non-BO Wave Functions Research Articles

Molecular Structure Theory without the Born-Oppenheimer Approximation: Rotationless Vibrational States of LiH.

Low-lying rotationless states of the lithium hydride molecule are studied in the framework of the variational method without assuming the Born-Oppenheimer (BO) approximation. Highly accurate solutions to the six-particle (two nuclei and four electrons) Schrödinger equation are obtained by means of expanding the wave functions of the considered states in terms of many thousands of all-particle explicitly correlated Gausssians. The basis functions are optimized independently for each state using the analytic energy gradient with respect to the nonlinear parameters. The non-BO wave functions obtained in the calculations are used to evaluate the leading-order relativistic and quantum electrodynamics energy corrections in the framework of the perturbation theory. The geometric structure of the molecule in the ground and excited states is discussed based on the analysis of the nucleus-nucleus correlation functions. The non-BO energies and structural parameters obtained of this work are compared with the most accurate BO results currently available.

Electron affinity of LiH-

In this work we present high-accuracy benchmark-quality calculations of the electron affinity (EA) of the LiH molecule in a framework that does not assume the Born–Oppenheimer (BO) approximation. The EA is calculated as a difference between the total energies of and LiH. The calculations of the energies are performed using the Rayleigh-Ritz variational method with large basis sets of all-particle explicitly correlated Gaussian functions (ECGs). Up to 14,000 ECGs are used in the calculations for each system. The nonlinear parameters of the ECGs are optimised by by the minimisation of the total non-relativistic energy of the system using an approach that employs the energy gradient determined with respect the parameters. The and LiH non-relativistic non-BO wave functions are subsequently used to calculate the leading relativistic corrections. The calculated EA is well converged in terms of the size of the basis sets and the obtained value falls within the uncertainty of the best available experimental result.

Lowest ^{2}S Electronic Excitations of the Boron Atom.

A theoretical abinitio approach for calculating bound states of small atoms is developed and implemented. The approach is based on finite-nuclear-mass [non-Born-Oppenheimer (non-BO)] nonrelativistic variational calculations performed with all-particle explicitly correlated Gaussian functions and includes the leading relativistic and quantum electrodynamics energy corrections determined using the non-BO wave functions. The approach is applied to determine the total and transition energies for the lowest four ^{2}S electronic excitations of the boron atom. The transition energies agree with the available experimental values within 0.2-0.3 cm^{-1}. Previously, such accuracy was achieved for three- and four-electron systems.

Non-Born-Oppenheimer calculations of the rovibrational spectrum of H2 excited to the second rotational level

Quantum mechanical, non-relativistic, non-Born-Oppenheimer (non-BO) calculations are performed for the rovibrational spectrum of H2 excited to the second rotational level. The non-BO wave functions of the considered states are expanded in terms of all-particle explicitly correlated Gaussian functions. The dissociation energies and rovibrational transition energies are calculated and compared with experimental values and values obtained in calculations performed by others. The average interparticle distances are calculated and compared with the corresponding values for HD. They show that H2 is a more “diffuse molecule”. The nuclear-nuclear correlation functions are calculated and plotted to visualize the “non-BO molecular structure” of H2.

Ortho–para nuclear-spin isomerization energies for all bound vibrational states of ditritium (T2) from non-Born–Oppenheimer variational calculations performed with explicitly correlated all-particle Gaussian functions

Direct variational calculations, where the Born–Oppenheimer (BO) approximation is not assumed, are performed for all 26 bound rovibrational states corresponding to the lowest rotational excitation (i.e. the N=1 states) of the tritium molecule (T2). The non-BO energies are used to determine the ortho–para isomerization energies. All-particle explicitly correlated Gaussian basis functions are employed in the calculations and over 11000 Gaussians independently generated for each state are used. The exponential parameters of the Gaussians are optimized with the aid of analytically calculated energy gradient determined with respect to these parameters. The non-BO wave functions are used to calculate expectation values of the inter-particle distances and the triton–triton correlation functions.

Para–ortho isomerization of H2+. Non-Born–Oppenheimer direct variational calculations with explicitly correlated all-particle Gaussian functions

Direct variational calculations are performed for all bound rovibrational states of the H2+ ion corresponding to the ground and first excited rotational levels (the N=0 and N=1 states). The Born–Oppenheimer (BO) approximation is not assumed in the calculations and all-particle explicitly correlated Gaussian basis functions are used for the wave-function expansion. The exponential parameters of the Gaussians are optimized with the aid of analytically calculated energy gradient determined with respect to these parameters. The non-BO energies are used to determine the ortho–para nuclear-spin isomerization energies and the non-BO wave functions are used to determine the expectation values of the interparticle distances.

Non-Born-Oppenheimer calculations of the pure vibrational spectrum of T₂ including relativistic corrections.

We report very accurate calculations of the complete pure vibrational spectrum of the T2 molecule with an approach where the Born-Oppenheimer (BO) approximation is not assumed. As the considered states correspond to the zero total angular momentum, their non-BO wave functions are spherically symmetric and are expanded in terms of all-particle, one-center, spherically symmetric explicitly correlated Gaussian functions multiplied by even nonnegative powers of the internuclear distance. The nonrelativistic energies of the states obtained in the non-BO calculations are corrected for the relativistic effects of the order of α(2) (where α is the fine structure constant) calculated as expectation values of the operators representing these effects.

Lifetimes of rovibrational levels ofHD+

A method for calculating the oscillator strengths for rovibrational transitions of a diatomic system within an approach that is not based on the Born-Oppenheimer (BO) approximation is presented. The non-BO wave functions representing the bound states of the system are expanded in terms of explicitly correlated Gaussian functions. The method is applied to calculate oscillator strengths for the ${\text{HD}}^{+}$ ion for transitions between rotationless vibrational states and vibrational states which are rotationally singly excited. The effect of the asymmetry of the ${\text{HD}}^{+}$ charge distribution on the oscillator strengths and on the lifetimes of the states is elucidated.

The two-component quantum theory of atoms in molecules (TC-QTAIM): the unified theory of localization/delocalization of electrons, nuclei, and exotic elementary particles

In this contribution, pursuing our research program by extending the atoms in molecules analysis into unorthodox domains, another key ingredient of the two-component quantum theory of atoms in molecules (TC-QTAIM) namely the theory of localization/delocalization of quantum particles, is disclosed. The unified proposed scheme is able not only to deal with the localization/delocalization of electrons in/between atomic basins, but also to treat nuclei as well as exotic particles such as positrons and muons equally. Based on the general reduced second-order density matrices for indistinguishable quantum particles, the quantum fluctuations of atomic basins are introduced and then used as a gauge to quantitate the localization/delocalization introducing proper indexes. The explicit mass dependence of the proposed indexes is demonstrated, and it is shown that a single localization/delocalization index is capable of being used for all kind of quantum particles regardless of their masses or charge content. For various non-Born–Oppenhiemer (non-BO) wavefunctions, including Hartree product as well as singlet and triplet determinants, the indices are calculated and then employed to rationalize the localization/delocalization of particles in a series of four-body model systems consist of two electrons and two positively charged particles with variable mass. The ab initio FV-MC_MO derived non-BO wavefunctions for the four-body series are used for a comprehensive computational TC-QTAIM analysis, including topological analysis as well as basin integrations, in a wide mass region, m = 10m e − 1013 m e (m e stands for electron mass), disclosing various traits in these series of species that are unique to the TC-QTAIM. On the other hand, it is demonstrated that in the large mass extreme, the TC-QTAIM analysis reduces to the one performed within context of the orthodox QTAIM with two clamped positive particles revealing the fact that the TC-QTAIM encompasses the orthodox QTAIM as an asymptote. Finally, it is concluded that the proposed localization/delocalization scheme is capable of quantitating quantum tunneling of nuclei for systems containing delocalized protons. Such capability promises novel applications for the TC-QTAIM as well as its extended multi-component version (MC-QTAIM) introduced recently.

Non-Born-Oppenheimer potential energy curve: Hydrogen molecular ion with highly accurate free complement method

Although the concept of a potential energy curve (PEC) originates from the outgrowth of the Born-Oppenheimer (BO) approximation, we propose the application of analysis methods for the physical PEC with non-Born-Oppenheimer (non-BO) wave functions. A numerical examination was performed with the highly accurate non-BO vibronic wave functions of hydrogen molecular ion, which were obtained in our previous studies with the free complement method. The reduced density function integrated over the electron coordinates plays an important role in understanding nuclear motion dynamics, since it corresponds to the wave function density of the vibrational and rotational motions. The maximum positions of this density indicate the high existence probability of nuclei and can be considered as a discrete representation of the PEC. Whereas an ordinary PEC with the BO approximation is obtained as a numeric curve after multiple electronic state calculations at fixed nuclear coordinates, we propose a new analytical expression of the PEC from a non-BO wave function.

Complete pure vibrational spectrum of HD calculated without the Born-Oppenheimer approximation and including relativistic corrections

All 18 bound pure vibrational levels of the HD molecule have been calculated within the framework that does not assume the Born-Oppenheimer (BO) approximation. The nonrelativistic energies of the states have been corrected for the relativistic effects of the order of ${\ensuremath{\alpha}}^{2}$ (where $\ensuremath{\alpha}$ is the fine structure constant), calculated using the perturbation theory with the nonrelativistic non-BO wave functions being the zero-order approximation. The calculations were performed by expanding the non-BO wave functions in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance and by performing extensive optimization of the Gaussian nonlinear parameters. Up to 10 000 basis functions were used for each state.

Lower vibrational transitions of the 3He 4He + ion calculated without the Born–Oppenheimer approximation and with leading relativistic corrections

Very accurate variational calculations of the five lowest vibrational states of the 3He 4He + ion are carried out within a framework that does not assume the Born–Oppenheimer (BO) approximation, i.e., treating the two nuclei and three electrons forming the system on an equal footing. The non-BO wave functions are expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. The wave functions are used to calculate the leading relativistic corrections. The approach reproduces the experimental 3He 4He + fundamental transition within 0.055 cm −1 and similar accuracy is expected for the higher yet unmeasured vibrational transitions determined in the present calculations.

Non-Born–Oppenheimer calculations of the BH molecule

Variational calculations employing explicitly correlated Gaussian basis functions have been performed for the ground state of the boron monohydride molecule (BH) and for the boron atom (B). Up to 2000 Gaussians were used for each system. The calculations did not assume the Born-Oppenheimer (BO) approximation. In the optimization of the wave function, we employed the analytical energy gradient with respect to the Gaussian exponential parameters. In addition to the total nonrelativistic energies, we computed scalar relativistic corrections (mass-velocity and Darwin). With those added to the total energies, we estimated the dissociation energy of BH. The non-BO wave functions were also used to compute some expectation values involving operators dependent on the interparticle distances.

Fundamental vibrational transitions of theH3e H4e+andL7iH+ions calculated without assuming the Born-Oppenheimer approximation and with including leading relativistic corrections

Very accurate variational calculations of the fundamental pure vibrational transitions of the $^{3}\text{H}\text{e}\text{ }{^{4}\text{H}\text{e}}^{+}$ and ${^{7}\text{L}\text{iH}}^{+}$ ions are performed within the framework that does not assume the Born-Oppenheimer (BO) approximation. The non-BO wave functions expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance are used to calculate the leading relativistic corrections. Up to $10\text{ }000$ Gaussian functions are used for each state. It is shown that the experimental $^{3}\text{H}\text{e}\text{ }{^{4}\text{H}\text{e}}^{+}$ fundamental transitions is reproduced within $0.06\text{ }{\text{cm}}^{\ensuremath{-}1}$ by the calculations. A similar precision is expected for the calculated, but still unmeasured, fundamental transition of ${^{7}\text{L}\text{iH}}^{+}$. Thus, three-electron diatomic systems are calculated with a similar accuracy as two-electron systems.

Charge asymmetry in pure vibrational states of the HD molecule

Very accurate variational calculations of all rotationless states (also called pure vibrational states) of the HD molecule have been performed within the framework that does not assume the Born-Oppenheimer (BO) approximation. The non-BO wave functions of the states describing the internal motion of the proton, the deuteron, and the two electrons were expanded in terms of one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. Up to 6000 functions were used for each state. Both linear and nonlinear parameters of the wave functions of all 18 states were optimized with a procedure that employs the analytical gradient of the energy with respect to the nonlinear parameters of the Gaussians. These wave functions were used to calculate expectation values of the interparticle distances and some other related quantities. The results allow elucidation of the charge asymmetry in HD as a function of the vibrational excitation.

Non-Born-Oppenheimer calculations of the lowest vibrational energy of HD including relativistic corrections

In this work we report variational calculations of the two lowest vibrational states of the HD molecule within the framework that does not assume the Born-Oppenheimer (BO) approximation. The nonrelativistic energies of the states were corrected for the relativistic effects of the order of ${\ensuremath{\alpha}}^{2}$ (where $\ensuremath{\alpha}=\frac{1}{c}$), calculated as expectation values of the operators representing these effects with the nonrelativistic non-BO wave functions. The non-BO wave functions were expanded in terms of the one-center explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. The $v=0\ensuremath{\rightarrow}1$ transition energy obtained in the calculations is compared with the previous calculations, as well as with the transition frequency obtained from the experimental spectra. The comparison shows the need to include corrections higher than second order in $\ensuremath{\alpha}$ to further improve the agreement between the theory and the experiment.

Lowest vibrational states ofHe4He+3: Non-Born-Oppenheimer calculations

Very accurate quantum mechanical calculations of the first five vibrational states of the {sup 4}He{sup 3}He{sup +} molecular ion are reported. The calculations have been performed explicitly including the coupling of the electronic and nuclear motions [i.e., without assuming the Born-Oppenheimer (BO) approximation]. The nonrelativistic non-BO wave functions were used to calculate the {alpha}{sup 2} relativistic mass velocity, Darwin, and spin-spin interaction corrections. For the lowest vibrational transition, whose experimental energy is established with high precision, the calculated and the experimental results differ by only 0.16 cm{sup -1}.

Electron affinity of Li7 calculated with the inclusion of nuclear motion and relativistic corrections

Explicitly correlated Gaussian functions have been used to perform very accurate variational calculations for the ground states of (7)Li and (7)Li(-). The nuclear motion has been explicitly included in the calculations (i.e., they have been done without assuming the Born-Oppenheimer (BO) approximation). An approach based on the analytical energy gradient calculated with respect to the Gaussian exponential parameters was employed. This led to a noticeable improvement of the previously determined variational upper bound to the nonrelativistic energy of Li(-). The Li energy obtained in the calculations matches those of the most accurate results obtained with Hylleraas functions. The finite-mass (non-BO) wave functions were used to calculate the alpha(2) relativistic corrections (alpha=1c). With those corrections and the alpha(3) and alpha(4) corrections taken from Pachucki and Komasa [J. Chem. Phys. 125, 204304 (2006)], the electron affinity (EA) of (7)Li was determined. It agrees very well with the most recent experimental EA.

Lowest Excitation Energy ofBe9

Variational calculations employing explicitly correlated Gaussian functions and explicitly including the nuclear motion [i.e., without assuming the Born-Oppenheimer (BO) approximation] have been performed to determine the lowest singlet transition energy in the 9Be atom. The non-BO wave functions were used to calculate the alpha2 relativistic corrections (alpha=1/137.035,999,679). With those corrections and with the alpha3 and alpha4 QED corrections determined previously by others, we obtained 54,677.35 cm(-1) for the 3(1)S-->2(1)S transition energy. This result falls within the error bracket for the experimental transition of 54,677.26(10) cm(-1). This is the first time an electronic transition of Be has been calculated from first principles with the experimental accuracy.

Ionization potential ofBe9calculated including nuclear motion and relativistic corrections

Variational calculations employing explicitly correlated Gaussian functions have been performed for the ground states of $^{9}\mathrm{Be}$ and $^{9}\mathrm{Be}^{+}$ including the nuclear motion [i.e., without assuming the Born-Oppenheimer (BO) approximation]. An approach based on the analytical energy gradient calculated with respect to the Gaussian exponential parameters was employed, leading to energies of the two systems noticeably improved over those found in the recent paper of Pachucki and Komasa [Phys. Rev. A 73, 052502 (2006)]. The non-BO wave functions were used to calculate the ${\ensuremath{\alpha}}^{2}$ relativistic corrections $(\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c)$. With those corrections and the ${\ensuremath{\alpha}}^{3}$ and ${\ensuremath{\alpha}}^{4}$ corrections taken from Pachucki and Komasa, a new value of the ionization potential (IP) of $^{9}\mathrm{Be}$ was determined. It agrees very well with the most recent experimental IP.