei, quantum mechanically at the same level. In the nuclear-electronic orbital (NEO) approach, all electrons and select nuclei, typically key protons, are treated quantum mechanically. For multicomponent DFT methods developed within the NEO framework, electron-proton correlation functionals based on explicitly correlated wavefunctions have been designed and used in conjunction with well-established electronic exchange-correlation functionals. Herein a general theory for multicomponent embedded DFT is developed to enable the accurate treatment of larger systems. In the general theory, the total electronic density is separated into two subsystem densities, denoted as regular and special, and different electron-proton correlation functionals are used for these two electronic densities. In the specific implementation, the special electron density is defined in terms of spatially localized Kohn-Sham electronic orbitals, and electron-proton correlation is included only for the special electron density. The electron-proton correlation functional depends on only the special electron density and the proton density, whereas the electronic exchange-correlation functional depends on the total electronic density. This scheme includes the essential electron-proton correlation, which is a relatively local effect, as well as the electronic exchange-correlation for the entire system. This multicomponent DFT-in-DFT embedding theory is applied to the HCN and FHF(-) molecules in conjunction with two different electron-proton correlation functionals and three different electronic exchange-correlation functionals. The results illustrate that this approach provides qualitatively accurate nuclear densities in a computationally tractable manner. The general theory is also easily extended to other types of partitioning schemes for multicomponent systems.

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.

Reduced explicitly correlated Hartree-Fock approach within the nuclear-electronic orbital framework: Applications to positronic molecular systems

In the application of the nuclear-electronic orbital (NEO) method to positronic systems, all electrons and the positron are treated quantum mechanically on the same level. Explicit electron-positron correlation can be included using Gaussian-type geminal functions within the variational self-consistent-field procedure. In this paper, we apply the recently developed reduced explicitly correlated Hartree-Fock (RXCHF) approach to positronic molecular systems. In the application of RXCHF to positronic systems, only a single electronic orbital is explicitly correlated to the positronic orbital. We apply NEO-RXCHF to three systems: positron-lithium, lithium positride, and positron-lithium hydride. For all three of these systems, the RXCHF approach provides accurate two-photon annihilation rates, average contact densities, electronic and positronic single-particle densities, and electron-positron contact densities. Moreover, the RXCHF approach is significantly more accurate than the original XCHF approach, in which all electronic orbitals are explicitly correlated to the positronic orbital in the same manner, because the RXCHF wavefunction is optimized to produce a highly accurate description of the short-ranged electron-positron interaction that dictates the annihilation rates and other local properties. Furthermore, RXCHF methods that neglect or approximate the electronic exchange interactions between the geminal-coupled electronic orbital and the regular electronic orbitals lead to virtually identical annihilation rates and densities as the fully antisymmetric RXCHF method but offer substantial advantages in computational tractability. Thus, NEO-RXCHF is a promising, computationally practical approach for studying larger positron-containing systems.

Analysis of electron-positron wavefunctions in the nuclear-electronic orbital framework

The nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach is extended and applied to the positronic systems PsH, LiPs, and e(+)LiH. In this implementation, all electrons and positrons are treated quantum mechanically, and all nuclei are treated classically. This approach utilizes molecular orbital techniques with Gaussian basis sets for the electrons and positrons and includes electron-positron correlation with explicitly correlated Gaussian-type geminal functions. An efficient strategy is developed to reduce the number of variational parameters in the NEO-XCHF calculations. The annihilation rates, electron and positron densities, and electron-positron contact densities are compared to available results from higher-level calculations. Our analysis illustrates that the NEO-XCHF method produces qualitative to semi-quantitative results for these properties at a relatively low computational cost by treating only the essential electron-positron correlation explicitly. The NEO-HF method, which does not include explicit correlation and therefore is extremely efficient, is found to provide qualitatively accurate electron-positron contact densities for the e(+)LiH system but not for the LiPs system. Thus, the utility of the NEO-HF method for determining where annihilation occurs is system dependent and not generally reliable. The NEO-XCHF method, however, provides a computationally practical and reliable approach for determining where annihilation will occur in positronic systems.

Localized Hartree product treatment of multiple protons in the nuclear-electronic orbital framework

An approximation for treating multiple quantum nuclei within the nuclear-electronic orbital (NEO) framework for molecular systems is presented. In the approximation to NEO-Hartree-Fock, the nuclear wave function is represented by a Hartree product rather than a Slater determinant, corresponding to the neglect of the nuclear exchange interactions. In the approximation to NEO-density functional theory, the nuclear exchange-correlation functional is chosen to be the diagonal nuclear exchange interaction terms, thereby eliminating the nuclear self-interaction terms. To further enhance the simplicity and computational efficiency, the nuclear molecular orbitals or Kohn-Sham orbitals are expanded in terms of localized nuclear basis sets. These approximations are valid because of the inherent localization of the nuclear orbitals and the numerical insignificance of the nuclear exchange interactions in molecular systems. Moreover, these approximations lead to substantial computational savings due to the reduction in both the number of integrals that must be calculated and the size of the matrices that must be diagonalized. These nuclear Hartree product approximation (HPA) methods scale linearly with the number of quantum protons and are highly parallelizable. Applications to a water hexamer, glycine dimer, and 32-water cluster, where all hydrogen nuclei are treated quantum mechanically, illustrate the accuracy and computational efficiency of the nuclear HPA methods. These strategies will facilitate the implementation of explicitly correlated NEO methods for molecular systems with multiple quantum protons.

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.