Density functional theory treatment of electron correlation in the nuclear-electronic orbital approach

This paper presents the nuclear-electronic orbital density functional theory [NEO-DFT(ee)] method for including electron-electron correlation and nuclear quantum effects self-consistently in quantum chemical calculations. The NEO approach is designed to treat a relatively small number of nuclei quantum mechanically, while the remaining nuclei are treated classically. In the NEO-DFT(ee) approach, the correlated electron density is used to obtain the nuclear molecular orbitals, and the resulting nuclear density is used to obtain the correlated electron density during an iterative procedure that continues until convergence of both the nuclear and electronic densities. This approach includes feedback between the correlated electron density and the nuclear wavefunction. The application of this approach to bihalides and acetylene indicates that the nuclear quantum effects do not significantly impact the electron correlation energy, but the quantum nuclear energy is enhanced in the NEO-DFT(ee) B3LYP method. The excellent agreement of the NEO-DFT(ee)-optimized bihalide structures with the vibrationally averaged geometries from grid-based quantum dynamical methods provides validation for the NEO-DFT(ee) approach. Electron-proton correlation could be included by the development of an electron-nucleus correlation functional. Alternatively, explicit electron-proton correlation could be included directly into the NEO self-consistent-field framework with Gaussian-type geminal functions.     

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+G?), where G? 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+G?) rather than (1+G?). 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.     

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.     

Analysis of the nuclear-electronic orbital method for model hydrogen transfer systems

Fundamental issues associated with the application of the nuclear-electronic orbital (NEO) approach to hydrogen transfer systems are addressed. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated with molecular orbital methods. The positions of the nuclear basis function centers are optimized variationally. In the application of the NEO approach to hydrogen transfer systems, the hydrogen nuclei and all electrons are treated quantum mechanically. Within the NEO framework, the transferring hydrogen atom can be represented by two basis function centers to allow delocalization of the proton vibrational wavefunction. In this paper, the NEO approach is applied to the [He-H-He]+ and [He-H-He]++ model systems. Analyses of technical issues pertaining to flexibility of the basis set to describe both single and double well proton potential energy surfaces, linear dependency of the hydrogen basis functions, multiple minima in the basis function center optimization, convergence of the number of hydrogen basis function centers, and basis set superposition error are presented. The accuracy of the NEO approach is tested by comparison to grid calculations for these model systems.     

Multi-component molecular orbital study of isotope effects on lithium hydride molecules with the configuration interaction scheme

The multi-component molecular orbital method, which can take account of the quantum effect of the electrons and nuclei, is applied to the calculation of lithium hydride isotope species with the configuration interaction (CI) scheme. The optimum basis set functions for quantum nuclei are proposed by the fully variational procedure under single electronic–single nuclear excitation CI level. The average internuclear distances and dipole moments for isotopic lithium hydride molecules calculated with small basis functions are reasonable agreement with the corresponding experimental values.     

Kinetic isotope effects calculated with the instanton method

The ring-opening reaction of the cyclopropylcarbinyl radical proceeds via heavy-atom tunneling at low temperature. We used instanton theory to calculate tunneling rates and kinetic isotope effects with on-the-fly calculation of energies by density functional theory (B3LYP). The accuracy was verified by explicitly correlated coupled-cluster calculations (UCCSD(T)-F12). At cryogenic temperatures, we found protium/deuterium KIEs up to 13 and inverse KIEs down to 0.2. We also studied an intramolecular tautomerization reaction. A simple and computationally efficient method is proposed to calculate KIEs with the instanton method: the instanton path is assumed to be independent of the atomic masses. This results in surprisingly good estimates of the KIEs for the cyclopropylcarbinyl radical and for the secondary KIEs of the tautomerization. Challenges and capabilities of the instanton method for calculating KIEs are discussed.     

Multi-component molecular orbital theory for electrons and nuclei including many-body effect with full configuration interaction treatment: isotope effects on hydrogen molecules

A multi-component molecular orbital (MC_MO) theory is developed for a combined quantum system of electrons and nuclei with the full configuration interaction (CI) scheme of Cartesian Gaussian-type functions. The technique of graphical unitary group approach (GUGA) is modified to obtain the CI matrix elements for many kinds of quantum particles efficiently. The optimum basis sets for quantum nuclei are proposed with the fully variational procedure. The average internuclear distances, dipole polarizabilities, and nuclear vibrational excitation energies for isotopic hydrogen molecules calculated with optimized basis sets are found to adequately reproduce the corresponding experimental values.     

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.     

A new interpretation of chlorine leaving group kinetic isotope effects; a theoretical approach

The chlorine leaving group kinetic isotope effects (KIEs) for the S(N)2 reactions between methyl chloride and a wide range of anionic, neutral, and radical anion nucleophiles were calculated in the gas phase and, in several cases, using a continuum solvent model. In contrast to the expected linear dependence of the chlorine KIEs on the C(alpha)-Cl bond order in the transition state, the KIEs fell in a very small range (1.0056-1.0091), even though the C(alpha)-Cl transition state bond orders varied widely from approximately 0.32 to 0.78, a range from reactant-like to very product-like. This renders chlorine KIEs, and possibly other leaving-group KIEs, less useful for studies of reaction mechanisms than commonly assumed. A partial explanation for this unexpected relationship between the C(alpha)-Cl transition state bond order and the magnitude of the chlorine KIE is presented.     

Evaluation of deuterium isotope effects in normal-phase LC-MS-MS separations using a molecular modeling approach

Molecular modeling of stationary phases presents a unique challenge because there is little available experimentally derived structural information. Verified interaction mechanisms at a molecular level with analytes are also rare. Molecular mechanics calculations using the Tripos force field were carried out to qualitatively and quantitatively assess stationary phase interactions. Binding energy values of -15.40, 15.28, -12.53, and -12.34 kcal/mol, respectively, are obtained for olanzapine (OLZ), OLZ-D3, des-methyl olanzapine (DES), and DES-D8 that corresponded to the retention behavior of the four compounds observed using liquid chromatography-mass spectrometry (MS)-MS. The model explains, semiquantitatively, the deuterium isotope effect in the normal-phase chromatographic separation of these compounds.     

An orbital perturbation approach of enzymatic catalysis

An orbital perturbation approach of enzymatic catalysis is proposed. Orbital symmetry restrictions to a fast reaction pathway is lifted by first order perturbation of the catalyst potential. The drastic decrease of the energy gap is a second order effect enhance by the band-like structure of the enzymes. The model incorporates conformational specify and signals the importance of the low symmetries of the active sites. It is proposed that allosteric effects are due to shifts of the enzyme band structure.     

Quantum-instanton evaluation of the kinetic isotope effects

A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis-Monte Carlo evaluation of the Boltzmann operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction H + H2 --> H2 + H. In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than approximately 10%.     

Large kinetic isotope effects with unsymmetrical transition states

The results of a series of calculations for a wide, unrestricted variation in the force constants for the making and breaking bonds and their interaction constant are presented for the abstraction reactions of CH₂D₂ with Cl atoms. A wide range of asymmetrical force constants leads to a high kinetic isotope effect as has been pointed out by others for a more restricted range of consideration. These results pointedly contradict the assumed connection between a high kinetic isotope effect and a symmetrical transition state. It is found by examining the atomic displacements of the normal mode that the motion of the H or D atom in the real stretch of the transition state will often have little influence on the isotope effect because the mode can be dominated by end group motions. It is further found that a 3-center model multiplied by a constant factor to account for the contributions of the other vibrations is capable of very satisfactorily reproducing the more rigorous 6-center calculations.     

Photodissociation laser isotope effects

Marked differences in the laser action (1.315 μm) observed following the flash photolysis of CD₃I and CH₃I are reported (substantially greater outputs are observed with CD₃I). These differences result from the significantly smaller cross section for quenching of I(5 ²P ) by CD₃I, relative to that for CH₃I. Absolute values for the quenching cross sections have been determined using time resolved atomic absorption spectrophotometry. These data were employed in a computer simulated model which satisfactorily reproduced the light output from CH₃I, CD₃I and CF₃I photochemical laser systems. It is concluded that isotopic substitution can markedly influence the cross section for quenching of an excited state and thus influence partitioning between the various available channels.     

Molecular orbital aspects of substituent effects

The charge densities at the exocyclic reaction centers of para-substituted benzoic acids calculated by the INDO-MO method correlate with the Hammett _p constants.     

A new approach to the molecular orbital theory of bonding

The Moffitt atoms in Molecules approach [8] and the Mulliken approximation [11] to two-centre integrals have been applied to M. O. Theory to give a simple equation for the energy of a diatomic molecule in terms of atomic spectral parameters and nuclear attraction integrals.The equation has been used to calculate the potential energy curves of 24 electronic states of the diatomic hydrides of second row elements and 24 states of the corresponding iso-electronic singly charged positive molecular ions. The agreement between the calculated and experimental values of the energies, the equilibrium internuclear separations, the force constants and the dipole moments is satisfactory.

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Zusammenfassung Die Moffittsche Näherung der Atome in Molekülen [8] und die Mullikensche Näherung für Zweizentrenintegrale [11] wurden auf die MO-Theorie angewandt, um eine einfache Gleichung für die Energie eines zweiatomigen Moleküls mit Termen aus atomaren Spektralparametern sowie Kernwechselwirkungsintegralen zu erhalten.Diese Gleichung wurde benutzt, um Potentialkurven für 24 elektronische Zustände zweiatomarer Hydride von Elementen der zweiten Hauptreihe zu berechnen sowie 24 Zustände der korrespondierenden isoelektronischen, positiv einfach geladenen Ionenmoleküle zu bestimmen. Die Übereinstimmung zwischen berechneten und experimentellen Werten der Energie, der Gleichgewichtsabstände, der Kraftkonstanten und der Dipolmomente ist befriedigend.

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Résumé L'approximation de Moffitt «atomes en molécules» [8] et l'approximation de Mulliken pour les intégrales bicentriques [11] sont utilisées dans la méthode des O. M. pour obtenir une équation simple pour l'énergie d'une molécule diatomique, contenant des termes en fonction de données de spectroscopie atomique et des intégrales d'attraction nucléaire.Cette équation est utilisée pour l'évaluation des courbes d'énergie potentielle pour 24 états électroniques des hydrures diatomiques des éléments de la deuxième période et de 24 états des mono-cations moléculaires correspondants isoélectroniques. L'agrément est satisfaisant entre les résultats calculés et les données expérimentales de l'énergie, des distances internucléaires d'équilibre, des constantes de force et des moments dipolaires.

     

Bound states of the positron with nitrile species with a configuration interaction multi-component molecular orbital approach

Characteristic features of the positron binding structure of some nitrile (-CN functional group) species such as acetonitrile, cyanoacetylene, acrylonitrile, and propionitrile are discussed with the configuration interaction scheme of multi-component molecular orbital calculations. This method can take the electron-positron correlation contribution into account through single electronic-single positronic excitation configurations. Our PA value of acetonitrile with the electronic 6-31++G(2df,2pd) and positronic [15s15p3d2f1g] basis set is calculated as 4.96 mhartree, which agrees to within 25% with the recent experimental value of 6.6 mhartree by Danielson et al. [Phys. Rev. Lett., 2010, 104, 233201]. Our PA values of acrylonitrile and propionitrile (5.70 and 6.04 mhartree) are the largest among these species, which is consistent with the relatively large dipole moments of the latter two systems.