A generalized any-particle propagator theory: Prediction of proton affinities and acidity properties with the proton propagator

Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory

Post-Kohn-Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's functionG W ${GW}$ approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn-Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, theG W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

Divide-and-Conquer Linear-Scaling Quantum Chemical Computations

Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee (J. Chem. Phys. 1995, 103, 5674-5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.

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-proton excitations, where two electrons and one proton are excited simultaneously. Our calculations show that this NEO-CCSD(eep) 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-CCSD(eep) 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 wide range of developments and applications within the NEO framework.

Multicomponent Orbital-Optimized Perturbation Theory with Density Fitting: Anharmonic Zero-Point Energies in Protonated Water Clusters

Nuclear quantum effects such as zero-point energy are important in a wide range of chemical and biological processes. The nuclear-electronic orbital (NEO) framework intrinsically includes such effects by treating electrons and specified nuclei quantum mechanically on the same level. Herein, we implement the NEO scaled-opposite-spin orbital-optimized second-order Møller-Plesset perturbation theory with electron-proton correlation scaling (NEO-SOS'-OOMP2) using density fitting. This efficient implementation allows applications to larger systems with multiple quantum protons. Both the NEO-SOS'-OOMP2 method and its counterpart without orbital optimization predict proton affinities to within experimental precision and relative energies of protonated water tetramer isomers in agreement with previous NEO coupled cluster calculations. Applications to protonated water hexamers and heptamers illustrate that anharmonicity is critical for computing accurate relative energies. The NEO-SOS'-OOMP2 approach captures anharmonic zero-point energies at any geometry in a computationally efficient manner and hence will be useful for investigating reaction paths and dynamics in chemical systems.

Quantum Proton Effects from Density Matrix Renormalization Group Calculations

We recently introduced [J. Chem. Phys. 2020, 152, 204103] the nuclear-electronic all-particle density matrix renormalization group (NEAP-DMRG) method to solve the molecular Schrödinger equation, based on a stochastically optimized orbital basis, without invoking the Born-Oppenheimer approximation. In this work, we combine the DMRG method with the nuclear-electronic Hartree-Fock (NEHF-DMRG) approach, treating nuclei and electrons on the same footing. Inter- and intraspecies correlations are described within the DMRG method without truncating the excitation degree of the full configuration interaction wave function. We extend the concept of orbital entanglement and mutual information to nuclear-electronic wave functions and demonstrate that they are reliable metrics to detect strong correlation effects. We apply the NEHF-DMRG method to the HeHHe⁺ molecular ion, to obtain accurate proton densities, ground-state total energies, and vibrational transition frequencies by comparison with state-of-the-art data obtained with grid-based approaches and modern configuration interaction methods. For HCN, we improve on the accuracy of the latter approaches with respect to both the ground-state absolute energy and proton density, which is a major challenge for multireference nuclear-electronic state-of-the-art methods.

Analytic Alchemical Derivatives for the Analysis of Differential Acidity Assisted by the h Function

Analytical calculation of alchemical derivatives based on auxiliary density perturbation theory is described, coded, and validated. For the case where the nucleus is a hydrogen atom and the nuclear charge is changed from 1 to 0, it turns out that a good estimate of the proton binding energies can be obtained very efficiently. First-order results correspond exactly to the molecular electrostatic potential evaluated at the hydrogen nucleus location (removing self-repulsion), in agreement with previously reported extensive studies. Therefore, the second-order results reported here are refinements in accuracy that finally allow a quantitative exploration of differential acidity. Furthermore, the recently reported h function is produced in its analytical form as a byproduct and local descriptor associated with the proton binding energy values found with this approach. In an example application, proton binding energies are computed for a family of imidazolium derivatives to demonstrate the capabilities and the stability of the method with respect to changes in basis set or exchange-correlation functional.

Multicomponent Coupled Cluster Singles and Doubles with Density Fitting: Protonated Water Tetramers with Quantized Protons

Nuclear quantum effects such as zero-point energy are important for describing a wide range of chemical properties. The nuclear-electronic orbital (NEO) approach incorporates such effects into quantum chemistry calculations by treating specified nuclei, typically protons, quantum mechanically on the same level as electrons. Herein, both the traditional and t₁-transformed NEO coupled cluster with singles and doubles (NEO-CCSD) methods are implemented with a density fitting (DF) scheme for approximating the four-center two-particle integrals. The enhanced computational efficiency enables calculations on larger molecules with multiple quantum protons. The NEO-DF-CCSD method predicts proton affinities within chemical accuracy. Its application to protonated water tetramers with all nine protons treated quantum mechanically produces the qualitatively correct ordering of the isomer energies, which are strongly influenced by the zero-point energy contributions inherently included in NEO energy calculations. This work showcases the capabilities of the NEO-DF-CCSD method and provides the foundation for future developments and applications.

Predicting Deprotonation Sites Using Alchemical Derivatives

An alchemical transformation is any process, physical or fictitious, that connects two points in the chemical space. A particularly important transformation is the vanishing of a proton, whose energy can be linked to the proton dissociation enthalpy of acids. In this work we assess the reliability of alchemical derivatives in predicting the proton dissociation enthalpy of a diverse series of mono- and polyprotic molecules. Alchemical derivatives perform remarkably well in ranking the proton affinity of all molecules. Additionally, alchemical derivatives could be use also as a predictive tool because their predictions correlate quite well with calculations based on energy differences and experimental values. Although second-order alchemical derivatives underestimate the dissociation enthalpy, the deviation seems to be almost constant. This makes alchemical derivatives extremely accurate to evaluate the difference in proton affinity between two acid sites of polyprotic molecule. Finally, we show that the reason for the underestimation of the dissociation enthalpy is most likely the contribution of higher-order derivatives.

The any particle molecular orbital/molecular mechanics approach

A computational scheme is proposed to broaden the range of applications of multicomponent methodologies for the study of local properties of big molecular systems existing in the gas phase and in solvated environments. This scheme extends the any particle molecular orbital (APMO) approach in the quantum mechanics/molecular mechanics (QM/MM) framework. As a first assessment of the performance of the proposed approach, we estimate the proton affinities (PAs) of seventy amines in the gas phase and the proton binding energies (PBEs) in the gas phase and in an explicitly solvated environment of the sixty-one protons present in the chignolin protein. These calculations are performed with the QM/MM versions of the APMO second-order proton propagator (APMO-PP2) and the APMO extended Koopmans' theorem (APMO-KT) approaches. Calculated PAs and PBEs show significant reductions in the computational effort with a reduced loss in accuracy. These results suggest that the APMO/MM scheme might be used as a low-cost multi-component alternative for studies of local properties in big molecular systems. Graphical Abstract QMMM regions and CPU times for the APMO/MM approach.

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.

Toward a muon-specific electronic structure theory: effective electronic Hartree-Fock equations for muonic molecules

An effective set of Hartree-Fock (HF) equations are derived for electrons of muonic systems, i.e., molecules containing a positively charged muon, conceiving the muon as a quantum oscillator, which are completely equivalent to the usual two-component HF equations used to derive stationary states of the muonic molecules. In these effective equations, a non-Coulombic potential is added to the orthodox coulomb and exchange potential energy terms, which describes the interaction of the muon and the electrons effectively and is optimized during the self-consistent field cycles. While in the two-component HF equations a muon is treated as a quantum particle, in the effective HF equations it is absorbed into the effective potential and practically transformed into an effective potential field experienced by electrons. The explicit form of the effective potential depends on the nature of muon's vibrations and is derivable from the basis set used to expand the muonic spatial orbital. The resulting effective Hartree-Fock equations are implemented computationally and used successfully, as a proof of concept, in a series of muonic molecules containing all atoms from the second and third rows of the Periodic Table. To solve the algebraic version of the equations muon-specific Gaussian basis sets are designed for both muon and surrounding electrons and it is demonstrated that the optimized exponents are quite distinct from those derived for the hydrogen isotopes. The developed effective HF theory is quite general and in principle can be used for any muonic system while it is the starting point for a general effective electronic structure theory that incorporates various types of quantum correlations into the muonic systems beyond the HF equations.

Prediction of proton affinities of organic molecules using the any-particle molecular-orbital second-order proton propagator approach

We assess the performance of the recently developed any-particle molecular-orbital second-order proton propagator (APMO/PP2) scheme [M. Díaz-Tinoco, J. Romero, J. V. Ortiz, A. Reyes and R. Flores-Moreno, J. Chem. Phys., 2013, 138, 194108] on the calculation of gas phase proton affinities (PAs) of a set of 150 organic molecules comprising several functional groups: amines, alcohols, aldehydes, amides, ketones, esters, ethers, carboxylic acids and carboxylate anions. APMO/PP2 PAs display an overall mean absolute error of 0.68 kcal mol^-1 with respect to experimental data. These results suggest that the APMO/PP2 method is an alternative approach for the quantitative prediction of gas phase proton affinities. One novel feature of the method is that a PA can be obtained from a single calculation of the optimized protonated molecule.

Fast and accurate prediction of proton affinities: revisiting the extended Koopmans' theorem for protons

In this work we propose schemes based on the extended Koopmans' theorem for quantum nuclei (eKT), in the framework of the any particle molecular orbital approach (APMO/KT), for the quantitative prediction of gas phase proton affinities (PAs). The performance of these schemes has been tested on a set of 300 organic molecules containing diverse functional groups. The APMO/KT scheme scaled by functional group (APMO/KT-SC-FG) displays an overall mean absolute error of 1.1 kcal mol^-1 with respect to experimental data. Its performance in PA calculations is similar to that of post-Hartree-Fock composite methods or that of the APMO second order proton propagator (APMO/PP2) approach. The APMO/KT-SC-FG scheme is also employed to predict PAs of polyfunctional molecules such as the Nerve Agent VX and the 20 common α-amino acids, finding excellent agreement with available theoretical and/or experimental data. The accuracy of the predictions demonstrates that the APMO/KT-SC-FG scheme is a low-cost alternative to adiabatic methods for the calculation of accurate PAs. One of the most appealing features of the APMO/KT-SC-FG scheme, is that PAs can be derived from one single-point APMO Hartree-Fock calculation.

Multicomponent Density Functional Theory: Impact of Nuclear Quantum Effects on Proton Affinities and Geometries

Nuclear quantum effects such as zero point energy play a critical role in computational chemistry and often are included as energetic corrections following geometry optimizations. The nuclear-electronic orbital (NEO) multicomponent density functional theory (DFT) method treats select nuclei, typically protons, quantum mechanically on the same level as the electrons. Electron-proton correlation is highly significant, and inadequate treatments lead to highly overlocalized nuclear densities. A recently developed electron-proton correlation functional, epc17, has been shown to provide accurate nuclear densities for molecular systems. Herein, the NEO-DFT/epc17 method is used to compute the proton affinities for a set of molecules and to examine the role of nuclear quantum effects on the equilibrium geometry of FHF^-. The agreement of the computed results with experimental and benchmark values demonstrates the promise of this approach for including nuclear quantum effects in calculations of proton affinities, pK_a's, optimized geometries, and reaction paths.

On the physical interpretation of the nuclear molecular orbital energy

Recently, several groups have extended and implemented molecular orbital (MO) schemes to simultaneously obtain wave functions for electrons and selected nuclei. Many of these schemes employ an extended Hartree-Fock approach as a first step to find approximate electron-nuclear wave functions and energies. Numerous studies conducted with these extended MO methodologies have explored various effects of quantum nuclei on physical and chemical properties. However, to the best of our knowledge no physical interpretation has been assigned to the nuclear molecular orbital energy (NMOE) resulting after solving extended Hartree-Fock equations. This study confirms that the NMOE is directly related to the molecular electrostatic potential at the position of the nucleus.

The divide-and-conquer second-order proton propagator method based on nuclear orbital plus molecular orbital theory for the efficient computation of proton binding energies

An efficient computational method to evaluate the binding energies of many protons in large systems was developed.