Accurate, two-state ab initio study of the ground and first-excited states of He2+, including exact treatment of all Born-Oppenheimer correction terms

Born-Oppenheimer (BO) potentials for the ground and first-excited electronic states of He2+ are determined using high level ab initio techniques for internuclear separations R of 1.2-100 bohrs and accurately fit to analytical functions. In the present formulation, the BO potentials are nuclear mass independent, and the corresponding BO approximation is obtained by ignoring four terms of the full rovibronic Hamiltonian. These four Born-Oppenheimer correction (BOC) terms are as follows: (1) mass polarization, (2) electronic orbital angular momentum, (3) first derivative with respect to R, and (4) second derivative with respect to R. In order to enable an exact rovibronic calculation, each of the four BOC terms are computed as a function of R, for the two electronic states and for their coupling, without any approximation or use of empirical parameters. Each of the BOC terms is found to make a contribution to the total energy over at least some portion of the range of R investigated. Interestingly, the most significant coupling contribution arises from the electronic orbital angular momentum term, which is evidently computed for the first time in this work. Although several BOC curves exhibit a nontrivial dependence on R, all are accurately fit to analytical functions. The resulting functions, together with the BO potentials, are used to compute exact rovibronic energy levels for 3He 3He+,3He 4He+) and 4He 4He+. Comparison to available high quality experimental data indicates that the present BOC potentials provide the most accurate representation currently available of both the low- and high-lying levels of the ground electronic state and the bound levels of the excited state.     

Born-Oppenheimer expansion at constant energy

The well-known Born-Oppenheimer procedure considers molecular properties in the limit of large values of the nuclear mass M (in units of electronic mass), with quantum numbers held constant. It is shown that the lowest-order correction to the molecular wave function is proportional to M(-34). Here we consider the same expansion, but with energy held constant instead of quantum numbers. We show that, if the singular coupling term near a conical intersection is removed via a quasidiabatic transformation, the lowest-order remaining correction is proportional to M(-12).     

Orbit-orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H(2)

We report the derivation of the orbit-orbit relativistic correction for calculating pure vibrational states of diatomic molecular systems with sigma electrons within the framework that does not assume the Born-Oppenheimer (BO) approximation. The correction is calculated as the expectation value of the orbit-orbit interaction operator with the non-BO wave function expressed in terms of explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. With that we can now calculate the complete relativistic correction of the order of alpha(2) (where alpha=1/c). The new algorithm is applied to determine the full set of the rotationless vibrational levels and the corresponding transition frequencies of the H(2) molecule. The results are compared with the previous calculations, as well as with the frequencies obtained from the experimental spectra. The comparison shows the need to include corrections higher than second order in alpha to further improve the agreement between the theory and the experiment.     

Towards an assessment of the accuracy of density functional theory for first principles simulations of water. II

A series of 20 ps ab initio molecular dynamics simulations of water at ambient density and temperatures ranging from 300 to 450 K are presented. Car-Parrinello (CP) and Born-Oppenheimer (BO) molecular dynamics techniques are compared for systems containing 54 and 64 water molecules. At 300 K, an excellent agreement is found between radial distribution functions (RDFs) obtained with BO and CP dynamics, provided an appropriately small value of the fictitious mass parameter is used in the CP simulation. However, we find that the diffusion coefficients computed from CP dynamics are approximately two times larger than those obtained with BO simulations for T>400 K, where statistically meaningful comparisons can be made. Overall, both BO and CP dynamics at 300 K yield overstructured RDFs and slow diffusion as compared to experiment. In order to understand these discrepancies, the effect of proton quantum motion is investigated with the use of empirical interaction potentials. We find that proton quantum effects may have a larger impact than previously thought on structure and diffusion of the liquid.     

Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with including relativistic corrections

In this work we report very accurate variational calculations of the complete pure vibrational spectrum of the D(2) molecule performed within the framework where the Born-Oppenheimer (BO) approximation is not assumed. After the elimination of the center-of-mass motion, D(2) becomes a three-particle problem in this framework. As the considered states correspond to the zero total angular momentum, their wave functions are expanded in terms of all-particle, one-center, spherically symmetric explicitly correlated Gaussian functions multiplied by even non-negative 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 = 1/c is the fine structure constant) calculated as expectation values of the operators representing these effects.     

Darwin and mass-velocity relativistic corrections in non-Born-Oppenheimer variational calculations

The Pauli approach to account for the mass-velocity and Darwin relativistic corrections has been applied to the formalism for quantum mechanical molecular calculations that does not assume the Born-Oppenheimer (BO) approximation regarding separability of the electronic and nuclear motions in molecular systems. The corrections are determined using the first order perturbation theory and are derived for the non-BO wave function of a diatomic system expressed in terms of explicitly correlated Gaussian functions with premultipliers in the form of even powers of the internuclear distance. As a numerical example we used calculations of the transition energies for pure vibrational states of the HD(+) ion.     

Ab initio Ehrenfest dynamics

We present an ab initio direct Ehrenfest dynamics scheme using a three time-step integrator. The three different time steps are implemented with nuclear velocity Verlet, nuclear-position-coupled midpoint Fock integrator, and time-dependent Hartree-Fock with a modified midpoint and unitary transformation algorithm. The computational cost of the ab initio direct Ehrenfest dynamics presented here is found to be only a factor of 2-4 larger than that of Born-Oppenheimer (BO) dynamics. As an example, we compute the vibration of the NaCl molecule and the intramolecular torsional motion of H2C=NH2+ by Ehrenfest dynamics compared with BO dynamics. For the vibration of NaCl with an initial kinetic energy of 1.16 eV, Ehrenfest dynamics converges to BO dynamics with the same vibrational frequency. The intramolecular rotation of H2C=NH2+ produces significant electronic excitation in the Ehrenfest trajectory. The amount of nonadiabaticity, suggested by the amplitude of the coherent progression of the excited and ground electronic states, is observed to be directly related to the strength of the electron-nuclear coupling. Such nonadiabaticity is seen to have a significant effect on the dynamics compared with the adiabatic approximation.     

A quantum dynamical treatment of symmetry-induced kinetic isotope effects in the formation of He2+

Kinetic isotope effects for He(2)(+) formation are calculated quantum dynamically using high-quality Born-Oppenheimer (BO) potentials for two electronic states of He(2)(+) and an accurate treatment of all nonadiabatic BO corrections. The two potentials are coupled only when the helium isotopes are different, and the calculations reveal that this coupling is sufficient to allow the two sets of distinguishable reactants, (4)He(+) + (3)He or (3)He(+) + (4)He, to yield He(2)(+) with comparable efficiency over a wide temperature range. Consequently, the potential coupling provides a significant formation rate enhancement for the low isotopic symmetry reactants, as compared to the symmetrical cases (e.g., (4)He(+) + (4)He or (3)He(+) + (3)He). The computed symmetry-induced kinetic isotope effects (SIKIEs) are in substantial agreement with the available experimental results and represent the first theoretical demonstration of this unusual kinetic phenomenon. Possible application of SIKIE to ozone formation and other chemical systems is discussed.     

Investigation of the vaporization of boric acid by transpiration thermogravimetry and knudsen effusion mass spectrometry

The vaporization of H3BO3(s) was studied by using a commercial thermogravimetric apparatus and a Knudsen effusion mass spectrometer. The thermogravimetric measurements involved use of argon as the carrier gas for vapor transport and derivation of vapor pressures of H3BO3(g) in the temperature range 315-352 K through many flow dependence and temperature dependence runs. The vapor pressures as well as the enthalpy of sublimation obtained in this study represent the first results from measurements at low temperatures that are in accord with the previously reported near-classical transpiration measurements (by Stackelberg et al. 70 years ago) at higher temperatures (382-413 K with steam as the carrier gas). The KEMS measurements performed for the first time on boric acid showed H3BO3(g) as the principal vapor species with no meaningful information discernible on H2O(g) though. The thermodynamic parameters, both p(H3BO3) and Delta sub H degrees m(H3BO3,g), deduced from KEMS results in the temperature range 295-342 K are in excellent agreement with the transpiration results lending further credibility to the latter. All this information points toward congruent vaporization at the H3BO3 composition in the H2O-B2O3 binary system. The vapor pressures obtained from transpiration (this study and that of Stackelberg et al.) as well as from KEMS measurements are combined to recommend the following: log [p(H3BO3)/Pa]=-(5199+/-74)/(T/K)+(15.65+/-0.23), valid for T=295-413 K; and Delta sub H degrees m=98.3+/-9.5 kJ mol (-1) at T=298 K for H3BO3(s)=H3BO3(g).     

Equilibrium structure and torsional barrier of BH3NH3

Born-Oppenheimer equilibrium structures, r(e)(BO), of the electronic ground state of the borazane (BH3NH3) molecule of C3v point-group symmetry are computed ab initio using the CCSD(T) method with basis sets up to quintuple-zeta quality. Inclusion of the counterpoise correction and extrapolation of the structural parameters to the complete basis set limit yield a best estimate of r(e)(BO) of BH3NH3. The anharmonic force field of BH3NH3, computed at the CCSD(T) level of theory with a basis set of triple-zeta quality, allows the determination of semi-experimental equilibrium rotational constants, which in turn result in a semi-experimental equilibrium structure, r(e)(SE). The r(e)(BO) and r(e)(SE) structures are in excellent agreement, indicating the validity of the methods used for their determination. The empirical mass-dependent structure, r(m)(1), of BH3NH3 is also determined. Although it is inferior in quality to the previous two structures, it is much more accurate than the standard empirical r0 and r(s) structures reported earlier for BH3NH3. The semi-experimental r(e)(SE) as well as the empirical r(m)(1) structures determined are based on experimental ground-state rotational constants available from the literature for nine isotopologues of borazane. The effective barrier to the internal rotation of BH3NH3, a molecule isoelectronic with CH3CH3, has been computed ab initio, employing the focal-point analysis (FPA) approach, to be 699 +/- 11 cm(-1). This compares favorably with an empirical redetermination of the effective barrier based on the above r(e)(SE) structure, V3 = 718(17) cm(-1).     

Darwin and mass-velocity relativistic corrections in the non-Born-Oppenheimer calculations of pure vibrational states of H2

The Darwin and mass-velocity relativistic corrections have been calculated for all pure vibrational states of the H2 using the perturbation theory and very accurate variational wave functions obtained without assuming the Born-Oppenheimer (BO) approximation. Expansions in terms of explicitly correlated Gaussians with premultipliers in the form of even powers of the internuclear distance were used for the wave functions. With the inclusion of the two relativistic corrections to the non-BO energies the transition energies for the highest states agree more with the experimental results.     

Systematic corrections to the Born-Oppenheimer approximation

A method for providing systematic diabatic corrections to the Born-Oppenheimer approximation is presented. We begin with an adiabatic expansion of the exact vibronic wavefunctions and, via the molecular Hamiltonian, develop expressions for the diabatic terms in the Schrödinger equation. We then derive recursion relations which allow one to introduce the diabatic interactions to any desired degree of approximation. As an example, the first approximation (beyond the Born-Oppenheimer approximation) is discussed explicitly. In passing, we assess some of the common misconceptions associated with the Born-Oppenheimer approximation.     

Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium

The adiabatic, relativistic, and quantum electrodynamics (QED) contributions to the pair potential of helium were computed, fitted separately, and applied, together with the nonrelativistic Born-Oppenheimer (BO) potential, in calculations of thermophysical properties of helium and of the properties of the helium dimer. An analysis of the convergence patterns of the calculations with increasing basis set sizes allowed us to estimate the uncertainties of the total interaction energy to be below 50 ppm for interatomic separations R smaller than 4 bohrs and for the distance R = 5.6 bohrs. For other separations, the relative uncertainties are up to an order of magnitude larger (and obviously still larger near R = 4.8 bohrs where the potential crosses zero) and are dominated by the uncertainties of the nonrelativistic BO component. These estimates also include the contributions from the neglected relativistic and QED terms proportional to the fourth and higher powers of the fine-structure constant α. To obtain such high accuracy, it was necessary to employ explicitly correlated Gaussian expansions containing up to 2400 terms for smaller R (all R in the case of a QED component) and optimized orbital bases up to the cardinal number X = 7 for larger R. Near-exact asymptotic constants were used to describe the large-R behavior of all components. The fitted potential, exhibiting the minimum of -10.996 ± 0.004 K at R = 5.608 0 ± 0.000 1 bohr, was used to determine properties of the very weakly bound (4)He(2) dimer and thermophysical properties of gaseous helium. It is shown that the Casimir-Polder retardation effect, increasing the dimer size by about 2 ? relative to the nonrelativistic BO value, is almost completely accounted for by the inclusion of the Breit-interaction and the Araki-Sucher contributions to the potential, of the order α(2) and α(3), respectively. The remaining retardation effect, of the order of α(4) and higher, is practically negligible for the bound state, but is important for the thermophysical properties of helium. Such properties computed from our potential have uncertainties that are generally significantly smaller (sometimes by nearly two orders of magnitude) than those of the most accurate measurements and can be used to establish new metrology standards based on properties of low-density helium.     

Extended Born-Oppenheimer equation for a three-state system

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.     

Automatic generation of potential energy and property surfaces of polyatomic molecules in normal coordinates

A procedure for the automatic construction of Born-Oppenheimer (BO) potential energy and molecular property surfaces in rectilinear normal coordinates is presented and its suitability and accuracy when combined with vibrational structure calculations are assessed. The procedure relies on a hierarchical n-mode representation of the BO potential energy or molecular property surface, where the n-mode term of the sequence of potentials/molecular properties includes only the couplings between n or less vibrational degrees of freedom. Each n-mode cut of the energy/molecular property surface is first evaluated in a grid of points with ab initio electronic structure methods. The ab initio data are then spline interpolated and a subsequent polynomial fitting provides an analytical semiglobal representation for use in vibrational structure programs. The implementation of the procedure is outlined and the accuracy of the method is tested on water and difluoromethane. Strategies for improving the proposed algorithm are also discussed.     

On equilibrium structures of the water molecule

Equilibrium structures are fundamental entities in molecular sciences. They can be inferred from experimental data by complicated inverse procedures which often rely on several assumptions, including the Born-Oppenheimer approximation. Theory provides a direct route to equilibrium geometries. A recent high-quality ab initio semiglobal adiabatic potential-energy surface (PES) of the electronic ground state of water, reported by Polyansky et al. [ ibid. 299, 539 (2003)] and called CVRQD here, is analyzed in this respect. The equilibrium geometries resulting from this direct route are deemed to be of higher accuracy than those that can be determined by analyzing experimental data. Detailed investigation of the effect of the breakdown of the Born-Oppenheimer approximation suggests that the concept of an isotope-independent equilibrium structure holds to about 3 x 10(-5) A and 0.02 degrees for water. The mass-independent [Born-Oppenheimer (BO)] equilibrium bond length and bond angle on the ground electronic state PES of water is r(e) (BO)=0.957 82 A and theta e (BO)=104.48(5) degrees , respectively. The related mass-dependent (adiabatic) equilibrium bond length and bond angle of H2 (16)O is r(e) (ad)=0.957 85 A and theta e (ad)=104.50(0) degrees , respectively, while those of D2 (16)O are r(e) (ad)=0.957 83 A and theta e (ad)=104.49(0) degrees . Pure ab initio prediction of J=1 and 2 rotational levels on the vibrational ground state by the CVRQD PESs is accurate to better than 0.002 cm(-1) for all isotopologs of water considered. Elaborate adjustment of the CVRQD PESs to reproduce all observed rovibrational transitions to better than 0.05 cm(-1) (or the lower ones to better than 0.0035 cm(-1)) does not result in noticeable changes in the adiabatic equilibrium structure parameters. The expectation values of the ground vibrational state rotational constants of the water isotopologs, computed in the Eckart frame using the CVRQD PESs and atomic masses, deviate from the experimentally measured ones only marginally, especially for A0 and B0. The small residual deviations in the effective rotational constants are due to centrifugal distortion, electronic, and non-Born-Oppenheimer effects. The spectroscopic (nonadiabatic) equilibrium structural parameters of H2 16O, obtained from experimentally determined A'0 and B'0 rotational constants corrected empirically to obtain equilibrium rotational constants, are r(e) (sp)=0.957 77 A and theta e (sp)=104.48 degrees .     

Electron affinity of (7)Li 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.     

Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer (BO) approximation, nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface (PES). However, this single PES picture should be reconsidered if a conical intersection (CI) is present, although the energy is well below the CI. The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation, i.e., the diagonal BO correction (DBOC) and the geometric phase (GP), which are divergent at the CI. At the same time, there are cusps in the adiabatic PESs. Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation. A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation (DVR) method. We examine the numerical accuracy of the Sinc DVR method for solving the Schr?dinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation. The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points, without special treatment of the divergence of the DBOC and the GP. The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP, whose accurate form usually is not easy to obtain.     

Physical origins of high-order Born-Oppenheimer electronic polarizabilities: Application to a 30th-order perturbation study of H2+

A method is developed for investigating the physical origins of high-order Born-Oppenheimer (BO) electronic polarizabilities for diatomic molecules; the procedure is based upon resolution of the polarizabilities in each order into their kinetic, internal-potential, and field-potential energetic components. The energetic components of BO polarizabilities are subject to a priori constraints on their relative magnitudes and signs which are similar to but less restrictive than.those for non-Born-Oppenheimer (NBO) polarizabilities. The procedure, implemented via the perturbational-variational Rayleigh-Ritz formalism, is illustrated by a 30th-order polarizability study of the ground state of BO H₂⁺. The BO results are compared to those for NBO H₂⁺ and their differing behaviour is discussed. Other diatomic molecules are considered.     

Velocity dependence of friction and Kramers relaxation rates

We study the influence of the velocity dependence of friction on the escape rate of a Brownian particle from the deep potential well (Eb>kBT, Eb is the barrier height, kB is the Boltzmann constant, and T is the bath temperature). The bath-induced relaxation is treated within the Rayleigh model (a heavy particle of mass M in the bath of light particles of mass m相似文献