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  • 1
    Electronic Resource
    Electronic Resource
    Wigner–Weyl correspondence and semiclassical quantization in spherical coordinates (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 6302-6318 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533094
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  • 2
    Electronic Resource
    Electronic Resource
    Efficient distributed Gaussian basis for rovibrational spectroscopy calculations (2000)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 113 (2000), S. 211-217 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: We examine the problem of choosing efficient basis sets for the calculation of vibrational states of molecules. An exact quantum functional is derived for optimizing the parameters of distributed Gaussian basis sets (DGBs). For a given Hamiltonian and energy range, the basis is optimized with respect to the accuracy of the computed eigenvalues. This procedure demonstrates that optimized DGBs are remarkably efficient, being essentially exact for the one-dimensional harmonic oscillator, and orders of magnitude more accurate for the 23-state Morse oscillator than previous DGB calculations of comparable size. Contrary to expectations however, the quantum optimized DGBs have large overlaps, resulting in nearly singular overlap matrices that may cause numerical instabilities in larger calculations. On the other hand, the optimized eigenvalue calculation is shown to be fairly robust with respect to DGB parameter variations, implying that accurate results are possible using more numerically stable DGBs. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.481787
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  • 3
    Electronic Resource
    Electronic Resource
    Quantum reactive scattering for three-body systems via optimized preconditioning, as applied to the O+HCl reaction (1998)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 108 (1998), S. 5216-5224 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The evaluation of quantum scattering quantities for three-body systems is explored in conjunction with the optimal separable basis methodology, which is utilized in two different ways. First, numerical results are obtained for the zero total angular momentum case using a three-dimensional discrete variable Hamiltonian with absorbing boundary conditions and optimized preconditioning. The J≠0 results are then estimated using helicity-conserving and J-shifting approximations, after minimizing the coriolis coupling via another application of the optimal basis method. An "effective potential" interpretation of the helicity-conserving approximation is employed, which obviates the requirement of a K-varying associated Legendre basis for the angular coordinate. This treatment also leads to an improved version of the J-shifting method that automatically incorporates centrifugal distortion and other effects. Fixed-energy cumulative reaction probabilities and thermal rate constants are presented for the O+HCl→OH+Cl reactive scattering system. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.475958
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  • 4
    Electronic Resource
    Electronic Resource
    Phase space optimization of quantum representations: Direct-product basis sets (1999)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 111 (1999), S. 4869-4885 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The quantitative phase space similarities between the uniformly mixed ensembles of eigenstates, and the quasiclassical Thomas–Fermi distribution, are exploited in order to generate a nearly optimal basis representation for an arbitrary quantum system. An exact quantum optimization functional is provided, and the minimum of the corresponding quasiclassical functional is proposed as an excellent approximation in the limit of large basis size. In particular, we derive a stationarity condition for the quasiclassical solution under the constraint of strong separability. The corresponding quantum result is the phase space optimized direct-product basis—customized with respect to the Hamiltonian itself, as well as the maximum energy of interest. For numerical implementations, an iterative, self-consistent-field-like algorithm based on optimal separable basis theory is suggested, typically requiring only a few reduced-dimensional integrals of the potential. Results are obtained for a coupled oscillator system, and also for the 2D Henon–Heiles system. In the latter case, a phase space optimized discrete variable representation (DVR) is used to calculate energy eigenvalues. Errors are reduced by several orders of magnitude, in comparison with an optimized sinc-function DVR of comparable size. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.479747
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  • 5
    Electronic Resource
    Electronic Resource
    A preconditioned inexact spectral transform method for calculating resonance energies and widths, as applied to HCO (2002)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 116 (2002), S. 1215-1227 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: We present a complex-symmetric version of the preconditioned inexact spectral transform (PIST) method, for calculating resonance energies and widths. The PIST method uses an iterative linear solver to compute inexact Lanczos vectors for (EI−H)−1, and then diagonalizes the Hamiltonian in the inexact Lanczos representation. Our new version requires complex-symmetric variants of: (1) the Lanczos algorithm, (2) the linear solver, (3) the preconditioner we introduced in a previous paper [J. Chem. Phys. 114, 9254 (2001)]. The new method works extremely well for HCO, enabling us to calculate the first 17 dissociative resonances in less then 90 second of CPU time. © 2002 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1428752
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  • 6
    Electronic Resource
    Electronic Resource
    Phase space optimization of quantum representations: Three-body systems and the bound states of HCO (2001)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 114 (2001), S. 6562-6571 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: In an earlier paper [J. Chem. Phys. 111, 4869 (1999)] we introduced a quasiclassical phase space approach for generating a nearly optimal direct-product basis for representing an arbitrary quantum Hamiltonian within a given energy range of interest. From a few reduced-dimensional integrals, the method determines the optimal one-dimensional marginal Hamiltonians, whose eigenstates comprise the direct-product basis. In this paper the method is applied to three-body vibrational systems expressed in radial and angular coordinates. Numerical results are obtained for the bound state eigenenergies of the nonrotating HCO molecule, determined to ∼0.01 cm−1 accuracy using a phase space optimized direct-product basis of 1972 functions. This represents a computational reduction of several orders of magnitude, in comparison with previous calculations. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1354181
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  • 7
    Electronic Resource
    Electronic Resource
    Accelerating the calculation of energy levels and wave functions using an efficient preconditioner with the inexact spectral transform method (2001)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 114 (2001), S. 9254-9264 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: In an earlier paper [J. Chem. Phys. 112, 8765 (2000)] our group introduced a preconditioned inexact spectral transform method for calculating energy levels and wave functions. Although we could calculate high-lying levels with far fewer matrix–vector products than with the filter diagonalization method of Mandelshtam and Taylor, even better performance can be achieved with a better preconditioner. In this paper, we develop an extremely efficient preconditioner consisting of two components: (1) transformation to an optimal separable basis, in which off-diagonal elements of the Hamiltonian matrix are minimized; and (2) removal of all off-diagonal coupling near the energies of interest. The new preconditioner works extremely well; it enables us to calculate high-lying vibrational states of H2O with orders of magnitude fewer matrix–vector products than for all other known methods. The new preconditioner should also accelerate the calculation of other quantities, such as photodissociation cross sections and rate constants. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1367396
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  • 8
    Electronic Resource
    Electronic Resource
    Algebraically Self-Consistent Quasiclassical Approximation on Phase Space (2000)
    Springer
    Foundations of physics 30 (2000), S. 1191-1226 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary “*” operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003632404712
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  • 9
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    Two-body Schrödinger wave functions in a plane-wave basis via separation of dimensions (2018)
    American Institute of Physics
    Details
    Publication Date: 2018-03-14
    Print ISSN: 0021-9606
    Electronic ISSN: 1089-7690
    Topics: Chemistry and Pharmacology , Physics
    Published by American Institute of Physics
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  • 10
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    Optimal separable bases and series expansions (1997)
    American Physical Society
    Details
    Publication Date: 1997-07-01
    Print ISSN: 1050-2947
    Electronic ISSN: 1094-1622
    Topics: Electrical Engineering, Measurement and Control Technology , Physics
    Published by American Physical Society
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