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Nakano K, Sorella S, Alfè D, Zen A. Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons. J Chem Theory Comput 2024; 20:4591-4604. [PMID: 38788330 PMCID: PMC11171267 DOI: 10.1021/acs.jctc.4c00139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2024] [Revised: 05/05/2024] [Accepted: 05/09/2024] [Indexed: 05/26/2024]
Abstract
Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.
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Affiliation(s)
- Kousuke Nakano
- Center
for Basic Research on Materials, National
Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan
- International
School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
| | - Sandro Sorella
- International
School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
| | - Dario Alfè
- Dipartimento
di Fisica Ettore Pancini, Università
di Napoli Federico II, Monte S. Angelo, 80126 Napoli, Italy
- Department
of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K.
- Thomas
Young Centre and London Centre for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, U.K.
| | - Andrea Zen
- Dipartimento
di Fisica Ettore Pancini, Università
di Napoli Federico II, Monte S. Angelo, 80126 Napoli, Italy
- Department
of Earth Sciences, University College London, Gower Street, London WC1E 6BT, U.K.
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Luo D, Clark BK. Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions. PHYSICAL REVIEW LETTERS 2019; 122:226401. [PMID: 31283262 DOI: 10.1103/physrevlett.122.226401] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2018] [Revised: 01/12/2019] [Indexed: 06/09/2023]
Abstract
Obtaining an accurate ground state wave function is one of the great challenges in the quantum many-body problem. In this Letter, we propose a new class of wave functions, neural network backflow (NNB). The backflow approach, pioneered originally by Feynman and Cohen [Phys. Rev. 102, 1189 (1956)10.1103/PhysRev.102.1189], adds correlation to a mean-field ground state by transforming the single-particle orbitals in a configuration-dependent way. NNB uses a feed-forward neural network to learn the optimal transformation via variational Monte Carlo calculations. NNB directly dresses a mean-field state, can be systematically improved, and directly alters the sign structure of the wave function. It generalizes the standard backflow [L. F. Tocchio et al., Phys. Rev. B 78, 041101(R) (2008)10.1103/PhysRevB.78.041101], which we show how to explicitly represent as a NNB. We benchmark the NNB on Hubbard models at intermediate doping, finding that it significantly decreases the relative error, restores the symmetry of both observables and single-particle orbitals, and decreases the double-occupancy density. Finally, we illustrate interesting patterns in the weights and bias of the optimized neural network.
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Affiliation(s)
- Di Luo
- Institute for Condensed Matter Theory and Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA
| | - Bryan K Clark
- Institute for Condensed Matter Theory and Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA
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3
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Liu YYF, Andrews B, Conduit GJ. Direct evaluation of the force constant matrix in quantum Monte Carlo. J Chem Phys 2019; 150:034104. [DOI: 10.1063/1.5070138] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Y. Y. F. Liu
- Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - B. Andrews
- Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - G. J. Conduit
- Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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4
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Azadi S, Cohen RE. Chemical accuracy from quantum Monte Carlo for the benzene dimer. J Chem Phys 2016; 143:104301. [PMID: 26374029 DOI: 10.1063/1.4930137] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We report an accurate study of interactions between benzene molecules using variational quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) methods. We compare these results with density functional theory using different van der Waals functionals. In our quantum Monte Carlo (QMC) calculations, we use accurate correlated trial wave functions including three-body Jastrow factors and backflow transformations. We consider two benzene molecules in the parallel displaced geometry, and find that by highly optimizing the wave function and introducing more dynamical correlation into the wave function, we compute the weak chemical binding energy between aromatic rings accurately. We find optimal VMC and DMC binding energies of -2.3(4) and -2.7(3) kcal/mol, respectively. The best estimate of the coupled-cluster theory through perturbative triplets/complete basis set limit is -2.65(2) kcal/mol [Miliordos et al., J. Phys. Chem. A 118, 7568 (2014)]. Our results indicate that QMC methods give chemical accuracy for weakly bound van der Waals molecular interactions, comparable to results from the best quantum chemistry methods.
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Affiliation(s)
- Sam Azadi
- Department of Earth Science and Thomas Young Centre, University College London, London WC1E 6BT, United Kingdom
| | - R E Cohen
- London Centre for Nanotechnology, University College London, London WC1E 6BT, United Kingdom and Extreme Materials Initiative, Geophysical Laboratory, Carnegie Institution of Washington, Washington, D.C. 20015, USA
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5
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Weerasinghe GL, Ríos PL, Needs RJ. Compression algorithm for multideterminant wave functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:023304. [PMID: 25353602 DOI: 10.1103/physreve.89.023304] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2013] [Indexed: 06/04/2023]
Abstract
A compression algorithm is introduced for multideterminant wave functions which can greatly reduce the number of determinants that need to be evaluated in quantum Monte Carlo calculations. We have devised an algorithm with three levels of compression, the least costly of which yields excellent results in polynomial time. We demonstrate the usefulness of the compression algorithm for evaluating multideterminant wave functions in quantum Monte Carlo calculations, whose computational cost is reduced by factors of between about 2 and over 25 for the examples studied. We have found evidence of sublinear scaling of quantum Monte Carlo calculations with the number of determinants when the compression algorithm is used.
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Affiliation(s)
- Gihan L Weerasinghe
- Theory of Condensed Matter Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Pablo López Ríos
- Theory of Condensed Matter Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Richard J Needs
- Theory of Condensed Matter Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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6
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Buendía E, Gálvez F, Maldonado P, Sarsa A. Quantum Monte Carlo ionization potential and electron affinity for transition metal atoms. Chem Phys Lett 2013. [DOI: 10.1016/j.cplett.2012.12.055] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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7
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Li HR, Cheng XL, Zhang H. Diffusion Monte Carlo Study of Bond Dissociation Energies for BH2, B(OH)2, BCl2, and BCl. CHINESE J CHEM PHYS 2012. [DOI: 10.1088/1674-0068/25/01/65-69] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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8
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Affiliation(s)
- Brian M. Austin
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
| | - Dmitry Yu. Zubarev
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
| | - William A. Lester
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
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9
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Lüchow A. Quantum Monte Carlo methods. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2011. [DOI: 10.1002/wcms.40] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
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Seth P, Ríos PL, Needs RJ. Quantum Monte Carlo study of the first-row atoms and ions. J Chem Phys 2011; 134:084105. [DOI: 10.1063/1.3554625] [Citation(s) in RCA: 40] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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11
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Berner R, Lüchow A. Isomerization of Bicyclo[1.1.0]butane by Means of the Diffusion Quantum Monte Carlo Method. J Phys Chem A 2010; 114:13222-7. [DOI: 10.1021/jp108605g] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Raphael Berner
- Institute of Physical Chemistry, RWTH Aachen University, 52056 Aachen, Germany
| | - Arne Lüchow
- Institute of Physical Chemistry, RWTH Aachen University, 52056 Aachen, Germany
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12
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Booth GH, Alavi A. Approaching chemical accuracy using full configuration-interaction quantum Monte Carlo: A study of ionization potentials. J Chem Phys 2010; 132:174104. [DOI: 10.1063/1.3407895] [Citation(s) in RCA: 99] [Impact Index Per Article: 7.1] [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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Gurtubay IG, Gaudoin R, Pitarke JM. Benchmark quantum Monte Carlo calculations of the ground-state kinetic, interaction and total energy of the three-dimensional electron gas. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:065501. [PMID: 21389370 DOI: 10.1088/0953-8984/22/6/065501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We report variational and diffusion quantum Monte Carlo ground-state energies of the three-dimensional electron gas using a model periodic Coulomb interaction and backflow corrections for N = 54, 102, 178, and 226 electrons. We remove finite-size effects by extrapolation and we find lower energies than previously reported. Using the Hellman-Feynman operator sampling method introduced in Gaudoin and Pitarke (2007 Phys. Rev. Lett. 99 126406), we compute accurately, within the fixed-node approximation, the separate kinetic and interaction contributions to the total ground-state energy. The difference between the interaction energies obtained from the original Slater-determinant nodes and the backflow-displaced nodes is found to be considerably larger than the difference between the corresponding kinetic energies.
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Affiliation(s)
- I G Gurtubay
- Materia Kondentsatuaren Fisika Saila, Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta kutxatila, E-48080 Bilbo, Basque Country, Spain.
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14
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Needs RJ, Towler MD, Drummond ND, López Ríos P. Continuum variational and diffusion quantum Monte Carlo calculations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:023201. [PMID: 21386247 DOI: 10.1088/0953-8984/22/2/023201] [Citation(s) in RCA: 163] [Impact Index Per Article: 11.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
This topical review describes the methodology of continuum variational and diffusion quantum Monte Carlo calculations. These stochastic methods are based on many-body wavefunctions and are capable of achieving very high accuracy. The algorithms are intrinsically parallel and well suited to implementation on petascale computers, and the computational cost scales as a polynomial in the number of particles. A guide to the systems and topics which have been investigated using these methods is given. The bulk of the article is devoted to an overview of the basic quantum Monte Carlo methods, the forms and optimization of wavefunctions, performing calculations under periodic boundary conditions, using pseudopotentials, excited-state calculations, sources of calculational inaccuracy, and calculating energy differences and forces.
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Affiliation(s)
- R J Needs
- Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge CB3 0HE, UK
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15
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Buendía E, Gálvez FJ, Maldonado P, Sarsa A. Quantum Monte Carlo ground state energies for the atoms Li through Ar. J Chem Phys 2009; 131:044115. [PMID: 19655845 DOI: 10.1063/1.3187526] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
All-electron quantum Monte Carlo energies are reported for the ground state of the atoms Li to Ar. The present work is mainly focused on the atoms Na to Ar as well as in those that have a stronger multiconfiguration nature, i.e., Be, B, and C and Mg, Al, and Si. Explicitly correlated wave functions with a single configuration model function times a Jastrow factor are employed for all of the atoms studied. The accuracy obtained for the atoms Na to Ar is similar to that reached for the atoms Li to Ne. In addition, a restricted multiconfiguration expansion has been employed for the atoms Be, B, and C and Mg, Al, and Si obtaining accurate results. Near degeneracy and the effect of other configurations are systematically analyzed for these systems, at both variational and diffusion Monte Carlo levels.
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Affiliation(s)
- E Buendía
- Departamento de Física Atómica Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain
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16
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Drummond ND, López Ríos P, Ma A, Trail JR, Spink GG, Towler MD, Needs RJ. Erratum: “Quantum Monte Carlo study of the Ne atom and the Ne+ ion” [J. Chem. Phys. 124, 224104 (2006)]. J Chem Phys 2009. [DOI: 10.1063/1.3245405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [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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Koseki J, Maezono R, Tachikawa M, Towler MD, Needs RJ. Quantum Monte Carlo study of porphyrin transition metal complexes. J Chem Phys 2009; 129:085103. [PMID: 19044853 DOI: 10.1063/1.2966003] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Diffusion quantum Monte Carlo (DMC) calculations for transition metal (M) porphyrin complexes (MPo, M=Ni,Cu,Zn) are reported. We calculate the binding energies of the transition metal atoms to the porphin molecule. Our DMC results are in reasonable agreement with those obtained from density functional theory calculations using the B3LYP hybrid exchange-correlation functional. Our study shows that such calculations are feasible with the DMC method.
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Affiliation(s)
- Jun Koseki
- Quantum Chemistry Division, Graduate School of Science, Yokohama City University, Seto 22-2, Kanazawa-ku, Yokohama 236-0027, Japan
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18
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Badinski A, Needs RJ. Accurate forces in quantum Monte Carlo calculations with nonlocal pseudopotentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:036707. [PMID: 17930361 DOI: 10.1103/physreve.76.036707] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2007] [Indexed: 05/25/2023]
Abstract
Calculating accurate forces within variational and diffusion Monte Carlo (VMC and DMC) methods is a very challenging problem. We derive expressions for the contribution to the Hellmann-Feynman force from nonlocal pseudopotentials for use within the VMC and DMC methods. Equilibrium bond lengths and harmonic vibrational frequencies are calculated from the Hellmann-Feynman forces and compared with those obtained from the energies at the Hartree-Fock, VMC, and pure DMC levels. Results for five small molecules show that the equilibrium bond lengths obtained from the force and energy calculations differ by less than 0.007 A at the DMC level.
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Affiliation(s)
- A Badinski
- Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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19
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Sorella S, Casula M, Rocca D. Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods. J Chem Phys 2007; 127:014105. [PMID: 17627335 DOI: 10.1063/1.2746035] [Citation(s) in RCA: 166] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We report a systematic study of the weak chemical bond between two benzene molecules. We first show that it is possible to obtain a very good description of the C(2) dimer and the benzene molecule by using pseudopotentials for the chemically inert 1s electrons and a resonating valence bond wave function as a variational ansatz, expanded on a relatively small Gaussian basis set. We employ an improved version of the stochastic reconfiguration technique to optimize the many-body wave function, which is the starting point for highly accurate simulations based on the lattice regularized diffusion Monte Carlo method. This projection technique provides a rigorous variational upper bound for the total energy, even in the presence of pseudopotentials, and substantially improves the accuracy of the trial wave function, which already yields a large fraction of the dynamical and nondynamical electron correlation. We show that the energy dispersion of two benzene molecules in the parallel displaced geometry is significantly deeper than the face-to-face configuration. However, contrary to previous studies based on post-Hartree-Fock methods, the binding energy remains weak ( approximately 2 kcal/mol) also in this geometry, and its value is in agreement with the most accurate and recent experimental findings [H. Krause et al., Chem. Phys. Lett. 184, 411 (1991)].
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Affiliation(s)
- Sandro Sorella
- International School for Advanced Studies (SISSA), 34014 Trieste, Italy.
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Brown MD, Trail JR, Ríos PL, Needs RJ. Energies of the first row atoms from quantum Monte Carlo. J Chem Phys 2007; 126:224110. [PMID: 17581047 DOI: 10.1063/1.2743972] [Citation(s) in RCA: 72] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.
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Affiliation(s)
- M D Brown
- Theory of Condensed Matter Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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21
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Lüchow A, Petz R, Scott TC. Direct optimization of nodal hypersurfaces in approximate wave functions. J Chem Phys 2007; 126:144110. [PMID: 17444704 DOI: 10.1063/1.2716640] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The fixed-node variant of the diffusion quantum Monte Carlo method (FN-DMC) is capable of obtaining the exact eigenvalues (albeit numerically with statistical error) of a many-electron Hamilton operator, provided that the nodal hypersurface of the exact wave function is given. The use of nodes of a trial wave function leads to the node location error. The authors have developed local criteria to assess the accuracy of the nodes based on the distances of the nodal hypersurfaces of PsiT, TPsiT, and HPsiT which coincide for the exact wave function. These criteria are used to develop direct optimization methods for the nodal hypersurface. The optimization of the nodes is demonstrated for simple wave functions of the Be atom and the C2 molecule and verified with FN-DMC calculations.
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Affiliation(s)
- Arne Lüchow
- Institut für Physikalische Chemie, RWTH Aachen University, 52056 Aachen, Germany.
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