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Nykänen A, Miller A, Talarico W, Knecht S, Kovyrshin A, Skogh M, Tornberg L, Broo A, Mensa S, Symons BCB, Sahin E, Crain J, Tavernelli I, Pavošević F. Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals. J Chem Theory Comput 2023; 19:9269-9277. [PMID: 38081802 DOI: 10.1021/acs.jctc.3c01091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2023]
Abstract
Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.
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Affiliation(s)
- Anton Nykänen
- Algorithmiq Ltd., Kanavakatu 3C, Helsinki FI-00160, Finland
| | - Aaron Miller
- Algorithmiq Ltd., Kanavakatu 3C, Helsinki FI-00160, Finland
- School of Physics, Trinity College Dublin, College Green Dublin 2, Ireland
| | - Walter Talarico
- Algorithmiq Ltd., Kanavakatu 3C, Helsinki FI-00160, Finland
- Department of Applied Physics, QTF Centre of Excellence, Center for Quantum Engineering, Aalto University School of Science, Aalto FIN-00076, Finland
| | - Stefan Knecht
- Algorithmiq Ltd., Kanavakatu 3C, Helsinki FI-00160, Finland
- ETH Zürich, Department of Chemistry and Applied Life Sciences Vladimir-Prelog-Weg 1-5/10, Zürich 8093, Switzerland
| | - Arseny Kovyrshin
- Data Science and Modelling, Pharmaceutical Sciences, R&D, AstraZeneca Gothenburg, Pepparedsleden 1, Molndal SE-431 83, Sweden
| | - Mårten Skogh
- Data Science and Modelling, Pharmaceutical Sciences, R&D, AstraZeneca Gothenburg, Pepparedsleden 1, Molndal SE-431 83, Sweden
- Department of Chemistry and Chemical Engineering, Chalmers University of Technology, Gothenburg 412 96, Sweden
| | - Lars Tornberg
- Data Science and Modelling, Pharmaceutical Sciences, R&D, AstraZeneca Gothenburg, Pepparedsleden 1, Molndal SE-431 83, Sweden
| | - Anders Broo
- Data Science and Modelling, Pharmaceutical Sciences, R&D, AstraZeneca Gothenburg, Pepparedsleden 1, Molndal SE-431 83, Sweden
| | - Stefano Mensa
- The Hartree Centre, STFC, Sci-Tech Daresbury, Warrington WA4 4AD, U.K
| | | | - Emre Sahin
- The Hartree Centre, STFC, Sci-Tech Daresbury, Warrington WA4 4AD, U.K
| | - Jason Crain
- IBM Research Europe, Hartree Centre STFC Laboratory, Sci-Tech Daresbury, Warrington WA4 4AD, U.K
- Department of Biochemistry, University of Oxford, Oxford OX1 3QU, U.K
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Chen Z, Yang J. Nucleus-electron correlation revising molecular bonding fingerprints from the exact wavefunction factorization. J Chem Phys 2021; 155:104111. [PMID: 34525813 DOI: 10.1063/5.0056773] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
We present a novel theory and implementation for computing coupled electronic and quantal nuclear subsystems on a single potential energy surface, moving beyond the standard Born-Oppenheimer (BO) separation of nuclei and electrons. We formulate an exact self-consistent nucleus-electron embedding potential from the single product molecular wavefunction and demonstrate that the fundamental behavior of the correlated nucleus-electron can be computed for mean-field electrons that are responsive to a quantal anharmonic vibration of selected nuclei in a discrete variable representation. Geometric gauge choices are discussed and necessary for formulating energy invariant biorthogonal electronic equations. Our method is further applied to characterize vibrationally averaged molecular bonding properties of molecular energetics, bond lengths, and protonic and electron densities. Moreover, post-Hartree-Fock electron correlation can be conveniently computed on the basis of nucleus-electron coupled molecular orbitals, as demonstrated for correlated models of second-order Møllet-Plesset perturbation and full configuration interaction theories. Our approach not only accurately quantifies non-classical nucleus-electron couplings for revising molecular bonding properties but also provides an alternative time-independent approach for deploying non-BO molecular quantum chemistry.
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Affiliation(s)
- Ziyong Chen
- Department of Chemistry, The University of Hong Kong, Hong Kong, People's Republic of China
| | - Jun Yang
- Department of Chemistry, The University of Hong Kong, Hong Kong, People's Republic of China
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3
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Muolo A, Mátyus E, Reiher M. Erratum: "Generalized elimination of the global translation from explicitly correlated Gaussian functions" [J. Chem. Phys. 148, 084112 (2018)]. J Chem Phys 2019; 151:039901. [PMID: 31325930 DOI: 10.1063/1.5113958] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Andrea Muolo
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Edit Mátyus
- Institute of Chemistry, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
| | - Markus Reiher
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
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Mátyus E, Teufel S. Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states. J Chem Phys 2019; 151:014113. [PMID: 31272174 DOI: 10.1063/1.5097899] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The quantum mechanical motion of the atomic nuclei is considered over a single- or a multidimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to O(εn+1) through a unitary transformation of the electronic subspace, and the corresponding nth-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulas are given for the second- and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multidimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual Born-Oppenheimer terms and nonadiabatic corrections, but generalized mass-correction terms appear as well. These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace.
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Affiliation(s)
- Edit Mátyus
- Institute of Chemistry, ELTE, Eötvös Loránd University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
| | - Stefan Teufel
- Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
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5
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Muolo A, Mátyus E, Reiher M. Generalized elimination of the global translation from explicitly correlated Gaussian functions. J Chem Phys 2018; 148:084112. [PMID: 29495776 DOI: 10.1063/1.5009465] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper presents the multi-channel generalization of the center-of-mass kinetic energy elimination approach [B. Simmen et al., Mol. Phys. 111, 2086 (2013)] when the Schrödinger equation is solved variationally with explicitly correlated Gaussian functions. The approach has immediate relevance in many-particle systems which are handled without the Born-Oppenheimer approximation and can be employed also for Dirac-type Hamiltonians. The practical realization and numerical properties of solving the Schrödinger equation in laboratory-frame Cartesian coordinates are demonstrated for the ground rovibronic state of the H2+={p+,p+,e-} ion and the H2 = {p+, p+, e-, e-} molecule.
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Affiliation(s)
- Andrea Muolo
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Edit Mátyus
- Institute of Chemistry, Eötvös Loránd University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
| | - Markus Reiher
- ETH Zürich, Laboratory of Physical Chemistry, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
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6
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Pachucki K, Komasa J. Nonadiabatic rotational states of the hydrogen molecule. Phys Chem Chem Phys 2018; 20:247-255. [PMID: 29200217 DOI: 10.1039/c7cp06516g] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
Abstract
We present a new computational method for the determination of energy levels in four-particle systems like H2, HD, and HeH+ using explicitly correlated exponential basis functions and analytic integration formulas. In solving the Schrödinger equation, no adiabatic separation of the nuclear and electronic degrees of freedom is introduced. We provide formulas for the coupling between the rotational and electronic angular momenta, which enable calculations of arbitrary rotationally excited energy levels. To illustrate the high numerical efficiency of the method, we present the results for various states of the hydrogen molecule. The relative accuracy to which we determined the nonrelativistic energy reached the level of 10-12-10-13, which corresponds to an uncertainty of 10-7-10-8 cm-1.
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Affiliation(s)
- Krzysztof Pachucki
- Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland.
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Cassam-Chenaï P, Suo B, Liu W. A quantum chemical definition of electron–nucleus correlation. Theor Chem Acc 2017. [DOI: 10.1007/s00214-017-2081-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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8
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Non-Born-Oppenheimer calculations of the rovibrational spectrum of H2 excited to the second rotational level. Chem Phys Lett 2017. [DOI: 10.1016/j.cplett.2016.12.049] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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9
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Pachucki K, Komasa J. Schrödinger equation solved for the hydrogen molecule with unprecedented accuracy. J Chem Phys 2016; 144:164306. [DOI: 10.1063/1.4948309] [Citation(s) in RCA: 34] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Krzysztof Pachucki
- Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Jacek Komasa
- Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland
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10
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Abstract
The Stochastic Variational Method (SVM) is used to show that the effective mass model correctly estimates the binding energies of excitons and trions but fails to predict the experimental binding energy of the biexciton. Using high-accuracy variational calculations, it is demonstrated that the biexciton binding energy in transition metal dichalcogenides is smaller than the trion binding energy, contradicting experimental findings. It is also shown that the biexciton has bound excited states and that the binding energy of the L = 0 excited state is in very good agreement with experimental data. This excited state corresponds to a hole attached to a negative trion and may be a possible resolution of the discrepancy between theory and experiment.
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Affiliation(s)
- David K Zhang
- Department of Physics and Astronomy, Vanderbilt University , Nashville, Tennessee 37235, United States
| | - Daniel W Kidd
- Department of Physics and Astronomy, Vanderbilt University , Nashville, Tennessee 37235, United States
| | - Kálmán Varga
- Department of Physics and Astronomy, Vanderbilt University , Nashville, Tennessee 37235, United States
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11
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Pachucki K, Komasa J. Leading order nonadiabatic corrections to rovibrational levels of H2, D2, and T2. J Chem Phys 2015. [DOI: 10.1063/1.4927079] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Krzysztof Pachucki
- Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Jacek Komasa
- Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland
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12
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Simmen B, Mátyus E, Reiher M. Electric transition dipole moment in pre-Born-Oppenheimer molecular structure theory. J Chem Phys 2014; 141:154105. [PMID: 25338879 DOI: 10.1063/1.4897632] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper presents the calculation of the electric transition dipole moment in a pre-Born-Oppenheimer framework. Electrons and nuclei are treated equally in terms of the parametrization of the non-relativistic total wave function, which is written as a linear combination of basis functions constructed from explicitly correlated Gaussian functions and the global vector representation. The integrals of the electric transition dipole moment are derived corresponding to these basis functions in both the length and the velocity representation. The calculations are performed in laboratory-fixed Cartesian coordinates without relying on coordinates which separate the center of mass from the translationally invariant degrees of freedom. The effect of the overall motion is eliminated through translationally invariant integral expressions. The electric transition dipole moment is calculated between two rovibronic levels of the H2 molecule assignable to the lowest rovibrational states of the X (1)Σ(g)(+) and B (1)Σ(u)(+) electronic states in the clamped-nuclei framework. This is the first evaluation of this quantity in a full quantum mechanical treatment without relying on the Born-Oppenheimer approximation.
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Affiliation(s)
- Benjamin Simmen
- ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
| | - Edit Mátyus
- Eövtös University, Institute of Chemistry, P.O. Box 32, H-1518, Budapest 112, Hungary
| | - Markus Reiher
- ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland
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13
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Aguirre NF, Villarreal P, Delgado-Barrio G, Posada E, Reyes A, Biczysko M, Mitrushchenkov AO, de Lara-Castells MP. Including nuclear quantum effects into highly correlated electronic structure calculations of weakly bound systems. J Chem Phys 2013; 138:184113. [DOI: 10.1063/1.4803546] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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14
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Diniz LG, Alijah A, Mohallem JR. Core-mass nonadiabatic corrections to molecules: H2, H2+, and isotopologues. J Chem Phys 2013; 137:164316. [PMID: 23126719 DOI: 10.1063/1.4762442] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/02/2023] Open
Abstract
For high-precision calculations of rovibrational states of light molecules, it is essential to include non-adiabatic corrections. In the absence of crossings of potential energy surfaces, they can be incorporated in a single surface picture through coordinate-dependent vibrational and rotational reduced masses. We present a compact method for their evaluation and relate in particular the vibrational mass to a well defined nuclear core mass derived from a Mulliken analysis of the electronic density. For the rotational mass we propose a simple, but very effective parametrization. The use of these masses in the nuclear Schrödinger equation yields numerical data for the corrections of a much higher quality than can be obtained with optimized constant masses, typically better than 0.1 cm(-1). We demonstrate the method for H(2), H(2)(+), and singly deuterated isotopologues. Isotopic asymmetry does not present any particular difficulty. Generalization to polyatomic molecules is straightforward.
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Affiliation(s)
- Leonardo G Diniz
- Laboratório de Átomos e Moléculas Especiais, Departamento de Física, ICEx, Universidade Federal de Minas Gerais, P. O. Box 702, 30123-970 Belo Horizonte, MG, Brazil.
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15
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Bubin S, Pavanello M, Tung WC, Sharkey KL, Adamowicz L. Born–Oppenheimer and Non-Born–Oppenheimer, Atomic and Molecular Calculations with Explicitly Correlated Gaussians. Chem Rev 2012; 113:36-79. [DOI: 10.1021/cr200419d] [Citation(s) in RCA: 115] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Sergiy Bubin
- Department
of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235,
United States
| | - Michele Pavanello
- Department
of Chemistry, Rutgers University Newark, Newark, New Jersey 07102,
United States
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Mátyus E, Reiher M. Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation. J Chem Phys 2012; 137:024104. [DOI: 10.1063/1.4731696] [Citation(s) in RCA: 53] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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17
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Goli M, Shahbazian S. The two-component quantum theory of atoms in molecules (TC-QTAIM): foundations. Theor Chem Acc 2012. [DOI: 10.1007/s00214-012-1208-9] [Citation(s) in RCA: 33] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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18
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Mátyus E, Hutter J, Müller-Herold U, Reiher M. Extracting elements of molecular structure from the all-particle wave function. J Chem Phys 2011; 135:204302. [DOI: 10.1063/1.3662487] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
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19
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Tennyson J. Accurate variational calculations for line lists to model the vibration-rotation spectra of hot astrophysical atmospheres. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2011. [DOI: 10.1002/wcms.94] [Citation(s) in RCA: 44] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
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20
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Bubin S, Stanke M, Adamowicz L. Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with including relativistic corrections. J Chem Phys 2011; 135:074110. [DOI: 10.1063/1.3625955] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
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21
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Accurate non-Born–Oppenheimer calculations of the lowest vibrational energies of D2 and T2 with including relativistic corrections. Chem Phys Lett 2010. [DOI: 10.1016/j.cplett.2010.05.081] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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22
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An empirical formula to estimate off-diagonal adiabatic corrections to rotation–vibrational energy levels. Theor Chem Acc 2009. [DOI: 10.1007/s00214-009-0710-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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23
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Piszczatowski K, Łach G, Przybytek M, Komasa J, Pachucki K, Jeziorski B. Theoretical Determination of the Dissociation Energy of Molecular Hydrogen. J Chem Theory Comput 2009; 5:3039-48. [DOI: 10.1021/ct900391p] [Citation(s) in RCA: 149] [Impact Index Per Article: 9.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Konrad Piszczatowski
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
| | - Grzegorz Łach
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
| | - Michal Przybytek
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
| | - Jacek Komasa
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
| | - Krzysztof Pachucki
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
| | - Bogumil Jeziorski
- Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland, Center for Theoretical and Computational Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway, Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland, and Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland
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