Energy Curves Of Diatomic Molecules Research Articles

Extrapolation properties of the Morse-Long Range potential at large internuclear distances

The accuracy of the potential energy curves for diatomic molecules determined from experimental data is confirmed empirically with numerous examples in the literature. Usually PECs are determined from a limited set of experimental data and this in turns limits the range of internuclear distances where the shape of the potential is unambiguously fixed. While the uncertainty for interpolation could be assessed, the extrapolation is usually questionable and needs careful analyses. The Morse/Long-Range potential (Le Roy and Henderson, 2007) has been reported to have a built in long-range asymptotic behavior and therefore it is plausible to expect that one can expect good extrapolation properties and even possibility to determine important molecular parameters like De or/and Cm from limited set of experimental data. In this contribution we undertake a systematic study which confirms these expectations in the case of Ca2 ground state.

Towards the analytic theory of potential energy curves for diatomic molecules: studying He2 + and LiH diatomics as illustration

Following the first principles, the elements of the analytic theory of potential curves for diatomic molecules (diatomics) are presented. It is based on matching the perturbation theory at small internuclear distances R and multipole expansion at large distances, modified by exponentially small terms for homonuclear case, with the addition of the phenomenologically described equilibrium configuration, if it exists. It leads to a new class of (generalised) rational potentials (modified by exponentially small terms) with a difference in degrees of polynomials in numerator and denominator equal to 4 (6) for positively charged (neutral) diatomics. For example, the He and LiH diatomics in Born–Oppenheimer approximation are considered. For He (He) diatomics it is found the approximate analytic expression for the potential energy curves (analytic PEC) for the ground state and the first excited state . It provides 3–4 s.d. correctly for distances a.u. with some irregularities for PEC at small distances (much are smaller than equilibrium distances) probably related to level (quasi)-crossings that may occur there. The analytic PEC for the ground state supports 829 (626) rovibrational states with 3–4 s.d. of accuracy in energy, which is only by 1 state less (more) than 830 (625) reported in the literature. The analytic PEC for the excited state supports all 9 reported weakly-bound rovibrational states. For LiH it is found the analytic expression for the ground state PEC in the form of a rational function, which supports 906 rovibrational states with 3–4 s.d. accuracy in energy, it is only of 5 states more than reported in the literature (901). For both diatomics, the difference in a number of rovibrational states is related to the non-existence/existence of weakly-bound states close to the threshold (to dissociation limit). Entire rovibrational spectra are found in a single calculation using the code based on the Lagrange mesh method.

Multivalued property of Rayleigh–Schrödinger perturbation series for vibrational energy levels of molecules

We have calculated 200 lower vibrational energy levels of the formaldehyde molecule using ab initio quartic force field, high-order Rayleigh–Schrödinger perturbation theory (up to 200th order) and algebraic Hermit–Padé approximants (up to sixth degree) to sum up divergent perturbation series, corresponding to each vibrational state. Both energy levels and algebraic Hermit–Padé approximants are, in fact, multivalued functions and consist of several branches. We find that branches of the approximant constructed using coefficients of the perturbation series for a particular vibrational state can reproduce not only the energy of this state, but also of other states which are close in energy to the considered one and are coupled with them by anharmonic resonances. The multivalued property of algebraic approximants has been previously demonstrated by Jordan in the case of two-state avoided crossings of the potential energy curves of diatomic molecules (Jordan 1975 Int. J. Quantum Chem. 9 325) and by Fernandez and Diaz in the case of one-dimensional periodic eigenvalue problems (Fernandéz and Diaz 2001 Eur. Phys. J. D 15 41). This paper extends the study of this nontrivial property of perturbation theory to the vibrational states of polyatomic molecules, a problem with many dimensions. Additionally, we have found that this property is useful in some special cases when perturbation series diverges so fast that most high-order approximants do not reproduce the correct value of the energy with the required accuracy.

Genetic based fitting techniques for high precision potential energy curves of diatomic molecules

We present development of a genetic algorithm for fitting potential energy curves of diatomic molecules to experimental data. Our approach does not involve any functional form for fitting, which makes it a general fitting procedure. In particular, it takes in a ‘trial’ potential, along with experimental measurements of vibrational binding energies, rotational constants, and their experimental uncertainties. The fitting procedure is able to converge to better than 1% uncertainty, as measured by or reproduce the experimental data to better than 0.03 cm−1. We present the details of this technique for the of lithium–rubidium.

Correlations of Rydberg orbitals of diatomic molecules

SynopsisCalculations of the potential energy curves of diatomic molecules are performed for the entire range of internuclear separation in order to correlate Rydberg orbitals at the united atom and separated atoms which are needed to properly assign highly excited Rydberg states.

MR-ccCA: A route for accurate ground and excited state potential energy curves and spectroscopic properties for third-row diatomic molecules

The potential energy curves for diatomic molecules containing third-row elements were examined using the multi-reference correlation consistent composite approach (MR-ccCA) to determine the applicability of the new approach for these systems. Remarkably smooth potential energy curves resulted, though the methodology was not designed, per se, for potential energy curves, particularly due to the additive nature of the methodology. A number of properties have been considered including the dissociation energy Do, well depth De, equilibrium bond length re, vibrational constants (ωe, ωexe) and rotational constants (Be, αe) and these properties have been compared with the experimental values for the ground state diatomics. Additionally, potential energy curves and spectroscopic properties of several of the lowest excited states of As2 have been determined, with the addition of transition energies Te and the determinations have been compared to experimental and theoretical values.

Excitation energies from range-separated time-dependent density and density matrix functional theory

Time-dependent density functional theory (TD-DFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the time-dependent theory based on a one-electron reduced density matrix functional (time-dependent density matrix functional theory, TD-DMFT) has proven accurate in determining single and double excitations of H(2) molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited state energies that relies on functionals of electron density and one-electron reduced density matrix, where the latter is applied in the long-range region of electron-electron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a time-dependent functional theory based on the range-separation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short- and long-range components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the short-range local density approximation (srLDA) and the long-range Buijse-Baerends density matrix functional (lrBB) is applied to H(2) molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TD-DFT-LDA or TD-DMFT-BB approximations if the range-separation parameter is properly chosen. The latter remains an open problem.

Fitting potential energy curves for diatomic molecules using experimental line intensities

The existing routines for construction of experimental potential energy curves for diatomic molecules are based mainly on transition frequencies while the line intensities are seldom used. Some of the reasons for this are experimental: the line intensities are usually measured with much higher uncertainty. Other are numerical and the main problem is to fix the sign of the overlap integrals since the line intensity depends on their square. In this contribution a simple approach is proposed for solving this problem.

Assessment of simple exchange-correlation energy functionals of the one-particle density matrix

An improved density matrix functional (DMF) combining the properties of the “corrected Hartree” (CH) and “corrected Hartree–Fock” (CHF) approximations is proposed. Functionals of the CH/CHF type and the closely related natural orbital functional of Goedecker and Umrigar (GU) are tested in fully variational finite basis set calculations of light atoms, the lowest energy singlet methylene, and, for the first time, potential energy curves of diatomic molecules. Although CH/CHF-style DMFs may give reasonable energies for atoms and molecules near equilibrium geometries, they predict unrealistically shallow minima in the potential energy curves for diatomic molecules with more than two electrons. The calculated CH and CHF molecular dissociation curves exhibit the same patterns of over- and under-correlations as the corresponding correlation energy plots for the homogeneous electron gas undergoing a transition from high to low densities. In contrast, the GU functional yields not only accurate atomic and molecular energies but also plausible dissociation curves. The reasons behind the observed performance are analyzed.

The estimation of potential energy curves for diatomic molecules

The Fourier grid method is used to deduce potential energy curves from spectroscopic data. The results are compared with those of ab initio calculations for H2 and with RKR method data for Na2.

An energy-consistent method for potential energy curves of diatomic molecules

A physically well behaved analytical potential of a diatomic system is constructed based on perturbation theory and a potential correction to the Murrell-Sorbie potential. Alternative formulae for rotational constants and ro-vibrational energies are suggested. An energy-consistent method is proposed for calculating accurate potential energy curves of diatomic states. Applications of this method to the electronic states X 1g+, C 1u, a 3g+, e 3u+ of H2, the ground state X 1g+ of N2, and the states X 3g-, C 1u-, B 3u- of the O2 molecule show that the present potential agrees excellently with the known Ryderberg-Klein-Rees data or accurate configuration-interaction (CI) studies, and is better than other analytical and some CI potentials, particularly for molecular electronic excited states.

Reduced potential energy curves for diatomic molecules and their respective cations

It is shown that the reduced potential energy curve for a diatomic molecule and its cation are nearly identical, particularly near the minimum, implying that the force constant in reduced variables is nearly the same for the two species. An explanation is given in terms of scaling of the valence-electron wavefunction with respect to the nuclear coordinates.

The computation of RKR potential energy curves of diatomic molecules using mathematica

A procedure is given for computing classical turning points in the potential energy curves of diatomic molecules according to the well known RKR method using the software package Mathematica.

FORTRAN routine to compute Born–Oppenheimer potential energy curves directly from spectroscopic data

AbstractA computer program that calculates Born–Oppenheimer potential energy curves of diatomic molecules directly from spectroscopic data is presented. Rather than performing the usual preliminary fitting of the experimental data to a single polynomial in the variable (v + 1/2), where v is the vibrational quantum number, the program contains an accurate built‐in interpolation routine by which each experimental input point belongs to its own polynomial. The program is tested on the ground state of the H2 molecule and results are compared with the most accurate ab initio calculations available. © 1993 John Wiley & Sons, Inc.

Construction of bound-state potential energy curves of diatomic molecules from vibration-rotational spectroscopic data

By means of a purely quantum mechanical method we construct accurate analytic adiabatic potential energy curves for diatomic molecules from vibration-rotational transition frequencies. Our potential energy functions for two sample molecules, CO and HCl, are in better agreement with available Rydberg-Klein-Rees (RKR) turning points and reproduce the experimental data more accurately than corresponding curves obtained by other authors. The perturbation series for the calculated line positions in terms of the small parameter 2B e/ω e, where B e, and ω e are the rotational and vibrational spectroscopic constants, respectively, exhibits better convergence properties when applied to our potential energy functions.

A numerical procedure to obtain accurate potential energy curves for diatomic molecules

The potential energy curves for the X 1Σ + state of 12C 16O and 6Li 1H were obtained by fitting the Rydberg-Klein-Rees potential in the Chebyshev sense (minimum-maximum approximation) to a simple functional form very similar to a perturbed-Morse-oscillator potential but with a minor number of parameters. Using Hermite orthogonal functions (eigenfunctions of the harmonic oscillator) as the basis set we solved variationally the radial Schrödinger equation to obtain the vibrational energies E v, and the rotational constants B v. Using the potential thus determined and the proposed basis set, the matrix elements of the Hamiltonian may be calculated analytically, presenting only round-off errors. The agreement between the calculated and experimentally determined E v, and B v, values is good, the self-consistency of our potentials being very similar to that obtained by other authors.

Construction of bound-state potential energy curves of diatomic molecules from spectroscopic data

We propose a method for the construction of potential energy curves of diatomic molecules from spectroscopic data. It is based on perturbation theory and an appropriate numerical integration algorithm and corrects the potential energy function iteratively until the eigenvalues of the Schrödinger equation agree with the experimental energies. The method is stable and converges fast enough to allow one to adjust many potential parameters. We choose the eigenvalues of the Kratzer oscillator as experimental data to test both the performance of the method and the accuracy of the Dunham expansion about equilibrium. The Dunham expansion obtained from the spectrum has a range of utility which is remarkably larger than the radius of convergence of the Taylor expansion about equilibrium. We also apply the present method to the ground electronic state of the CO molecule and show that the Dunham expansion agrees with the RKR turning points. We discuss further applications of the method.

Construction of molecular potential energy curves by an optimization method

A technique for determining the potential energy curves for diatomic molecules from measurements of diffused or continuum spectra is presented. It is based on a numerical procedure which minimizes the difference between the calculated spectra and the experimental measurements and can be used in cases where other techniques, such as the conventional RKR method, are not applicable. With the aid of suitable spectral data, the associated dipole electronic transition moments can be simultaneously obtained. The method is illustrated by modeling the “longest band” of molecular oxygen to extract the E 3Σ u − and B 3Σ u − potential curves in analytical form.

Potential energy curves of diatomic molecules from the Hill determinant method

The Hill determinant method is shown to be suitable for constructing potential energy curves of diatomic molecules. Both the Dunham and the perturbed Morse oscillator potentials are used to fit spectroscopic data. Results are shown for ionic and covalent molecules.

Further contributions to the energy levels of a perturbed anharmonic oscillator: Application to adiabatic corrections

The investigation of further contributions to the energy levels of a perturbed anharmonic oscillator is continued. It is shown that, by using a Taylor series for several variables, the contributions to the energy levels due to an additional perturbing interaction can be directly derived from the Yun Dunham-type coefficients by means of analytic relations involving products of differential operators. Consequently, the analytic expressions for the whole set of these contributions are generated from the Yu0 coefficients alone and are easily calculated by using a computer algebraic manipulation language. As an illustrative application, the case of the adiabatic corrections for nuclear motion to the potential energy curves of diatomic molecules in 1Σ+ states, such as lithium hydrides and hydrogen chlorides, is considered.