Self‐consistent field energies and wavefunctions for coupled oscillators
15 Jan 1978-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 68, Iss: 2, pp 608-610
TL;DR: In this paper, a self-consistent field description of coupled anharmonic oscillators was formulated for a model problem of two coupled an-harmonic OO and energy eigenvalues compared with previous exact quantum and semiclassical ones.
Abstract: We formulate a self‐consistent field description of coupled anharmonic oscillators. The SCF equations are solved numerically for a model problem of two coupled anharmonic oscillators and energy eigenvalues compared with previously published exact quantum and semiclassical ones.
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TL;DR: The traditional view of vibrational motion is based on an harmonic approximation to the full nuclear potential as mentioned in this paper, which gives rise to the complexity of this potential and the richness of much of chemistry.
Abstract: on the Born-Oppenheimer approximation which separates the motion of the electrons from that of the nuclei. The electronic motion produces an effective potential which holds molecules together and governs their vibrational motion. The complexity of this potential gives rise to the richness of much of chemistry. Thus, a theoretical picture of the vibrations of molecules is at the heart of many chemical questions. The traditional view of vibrational motion is based on an harmonic approximation to the full nuclear potential. This very simple approximation gives rise to
523 citations
TL;DR: In this paper, the vibrational energies of CO-Cu(100) were calculated using a new code to perform vibrational self-consistent field (VSCF) and state-mixing calculations for many-mode systems.
Abstract: We report calculations of the vibrational energies of CO–Cu(100) using a new code to perform vibrational self-consistent field (VSCF) and state-mixing calculations for many-mode systems. The major new feature of the code is the representation of the potential. Unlike recent implementations of the VSCF method, the potential is not expanded in terms of normal coordinates as a multinomial series about a minimum. The full potential, in normal coordinates, is used in the Watson Hamiltonian. This approach, while rigorous, can lead to prohibitively large numerical quadratures, and so we suggest a novel representation of the potential as an expansion in all two-mode, or all three-mode, or all four-mode coupling terms. The new code is tested against previous exact calculations of vibrational states of HCO, and also against previous VSCF calculations that used a fourth-order, normal coordinate force field representation of the global HCO potential. The new code is applied to calculations of the vibrations of CO ads...
505 citations
TL;DR: In this article, an algorithm for first-principles calculation of vibrational spectroscopy of polyatomic molecules is proposed, which combines electronic ab initio codes with the vibrational self-consistent field (VSCF) method, and with a perturbation-theoretic extension of VSCF.
Abstract: An algorithm for first-principles calculation of vibrational spectroscopy of polyatomic molecules is proposed, which combines electronic ab initio codes with the vibrational self-consistent field (VSCF) method, and with a perturbation-theoretic extension of VSCF. The integrated method directly uses points on the potential energy surface, computed from the electronic ab initio code, in the VSCF part. No fitting of an analytic potential function is involved. A key element in the approach is the approximation that only interactions between pairs of normal modes are important, while interactions of triples or more can be neglected. This assumption was found to hold well in applications. The new algorithm was applied to the fundamental vibrational excitations of H2O, Cl−(H2O), and (H2O)2, using the Moller–Plesset method for the electronic structure. The vibrational frequencies found are in very good accord with experiments. Estimates suggest that this electronic ab initio/VSCF approach should be feasible, with...
421 citations
TL;DR: In this article, a review of state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches is presented.
Abstract: In this article, we review state-of-the-art methods for computing vibrational energies of polyatomic molecules using quantum mechanical, variationally-based approaches. We illustrate the power of those methods by presenting applications to molecules with more than four atoms. This demonstrates the great progress that has been made in this field in the last decade in dealing with the exponential scaling with the number of vibrational degrees of freedom. In this review we present three methods that effectively obviate this bottleneck. The first important idea is the n-mode representation of the Hamiltonian and notably the potential. The potential (and other functions) is represented as a sum of terms that depend on a subset of the coordinates. This makes it possible to compute matrix elements, form a Hamiltonian matrix, and compute its eigenvalues and eigenfunctions. Another approach takes advantage of this multimode representation and represents the terms as a sum of products. It then exploits the powerful...
410 citations
TL;DR: In this article, a discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis for the expansion of other, radial coordinates.
Abstract: A novel, efficient, and accurate quantum method for the calculation of highly excited vibrational levels of triatomic molecules is presented. The method is particularly well suited for applications to ‘‘floppy’’ molecules, having large amplitude motion, on potential surfaces which may have more than one local minimum. The discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis (DGB) for the expansion of other, radial coordinates. The DGB is tailored to the potential, covering only those regions where V(r)
380 citations
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4,691 citations
386 citations
TL;DR: In this paper, an extended concept of quantum ergodicity is proposed for relaxation processes in isolated molecules, and it allows straightforward numerical investigation of molecular models, such as triatomic models and linear molecules with four atoms.
Abstract: Recent advances in classical ergodic theory have produced qualitative information about the dynamics of anharmonic systems that is relevant to internal molecular dynamics and to unimolecular reaction rate theories. We describe here the results of an attempt to translate this progress into quantum molecular dynamics. Existing theories of quantum ergodicity are found to be ill suited to our task, so an extended concept of quantum ergodicity closely adhering to the well established classical concept is suggested. The new concept has the desired relevance to relaxation processes in isolated molecules, and it allows straightforward numerical investigation of molecular models. We report results for several triatomic models and one model of a linear molecule with four atoms. A prevalence of nonergodic behavior is observed for energies below the minimum required for dissociation.
174 citations
TL;DR: In this article, the Hamilton-Jacobi equation for a multidimensional non-separable system can be efficiently solved directly in action-angle variables, which allows one to construct the total (classical) Hamiltonian as a function of the "good" action variables which are the complete set of constants of the motion of the system.
Abstract: It is shown how the Hamilton–Jacobi equation for a multidimensional nonseparable system can be efficiently solved directly in action‐angle variables. This allows one to construct the total (classical) Hamiltonian as a function of the ’’good’’ action‐angle variables which are the complete set of constants of the motion of the system; requiring the action variables to be integers then provides the semiclassical eigenvalues. Numerical results are presented for a two‐dimensional potential well, and one sees that the semiclassical eigenvalues are in good agreement with the exact quantum mechanical values even for the case of large nonseparable coupling.
158 citations
TL;DR: The finite difference boundary value method for obtaining eigenvalues and eigenfunctions of the one-dimensional Schroedinger equation is discussed in this article, where the method is noniterative and may be applied to one dimensional problems on (- ∞, ∞) or to the radial equation on (0, ∾).
139 citations