Discrete variable representations of differential operators

Szalay, Viktor

doi:10.1063/1.465258

Cited by 97 publications

(75 citation statements)

References 25 publications

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“…This also leads to compact expressions for the kinetic-energy matrix elements that were missing in references [2,14]. These sum rules were already used by Szalay in a similar context [19]. Finally, new sum rules involving Laguerre or Jacobi zeros and weights are derived from these expressions.…”

Section: Introductionmentioning

confidence: 93%

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Integrals of Lagrange functions and sum rules

Baye¹

2011

J. Phys. A: Math. Theor.

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Section: Introductionmentioning

confidence: 93%

“…In order to make use of (10) and (12), one needs explicit expressions of the derivatives of the Lagrange functions at mesh points. A direct differentiation provides [19] …”

Section: Definitions and Formulasmentioning

confidence: 99%

Integrals of Lagrange functions and sum rules

Baye¹

2011

J. Phys. A: Math. Theor.

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“…Depending on the underlying basis, DVR matrix elements of differential operators can be calculated straightforwardly either via basis transformations to and from a finite basis representation 34 , e.g. a harmonic oscillator, or by explicit formulas 35 , most commonly based on Fourier functions [36][37][38] . The selection of an appropriate DVR basis consists primarily of choosing one whose underlying basis functions have the same boundary conditions as the eigenfunctions to be obtained 37 .…”

Section: The Kinetic Energy Operatormentioning

confidence: 99%

Reduced dimension discrete variable representation study of cis–trans isomerization in the S1 state of C2H2

Baraban

Beck

Steeves

et al. 2011

The Journal of Chemical Physics

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Isomerization between the cis and trans conformers of the S 1 state of acetylene is studied using a reduced dimension DVR calculation. Existing DVR techniques are combined with a high accuracy potential energy surface and a kinetic energy operator derived from FG theory to yield an effective but simple Hamiltonian for treating large amplitude motions. The spectroscopic signatures of the S 1 isomerization are discussed, with emphasis on the vibrational aspects. The presence of a low barrier to isomerization causes distortion of the trans vibrational level structure and the appearance of nominally electronically forbiddenÃ 1 A 2 ←X 1 Σ + g transitions to vibrational levels of the cis conformer. Both of these effects are modeled in agreement with experimental results, and the underlying mechanisms of tunneling and state mixing are elucidated by use of the calculated vibrational wavefunctions.

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“…The calculations were performed using a discrete variable representation ͑DVR͒ based on Gauss-Hermite quadrature, 54 i.e., corresponding to a finite basis representation of Hermite polynomials.…”

Section: Calculationsmentioning

confidence: 99%

“…The exact density of states was calculated by binning the eigenvalues obtained from direct diagonalization of the full Hamiltonian in a twodimensional DVR based on Gauss-Hermite quadrature. 54 The density correlation calculations utilized a 50-iteration Lanczos scheme. The figure highlights a typical difficulty with testing the accuracy of approximate convolution methods by comparison with the results of a variational basis set/direct diagonalization calculation.…”

Section: B Hé Non-heiles Potentialmentioning

confidence: 99%

On the calculation of absolute spectral densities

Smith

Jeffrey

1996

The Journal of Chemical Physics

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On the relation of protein dynamics and exciton relaxation in pigment-protein complexes: An estimation of the spectral density and a theory for the calculation of optical spectra J. Chem. Phys. 116, 9997 (2002) A new method of calculating the absolute spectral density of a Hamiltonian operator is derived and discussed. The spectral density is expressed as the solution of an integral equation in which the kernel is a renormalized one-sided energy correlation function of the full microcanonical density operator and a microcanonical density operator for a reference Hamiltonian. The integral operator associated with this equation transforms a known spectral density function for the reference Hamiltonian into the spectral density of the full Hamiltonian. The integral equation, by virtue of its formulation in energy space, is inherently one-dimensional and offers no storage difficulties, and the elements of its kernel may be computed by applying the Lanczos algorithm to randomly selected eigenfunctions of the reference Hamiltonian. This spectral density correlation method offers a number of advantages over variational methods. In particular, it has the potential for overcoming the hitherto largely insurmountable problem of tracing over a multidimensional Hilbert space in order to compute the spectral density of a nonseparable molecular Hamiltonian.

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Discrete variable representations of differential operators

Cited by 97 publications

References 25 publications

Integrals of Lagrange functions and sum rules

Integrals of Lagrange functions and sum rules

Reduced dimension discrete variable representation study of cis–trans isomerization in the S1 state of C2H2

On the calculation of absolute spectral densities

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