1
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Zhang Q, Wang RS, Wang L. Neural canonical transformations for vibrational spectra of molecules. J Chem Phys 2024; 161:024103. [PMID: 38979703 DOI: 10.1063/5.0209255] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2024] [Accepted: 06/19/2024] [Indexed: 07/10/2024] Open
Abstract
The behavior of polyatomic molecules around their equilibrium positions can be regarded as that of quantum-coupled anharmonic oscillators. Solving the corresponding Schrödinger equations enables the interpretation or prediction of the experimental spectra of molecules. In this study, we developed a novel approach for solving the excited states of anharmonic vibrational systems. The normal coordinates of the molecules are transformed into new coordinates through a normalizing flow parameterized by a neural network. This facilitates the construction of a set of orthogonal many-body variational wavefunctions. This methodology has been validated on an exactly solvable 64-dimensional coupled harmonic oscillator, yielding numerical results with a relative error of 10-6. The neural canonical transformations are also applied to calculate the energy levels of two specific molecules, acetonitrile (CH3CN) and ethylene oxide (C2H4O). These molecules involve 12 and 15 vibrational modes, respectively. A key advantage of this approach is its flexibility concerning the potential energy surface, as it requires no specific form. Furthermore, this method can be readily implemented on large-scale distributed computing platforms, making it easy to extend to investigating complex vibrational structures.
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Affiliation(s)
- Qi Zhang
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Rui-Si Wang
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- University of Chinese Academy of Sciences, Beijing 100049, China
| | - Lei Wang
- Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
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2
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Artiukhin DG, Godtliebsen IH, Schmitz G, Christiansen O. Gaussian process regression adaptive density-guided approach: Toward calculations of potential energy surfaces for larger molecules. J Chem Phys 2023; 159:024102. [PMID: 37428042 DOI: 10.1063/5.0152367] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2023] [Accepted: 06/15/2023] [Indexed: 07/11/2023] Open
Abstract
We present a new program implementation of the Gaussian process regression adaptive density-guided approach [Schmitz et al., J. Chem. Phys. 153, 064105 (2020)] for automatic and cost-efficient potential energy surface construction in the MidasCpp program. A number of technical and methodological improvements made allowed us to extend this approach toward calculations of larger molecular systems than those previously accessible and maintain the very high accuracy of constructed potential energy surfaces. On the methodological side, improvements were made by using a Δ-learning approach, predicting the difference against a fully harmonic potential, and employing a computationally more efficient hyperparameter optimization procedure. We demonstrate the performance of this method on a test set of molecules of growing size and show that up to 80% of single point calculations could be avoided, introducing a root mean square deviation in fundamental excitations of about 3 cm-1. A much higher accuracy with errors below 1 cm-1 could be achieved with tighter convergence thresholds still reducing the number of single point computations by up to 68%. We further support our findings with a detailed analysis of wall times measured while employing different electronic structure methods. Our results demonstrate that GPR-ADGA is an effective tool, which could be applied for cost-efficient calculations of potential energy surfaces suitable for highly accurate vibrational spectra simulations.
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Affiliation(s)
- Denis G Artiukhin
- Institut für Chemie und Biochemie, Freie Universität Berlin, Arnimallee 22, 14195 Berlin, Germany
| | - Ian H Godtliebsen
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
| | - Gunnar Schmitz
- Lehrstuhl für Theoretische Chemie II, Ruhr-Universität Bochum, Universitätstraße 150, 44801 Bochum, Germany
| | - Ove Christiansen
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
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3
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Carvalho JR, Vidal LN. Calculation of absolute Raman scattering cross-sections using vibrational self-consistent field/vibrational configuration interaction wave functions. J Comput Chem 2022; 43:1484-1494. [PMID: 35731622 DOI: 10.1002/jcc.26951] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Revised: 05/21/2022] [Accepted: 06/02/2022] [Indexed: 11/12/2022]
Abstract
In the present study, the differential scattering cross-sections, depolarization ratios and Raman shifts of small molecular systems are obtained from configuration iteration wave functions of vibrational self-consistent field (VSCF) states. The transition polarizabilities were modeled using the Placzek approximation, neglecting those contributions not arising from the electric dipole mechanism. This theoretical approach is considered a good approximation for samples that absorb in the UV range if the excitation radiation falls in the visible region, as is the case of the molecules selected for the present study, namely: water, methane, and acetylene. Potential energy and electronic polarizability surfaces are calculated by the CCSD(T) and CC3 methods with aug-cc-p(C)V(T,Q,5)Z basis sets. The vibrational Hamiltonian includes the vibrational angular momentum contribution of the Watson kinetic energy operator. As expected, due to the variational nature of the VSCF and vibrational configuration interaction (VCI) methods, the Raman transition wavenumbers are substantially improved over the harmonic predictions. Surprisingly, the scattering cross-sections obtained using the harmonic approximation or the VSCF method better agrees with the experimental values than those cross-sections predicted using VCI wave functions. The more significant deviations of the VCI results from the experimental reference may be related to the significant uncertainties of the measured cross-sections. Still, it may also indicate that the VCI Raman transition moments may require a more accurate description of the electronic polarizability surface. Finally, the depolarization ratios calculated for H2 O and C2 D2 using harmonic and VCI wave functions have similar accuracy, whereas, for C2 H2 and C2 HD, the VCI results are more accurate.
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Affiliation(s)
- Jhonatas R Carvalho
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas, USA
| | - Luciano N Vidal
- Departamento Acadêmico de Química e Biologia, Universidade Tecnológica Federal do Paraná, Curitiba, Paraná, Brazil
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4
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Madsen NK, Jensen RB, Christiansen O. Calculating vibrational excitation energies using tensor-decomposed vibrational coupled-cluster response theory. J Chem Phys 2021; 154:054113. [DOI: 10.1063/5.0037240] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Rasmus Berg Jensen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
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5
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Madsen NK, Jensen AB, Hansen MB, Christiansen O. A general implementation of time-dependent vibrational coupled-cluster theory. J Chem Phys 2020; 153:234109. [PMID: 33353317 DOI: 10.1063/5.0034013] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Abstract
The first general excitation level implementation of the time-dependent vibrational coupled cluster (TDVCC) method introduced in a recent publication [J. Chem. Phys. 151, 154116 (2019)] is presented. The general framework developed for time-independent vibrational coupled cluster (VCC) calculations has been extended to the time-dependent context. This results in an efficient implementation of TDVCC with general coupling levels in the cluster operator and Hamiltonian. Thus, the convergence of the TDVCC[k] hierarchy toward the complete-space limit can be studied for any sum-of-product Hamiltonian. Furthermore, a scheme for including selected higher-order excitations for a subset of modes is introduced and studied numerically. Three different definitions of the TDVCC autocorrelation function (ACF) are introduced and analyzed in both theory and numerical experiments. Example calculations are presented for an array of systems including imidazole, formyl fluoride, formaldehyde, and a reduced-dimensionality bithiophene model. The results show that the TDVCC[k] hierarchy converges systematically toward the full-TDVCC limit and that the implementation allows accurate quantum-dynamics simulations of large systems to be performed. Specifically, the intramolecular vibrational-energy redistribution of the 21-dimensional imidazole molecule is studied in terms of the decay of the ACF. Furthermore, the importance of product separability in the definition of the ACF is highlighted when studying non-interacting subsystems.
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Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | | | - Mads Bøttger Hansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
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6
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Schmitz G, Klinting EL, Christiansen O. A Gaussian process regression adaptive density guided approach for potential energy surface construction. J Chem Phys 2020; 153:064105. [DOI: 10.1063/5.0015344] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Gunnar Schmitz
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
| | | | - Ove Christiansen
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
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7
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Hansen MB, Madsen NK, Christiansen O. Extended vibrational coupled cluster: Stationary states and dynamics. J Chem Phys 2020; 153:044133. [DOI: 10.1063/5.0015413] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Mads Bøttger Hansen
- Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Niels Kristian Madsen
- Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Ove Christiansen
- Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark
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Ollitrault PJ, Baiardi A, Reiher M, Tavernelli I. Hardware efficient quantum algorithms for vibrational structure calculations. Chem Sci 2020; 11:6842-6855. [PMID: 32874524 PMCID: PMC7448527 DOI: 10.1039/d0sc01908a] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2020] [Accepted: 06/02/2020] [Indexed: 11/21/2022] Open
Abstract
We introduce a framework for the calculation of ground and excited state energies of bosonic systems suitable for near-term quantum devices and apply it to molecular vibrational anharmonic Hamiltonians. Our method supports generic reference modal bases and Hamiltonian representations, including the ones that are routinely used in classical vibrational structure calculations. We test different parametrizations of the vibrational wavefunction, which can be encoded in quantum hardware, based either on heuristic circuits or on the bosonic Unitary Coupled Cluster Ansatz. In particular, we define a novel compact heuristic circuit and demonstrate that it provides a good compromise in terms of circuit depth, optimization costs, and accuracy. We evaluate the requirements, number of qubits and circuit depth, for the calculation of vibrational energies on quantum hardware and compare them with state-of-the-art classical vibrational structure algorithms for molecules with up to seven atoms.
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Affiliation(s)
- Pauline J Ollitrault
- IBM Quantum , IBM Research - Zurich , Säumerstrasse 4 , 8803 Rüschlikon , Switzerland .
- Laboratory of Physical Chemistry , ETH Zürich , Vladimir-Prelog-Weg 2 , 8093 Zürich , Switzerland .
| | - Alberto Baiardi
- Laboratory of Physical Chemistry , ETH Zürich , Vladimir-Prelog-Weg 2 , 8093 Zürich , Switzerland .
| | - Markus Reiher
- Laboratory of Physical Chemistry , ETH Zürich , Vladimir-Prelog-Weg 2 , 8093 Zürich , Switzerland .
| | - Ivano Tavernelli
- IBM Quantum , IBM Research - Zurich , Säumerstrasse 4 , 8803 Rüschlikon , Switzerland .
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9
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Madsen NK, Hansen MB, Worth GA, Christiansen O. MR-MCTDH[n]: Flexible Configuration Spaces and Nonadiabatic Dynamics within the MCTDH[n] Framework. J Chem Theory Comput 2020; 16:4087-4097. [DOI: 10.1021/acs.jctc.0c00379] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK−8000 Aarhus C, Denmark
| | - Mads Bøttger Hansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK−8000 Aarhus C, Denmark
| | - Graham A. Worth
- Department of Chemistry, University College London, 20, Gordon St., WC1H 0AJ London, United Kingdom
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK−8000 Aarhus C, Denmark
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10
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Artiukhin DG, Klinting EL, König C, Christiansen O. Adaptive density-guided approach to double incremental potential energy surface construction. J Chem Phys 2020; 152:194105. [PMID: 33687258 DOI: 10.1063/5.0004686] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
We present a combination of the recently developed double incremental expansion of potential energy surfaces with the well-established adaptive density-guided approach to grid construction. This unique methodology is based on the use of an incremental expansion for potential energy surfaces, known as n-mode expansion; an incremental many-body representation of the electronic energy; and an efficient vibrational density-guided approach to automated determination of grid dimensions and granularity. The reliability of the method is validated calculating potential energy surfaces and obtaining fundamental excitation energies for three moderate-size chain-like molecular systems. The use of our methodology leads to considerable computational savings for potential energy surface construction compared to standard approaches while maintaining a high level of accuracy in the resulting potential energy surfaces. Additional investigations indicate that our method can be applied to covalently bound and strongly interacting molecular systems, even though these cases are known to be very unfavorable for fragmentation schemes. We therefore conclude that the presented methodology is a robust and flexible approach to potential energy surface construction, which introduces considerable computational savings without compromising the accuracy of vibrational spectra calculations.
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Affiliation(s)
- Denis G Artiukhin
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
| | | | - Carolin König
- Institute of Physical Chemistry, Christian-Albrechts-University Kiel, Max-Eyth-Straße 1, D-24118 Kiel, Germany
| | - Ove Christiansen
- Department of Chemistry, Aarhus Universitet, DK-8000 Aarhus, Denmark
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11
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Klinting EL, Lauvergnat D, Christiansen O. Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator. J Chem Theory Comput 2020; 16:4505-4520. [DOI: 10.1021/acs.jctc.0c00261] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Affiliation(s)
| | - David Lauvergnat
- Université Paris-Saclay, CNRS, Institut de Chimie Physique, UMR8000, 91405 Orsay, France
| | - Ove Christiansen
- Department of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark
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12
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Klinting EL, Christiansen O, König C. Toward Accurate Theoretical Vibrational Spectra: A Case Study for Maleimide. J Phys Chem A 2020; 124:2616-2627. [DOI: 10.1021/acs.jpca.9b11915] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Affiliation(s)
| | - Ove Christiansen
- Department of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark
| | - Carolin König
- Institute of Physical Chemistry, Kiel University, Max-Eyth-Straße 1, D-24118 Kiel, Germany
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13
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Madsen NK, Hansen MB, Worth GA, Christiansen O. Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy. J Chem Phys 2020; 152:084101. [DOI: 10.1063/1.5142459] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Mads Bøttger Hansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
| | - Graham A. Worth
- Department of Chemistry, University College London, 20, Gordon St., WC1H 0AJ London, United Kingdom
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK–8000 Aarhus C, Denmark
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14
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Avila G, Papp D, Czakó G, Mátyus E. Exact quantum dynamics background of dispersion interactions: case study for CH 4·Ar in full (12) dimensions. Phys Chem Chem Phys 2020; 22:2792-2802. [PMID: 31957778 DOI: 10.1039/c9cp04426d] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A full-dimensional ab initio potential energy surface of spectroscopic quality is developed for the van-der-Waals complex of a methane molecule and an argon atom. Variational vibrational states are computed on this surface including all twelve (12) vibrational degrees of freedom of the methane-argon complex using the GENIUSH computer program and the Smolyak sparse grid method. The full-dimensional computations make it possible to study the fine details of the interaction and distortion effects and to make a direct assessment of the reduced-dimensionality models often used in the quantum dynamics study of weakly-bound complexes. A 12-dimensional (12D) vibrational computation including only a single harmonic oscillator basis function (9D) to describe the methane fragment (for which we use the ground-state effective structure as the reference structure) has a 0.40 cm-1 root-mean-square error (rms) with respect to the converged 12D bound-state excitation energies, which is less than half of the rms of the 3D model set up with the 〈r〉0 methane structure. Allowing 10 basis functions for the methane fragment in a 12D computation performs much better than the 3D models by reducing the rms of the bound state vibrational energies to 0.07 cm-1. The full-dimensional potential energy surface correctly describes the dissociation of the system, which together with further development of the variational (ro)vibrational methodology opens a route to the study of the role of dispersion forces in the excited methane vibrations and the energy transfer from the intra- to the intermolecular vibrational modes.
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Affiliation(s)
- Gustavo Avila
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary.
| | - Dóra Papp
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Rerrich Béla tér 1, Szeged H-6720, Hungary.
| | - Gábor Czakó
- MTA-SZTE Lendület Computational Reaction Dynamics Research Group, Interdisciplinary Excellence Centre and Department of Physical Chemistry and Materials Science, Institute of Chemistry, University of Szeged, Rerrich Béla tér 1, Szeged H-6720, Hungary.
| | - Edit Mátyus
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary.
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Pokhilko P, Izmodenov D, Krylov AI. Extension of frozen natural orbital approximation to open-shell references: Theory, implementation, and application to single-molecule magnets. J Chem Phys 2020; 152:034105. [DOI: 10.1063/1.5138643] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Affiliation(s)
- Pavel Pokhilko
- Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA
| | - Daniil Izmodenov
- Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA
- Department of Chemistry, Lomonosov Moscow State University, Moscow 119991, Russia
| | - Anna I. Krylov
- Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA
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16
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Madsen NK, Hansen MB, Zoccante A, Monrad K, Hansen MB, Christiansen O. Exponential parameterization of wave functions for quantum dynamics: Time-dependent Hartree in second quantization. J Chem Phys 2018; 149:134110. [DOI: 10.1063/1.5049344] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Niels Kristian Madsen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Mads Bøttger Hansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Alberto Zoccante
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Kasper Monrad
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Mikkel Bo Hansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
| | - Ove Christiansen
- Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark
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17
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Thomas PS, Carrington T, Agarwal J, Schaefer HF. Using an iterative eigensolver and intertwined rank reduction to compute vibrational spectra of molecules with more than a dozen atoms: Uracil and naphthalene. J Chem Phys 2018; 149:064108. [DOI: 10.1063/1.5039147] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
Affiliation(s)
- Phillip S. Thomas
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Jay Agarwal
- Center for Computational Chemistry, University of Georgia, Athens, Georgia 30602-0525, USA
| | - Henry F. Schaefer
- Center for Computational Chemistry, University of Georgia, Athens, Georgia 30602-0525, USA
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