Energy Consistent Method Research Articles

Analytical potential energy function study for the 41∏ electronic state of Na85Rb molecule

In this paper, a new spectroscopic and analytical potential energy function (APEF) investigation on the electronic state of the Na85Rb molecule is reported. This research is conducted by using the 4-terms variation algebraic energy consistent method (VAECM(4)) with a variable parameter λ. The calculated full vibrational energies of 78 levels are used in a global linear reduction to molecular vibrational constants. With these new and improved vibrational constants, the potential energy curve and the vibrational force constants have been calculated. Precise values for the long-range parameters and the dissociation energy have been derived. The results obtained in this study also provide valuable reference data for other areas of researches on Na85Rb.

Regularized vibrational energies prediction and potential energy function study for some interhalogen diatomic molecules

Full vibrational energies and corresponding analytical potential energy functions (APEFs) for [Formula: see text] of ClF, [Formula: see text] of ICl, [Formula: see text] of IBr, and [Formula: see text]of IF are obtained by four-term variational algebraic energy-consistent method [VAECM(4)]. Three major methods are used to handle the regularization problems and make the results more reliable: (1) physical constraints like progressive dissociation behavior are used to overcome overfitting problems; (2) reasonable variational method is used to expand the experimental dataset and make the model optimization much faster; and (3) validation dataset is used to further enhance the reliability. After these treatments, our results agree well with experiment and have a good physical converged behavior in molecular dissociation and asymptotic region. This study provides important reference data for the related molecules.

Vibrational spectra and analytical potential energy functions of some electronic states of Na2

The improved variational algebraic energy consistent method (VAECM) is suggested to study the vibrational spectra and analytical potential energy functions of six excited electronic states [Formula: see text], 21Δg, (5d)1Δg, (6d)1Δg, (7d)1Δg, and (8d)1Δg of Na2. The full vibrational energies, the vibrational spectroscopic constants, the force constants fn, and the expansion coefficients an of the potential are tabulated. The VAECM analytical potential energy function with adjustable parameter λ for each electronic state is determined. The full vibrational energies of each of these electronic states correctly converge to its dissociation energy and have no artificial barrier in all the calculation ranges. The VAECM analytical potentials excellently agree with the Rydberg–Klein–Rees potentials.

Energy-consistent simulation of frictional contact in rigid multibody systems using implicit surfaces and penalty method

We present a conservative formulation for the frictional contact forces developed in the framework of an energy-consistent method. Bilateral constrains, that is, joints and rigid links, are imposed using the augmented Lagrange technique, provided that the constraints are exactly satisfied. We propose a formulation based on a penalty method to deal with the contact problem including damping and friction. The numerical scheme guarantees that the energy is consistently preserved or unconditionally dissipated. The treatment of the contact constraint also constitutes a novel aspect of this investigation. It is based on the description of implicit surfaces given an efficient strategy for detecting and evaluating the contact. The numerical tests exhibit a good energy preservation performance. Moreover, we use the proposed formulation to analyze a mechanism including a revolute joint with frictional clearance.

Studies on the Analytical Potential Energies for Partial Electronic States of Li(2) with Variational Algebraic Energy Consistent Method

The full vibrational spectra especially those high-lying vibrational energies in the dissociation region of four specific electronic states 1(3)Δ(g), 33Σ(+)(g), 1(3)Σ-(g) and b(3)Π(u) have been obtained by using the improved variational algebraic method (VAM). The analytical potential energy functions (APEFs) of these electronic states are also determined with corresponding adjustable parameter λ by using the variational algebraic energy consistent method (VAECM) based on the VAM vibrational spectra. The full vibrational energies, vibrational spectroscopic constants, force constants f(n), and expansion coefficients a(n) of the VAECM potential are also tabulated for each electronic state in this study. The results show that the VAECM analytical potentials are superior to some other widely used analytical ones, and do not have the unphysical tiny barriers existing in the precious AECM potentials.

High-lying vibrational energies and analytical potential energy functions for some electronic states of diatomic ions

The full vibrational spectra especially those high-lying vibrational energies in the dissociation region of the electronic state [Formula: see text] of [Formula: see text], the [Formula: see text] state of [Formula: see text], the [Formula: see text] and [Formula: see text] states of [Formula: see text] are obtained using the variational algebraic method (VAM). Then, an analytical potential energy function (APEF) with adjustable parameter [Formula: see text] for each electronic state is determined by the 4-terms variational algebraic energy consistent method (VAECM(4)) based on the VAM vibrational energies. The full vibrational energies, the vibrational spectroscopic constants, the force constants [Formula: see text], and the expansion coefficients [Formula: see text] of the potential are tabulated. Compared with experimental and other calculated results, accurate APEFs, vibrational energy levels and spectroscopic parameters are obtained with the VAECM(4) for the four electronic states of diatomic ions. The results show that the VAECM(4) method also applies to diatomic ion systems.

Analytical potential energy functions for some interhalogen diatomic electronic states

The studies of vibrational energies and analytical potential energy functions (APEFs) have been carried out for four interhalogen diatomic electronic states B(3 Π 0 +) and A(3 Π 1) of ClF, A′(3 Π 2u ) of Cl2, and the ground state X 1 Σ + of Br2 by using an improved variational algebraic energy-consistent method (VAECM(4)). The full vibrational energies, the vibrational spectroscopic constants, the force constants f n , and the expansion coefficients a n of the ECM (energy-consistent method) potential are tabulated. The VAECM(4) APEF with adjustable variational parameter λ for each electronic state is determined, and is shown to be in excellent agreement with available experimental data and has no any artificial barrier in all the calculation ranges that may appear in some other analytical potentials.

Analytical potential energy functions for the ground and some excited states of N2

The analytical potential energy functions have been calculated for the ground state X1Σ+g and four excited electronic states a1Πg, A3Σ+u, B′3Σ−u and B3Πg of N2 molecule using the algebraic and energy-consistent methods (AM-ECM). Based on our previously published full AM vibrational energies and spectroscopic constants, the low-lying force constants fn, the expansion coefficients an and the variational parameters λ in the AM–ECM potentials are determined for these states. The computed AM–ECM potential energy curve of each state is in excellent agreement with the experimental data and better than other analytical potentials.

Analytical potential curves of some hydride molecules using algebraic and energy-consistent method

Based on the algebraic method (AM) and the energy consistent method (ECM), an AM-ECM protocol for analytical potential energy curves of stable diatomic electronic states is proposed as functions of the internuclear distance. Applications of the AM-ECM to the 6 hydride electronic states of HF-X 1 Σ +, DF-X 1 Σ +, D35Cl-X 1 Σ +, 6LiH-X 1 Σ +, 7LiH-X 1 Σ +, and 7LiD-X 1 Σ + show that the AM-ECM potentials are in excellent agreement with the experimental RKR data and the full AM-RKR data, and that the AM-ECM can obtain reliable analytical potential energies in the molecular asymptotic and dissociation region for these molecular electronic states.

Studies on the analytical curves of diatomic molecules using algebraic and energy-consistent method

The analytical potential energy curves of stable diatomic electronic states are investigated based on the algebraic method (AM) and energy consistent method (ECM). The AM-ECM method is applied to 4 electronic states of 7Li2-23∑g+, KH-X1∑+, NaLi-X1∑+ and NaLi-A1∑+. The results show that the AM-ECM potentials are superior to some other analytical potentials, and can generate accurate analytical potential expressions of these electronic states in the molecular asymptotic and dissociation region.

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.

Studies on the analytical potential energy function of diatomic molecular ion XY+ using variational method

A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole corrections, converges quickly and is variationally, changeable. The new potential and the ECM are applied to variationally studying the potential energies of eight electronic states of several diatomic molecular ions: the A2π state of CO+, the X2∑g+ state of Li2+, the X2∑g+ state of He2+, the 12∏u state of Na2+, the A2∏u state of N2+, the X1∑+ state of KrH+, the X2∑+ state of SiO+ and the A2π state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanicalab initio potentials in the ionic asymptotic and dissociation regions.

Studies on the potential energy curves of hydride diatomic molecules using energy consistent method

By using QCISD/6-311G（d）method, the ground state of SO-2, and its energy, harmonic frequencies, force constants have been calculated. The calculated results are in good agreement with the experimental ones. The analytical potential energy function of SO-2 has been derived based on the many-body expansion theory. The structure and energy of SO-2 can correctly reappear on the potential surface.

Studies on the potential energy curves of heteronuclear diatomic molecules using energy consistent method

The potential energy curves of some heteronuclear diatomic molecules are studies using the energy consistent method (ECM) for the electronic states of Clf-A３Π1, ClF-B0＋(３Π)，CH-X２Π，BH-X1Σ＋，XeO-d1Σ＋，LaF-X1Σ＋，Li7D-X1Σ＋，NaRb-X1Σ＋，KRb-(2)3Σ＋ and KRb-21Π. The results show that the ECM potentials are in excellent agreement with the known Rydberg-Klein-Ress(RKR) data, and are superior to the frequently used Morse potential and Huxley-Murrell-Sorbie(HMS) potential. The EGM potential can give reliable potential data at the molecular asymptote and the dissociation region where the RKR results may not be available.

Solution of the Energy Level of Hydrogen-Like Atom for the Debye Shielding Potential**The project supported by Natural Science Foundation of Chongqing Education Committee (Grant No. 20011075) and the Basic Research of Application of Chongqing Science and Technology Committee (Grant No. 199966)

The first-order revision and the approximation analytical formula of the energy levels for hydrogen-like atoms under the condition of Debye shielding potential are achieved by means of the Rayleigh–Schrödinger perturbation theory; meanwhile, the corresponding recurrence relations are obtained from the use of the solution of power series. Based on the above solutions and with the use of energy consistent method the equivalent value of second-order reversion under the condition of Debye shielding potential is produced as well and the result is compared with the data obtained by the numerical method. Besides, the critical bond-state and corresponding cut-off conditions are discussed.

Potential Energy Curves of Some Electronic Excited States of Metal Diatomic Molecules Using the Energy Consistent Method

An analytical five-term expansion potential is suggested to improve the recently proposed energy consistent method (ECM) potential based on a three-term expansion [Weiguo Sun, Mol. Phys. 92, 105 (1997) and Weiguo Sun and Hao Feng, J. Phys. B: At., Mol. Opt. Phys. 32, 5109 (1999)]. Applications of the new ECM to the electronic excited state 31Π of NaK, the excited state 23Σ+g of Na2, and the excited states 31Σ+u and 31Πu of K2 molecules show that the new ECM potential not only agrees excellently with the known Rydberg–Klein–Rees (RKR) data or accurate configuration interaction (CI) studies but also gives the potential data at the molecular asymptote and the dissociation region where experimental and quantum mechanical studies may have difficulties. The new ECM can correct the artifact which appeared in the three-term expansion potentials of both the previous ECM and the Huxley–Murrell–Sorbie (HMS) potential and can describe some states to which both the previous ECM and the HMS fail.

STUDY ON THE POTENTIAL ENERGY CURVES OF ALKALI DIATOMIC MOLECULES WITH ENERGY CO NSISTENT METHOD

The studies on the potential energy curves of the electronic excited states 21Πg, 43Πg, b3Πu of Na2, the states a3Σ+u, 21 Πg, B1Πu, A1Σ+u of K2 and the state 11Πg of Cs 2 molecules by using the energy consistent method (ECM) are reported here . The results show that the present ECM potentials agree very well with the know n Rydberg-Klein-Rees data or the inverted perturbation approach data, and that they are much better than other analytical potentials such as the Morse and the Huxley-Murrell-Sorbie potentials for the electronic excited states of alkali dia tomic molecules.

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.

The energy-consistent method for the potential energy curves and the vibrational eigenfunctions of stable diatomic states

The energy-consistent method for calculating accurate potential energy curves of diatomic states is proposed based on a modified Murrell–Sorbie (MS) potential. The accurate vibrational eigenvalues and eigenfunctions for the physical potentials of stable diatomic molecules are obtained using this method. The artefact of the MS potential shown in the Huxley–Murrell formulation disappears in the present method.

The energy-consistent method for the potential energy curves and the vibrational eigenfunctions of stable diatomic states

The energy-consistent method for calculating accurate potential energy curves of diatomic states is proposed based on a modified Murrell–Sorbie (MS) potential. The accurate vibrational eigenvalues and eigenfunctions for the physical potentials of stable diatomic molecules are obtained using this method. The artefact of the MS potential shown in the Huxley–Murrell formulation disappears in the present method.