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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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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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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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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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5
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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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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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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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9
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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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Shimshovitz A, Bačić Z, Tannor DJ. The von Neumann basis in non-Cartesian coordinates: Application to floppy triatomic molecules. J Chem Phys 2014; 141:234106. [DOI: 10.1063/1.4902553] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Affiliation(s)
- Asaf Shimshovitz
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Zlatko Bačić
- Department of Chemistry, New York University, New York, New York 10003, USA
- NYU-ECNU Center of Computational Chemistry at NYU Shanghai, Shanghai 200062, China
| | - David J. Tannor
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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11
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Phase Space Approach to Solving the Schrödinger Equation: Thinking Inside the Box. ADVANCES IN CHEMICAL PHYSICS 2014. [DOI: 10.1002/9781118949702.ch1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register]
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12
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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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13
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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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Shimshovitz A, Tannor DJ. Phase-space approach to solving the time-independent Schrödinger equation. PHYSICAL REVIEW LETTERS 2012; 109:070402. [PMID: 23006346 DOI: 10.1103/physrevlett.109.070402] [Citation(s) in RCA: 38] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2010] [Revised: 12/13/2011] [Indexed: 06/01/2023]
Abstract
We propose a method for solving the time-independent Schrödinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)], we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit, the method reaches the remarkable efficiency of one basis function per one eigenstate. We illustrate the method for a challenging two-dimensional potential where the Fourier grid method breaks down.
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Affiliation(s)
- Asaf Shimshovitz
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel
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