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Sarka J, Poirier B. Hitting the Trifecta: How to Simultaneously Push the Limits of Schrödinger Solution with Respect to System Size, Convergence Accuracy, and Number of Computed States. J Chem Theory Comput 2021; 17:7732-7744. [PMID: 34761945 DOI: 10.1021/acs.jctc.1c00824] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Methods for solving the Schrödinger equation without approximation are in high demand but are notoriously computationally expensive. In practical terms, there are just three primary factors that currently limit what can be achieved: 1) system size/dimensionality; 2) energy level excitation; and 3) numerical convergence accuracy. Broadly speaking, current methods can deliver on any two of these three goals, but achieving all three at once remains an enormous challenge. In this paper, we shall demonstrate how to "hit the trifecta" in the context of molecular vibrational spectroscopy calculations. In particular, we compute the lowest 1000 vibrational states for the six-atom acetonitrile molecule (CH3CN), to a numerical convergence of accuracy 10-2 cm-1 or better. These calculations encompass all vibrational states throughout most of the dynamically relevant range (i.e., up to ∼4250 cm-1 above the ground state), computed in full quantum dimensionality (12 dimensions), to near spectroscopic accuracy. To our knowledge, no such vibrational spectroscopy calculation has ever previously been performed.
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Affiliation(s)
- János Sarka
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, United States
| | - Bill Poirier
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, United States
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2
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Revuelta F, Vergini E, Benito RM, Borondo F. Short-periodic-orbit method for excited chaotic eigenfunctions. Phys Rev E 2020; 102:042210. [PMID: 33212620 DOI: 10.1103/physreve.102.042210] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2019] [Accepted: 09/18/2020] [Indexed: 11/07/2022]
Abstract
An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - E Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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3
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Pandey A, Poirier B. An algorithm to find (and plug) “holes” in multi-dimensional surfaces. J Chem Phys 2020; 152:214102. [DOI: 10.1063/5.0005681] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Affiliation(s)
- Ankit Pandey
- Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
| | - Bill Poirier
- Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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4
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Karmakar S, Keshavamurthy S. Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective. Phys Chem Chem Phys 2020; 22:11139-11173. [DOI: 10.1039/d0cp01413c] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
The onset of facile intramolecular vibrational energy flow can be related to features in the connected network of anharmonic resonances in the classical phase space.
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Affiliation(s)
- Sourav Karmakar
- Department of Chemistry
- Indian Institute of Technology
- Kanpur
- India
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5
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Pandey A, Poirier B. Using phase-space Gaussians to compute the vibrational states of OCHCO+. J Chem Phys 2019; 151:014114. [DOI: 10.1063/1.5096770] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ankit Pandey
- Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
| | - Bill Poirier
- Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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6
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Large Scale Exact Quantum Dynamics Calculations: Using Phase Space to Truncate the Basis Effectively. ADVANCES IN CHEMICAL PHYSICS 2018. [DOI: 10.1002/9781119374978.ch9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register]
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7
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Jerke J, Poirier B. Two-body Schrödinger wave functions in a plane-wave basis via separation of dimensions. J Chem Phys 2018; 148:104101. [DOI: 10.1063/1.5017621] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jonathan Jerke
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
- Department of Physics, Texas Southern University, Houston, Texas 77004, USA
| | - Bill Poirier
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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9
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Larsson HR, Tannor DJ. Dynamical pruning of the multiconfiguration time-dependent Hartree (DP-MCTDH) method: An efficient approach for multidimensional quantum dynamics. J Chem Phys 2017; 147:044103. [DOI: 10.1063/1.4993219] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- H. R. Larsson
- Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
- Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
| | - D. J. Tannor
- Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
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Kay KG. Applying Bogomolny's quantization method to generic classical systems. J Chem Phys 2017; 146:204111. [PMID: 28571363 PMCID: PMC5451315 DOI: 10.1063/1.4983748] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2017] [Accepted: 05/05/2017] [Indexed: 11/15/2022] Open
Abstract
The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix TE as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for TE that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for TE produces accurate semiclassical energy levels from small transfer matrices.
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Affiliation(s)
- Kenneth G Kay
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
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11
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Larsson HR, Hartke B, Tannor DJ. Efficient molecular quantum dynamics in coordinate and phase space using pruned bases. J Chem Phys 2016; 145:204108. [DOI: 10.1063/1.4967432] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- H. R. Larsson
- Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
| | - B. Hartke
- Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
| | - D. J. Tannor
- Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
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Brown J, Carrington T. Assessing the utility of phase-space-localized basis functions: Exploiting direct product structure and a new basis function selection procedure. J Chem Phys 2016; 144:244115. [DOI: 10.1063/1.4954721] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- James Brown
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Machnes S, Assémat E, Larsson HR, Tannor DJ. Quantum Dynamics in Phase Space using Projected von Neumann Bases. J Phys Chem A 2016; 120:3296-308. [PMID: 26977715 DOI: 10.1021/acs.jpca.5b12370] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schrödinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.
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Affiliation(s)
- Shai Machnes
- Department of Chemical Physics, Weizmann Institute of Science , 76100 Rehovot, Israel
| | - Elie Assémat
- Department of Chemical Physics, Weizmann Institute of Science , 76100 Rehovot, Israel
| | - Henrik R Larsson
- Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel , Olshausenstraße 40, D-24098 Kiel, Germany
| | - David J Tannor
- Department of Chemical Physics, Weizmann Institute of Science , 76100 Rehovot, Israel
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14
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Affiliation(s)
- Thomas Halverson
- Department of Chemistry and
Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, United States
| | - Bill Poirier
- Department of Chemistry and
Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, United States
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15
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Brown J, Carrington T. Using an iterative eigensolver to compute vibrational energies with phase-spaced localized basis functions. J Chem Phys 2015; 143:044104. [DOI: 10.1063/1.4926805] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- James Brown
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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16
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Large scale exact quantum dynamics calculations: Ten thousand quantum states of acetonitrile. Chem Phys Lett 2015. [DOI: 10.1016/j.cplett.2015.02.004] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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17
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Halverson T, Poirier B. Calculation of exact vibrational spectra for P2O and CH2NH using a phase space wavelet basis. J Chem Phys 2014; 140:204112. [DOI: 10.1063/1.4879216] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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18
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Wang XG, Carrington T. Computing rovibrational levels of methane with curvilinear internal vibrational coordinates and an Eckart frame. J Chem Phys 2013; 138:104106. [DOI: 10.1063/1.4793474] [Citation(s) in RCA: 66] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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19
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Halverson T, Poirier B. Accurate quantum dynamics calculations using symmetrized Gaussians on a doubly dense Von Neumann lattice. J Chem Phys 2012; 137:224101. [DOI: 10.1063/1.4769402] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
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20
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LOMBARDINI RICHARD, POIRIER BILL. PARALLEL SUBSPACE ITERATION METHOD FOR THE SPARSE SYMMETRIC EIGENVALUE PROBLEM. JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2011. [DOI: 10.1142/s0219633606002738] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A new parallel iterative algorithm for the diagonalization of real sparse symmetric matrices is introduced, which uses a modified subspace iteration method. A novel feature is the preprocessing of the matrix prior to iteration, which allows for a natural parallelization resulting in a great speedup and scalability of the method with respect to the number of compute nodes. The method is applied to Hamiltonian matrices of model systems up to six degrees of freedom, represented in a truncated Weyl–Heisenberg wavelet (or "weylet") basis developed by one of the authors (Poirier). It is shown to accurately determine many thousands of eigenvalues for sparse matrices of the order N ≈ 106, though much larger matrices may also be considered.
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Affiliation(s)
- RICHARD LOMBARDINI
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Box 41061, Lubbock, Texas 79409-1061, USA
| | - BILL POIRIER
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Box 41061, Lubbock, Texas 79409-1061, USA
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21
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Morgan D, Meijer AJHM, Doyle RJ. Spectral difference methods in bound state calculations. J Chem Phys 2009; 130:084114. [PMID: 19256604 DOI: 10.1063/1.3080616] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The use of discrete variable representations is now commonplace in chemical dynamics calculations. In this paper, we employ spectral difference methods to speed up these calculations. We present five new spectral difference weight functions and compare them with those that already exist in the literature for two different bound state problems. We find that one particular weight we propose, based on a Gaussian function, outperforms all other weights.
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Affiliation(s)
- Dean Morgan
- Department of Chemistry, University of Sheffield, Sheffield, S3 7HF, United Kingdom
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Lombardini R, Poirier B. Improving the accuracy of Weyl-Heisenberg wavelet and symmetrized Gaussian representations using customized phase-space-region operators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:036705. [PMID: 17025784 DOI: 10.1103/physreve.74.036705] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2006] [Indexed: 05/12/2023]
Abstract
A particular basis set method developed by one of the authors, involving maximally localized orthogonal Weyl-Heisenberg wavelets (or "weylets") and a phase space truncation scheme, has been successfully applied to exact quantum calculations for many degrees of freedom (DOF's) [B. Poirier and A. Salam, J. Chem. Phys. 121, 1740 (2004)]. However, limitations in accuracy arise in the many-DOF case, owing to memory limits on conventional computers. This paper addresses this accuracy limitation by introducing phase space region operators (PSRO's) that customize individual weylet basis functions for the problem of interest. The construction of the PSRO's is straightforward, and does not require a priori knowledge of the desired eigenstates. The PSRO, when applied to weylets, as well as to simple phase space Gaussian basis functions, exhibits remarkable improvements in accuracy, reducing computed eigenvalue errors by orders of magnitude. The method is applied to various model systems at varying DOF's.
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Affiliation(s)
- Richard Lombardini
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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McCormack DA. Dynamical pruning of static localized basis sets in time-dependent quantum dynamics. J Chem Phys 2006; 124:204101. [PMID: 16774313 DOI: 10.1063/1.2196889] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the viability of dynamical pruning of localized basis sets in time-dependent quantum wave packet methods. Basis functions that have a very small population at any given time are removed from the active set. The basis functions themselves are time independent, but the set of active functions changes in time. Two different types of localized basis functions are tested: discrete variable representation (DVR) functions, which are localized in position space, and phase-space localized (PSL) functions, which are localized in both position and momentum. The number of functions active at each point in time can be as much as an order of magnitude less for dynamical pruning than for static pruning, in reactive scattering calculations of H2 on the Pt(211) stepped surface. Scaling of the dynamically pruned PSL (DP-PSL) bases with dimension is considerably more favorable than for either the primitive (direct product) or DVR bases, and the DP-PSL basis set is predicted to be three orders of magnitude smaller than the primitive basis set in the current state-of-the-art six-dimensional reactive scattering calculations.
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Affiliation(s)
- Drew A McCormack
- Theoretische Chemie, Faculteit Exacte Wetenschappen, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands.
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Lombardini R, Poirier B. Rovibrational spectroscopy calculations of neon dimer using a phase space truncated Weyl-Heisenberg wavelet basis. J Chem Phys 2006; 124:144107. [PMID: 16626180 DOI: 10.1063/1.2187473] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
In a series of earlier articles [B. Poirier J. Theor. Comput. Chem. 2, 65 (2003); B. Poirier and A. Salam J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam J. Chem. Phys. 121, 1740 (2004)], a new method was introduced for performing exact quantum dynamics calculations in a manner that formally defeats exponential scaling with system dimensionality. The method combines an optimally localized, orthogonal Weyl-Heisenberg wavelet basis set with a simple phase space truncation scheme, and has already been applied to model systems up to 17 degrees of freedom (DOF's). In this paper, the approach is applied for the first time to a real molecular system (neon dimer), necessitating the development of an efficient numerical scheme for representing arbitrary potential energy functions in the wavelet representation. All bound rovibrational energy levels of neon dimer are computed, using both one DOF radial coordinate calculations and a three DOF Cartesian coordinate calculation. Even at such low dimensionalities, the approach is found to be competitive with another state-of-the-art method applied to the same system [J. Montgomery and B. Poirier J. Chem. Phys. 119, 6609 (2003)].
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Affiliation(s)
- Richard Lombardini
- Department of Chemistry and Biochemistry and Department of Physics, Texas Tech University, Box 41061, Lubbock, Texas 79409-1061
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Griffin CD, Acevedo R, Massey DW, Kinsey JL, Johnson BR. Multimode wavelet basis calculations via the molecular self-consistent-field plus configuration-interaction method. J Chem Phys 2006; 124:134105. [PMID: 16613447 DOI: 10.1063/1.2183306] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Wavelets provide potentially useful quantum bases for coupled anharmonic vibrational modes in polyatomic molecules as well as many other problems. A single compact support wavelet family provides a flexible basis with properties of orthogonality, localization, customizable resolution, and systematic improvability for general types of one-dimensional and separable systems. While direct product wavelet bases can be used in coupled multidimensional problems, exponential scaling of basis size with dimensionality ultimately provides limits on the number of coupled modes that can be treated simultaneously in exact quantum calculations. The molecular self-consistent-field plus configuration-interaction method is used here in multimode wavelet calculations to reduce the basis size without sacrificing flexibility or the ability to systematically control errors. Both two-dimensional Cartesian coordinate and three-dimensional curvilinear coordinate systems are examined with wavelets serving as universal bases in each case. The first example uses standard Daubechies [Ten Lectures on Wavelets (SIAM, Philadelphia (1992)] wavelets for each mode and the second adapts symmlet wavelets to intervals for each of the curvilinear coordinates.
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Dawes R, Carrington T. Using simultaneous diagonalization and trace minimization to make an efficient and simple multidimensional basis for solving the vibrational Schrödinger equation. J Chem Phys 2006; 124:054102. [PMID: 16468846 DOI: 10.1063/1.2162168] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper we improve the product simultaneous diagonalization (SD) basis method we previously proposed [J. Chem. Phys. 122, 134101 (2005)] and applied to solve the Schrodinger equation for the motion of nuclei on a potential surface. The improved method is tested using coupled complicated Hamiltonians with as many as 16 coordinates for which we can easily find numerically exact solutions. In a basis of sorted products of one-dimensional (1D) SD functions the Hamiltonian matrix is nearly diagonal. The localization of the 1D SD functions for coordinate qc depends on a parameter we denote alphac. In this paper we present a trace minimization scheme for choosing alphac to nearly block diagonalize the Hamiltonian matrix. Near-block diagonality makes it possible to truncate the matrix without degrading the accuracy of the lowest energy levels. We show that in the sorted product SD basis perturbation theory works extremely well. The trace minimization scheme is general and easy to implement.
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Affiliation(s)
- Richard Dawes
- Département de chimie, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal (Québec) H3C 3J7, Canada.
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Trahan C, Poirier B. Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials. J Chem Phys 2006; 124:034115. [PMID: 16438575 DOI: 10.1063/1.2145883] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].
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Affiliation(s)
- Corey Trahan
- Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061, USA
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Trahan CJ, Wyatt RE, Poirier B. Multidimensional quantum trajectories: Applications of the derivative propagation method. J Chem Phys 2005; 122:164104. [PMID: 15945669 DOI: 10.1063/1.1884606] [Citation(s) in RCA: 37] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.
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Affiliation(s)
- Corey J Trahan
- Department of Chemistry and Biochemistry, Texas Tech University, Box 41061 Lubbock, Texas 79409-1061, USA.
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Dawes R, Carrington T. How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates. J Chem Phys 2005; 122:134101. [PMID: 15847449 DOI: 10.1063/1.1863935] [Citation(s) in RCA: 77] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.
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Affiliation(s)
- Richard Dawes
- Département de Chimie, Université de Montréal, Case Postale 6128, Succursale Centre-ville, Montréal, Québec H3C 3J7, Canada.
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Abstract
The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.
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Affiliation(s)
- Bill Poirier
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock, Texas 79409-1061, USA.
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Poirier B, Salam A. Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization. J Chem Phys 2004; 121:1690-703. [PMID: 15260720 DOI: 10.1063/1.1767511] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
In this paper, we extend and elaborate upon a wavelet method first presented in a previous publication [B. Poirier, J. Theo. Comput. Chem. 2, 65 (2003)]. In particular, we focus on construction and optimization of the wavelet functions, from theoretical and numerical viewpoints, and also examine their localization properties. The wavelets used are modified Wilson-Daubechies wavelets, which in conjunction with a simple phase space truncation scheme, enable one to solve the multidimensional Schrodinger equation. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.
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Affiliation(s)
- Bill Poirier
- Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock, Texas 79409-1061, USA.
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