By rewriting the formulas for 3-j and 6-j symbols in terms of several possible alternating binomial sums, it is possible to calculate these quantities quickly and accurately, often exactly, using floating point operations. The binomial sums can be calculated by direct summation or by recursion. A simple method for uniquely parameterizing the well-known Regge symmetries of the 3-j and 6-j symbols makes it possible to systematize the choice of the smallest magnitude binomial sum (which enhances the accuracy of floating point calculations and speeds up exact calculations using large integer routines). Formulas for special cases of the 3-j symbols enable the construction of recursion sequences which are often substantially faster than direct summation, especially for very large angular momentum arguments. For both 3-j and 6-j symbols, recursion offers several advantages over direct summation in exact calculations and for calculating tables.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTOsmotic Pressure and the Effects of Gravity on SolutionsDon Secrest View Author Information University of Illinois, Urbana, IL 61801Cite this: J. Chem. Educ. 1996, 73, 10, 998Publication Date (Web):October 1, 1996Publication History Received3 August 2009Published online1 October 1996Published inissue 1 October 1996https://pubs.acs.org/doi/10.1021/ed073p998https://doi.org/10.1021/ed073p998research-articleACS PublicationsRequest reuse permissionsArticle Views343Altmetric-Citations4LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose SUBJECTS:Osmotic pressure,Seawater,Solution chemistry,Thermodynamics Get e-Alerts

The recoupled states approximation is a distorted-wave approximation and as such leads to a nonunitary S matrix. Several unitarization approaches are known for systems for which the S matrix of the distorted wave is diagonal. In particular, the exponential distorted-wave method has been extremely successful. We develop in this paper a general unitarization technique, similar to the exponential distorted-wave method, useful when the S matrix of the distorted wave is nondiagonal. This approach is applied to the case of the recoupled states for which the distorted waves are the coupled states wave functions. For this particular example, unitarization does not improve the results.

The angular momentum decoupling approximations, used extensively in approximate calculations for complex systems, are evaluated here in extensive comparisons with exact calculations for a heteronuclear system with a moderately large dipole moment at high enough energy that there are many open rotational states. Orientation-dependent differential cross sections for He–SiO rigid-rotor scattering calculations at high translational energy are presented. Coupled states and recoupled states results are compared with those of close coupling at a total energy of 27 meV. Although the coupled states (CS) method approximates exact calculations by ignoring m coupling in the body frame, the l-averaged CS results are good for orientation-dependent differential cross sections. The l-averaged form of recoupled states makes significant improvements over the l-averaged form of coupled states for orientationally elastic transitions and for large-j transitions. As in past studies, the l-initial and l-final forms of coupled and recoupled states are found to be inferior to the l-averaged form. Orientation-dependent integral and degeneracy-averaged differential and integral cross sections are also discussed.

A subroutine that calculates Kratzer oscillator matrix elements of products of exponentials and integer powers of position is presented. The closed form expression for any such matrix element includes a terminating F2 hypergeometric function. Stable recursion relations used to evaluate a quantity related to F2 are used while calculating matrix elements.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTA computational study of Kratzer oscillator basis sets [Erratum to document cited in CA114(8):69191c]Don SecrestCite this: J. Phys. Chem. 1991, 95, 17, 6746Publication Date (Print):August 1, 1991Publication History Published online1 May 2002Published inissue 1 August 1991https://pubs.acs.org/doi/10.1021/j100170a067https://doi.org/10.1021/j100170a067research-articleACS PublicationsRequest reuse permissionsArticle Views14Altmetric-Citations-LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access options Get e-Alerts

It is shown that by reexpanding the interaction potential for a molecule as a product of functions of the internal coordinates, the Hamiltonian matrix may be rapidly and efficiently computed and the rotation–vibration eigenvalues of the molecule may be easily computed for a number of rotational states. The Kratzer oscillator functions are found to be a rapidly converging set for this purpose. The method was tested on the HCN molecule using a potential for which accurate earlier calculations are available for comparison.

The behavior of an atom-rigid rotor scattering system is investigated for the case when much energy is in internal states and only a little is in translational motion. The l-average coupled and recoupled states approximations are shown to fail completely for inelastic collisions in this regime. These approximations are also shown to be very inaccurate for transitions between highly excited states and slightly excited states and vice versa. L-labeled coupled states cross sections for a transition are proven to be directly proportional to those for the reverse transition, a symmetry not shown by exact cross sections.

A formula is given for computing integrals of exponentials and powers of r between eigenfunctions of two different Kratzer oscillators. Such integrals would be of use in using mixed basis sets of Kratzer oscillator functions. A practical, numerically stable recursion relation is given for computing these integrals for both mixed and pure basis sets. The use of Kratzer functions is illustrated in the computation of the eigenstates of a rotating-vibrating Morse oscillator. The eigenstates are also computed by the use of a harmonic oscillator basis set. The Kratzer oscillator functions have the advantage over the harmonic oscillator basis in that all integrals can be performed analytically. It is found that the Kratzer basis set size is only 2/3 as large as that of the harmonic basis set required to give the same accuracy.

An investigation is made of several manifestations of the orbiting of two scattering partners. One focus is the effect on the l-average coupled states approximation and its first-order perturbational improvement, the l-average recoupled states approximation. Both methods are shown to fail completely when applied to m-dependent and degeneracy averaged differential and integral cross sections for inelastic collisions of an atom and a rigid rotor when there is significant orbiting behavior. A series of calculations is made for the He–SiO system scattering at 2.25, 9.01, 18.0, and 27.0 meV. The gradual improvement of both approximations with energy is clearly demonstrated. Additionally, close coupling differential and integral cross sections are shown to be extremely sensitive to slight energy changes in the orbiting regime. Moreover, single energy results in this regime are shown to differ significantly from results averaged over a molecular beam’s energy distribution.

An exhaustive search of the parameter space for the Ne–CH4 interaction potential is made in an attempt to determine a potential which will reproduce the results of molecular beam scattering experiments. The experiments were performed at low energy, well below the threshold for vibrational excitation. It is shown that the effects of the long-range part of the potential were not seen in the experiment. It is further shown that the anisotropy is well represented as far as the experiments are concerned by a fourth rank expansion. In the light of this and earlier studies, it is concluded that a rigid top model for methane is not capable of describing the experimental results even though the collision energy is well below the threshold for vibrational excitation. This is in marked contrast to atom–diatomic molecule scattering.

Exact close-coupling calculations are used to evaluate the effectiveness of the coupled states approximation in the interpretation of low energy, noble gas methane collisions. The effect of the higher order angle dependent terms of the potential on the inelastic rotational transition was investigated. Calculations using only open channel basis functions were clearly different from converged calculations, but the differences were small enough to be negligible in comparison with experiment. It was shown that open channel coupled states calculations are of sufficient accuracy to evaluate the agreement of the potential model with experiment. The use of the coupled states approximation does not fully account for discrepancies between theory and experiment observed earlier.

We describe a method for obtaining the bound ro-vibrational states of polyatomic molecules through the expansion of the potential in terms of appropriate functions of internal coordinates. Using this method, the calculation of potential matrix elements in greatly simplified and the algorithm is ideally suited to the vector machine like Cray. Programs for potential matrix calculation and matrix diagonalization are given with the results of a test run.

The use of Kratzer oscillator functions as a basis set for calculations of diatomic molecule eigenfunctions is discussed. Exact expressions are given for integrals of these functions over general potential types.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTRotational-vibrational structure of a quasi-linear molecule: methyliumylJae Shin Lee and Don SecrestCite this: J. Phys. Chem. 1988, 92, 7, 1821–1830Publication Date (Print):April 1, 1988Publication History Published online1 May 2002Published inissue 1 April 1988https://pubs.acs.org/doi/10.1021/j100318a026https://doi.org/10.1021/j100318a026research-articleACS PublicationsRequest reuse permissionsArticle Views62Altmetric-Citations13LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access options Get e-Alerts

Two aspects of the coupled states (CS) and the recoupled states (RCS) formalism of K. McLenithan and D. Secrest [J. Chem. Phys. 80, 2480 (1984)] are discussed. First, there is an inconsistency, in that the Lippmann–Schwinger equation (51) implies, contrary to Eq. (21), that the CS wave function satisfies an inhomogeneous differential equation. This inconsistency is removed, and Eq. (54) is made exact, by the addition of a suitable term to the right-hand side of Eq. (51). Second, the CS-RCS formalism of McLenithan and Secrest defines zeroth- and first-order approximate transition operators; it is argued that the proper functioning of their formalism depends essentially on the use of a physically obscure l-labeling prescription for the determination of S matrices, and lacks uniqueness in that an alternative formalism exists that hews more closely to a strict perturbation theory approach for the zeroth- and first-order approximations to the exact transition operator.

A fit to a recent ab initio potential surface for HeHHe+ is presented and discussed. Calculations of the rotation–vibration spectra are made using the variation method with this potential surface. The calculations are carried out for both the 1H and 2H(D) isotope of H and comparisons of the spectra and vibrationally averaged bond length are discussed for the two species. A study of the dissociation to He+HeH+ is made and ‘‘vibrational bonding’’ is investigated. It is found that this system is not vibrationally bound. The deuterium substituted species is more strongly bound.

A fit to a recent ab initio potential surface for HeHHe+ is presented and discussed. Calculations of the rotation–vibration spectra are made using the variation method with this potential surface. The calculations are carried out for both the 1H and 2H(D) isotope of H and comparisons of the spectra and vibrationally averaged bond length are discussed for the two species. A study of the dissociation to He+HeH+ is made and ‘‘vibrational bonding’’ is investigated. It is found that this system is not vibrationally bound. The deuterium substituted species is more strongly bound.

A method is given for deriving a hamiltonian for a polyatomic molecule in generalized coordinates. By this technique, one may treat linear and nonlinear molecules in the same way. It is shown that the proof which asserts that the determinant of the moment of inertia tensor for a linear molecule vanishes is fallacious and, as an example, a hamiltonian for a linear triatomic molecule is derived and discussed. Notes This work was supported in part by a grant from the National Science Foundation and the Petroleum Research Fund administered by the American Chemical Society.