Born-Oppenheimer approximation and beyond for time-dependent electronic processes

Explicit computations of electronic motion in time and space are gradually becoming feasible and available. The knowledge of this motion is of relevance by itself but is also important for understanding available and predicting future experiments on the electronic time scale. In electronic processes of interest, usually several and even many stationary electronic states participate and the obvious question arises on how to describe the accompanying quantum nuclear dynamics at least on the time scale of the process. In this work, we attempt to study the nuclear dynamics in the framework of a fully time-dependent Born-Oppenheimer approximation. Additionally, we attempt to go beyond this approximation by introducing the coupling of several electronic wavepackets by the nuclear wavepackets. In this context, we also discuss a time-dependent transformation to diabatic electronic wavepackets. A simple but critical model of charge transfer is analyzed in some detail on various levels of approximation and also solved exactly.     

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.     

Semi-classical eigenvalues using the classical Born-Oppenheimer adiabatic approximation

The classical Born-Oppenheimer method is applied to the two-degrees-of-freedom Hénon-Heiles potential. The resulting one-dimensional potentials are then treated semi-classically to give the eigenvalues for the regular states. No trajectories are needed and the results are in good agreement with the exact results.     

Quantifying the breakdown of the Born-Oppenheimer approximation in surface chemistry

The Born-Oppenheimer Approximation (BOA) forms the basis for calculating electronically adiabatic potential energy surfaces, thus providing the framework for developing a molecular level understanding of a variety of important chemical problems. For surface chemistry at metal surfaces, it is now clear that for some processes electronically nonadiabatic effects can be important, even dominant; however, the magnitude of BOA breakdown may vary widely from one chemical system to another. In this paper we show that molecular-beam surface scattering experiments can be used to derive quantitative information about the magnitude of BOA breakdown. A state-to-state rate model is used to interpret the pre-exponential factor of the well-known Arrhenius surface temperature dependence of the electronically nonadiabatic vibrational excitation. We also show that reference to a "thermal limit" provides a quick and simple rule of thumb for quantifying BOA breakdown. We demonstrate this approach by comparing electronically nonadiabatic vibrational inelasticity for NO(ν = 0 → 1) to NO(ν = 15 →ν'? 15) and show that the electronically nonadiabatic coupling strengths are of a similar magnitude. We compare experiments for NO and HCl scattering from Au(111) and derive the quantitative relative magnitude for the electronically nonadiabatic influences in each system. The electronically nonadiabatic influences are 300-400 times larger for NO than for HCl, for incidence energies near 0.9 eV.     

Electron energy level crossings in the time-dependent Born-Oppenheimer approximation

Summary We summarize the results of a mathematical study of the time-dependent Born-Oppenheimer approximation near crossings of two non-degenerate electron energy surfaces. We illustrate our techniques by relatively simple examples that contain the essential ingredients of the general cases. We discuss all generic types of crossings of two non-degenerate electron energy surfaces.Supported in part by the National Science Foundation under Grant number DMS-8801360     

Calculations of the ground states of BeH and BeH+ without the Born-Oppenheimer approximation

Non-Born-Oppenheimer variational calculations employing explicitly correlated Gaussian basis functions have been performed for the ground states of the beryllium monohydride molecule (BeH) and its ion (BeH+), as well as for the beryllium atom (Be) and its ion (Be+). An approach based on the analytical energy gradient calculated with respect to the Gaussian exponential parameters was employed. The calculated energies were used to determine the ionization potential of BeH and the dissociation energies of BeH and BeH+. Also, the generated wave functions were used to compute various expectation values, such as the average interparticle distances and the nucleus-nucleus correlation functions.     

On the validity range of the Born-Oppenheimer approximation: a semiclassical study for all-particle quantization of three-body Coulomb systems

The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (mu) and antiproton (p) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppmu), and (ppp) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 3/2 to the mass of negative particles, that is, m(1.5). The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m(5/4) is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m(6/4). Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m(1/4).     

Path-integral simulations beyond the adiabatic approximation

Within the adiabatic approximation, it is trivial to generalize existing imaginary time path-integral techniques to the case of multiple electronic surfaces. However, there are many times where nonadiabatic effects can play an important role. To this end, we reformulate the well-known path-integral expressions to incorporate multiple potential surfaces, without necessitating the adiabatic approximation. We show that the resulting expression, like its adiabatic counterpart, can be interpreted in terms of a simple classical isomorphic system and thus is amenable to simulation through Monte Carlo techniques. We derive simple expressions to compute expectation values of a general operator in both the nuclear coordinate and electronic state, and demonstrate the existence of a simple internal diagnostic that can be used to evaluate the magnitude of equilibrium nonadiabatic effects.     

Some efforts beyond the local density approximation

The exchange-correlation density functional can be expressed as a many-body perturbation series in terms of the Coulomb interaction using the exact Kohn-Sham orbitals as the basis. A self-consistent equation is derived for the exact exchangecorrelation potential. This perturbation approach forms a basis for going beyond the local density approximation (LDA ). The discontinuity in the exchange-correlation potential for semiconductors calculated by the perturbative approach gives a good account of the discrepancy of the band gap calculated in LDA . The discontinuity also plays an important role in the interface band diagrams. A theory to account for the interaction effects of localized d or f orbitals is reviewed and the physics of the applications to a model test, to some 3d transition metals, and to heavy fermions is discussed. The perturbative approach to improvement beyond LDA tends to be computation-intensive and to be system-specific. © 1995 John Wiley & Sons, Inc.     

Structural and orthoselectivity study of 2-hydroxybenzaldehyde using spectroscopic analysis

The orthoselectivity and high yield are two significant subjects which should be studied more in the process of hydroxybenzaldehydes or salicylaldehyde production. In this work, salicylaldehyde was synthesized by the reaction of formaldehyde and phenol magnesium methoxide complex, in an anhydrous medium. In order to achieve a selectively orthoformylated product, at first the hydroxyl group of phenol was rearranged by magnesium methoxide. The phenol magnesium salt was then formylated by paraformaldehyde. Impurities of the resulted salicylaldehyde were removed by several steps of liquid extracting via water and acid washing. The spectroscopic data of FT-IR, ¹H NMR (500 MHz), and GC/MS on the final product were recorded and interpreted. The results of FT-IR spectrum and integration value of ¹H NMR spectrum imply on the high conversion of reaction. The GC/MS spectrum also shows that the amounts of by products are low enough.     

Theory of capillary-induced interactions beyond the superposition approximation

Within a general theoretical framework, we study the effective, deformation-induced interaction between two colloidal particles trapped at a fluid interface in the regime of small deformations. In many studies, this interaction has been computed with the ansatz that the actual interface configuration for the pair is given by the linear superposition of the interface deformations around the single particles. Here, we assess the validity of this approach and compute the leading term of the effective interaction for a large interparticle separation beyond this so-called superposition approximation. As an application, we consider the experimentally relevant case of interface deformations owing to the electrostatic field emanating from charged colloidal particles. In mechanical isolation, i.e., if the net force acting on the total system consisting of the particles plus the interface vanishes, the superposition approximation is actually invalid. The effective capillary interaction is governed by contributions beyond this approximation and turns out to be attractive. For sufficiently small surface charges on the colloids, such that linearization is strictly valid, and at asymptotically large separations, the effective interaction does not overcome the direct electrostatic repulsion between the colloidal particles.     

Single-substrate enzyme kinetics: the quasi-steady-state approximation and beyond

We analyze the standard model of enzyme-catalyzed reactions at various substrate-enzyme ratios by adopting a different scaling scheme and computational procedure. The regions of validity of the quasi-steady-state approximation are noted. Certain prevalent conditions are checked and compared against the actual findings. Efficacies of a few other measures, obtained from the present work, are highlighted. Some recent observations are rationalized, particularly at moderate and high enzyme concentrations.     

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.     

High precision variational calculations for the Born-Oppenheimer energies of the ground state of the hydrogen molecule

Born-Oppenheimer approximation Hylleraas variational calculations with up to 7034 expansion terms are reported for the 1sigma(g)+ ground state of neutral hydrogen at various internuclear distances. The nonrelativistic energy is calculated to be -1.174 475 714 220(1) hartree at R = 1.4 bohr, which is four orders of magnitude better than the best previous Hylleraas calculation, that of Wolniewicz [J. Chem. Phys. 103, 1792 (1995)]. This result agrees well with the best previous variational energy, -1.174 475 714 216 hartree, of Cencek (personal communication), obtained using explicitly correlated Gaussians (ECGs) [Cencek and Rychlewski, J. Chem. Phys. 98, 1252 (1993); Cencek et al., ibid. 95, 2572 (1995); Rychlewski, Adv. Quantum Chem. 31, 173 (1998)]. The uncertainty in our result is also discussed. The nonrelativistic energy is calculated to be -1.174 475 931 399(1) hartree at the equilibrium R = 1.4011 bohr distance. This result also agrees well with the best previous variational energy, -1.174 475 931 389 hartree, of Cencek and Rychlewski [Rychlewski, Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson (Wiley, New York, 2003), Vol. 2, pp. 199-218; Rychlewski, Explicitly Correlated Wave Functions in Chemistry and Physics Theory and Applications, edited by J. Rychlewski (Kluwer Academic, Dordrecht, 2003), pp. 91-147.], obtained using ECGs.     

Vibrational transitions of the 7LiH+ ion calculated without the Born-Oppenheimer approximation and with leading relativistic corrections

We recently presented very accurate calculations of the fundamental vibrational frequency of the (7)LiH(+) and (3)He(4)He(+) ions [Stanke et al. Phys. Rev. A 79, 060501(R) (2009)] performed without the Born-Oppenheimer approximation and included leading relativistic corrections. The accuracy of those calculations was estimated to be of the order of 0.06 cm(-1). In the present work we extend the calculations to the remaining pure vibrational states of (7)LiH(+) and similarly accurate results are generated. They may lead to the experimental search for still unidentified lines corresponding to those transitions.     

CN~+分子势能函数和光谱常数的多参考组态相互作用方法研究

采用多参考组态相互作用方法和aug-cc-p V5Z基函数组计算了CN+分子11∑+,21∑+,13∑+和13Π电子态的势能曲线。利用MS势能函数拟合得到了相应的解析势能表达式。在此基础上求解CN+分子的核运动薛定谔方程,获得了全部振动和转动能级,并用Dunham系数展开式拟合出了光谱常数,与目前仅有的11∑+,21∑+态的文献报道结果进行了比较。结果可对航天尾气及工业过程光谱方法监控提供参考。     

Small molecule microarrays: the first decade and beyond

The turn of the century witnessed the development of small molecule, protein, cell and tissue microarrays, heralding a new era of discovery-driven research using a host of different chemical libraries, biomolecules and tissue types. This highlight takes stock of this first decade of small molecule microarrays (SMMs) and describes how the technology has matured into a robust screening platform. We also highlight the many interesting and unique applications appearing using small molecule microarrays, in order to project forward to possible areas in which SMMs could contribute further to over the next five to ten years.