Born Oppenheimer Potential Research Articles

Orbital angular momentum in triatomic molecules

The effects of orbital angular momentum on the details of the spin and rotational fine structure of the [Xtilde] 2 B 1 and à 2 A 1 states of NH2 and H2O+ are considered. It is found that the erratic spin-orbit splittings and asymmetry parameters of the à 2 A 1 states can be reproduced with good accuracy by calculations that also account for the regular spin and rotational structures of the ground states; the only input parameters are the shapes of the Born-Oppenheimer potential curves, the bond lengths and the spin-orbit coupling constants. The same calculations give an almost quantitative explanation of various perturbations in the à 2 A 1 state of NH2. These perturbations are caused by the ground state and result from the presence of orbital angular momentum. Three types have been documented; one is direct spin-orbit interaction between the à and [Xtilde] states, and the others involve rotational asymmetry, which is found to be an important mechanism for causing perturbations between electronic states.

A new functional form for representing vibrational eigenenergies of diatomic molecules. Application to H2+ ground statea)

We propose representing the vibrational eigenenergies Ev in a form Ev=D−(vD−v)m [L/N] where v is the quantum number, D is the energy at the dissociation limit, and vD and m are state dependent parameters. The symbol [L/N] denotes a rational fraction with polynomials in (vD−v) of degree L and N in the numerator and denominator, respectively. This permits interpolation between near equilibrium, where power series expansions in v have validity, and near dissociation, where (vD−v)m forms have theoretical basis. We apply this Ev expression with a variety of [L/N] to the Born–Oppenheimer potential of the H2+ ground state. A [2/2] fit yields more accurate first energy differences ΔE (v+1/2) than a polynomial or any other L+N=4 fit. For a given number of variable parameters, the best fits have terms of degree less than L or N missing in either the numerator or denominator. Our best overall fit has vD=19.76, m=6.08, L=2, N=4, and rms error in calculated ΔE (v+1/2) of only 0.006 cm−1. Difficulties that might be encountered in the use of the above Ev expression are discussed in some detail.

Diatomic-molecule vibrational potentials. II. New representationsa)

Two new representations of diatomic-molecule vibrational potentials are presented. One of these includes most of the previously employed series approximations as special cases. The new representations are tested against older ones by using Peek’s ’’exact’’ numerical Born–Oppenheimer potentials for the 1sσg, 2pπu, and 3dσg states of H2+ as test problems. Accuracy comparisons are made with Dunham, Thakkar, Ogilvie–Tipping, Coulomb-subtracted Ogilvie–Tipping, and Padé representations. A central idea of the new treatment is that it is not necessary to use a global representation of the potential over the region 0?R/Re<∞. Rather one can choose separate representations suitable for the subregions R/Re?1 and R/Re?1 and then match them smoothly to each other and to the Dunham expansion at R/Re=1. Attention is focused on finding improved approximations for R/Re≳1, since this region exerts strong control on the vibrational eigenvalue spectrum and it is here, perhaps, where the older techniques are at their weakest. Known properties of the exact potential such as the R→∞ behavior and the dissociation energy can be built into the new forms a priori. If the dissociation energy is not known, the new methods allow it to be estimated with better accuracy then could be done previously. If one knows only Dunham coefficients, the Coulomb-subtracted Ogilvie–Tipping series is the superior representation. If one knows, in addition, the dissociation energy, the new representations are superior and give more accurate results on the interval R/Re?1.

Diatomic molecule vibrational potentials: Accuracy of representations

A method is presented for increasing the radius of convergence of certain representations of diatomic molecule vibrational potentials. The method relies on using knowledge of the analytic structure of such potentials to the maximum when attempting to approximate them. The known singular point (due to the centrifugal and/or Coulomb potentials) at zero internuclear separation should be included in its exact form in an approximate representation. The efficacy of this idea is tested [using Peek’s ’’exact’’ numerical Born–Oppenheimer potential for the (1sσg)2Σ+g state of H+2 as a test problem] when the representational form is the series of (1) Dunham, (2) Simons, Parr, and Finlan, (3) Thakkar, and (4) Ogilvie–Tipping, and also (5) when the form is a [2, 2] or a [3, 3] Padé approximant. Significant improvements in accuracy are obtained in some of these cases, particularly on the inner wall of the potential. A comparison of the effectiveness of the five methods is made both with and without the origin behavior being included exactly. This is useful in itself as no comprehensive accuracy comparison of the standard representations seems to have appeared in the literature. The Ogilvie–Tipping series, corrected at the origin for singular behavior, is the best representation presently available for states analogous to the (1sσg)2Σ+g state of H+2.

Some implications of the virial theorem for molecular force fields

Using two alternative integral forms of the virial theorem, the simple equality ∫ ∞o[T(∞) − T(sQ)] ds = −Vnn (Q) is derived, where T and Vnn represent respectively the kinetic and nuclear—nuclear contributions to the Born—Oppenheimer potential, s is a scale factor, and Q stands for nuclear coordinates. Illustrations for H2 and H+2 are given and consequences for molecular systems having the same nuclear framework are discussed. Use of the above equation as an overall criterion of the fulfilment of the virial theorem within the ab initio and well-behaved model kinetic fields is suggested.

Orthogonal trajectories of the electron density

AbstractFunctions of several variables, such as the electron density and the nuclear Born–Oppenheimer potential, may be analysed very effectively using orthogonal trajectories. In particular the behaviour of these trajectories at critical points of the functions is investigated in detail. The use of orthogonal trajectories in the virial partitioning of Bader and in the definition of reaction coordinates is extended. A chemical interpretation of the critical points is suggested. The relevance of catastrophe theory is illustrated.

Long-range interactions of ions with atoms having partially filled p subshells

We derive analytic Born–Oppenheimer potential energy functions, including the charge–quadrupole and charge–induced dipole contributions, for the long-range interaction of a positive or negative ion with an atom having a partially filled p subshell. The charge–quadrupole term vanishes at large separations for the 2P1/2 states of p1 and p5, the 3P0 state of p2 and p4, and all states of p3. For 2P and 3P states the potentials at very large separations are characterized by MJ and at smaller separations by ML. The switchover is controlled by the ratio of the spin–orbit energy to the total anisotropy of the interaction potential. Plots of the positive and negative ion interactions with C(3P), O(3P), and F(2P) are presented as illustrations.

The computation of nuclear motion and mass polarization adiabatic energy corrections for several states of the hydrogen molecule

This paper reports the computation of the diagonal matrix elements of the nuclear motion and mass polarization operators (often referred to as adiabatic corrections) for several electronic states of the hydrogen molecule. It is observed that reasonably simple wavefunctions will yield accurate expectation values for the nuclear motion operators. The present calculation covers a greater range of R for the X 1Σ+g and B 1Σ+u states than is covered in the very accurate work of Kolos and Wolniewicz, and includes computations for the C 1Πu and D 1Πu states for which no theoretical results have been previously reported. For the latter states the present diagonal corrections have been used in conjunction with the very recent Born–Oppenheimer potential energy curves of Kolos and Rychlewski [J. Mol. Spectrosc. 62, 109 (1976)] to compute vibrational level energies for both H2 and D2. The diagonal corrections largely remove the mass-dependent part of the discrepancy between theory and experiment for these vibrational level energies.

G+K 1Σ+g double-minimum excited state of H2

We have obtained a Born–Oppenheimer potential curve for the previously uncharacterized third 1Σ+g state of H2, using a correlated 20-term wavefunction of generalized James–Coolidge type. We find this potential curve to have a double-minimum character, with the inner (Rydberg-like) and outer (’’ionic’’) wells having minima at about 1.99 and 3.30 bohr, respectively, and an intervening maximum at 2.76 bohr. Unlike the extensively studied E+F double-minimum state, the outer well here appears to be the deeper, by some 450 cm−1 in our calculation. The inner and outer minima can apparently be associated with spectral lines that in experimental tables have previously been attributed to distinct G and K electronic states. The appropriate spectroscopic term symbol of this combined state is accordingly G+K 1Σ+g (1sσ3dσ+2pπ2).

Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory

A method is presented for accurately solving the Schrödinger equation for the reactive collision of an atom with a diatomic molecule in three dimensions on a single Born–Oppenheimer potential energy surface. The Schrödinger equation is first expressed in body-fixed coordinates. The wavefunction is then expanded in a set of vibration–rotation functions, and the resulting coupled equations are integrated in each of the three arrangement channel regions to generate primitive solutions. Next, these are smoothly matched to each other on three matching surfaces which appropriately separate the arrangement channel regions. The resulting matched solutions are linearly combined to generate wavefunctions which satisfy the reactance and scattering matrix boundary conditions, from which the corresponding R and S matrices are obtained. The scattering amplitudes in the helicity representation are easily calculated from the body fixed S matrices, and from these scattering amplitudes several types of differential and integral cross sections are obtained. Simplifications arising from the use of parity symmetry to decouple the coupled-channel equations, the matching procedures and the asymptotic analysis are discussed in detail. Relations between certain important angular momentum operators in body-fixed coordinate systems are derived and the asymptotic solutions to the body-fixed Schrödinger equation are analyzed extensively. Application of this formalism to the three-dimensional H+H2 reaction is considered including the use of arrangement channel permutation symmetry, even–odd rotational decoupling and postantisymmetrization. The range of applicability and limitations of the method are discussed.

Application of the Simons—Parr—Finlan potential to chemisorption

Infrared absorption spectra of chemisorbed diatom species on metal surfaces exhibit features which are sensitive to the nature of the metal surface, to the nature of the diatomic molecule, and to their interaction. To interpret such features, it is necessary to model the force field in order to describe the molecule—metal interaction. As an alternative to the usual Dunham Taylor series expansion of the Born—Oppenheimer potential function we investigate the use of Simons—Parr—Finlan expansion functions. Both types of expansions are compared with a recent theoretical model for molecule—surface interactions. With fewer parameters, the Simons—Parr—Finlan expansions are shown to provide a more realistic approximation to the adsorbed molecule—metal theoretical interaction potential than do the Dunham expansions.

New ab initio Potential Energy Curve and Vibrational Levels for the B1Σu+ State of the Hydrogen Molecule

The Born–Oppenheimer potential energy curve for the B1Σu+ state of the hydrogen molecule has been computed using a wave-function in the form of an 88 term expansion in elliptic coordinates and including the interelectronic distance. At R = Re the computed energy is 5.2 cm−1 lower than the previous most accurate value, in agreement with the prediction by Dabrowski and Herzberg. The new potential energy curve, with the previously computed adiabatic corrections, has been used to calculate the vibrational levels for H2, HD, and D2. The resulting dissociation energies differ from the experimental values by less than 1 cm−1. The discrepancies between the theoretical and experimental energies for various vibrational levels amount up to 12 cm−1 for H2 and 8 cm−1 for D2. Their analysis suggests that most of the discrepancy is due to the nonadiabatic effects, but partly also to incomplete convergence of the Born–Oppenheimer potential energy curve, especially at large internuclear separations.

Non-adiabatic interactions in unimolecular decay. I

A semi-classical treatment of the non-adiabatic among a set of several interacting states is presented. The electrons are described by a wave function obtained by the integration of a set of coupled differential equations. An expression for the force acting on the nuclei is obtained by requiring the total energy to remain constant. This force is found to be equal to the sum of two contributions. The first one is the average of the forces deriving from each Born-Oppenheimer potential. The second one is an interference term which exhibits oscillations in a region of strong non-adiabatic coupling and vanishes outside. An application to a set of fourteen states of a diatomic molecule is presented. The results are sometimes at variance with those predicted by the Landau-Zener equation.

Vibronic interaction in the lower electronic states of benzene

Results of further calculations on the dynamic coupling between the lower B 1u and E 1u electronic states of benzene via a single e 2g mode are presented. In part I we gave the ‘pseudo-cylindrical’ solutions that result when the leading coupling term, linear in the nuclear displacements, is considered (cylindrical Born-Oppenheimer potential). Here we investigate the effect of 1. (1) the next higher terms in the expansion of the coupling hamiltonian for the free molecule (hexagonal Born-Oppenheimer potential); 2. (2) a crystal field anisotropy. From the scanty data on the coupling constants available a priori it would follow that the refinement introduced by (1) is small for benzene. On the contrary, the effect of even a small crystal field anisotropy is predicted to be appreciable. The results of the calculations are analysed by comparison with experimental data on the benzene crystal at low temperature. A consistent interpretation of both magnetic resonance results and the triplet ← singlet absorption sp...

Static Limit of Field Theories. I

The field equations for two nonidentical massive isospinor fields, interacting through the exchange of massive isovector mesons, are solved in the static limit. The solution describes two infinitely heavy isospinor particles, fixed a distance $\mathcal{r}$ apart, exchanging isovector mesons with mass $\ensuremath{\mu}$. The effective Hamiltonian is the Born-Oppenheimer potential. It shows significant departure from the Yukawa potential when $\ensuremath{\mu}\mathcal{r}$ is of the order of the coupling strength or less. The case of spinor particles with gradient coupling to pseudoscalar mesons is also solved.

Point Charge-Point Dipole Model for Vibrating Triatomic Molecules

A point charge-point dipole model for the electronic potential energy contribution to the Born-Oppenheimer potential of triatomic molecules is proposed. The polyatomic virial theorem, with appropriate approximations, is shown to allow one to proceed by implicit rather than explicit inclusion of the kinetic energy. Bending force constants and cubic stretching constants for XY2 molecules are predicted.

Semiclassical Calculation of ElasticH+-He Differential Scattering Cross Sections

Elastic differential cross sections for proton-helium scattering are calculated in the semi-classical approximation for two assumed interaction potentials. Both potentials are of the form $V(r;A,B,C)=(\frac{2}{r}){e}^{\frac{\ensuremath{-}r}{A}}[1+\frac{r}{A}+\frac{1}{2}{r}^{2}(\frac{1}{{A}^{2}}\ensuremath{-}\frac{U}{B})]\ensuremath{-}U{[1+\frac{r}{B}+{(\frac{r}{C})}^{2}+\frac{2U{r}^{4}}{\ensuremath{\alpha}}]}^{\ensuremath{-}1},$ where $U$ is the difference between the ground-state energies of He and ${\mathrm{Li}}^{+}$ (4.373 11 hartree) and $\ensuremath{\alpha}$ is the polarizability of He (1.3835 ${\mathrm{bohr}}^{3}$). The first potential ${V}_{M}\ensuremath{\equiv}V(r;0.423,0.483,0.441)$ fits the He-${\mathrm{H}}^{+}$ ground-state energies which Michels has calculated. The second, ${V}_{W}\ensuremath{\equiv}V(r;0.442,0.505,0.451)$, is similar to ${V}_{M}$ except that its minimum is decreased by 10% to agree with the value obtained by Wolniewicz. The cross sections for these two potentials are shown for protons incident at energies $T$ of 7, 19, 58, and 116 eV in the laboratory frame and for scattering angles, at each energy, out to the rainbow angle ${\ensuremath{\theta}}_{R}$. ${\ensuremath{\theta}}_{R}$ is given in center-of-mass coordinates by the expression ${\ensuremath{\theta}}_{R}T\ensuremath{\cong}0.1$ rad hartree. As the collision energy decreases, the cross sections develop oscillatory structure not present in the classical cross sections. This structure and the rainbow angle are sensitive to the choice of potential, which suggests that measurements of ${\mathrm{H}}^{+}$-He cross sections may be used to test the suitability of, e.g., the Born-Oppenheimer potential for scattering phenomena. It is also suggested that many-body calculations of these cross sections would allow, by comparison with the present results, an evaluation of the potential scattering model.

Calculated Spectrum of Quasibound States for H2(1Σg+) and Resonances in H + H Scattering

The Born—Oppenheimer potential of Koℏos and Wolniewicz for H2(1Σg+) was fitted (±0.7 cm−1) by a polynomial, and the complete spectrum of bound and quasibound states was computed by direct numerical integration of the radial Schrödinger equation. The short-lived virtual states were studied via the energy dependence of the resonant phase shifts, following Buckingham, Fox, and Gal. Of the total of 301 bound states, the 108 levels observed by Herzberg and Howe were reproduced by the computations with an average deviation of ∼3.6 cm−1 (maximum deviation of 5.3 cm−1). Of the two observed quasibound states, the lower was satisfactorily reproduced, but for the upper one, the computed value deviated by 5.3 cm−1; in addition the calculated width was inconsistent with the sharpness of the observed level. Thus it is inferred that a small change in the potential (e.g., a drop of ∼5 cm−1 near R=3 a.u.) is required. Of the shortest-lived quasibound states, 10 have fractional widths Γ/E in the range 0.01–0.1, and are thus possibly suitable for observation as resonances in singlet atom—atom scattering. Collision energies for these range from 0.011 to 0.93 eV.

Theory of Slow Atomic Collisions. I. H2+

A theory of zero-order elastic and first-order inelastic scattering for slow atomic collisions in the system H2+ is presented. The method of stationary states and partial wave analysis is used and is the first such treatment which meets the formal requirement of scattering theory that the zero-order eigenfunctions become correct eigenfunctions at R→∞; the ``perturbed-stationary-states'' method, for example, fails to meet this criterion. The method is applicable only to electronic states which are ``tightly bound'' (≳0.25 eV below the ionization limit); no treatment of scattering to ``continuum'' states of the electron is given. The use of partial wave analysis makes possible the elimination of practical difficulties which appear in the time-dependent treatments due to nonorthogonality of the electronic basis functions. The zero-order eigenfunctions may properly be termed ``adiabatic'' solutions, since they are physically the logical extension of the Born—Oppenheimer concept to unbound states, and for many problems zero-order elastic scattering is indeed just potential scattering by the Born—Oppenheimer potential. First-order inelastic scattering cross sections can be computed from electronic transition matrix elements which differ in significant respects from those of the ``perturbed stationary states'' method.