1
|
Sun K, Ribeiro RF. Theoretical formulation of chemical equilibrium under vibrational strong coupling. Nat Commun 2024; 15:2405. [PMID: 38493189 PMCID: PMC10944518 DOI: 10.1038/s41467-024-46442-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2023] [Accepted: 02/28/2024] [Indexed: 03/18/2024] Open
Abstract
Experiments have suggested that strong interactions between molecular ensembles and infrared microcavities can be employed to control chemical equilibria. Nevertheless, the primary mechanism and key features of the effect remain largely unexplored. In this work, we develop a theory of chemical equilibrium in optical microcavities, which allows us to relate the equilibrium composition of a mixture in different electromagnetic environments. Our theory shows that in planar microcavities under strong coupling with polyatomic molecules, hybrid modes formed between all dipole-active vibrations and cavity resonances contribute to polariton-assisted chemical equilibrium shifts. To illustrate key aspects of our formalism, we explore a model SN2 reaction within a single-mode infrared resonator. Our findings reveal that chemical equilibria can be shifted towards either direction of a chemical reaction, depending on the oscillator strength and frequencies of reactant and product normal modes. Polariton-induced zero-point energy changes provide the dominant contributions, though the effects in idealized single-mode cavities tend to diminish quickly as the temperature and number of molecules increase. Our approach is valid in generic electromagnetic environments and paves the way for understanding and controlling chemical equilibria with microcavities.
Collapse
Affiliation(s)
- Kaihong Sun
- Department of Chemistry and Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, GA, 30322, USA
| | - Raphael F Ribeiro
- Department of Chemistry and Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, GA, 30322, USA.
| |
Collapse
|
2
|
Sidler D, Ruggenthaler M, Rubio A. Numerically Exact Solution for a Real Polaritonic System under Vibrational Strong Coupling in Thermodynamic Equilibrium: Loss of Light-Matter Entanglement and Enhanced Fluctuations. J Chem Theory Comput 2023; 19:8801-8814. [PMID: 37972347 PMCID: PMC10720342 DOI: 10.1021/acs.jctc.3c00092] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2023] [Revised: 10/25/2023] [Accepted: 10/25/2023] [Indexed: 11/19/2023]
Abstract
The first numerically exact simulation of a full ab initio molecular quantum system (HD+) under strong ro-vibrational coupling to a quantized optical cavity mode in thermal equilibrium is presented. Theoretical challenges in describing strongly coupled systems of mixed quantum statistics (bosons and Fermions) are discussed and circumvented by the specific choice of our molecular system. Our numerically exact simulations highlight the absence of zero temperature for the strongly coupled matter and light subsystems, due to cavity-induced noncanonical conditions. Furthermore, we explore the temperature dependency of light-matter quantum entanglement, which emerges for the ground state but is quickly lost already in the deep cryogenic regime. This is in contrast to predictions from the Jaynes-Cummings model, which is the standard starting point to model collective strong-coupling chemistry phenomenologically. Moreover, we find that the fluctuations of matter remain modified by the quantum nature of the thermal and vacuum-field fluctuations for significant temperatures, e.g., at ambient conditions. These observations (loss of entanglement and coupling to quantum fluctuations) have implications for the understanding and control of polaritonic chemistry and materials science, since a semiclassical theoretical description of light-matter interaction becomes reasonable, but the typical (classical) canonical equilibrium assumption for the nuclear subsystem remains violated. This opens the door for quantum fluctuation-induced stochastic resonance phenomena under vibrational strong coupling, which have been suggested as a plausible theoretical mechanism to explain the experimentally observed resonance phenomena in the absence of periodic driving that has not yet been fully understood.
Collapse
Affiliation(s)
- Dominik Sidler
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, Hamburg 22761, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany
| | - Michael Ruggenthaler
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, Hamburg 22761, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany
| | - Angel Rubio
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, Hamburg 22761, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany
- Center
for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United States
- Nano-Bio
Spectroscopy Group, University of the Basque Country (UPV/EHU), San Sebastián 20018, Spain
| |
Collapse
|
3
|
Ruggenthaler M, Sidler D, Rubio A. Understanding Polaritonic Chemistry from Ab Initio Quantum Electrodynamics. Chem Rev 2023; 123:11191-11229. [PMID: 37729114 PMCID: PMC10571044 DOI: 10.1021/acs.chemrev.2c00788] [Citation(s) in RCA: 21] [Impact Index Per Article: 21.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2022] [Indexed: 09/22/2023]
Abstract
In this review, we present the theoretical foundations and first-principles frameworks to describe quantum matter within quantum electrodynamics (QED) in the low-energy regime, with a focus on polaritonic chemistry. By starting from fundamental physical and mathematical principles, we first review in great detail ab initio nonrelativistic QED. The resulting Pauli-Fierz quantum field theory serves as a cornerstone for the development of (in principle exact but in practice) approximate computational methods such as quantum-electrodynamical density functional theory, QED coupled cluster, or cavity Born-Oppenheimer molecular dynamics. These methods treat light and matter on equal footing and, at the same time, have the same level of accuracy and reliability as established methods of computational chemistry and electronic structure theory. After an overview of the key ideas behind those ab initio QED methods, we highlight their benefits for understanding photon-induced changes of chemical properties and reactions. Based on results obtained by ab initio QED methods, we identify open theoretical questions and how a so far missing detailed understanding of polaritonic chemistry can be established. We finally give an outlook on future directions within polaritonic chemistry and first-principles QED.
Collapse
Affiliation(s)
- Michael Ruggenthaler
- Max-Planck-Institut
für Struktur und Dynamik der Materie, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Dominik Sidler
- Max-Planck-Institut
für Struktur und Dynamik der Materie, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Angel Rubio
- Max-Planck-Institut
für Struktur und Dynamik der Materie, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United States
| |
Collapse
|
4
|
Mandal A, Taylor MA, Weight BM, Koessler ER, Li X, Huo P. Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrodynamics. Chem Rev 2023; 123:9786-9879. [PMID: 37552606 PMCID: PMC10450711 DOI: 10.1021/acs.chemrev.2c00855] [Citation(s) in RCA: 31] [Impact Index Per Article: 31.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2022] [Indexed: 08/10/2023]
Abstract
When molecules are coupled to an optical cavity, new light-matter hybrid states, so-called polaritons, are formed due to quantum light-matter interactions. With the experimental demonstrations of modifying chemical reactivities by forming polaritons under strong light-matter interactions, theorists have been encouraged to develop new methods to simulate these systems and discover new strategies to tune and control reactions. This review summarizes some of these exciting theoretical advances in polariton chemistry, in methods ranging from the fundamental framework to computational techniques and applications spanning from photochemistry to vibrational strong coupling. Even though the theory of quantum light-matter interactions goes back to the midtwentieth century, the gaps in the knowledge of molecular quantum electrodynamics (QED) have only recently been filled. We review recent advances made in resolving gauge ambiguities, the correct form of different QED Hamiltonians under different gauges, and their connections to various quantum optics models. Then, we review recently developed ab initio QED approaches which can accurately describe polariton states in a realistic molecule-cavity hybrid system. We then discuss applications using these method advancements. We review advancements in polariton photochemistry where the cavity is made resonant to electronic transitions to control molecular nonadiabatic excited state dynamics and enable new photochemical reactivities. When the cavity resonance is tuned to the molecular vibrations instead, ground-state chemical reaction modifications have been demonstrated experimentally, though its mechanistic principle remains unclear. We present some recent theoretical progress in resolving this mystery. Finally, we review the recent advances in understanding the collective coupling regime between light and matter, where many molecules can collectively couple to a single cavity mode or many cavity modes. We also lay out the current challenges in theory to explain the observed experimental results. We hope that this review will serve as a useful document for anyone who wants to become familiar with the context of polariton chemistry and molecular cavity QED and thus significantly benefit the entire community.
Collapse
Affiliation(s)
- Arkajit Mandal
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
- Department
of Chemistry, Columbia University, New York, New York 10027, United States
| | - Michael A.D. Taylor
- The
Institute of Optics, Hajim School of Engineering, University of Rochester, Rochester, New York 14627, United States
| | - Braden M. Weight
- Department
of Physics and Astronomy, University of
Rochester, Rochester, New York 14627, United
States
| | - Eric R. Koessler
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
| | - Xinyang Li
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
- Theoretical
Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States
| | - Pengfei Huo
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
- The
Institute of Optics, Hajim School of Engineering, University of Rochester, Rochester, New York 14627, United States
| |
Collapse
|
5
|
Mandal A, Taylor MAD, Huo P. Theory for Cavity-Modified Ground-State Reactivities via Electron-Photon Interactions. J Phys Chem A 2023; 127:6830-6841. [PMID: 37499090 PMCID: PMC10440810 DOI: 10.1021/acs.jpca.3c01421] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2023] [Revised: 06/27/2023] [Indexed: 07/29/2023]
Abstract
We provide a simple and intuitive theory to explain how coupling a molecule to an optical cavity can modify ground-state chemical reactivity by exploiting intrinsic quantum behaviors of light-matter interactions. Using the recently developed polarized Fock states representation, we demonstrate that the change of the ground-state potential is achieved due to the scaling of diabatic electronic couplings with the overlap of the polarized Fock states. Our theory predicts that for a proton-transfer model system, the ground-state barrier height can be modified through light-matter interactions when the cavity frequency is in the electronic excitation range. Our simple theory explains several recent computational investigations that discovered the same effect. We further demonstrate that under the deep strong coupling limit of the light and matter, the polaritonic ground and first excited eigenstates become the Mulliken-Hush diabatic states, which are the eigenstates of the dipole operator. This work provides a simple but powerful theoretical framework to understand how strong coupling between the molecule and the cavity can modify ground-state reactivities.
Collapse
Affiliation(s)
- Arkajit Mandal
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
- Department
of Chemistry, Columbia University, New York, New York 10027, United States
| | - Michael A. D. Taylor
- Institute
of Optics, Hajim School of Engineering, University of Rochester, Rochester, New York 14627, United States
| | - Pengfei Huo
- Department
of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, United States
- Institute
of Optics, Hajim School of Engineering, University of Rochester, Rochester, New York 14627, United States
| |
Collapse
|
6
|
Sidler D, Ruggenthaler M, Schäfer C, Ronca E, Rubio A. A perspective on ab initio modeling of polaritonic chemistry: The role of non-equilibrium effects and quantum collectivity. J Chem Phys 2022; 156:230901. [PMID: 35732522 DOI: 10.1063/5.0094956] [Citation(s) in RCA: 20] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
This Perspective provides a brief introduction into the theoretical complexity of polaritonic chemistry, which emerges from the hybrid nature of strongly coupled light-matter states. To tackle this complexity, the importance of ab initio methods is highlighted. Based on those, novel ideas and research avenues are developed with respect to quantum collectivity, as well as for resonance phenomena immanent in reaction rates under vibrational strong coupling. Indeed, fundamental theoretical questions arise about the mesoscopic scale of quantum-collectively coupled molecules when considering the depolarization shift in the interpretation of experimental data. Furthermore, to rationalize recent findings based on quantum electrodynamical density-functional theory (QEDFT), a simple, but computationally efficient, Langevin framework is proposed based on well-established methods from molecular dynamics. It suggests the emergence of cavity-induced non-equilibrium nuclear dynamics, where thermal (stochastic) resonance phenomena could emerge in the absence of external periodic driving. Overall, we believe that the latest ab initio results indeed suggest a paradigmatic shift for ground-state chemical reactions under vibrational strong coupling from the collective quantum interpretation toward a more local, (semi)-classically and non-equilibrium dominated perspective. Finally, various extensions toward a refined description of cavity-modified chemistry are introduced in the context of QEDFT, and future directions of the field are sketched.
Collapse
Affiliation(s)
- Dominik Sidler
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Michael Ruggenthaler
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Christian Schäfer
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Enrico Ronca
- Istituto per i Processi Chimico Fisici del CNR (IPCF-CNR), Via G. Moruzzi, 1, 56124 Pisa, Italy
| | - Angel Rubio
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| |
Collapse
|
7
|
Yang J, Pei Z, Leon EC, Wickizer C, Weng B, Mao Y, Ou Q, Shao Y. Cavity quantum-electrodynamical time-dependent density functional theory within Gaussian atomic basis. II. Analytic energy gradient. J Chem Phys 2022; 156:124104. [DOI: 10.1063/5.0082386] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Abstract
Following the formulation of cavity quantum-electrodynamical time-dependent density functional theory (cQED-TDDFT) models [Flick et al., ACS Photonics 6, 2757–2778 (2019) and Yang et al., J. Chem. Phys. 155, 064107 (2021)], here, we report the derivation and implementation of the analytic energy gradient for polaritonic states of a single photochrome within the cQED-TDDFT models. Such gradient evaluation is also applicable to a complex of explicitly specified photochromes or, with proper scaling, a set of parallel-oriented, identical-geometry, and non-interacting molecules in the microcavity.
Collapse
Affiliation(s)
- Junjie Yang
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019, USA
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - Zheng Pei
- State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, People’s Republic of China
| | - Erick Calderon Leon
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019, USA
| | - Carly Wickizer
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019, USA
| | - Binbin Weng
- Microfabrication Research and Education Center and School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma 73019, USA
| | - Yuezhi Mao
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
| | - Qi Ou
- MOE Key Laboratory of Organic OptoElectronics and Molecular Engineering, Department of Chemistry, Tsinghua University, Beijing 100084, China
- AI for Science Institute, Beijing 100080, China
| | - Yihan Shao
- Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73019, USA
| |
Collapse
|
8
|
Riso RR, Haugland TS, Ronca E, Koch H. Molecular orbital theory in cavity QED environments. Nat Commun 2022; 13:1368. [PMID: 35292631 PMCID: PMC8924263 DOI: 10.1038/s41467-022-29003-2] [Citation(s) in RCA: 21] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2021] [Accepted: 02/23/2022] [Indexed: 02/02/2023] Open
Abstract
Coupling between molecules and vacuum photon fields inside an optical cavity has proven to be an effective way to engineer molecular properties, in particular reactivity. To ease the rationalization of cavity induced effects we introduce an ab initio method leading to the first fully consistent molecular orbital theory for quantum electrodynamics environments. Our framework is non-perturbative and explains modifications of the electronic structure due to the interaction with the photon field. In this work, we show that the newly developed orbital theory can be used to predict cavity induced modifications of molecular reactivity and pinpoint classes of systems with significant cavity effects. We also investigate electronic cavity-induced modifications of reaction mechanisms in vibrational strong coupling regimes.
Collapse
Affiliation(s)
- Rosario R Riso
- Department of Chemistry, Norwegian University of Science and Technology, 7491, Trondheim, Norway
| | - Tor S Haugland
- Department of Chemistry, Norwegian University of Science and Technology, 7491, Trondheim, Norway
| | - Enrico Ronca
- Istituto per i Processi Chimico Fisici del CNR (IPCF-CNR), Via G. Moruzzi, 1, 56124, Pisa, Italy
| | - Henrik Koch
- Department of Chemistry, Norwegian University of Science and Technology, 7491, Trondheim, Norway.
- Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy.
| |
Collapse
|
9
|
Liebenthal M, Vu NH, DePrince E. Equation-of-motion cavity quantum electrodynamics coupled-cluster theory for electron attachment. J Chem Phys 2022; 156:054105. [DOI: 10.1063/5.0078795] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - Nam Hoang Vu
- Chemistry & Biochemistry, Florida State University, United States of America
| | - Eugene DePrince
- Chemistry and Biochemistry, Florida State University, United States of America
| |
Collapse
|
10
|
Ahrens A, Huang C, Beutel M, Covington C, Varga K. Stochastic Variational Approach to Small Atoms and Molecules Coupled to Quantum Field Modes in Cavity QED. PHYSICAL REVIEW LETTERS 2021; 127:273601. [PMID: 35061426 DOI: 10.1103/physrevlett.127.273601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2021] [Accepted: 12/02/2021] [Indexed: 06/14/2023]
Abstract
In this work, we present a stochastic variational calculation (SVM) of energies and wave functions of few particle systems coupled to quantum fields in cavity QED. The spatial wave function and the photon spaces are optimized by a random selection process. Using correlated basis functions, the SVM approach solves the problem accurately and opens the way to the same precision that is reached the nonlight coupled quantum systems. Examples for a two-dimensional trion and confined electrons as well as for the He atom and the H_{2} molecule are presented showing that the light-matter coupling drastically changes the electronic states.
Collapse
Affiliation(s)
- Alexander Ahrens
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
| | - Chenhang Huang
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
| | - Matt Beutel
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
| | - Cody Covington
- Department of Chemistry, Austin Peay State University, Clarksville, Tennessee 37044, USA
| | - Kálmán Varga
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
| |
Collapse
|
11
|
Theophilou I, Penz M, Ruggenthaler M, Rubio A. Virial Relations for Electrons Coupled to Quantum Field Modes. J Chem Theory Comput 2020; 16:6236-6243. [PMID: 32816479 PMCID: PMC7558318 DOI: 10.1021/acs.jctc.0c00618] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Indexed: 11/28/2022]
Abstract
In this work, we present a set of virial relations for many electron systems coupled to both classical and quantum fields, described by the Pauli-Fierz Hamiltonian in dipole approximation and using length gauge. Currently, there is growing interest in solutions of this Hamiltonian because of its relevance for describing molecular systems strongly coupled to photonic modes in cavities and in the possible modification of chemical properties of such systems compared to the ones in free space. The relevance of such virial relations is demonstrated by showing a connection to mass renormalization and by providing an exact way to obtain total energies from potentials in the framework of quantum electrodynamical density functional theory.
Collapse
Affiliation(s)
- Iris Theophilou
- Theory
Department, Max Planck Institute for the
Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Markus Penz
- Theory
Department, Max Planck Institute for the
Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Michael Ruggenthaler
- Theory
Department, Max Planck Institute for the
Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Angel Rubio
- Theory
Department, Max Planck Institute for the
Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United
States
| |
Collapse
|