Full Vibrational Energies Research Articles

Reduced-dimensional vibrational models of the water dimer.

A model based on the finite-basis representation of a vibrational Hamiltonian expressed in internal coordinates is developed. The model relies on a many-mode, low-order expansion of both the kinetic energy operator and the potential energy surface (PES). Polyad truncations and energy ceilings are used to control the size of the vibrational basis to facilitate accurate computations of the OH stretch and HOH bend intramolecular transitions of the water dimer (H2 16O)2. Advantages and potential pitfalls of the applied approximations are highlighted. The importance of choices related to the treatment of the kinetic energy operator in reduced-dimensional calculations and the accuracy of different water dimer PESs are discussed. A range of different reduced-dimensional computations are performed to investigate the wavenumber shifts in the intramolecular transitions caused by the coupling between the intra- and intermolecular modes. With the use of symmetry, full 12-dimensional vibrational energy levels of the water dimer are calculated, predicting accurately the experimentally observed intramolecular fundamentals. It is found that one can also predict accurate intramolecular transition wavenumbers for the water dimer by combining a set of computationally inexpensive reduced-dimensional calculations, thereby guiding future effective-Hamiltonian treatments.

A method for predicting the molar heat capacities of HBr and HCl gases based on the full set of molecular rovibrational energies

A new method is presented for one to obtain the molar heat capacities of diatomic macroscopic gas with a full set of microscopic molecular rovibrational energies. Based on an accurate experimental vibrational energies subset of a diatomic electronic ground state, the full vibrational energies can be obtained by using the variational algebraic method (VAM), the potential energy curves (PECs) will be constructed by the Rydberg-Klein-Rees (RKR) method, the full set of rovibrational energies will be calculated by the LEVEL program, and then the partition functions and the molar heat capacities of macroscopic gas can be calculated with the help of the quantum statistical ensemble theory. Applying the method to the ground state HBr and HCl gases, it is found that the relative errors of the partition functions calculated in the temperature range of 300 ∼ 6000 K are in excellent agreement with those obtained from TIPS database, and the calculated molar heat capacities are closer to the experimental values than those calculated by other methods without considering the energy levels of highly excited quantum states. The present method provides an effective new way for one to obtain the full set of molecular rovibrational energies and the molar heat capacities of macroscopic gas through the microscopic spectral information of a diatomic system.

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.

Theoretical study on thermodynamic properties of NO gas

<sec>Nitric oxide (NO) is one of atmospheric molecules of interest and has attracted considerable attention due to its important role in the chemical process taking place in a flow field of hypersonic vehicle, in which the thermodynamic properties are required in the calculation of the aerothermodynamic flow field. Moreover, the total internal partition function is the key to calculating the thermodynamic properties of high-temperature gases. For diatomic molecules, according to the product approximation, the total internal partition function is split into three parts: electronic, vibration and rotation partition function. In this paper, by using the quantum statistical ensemble theory based on some classical thermodynamic and statistical formulae, the thermodynamic properties of NO are analyzed and discussed.</sec><sec>Firstly, in order to obtain an accurate energy of molecule, the variational algebraic method (VAM) is employed to calculate the full vibrational energy, the resultis in good agreement with the experimental result and thus yielding the realistic predictions of the unobserved higher vibrational energy that converges to the dissociation limit. Secondly, an attempt is to use the full VAM vibrational energy, the Rydberg-Klein-Rees (RKR) vibrational energy, the simple Harmonic oscillator (SHO) model and the quantum-mechanical vibrational energy obtained by the multiconfiguration self-consistent-field (MCSCF) to calculate the vibrational partition function. Then, with the rotational contributions from the Müller-McDowell formula, the internal partition function can be determined by combining the product of electronic, vibration and rotation partition functions. Thirdly, according to the thermodynamic and statistical formulae, it is easy to calculate the internal energy, entropy and heat capacity for the NO molecule in a range of 1000－5000 K. Comparison of different calculated heat capacities with the experimental ones reveals the heat capacity, of which vibrational contributions determined by the full VAM vibrational energy accord better with the experimental ones, with the maximum relative error being no more than 2.4%, whereas it can be seen that those thermodynamic results evaluated from the SHO model attest to a failure for the summation of infinite vibrational energy. The thermodynamic results of NO may have proper applications in areas that can be of great importance in theoretical and (or) experimental aspects.</sec>

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.

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.

A variational algebraic method used to study the full vibrational spectra and dissociation energies of some specific diatomic systems

The algebraic method (AM) proposed by Sun et al. is improved to be a variational AM (VAM) to offset the possible experimental errors and to adapt to the individual energy expansion nature of different molecular systems. The VAM is used to study the full vibrational spectra {Eυ} and the dissociation energies De of 4HeH+–X1Σ+, 7Li2–13Δg,Na2–C1Πu,NaK–71Π, Cs2–B1Πu and 79Br2–β1g(3P2) diatomic electronic states. The results not only precisely reproduce all known experimental vibrational energies, but also predict correct dissociation energies and all unknown high-lying levels that may not be given by the original AM or other numerical methods or experimental methods. The analyses and the skill suggested here might be useful for other numerical simulations and theoretical fittings using known data that may carry inevitable errors.

Studies on the full vibrational energies and dissociation energies of some halogen diatomic molecules

Full vibrational spectra and dissociation energies are studied for the 0u−, B″Iu states of I2, X of Br2 and of Cl2. The full vibrational spectra were computed using a standard algebraic method (AM) with available experimental vibrational energies as input parameters. The dissociation energies were estimated using an analytical formula and the three highest AM vibrational energies. Prior to the present work, reliable high-lying vibrational energies and dissociation energies of these states were difficult to obtain experimentally and theoretically.

Studies on the full vibrational energies and dissociation limits of some electronic states of Kr2+ and Xe2+ diatomic molecular ions

In addition to the algebraic method (AM) (Sun et al 2002 J. Mol. Spectrosc.215 93) that was proposed to generate accurate full vibrational energies for a stable diatomic electronic state that has regular (Morse-like) potentials, a parameter-free formula is suggested in this work in order to study the high-lying vibrational spectra and dissociation limits of four electronic states of Kr2+ and Xe2+ diatomic ions. Those reliable AM energies EυAM of high-lying vibrational states and dissociation limits DLAM for each electronic state that we have predicted in this study were probably difficult to obtain experimentally or theoretically before.

Accurate studies on the full vibrational energy spectra and molecular dissociation energies for some electronic states of halogen molecule

This paper obtains accurate vibrational spectroscopic constants and full vibrational energy spectrum by the algebraic method (AM) for some electronic states of halogen diatomic molecules. Motivated by the recent success of obtaining the dissociation energies of Li2 molecule by using a new analytical formula, it further extends the formula to study the dissociation energies of halogen diatomic molecules. The results show that the AM spectrum and the theoretical dissociation energies agree well with RKR data and experimental data respectively.

Full vibrational energy spectra and dissociation energies for some electronic states of diatomic alkali-metal molecules

It is difficult to obtain the accurate high-lying vibrational energies for most of the diatomic electronic states on modern experiments or theoretical computations based on quantum mechanics. Based on the new analytical formula for dissociation energy and algebraic method (AM) generated by Sun et al., the second order perturbation theory are used to study the full vibritional energies{EAMυ} and dissociation energies of the Li2－33Σ+g,Li2－13Δg,Li2－23Πg,Na2－B1Πu and K2－41Σ+g electronic states. The obtained results not only agree well with the experimental data for the low-lying vibrational energies, but also give all high-lying vibrational energies which are still difficult to obtain by experiment at present. These results supply necessary data for the studies which need high-lying vibrational energies and dissociation energies of diatomic alkali-metal molecule.

Studies on the full vibrational spectra and dissociation energies of some diatomic ions

A parameter-free analytical formula for dissociation energy of diatomic system is suggested recently by Sun et al. Together with the algebraic method (AM) which was presented to obtain accurate full vibrational energies for diatomic molecules in their previous work，the new formula is applied to study the dissociation energies and the full vibrational spectra of some diatomic ions in this work. The results show that the AM vibrational spectra and dissociation energies have excellent agreement with experimental values.

The full vibrational spectra and dissociation energies of Li2 molecule

Based on the algebraic method (AM) proposed to calculate accurate full vibrational energies for diatomic molecules in Sun’s previous work, the parameter-free analytical formula for dissociation energy of diatomic system is used to study the dissociation energies and the full vibrational spectra of 5 electronic states of Li2 molecule in this work. The results show that the AM vibrational spectra and dissociation energies are in excellent agreement with experimental results.

Studies on the full vibrational energies and dissociation energies of some heteronuclear diatomic molecules

A parameter-free analytical formula for dissociation energies of diatomic molecules is proposed by Fan and Sun (2009) [20] based on LeRoy and Bernstein's vibrational energy expression near dissociation limit. Using three highest vibrational energies which may be generated by the algebraic method (AM) presented in our previous study and by some other physical methods, the new formula is applied to study the molecular dissociation energies of 10 electronic states of KH, 7LiD, 7LiH, 6LiH, NaK, NaLI and NaRb heteronuclear diatomic molecules which have regular (Morse-like) potentials in this work. The results show that the AM energies E υ A M and dissociation energies D e A M have excellent agreement with experimental values.

Full Vibrational Energy Spectra and Dissociation Energies for Some Electronic States of Diatomic Halogen Molecules

It is difficult to obtain the accurate high-lying vibrational energies for most of the diatomic electronic states on modern experiments or theoretical computations based on quantum mechanics. Based on the new analytical formula for dissociation energy and algebraic method (AM) generated by Sun et al., the second order perturbation theory are used to study the full vibritional energies{E AM υ } and dissociation energies of the Li 2 －3 3 Σ + g ,Li 2 －1 3 Δ g ,Li 2 －2 3 Π g ,Na 2 －B 1 Π u and K 2 －4 1 Σ + g electronic states. The obtained results not only agree well with the experimental data for the low-lying vibrational energies, but also give all high-lying vibrational energies which are still difficult to obtain by experiment at present. These results supply necessary data for the studies which need high-lying vibrational energies and dissociation energies of diatomic alkali-metal molecule.

Accurate vibrational energy spectra and dissociation energies of some diatomic electronic states

An algebraic method (AM) used to study the full vibrational spectra of diatomic systems, and an analytical formula used to calculate accurate molecular dissociation energies are applied to study the full vibrational spectra and molecular dissociation energies of some electronic states of homonuclear and heteronuclear diatomic molecules and diatomic ions. Studies show that the AM method and the analytical expression are reliable and economical physical methods for studying full vibrational spectra and molecular dissociation energies of diatomic electronic systems theoretically. They are particularly useful for those diatomic systems whose high-lying vibrational energies may not be available.