179,808 research outputs found
Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge
No full textThe Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple, Proc. R. Soc. A Math. Phys. Eng. Sci. 119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry
Semiclassical reaction rate constant calculations: investigation of anharmonicity and quantum effects
No full textSemiclassical transition state theory (SCTST) is a relatively simple method for the computation from first principles of reactive rate constants, including quantum effects while accounting for anharmonicity and the coupling between reactive and bound modes.[1-3] In this talk, I will illustrate how we have developed this technique for practical applications[4-7] involving the study of phenomena like kinetic isotope effects, heavy atom tunneling, and elusive conformer lifetimes.[5,6,8]
While many approximate reaction rate theories reduce to the parabolic barrier estimate for the tunneling correction at high temperatures, SCTST, which is based on vibrational perturbation theory (VPT2), gives the exact limit when one considers the leading order term in an expansion of powers of ħ2 of the tunneling transmission coefficient.[9-11] Our investigation of molecular reactive systems assesses the importance of the non-linear corrections to the parabolic barrier estimate of the transmission coefficient. When the reaction barrier is significantly anharmonic, it is mandatory to account for non-linear corrections; otherwise, the transmission coefficient overlooks a high-temperature regime which may be dominated by quantum reflection.[12] These results highlight the importance of having a theory such as SCTST that includes the correct high-temperature limit.
[1] W.H. Miller Faraday Discuss. Chem. Soc. 62, 40 (1977).
[2] W.H. Miller J. Chem. Phys. 62, 1899 (1975)
[3] R. Hernandez et al., Chem. Phys. Lett. 214, 129 (1993).
[4] C. Aieta, F. Gabas, M. Ceotto, J. Phys. Chem. A 120, 4853 (2016).
[5] C. Aieta F. Gabas, M. Ceotto, J. Chem. Theory Comput. 15, 2142 (2019).
[6] G. Mandelli, C. Aieta, M. Ceotto J. Chem. Theory Comput. 18, 623 (2022).
[7] J.R. Barker, MultiWell-2023 software suite; University of Michigan: Ann Arbor, Michigan, USA, 2023; http://clasp-research.engin.umich.edu/multiwell/
[8] G. Mandelli, L. Corneo, C. Aieta J. Phys. Chem. Lett. 14, 9996 (2023).
[9] E. Pollak, J. Cao, Phys. Rev. A, 107, 022203 (2023).
[10] E. Pollak, S Upadhyayula J. Chem. Phys. 160, (2024).
[11] E. Pollak J. Chem. Phys. 160, 150902 (2024).
[12] C. Aieta, M. Ceotto, E. Pollak, in preparation
Lower Bounds for Coulombic Systems
No full textAs of the writing of this paper, lower bounds are not a staple of quantum chemistry computations and for good reason. All previous attempts at applying lower bound theory to Coulombic systems led to lower bounds whose quality was inferior to the Ritz upper bounds so that their added value was minimal. Even our recent improvements upon Temple's lower bound theory were limited to Lanczos basis sets and these are not available to atoms and molecules due to the Coulomb singularity. In the present paper, we overcome these problems by deriving a rather simple eigenvalue equation whose roots, under appropriate conditions, give lower bounds which are competitive with the Ritz upper bounds. The input for the theory is the Ritz eigenvalues and their variances; there is no need to compute the full matrix of the squared Hamiltonian. Along the way, we present a Cauchy-Schwartz inequality which underlies many aspects of lower bound theory. We also show that within the matrix Hamiltonian theory used here, the methods of Lehmann and our recent self-consistent lower bound theory (J. Chem. Phys. 2020, 115, 244110) are identical. Examples include implementation to the hydrogen and helium atoms
Continuum limit frozen Gaussian approximation for the reduced thermal density matrix of dissipative systems
No full textA continuum limit frozen Gaussian approximation is formulated for the reduced thermal density matrix for dissipative systems. The imaginary time dynamics is obtained from a novel generalized Langevin equation for the system coordinates. The method is applied to study the thermal density in a double well potential in the presence of Ohmic-like friction. We find that the approximation describes correctly the delocalization of the density due to quantization of the vibrations in the well. It also accounts for the friction induced reduction of the tunneling density in the barrier region. © 2012 American Institute of Physics
Comparison between different Gaussian series representations of the imaginary time propagator
No full textA useful approximation for the thermal operator exp (-β Ĥ) is based on its representation in terms of either frozen or thawed Gaussian states. Such approximate representations are leading-order terms in respective series representations of the thermal operator. A numerical study of the convergence properties of the frozen Gaussian series representation has been recently published. In this paper, we extend the previous study to include also the convergence properties of the more expensive thawed Gaussian series representation of the thermal operator. We consider three different formulations for the series representation and apply them to a quartic double-well potential to find that the thawed Gaussian series representation converges faster than the frozen Gaussian one. Further analysis is presented as to the convergence properties and the numerical efficiency of three different thawed Gaussian series representation. The unsymmetrized form converges most rapidly, however, the lower order approximations of the symmetrized forms are more accurate. Comparison with a standard discretized path-integral evaluation demonstrates that the Gaussian based perturbation series representation converges much faster. © 2010 The American Physical Society
Self-consistent theory of lower bounds for eigenvalues
No full textA rigorous practically applicable theory is presented for obtaining lower bounds to eigenvalues of Hermitian operators, whether the ground state or excited states. Algorithms are presented for computing "residual energies"whose magnitude is essential for the computation of the eigenvalues. Their practical application is possible due to the usage of the Lanczos method for creating a tridiagonal representation of the operator under study. The theory is self-consistent, in the sense that a lower bound for one state may be used to improve the lower bounds for others, and this is then used self-consistently until convergence. The theory is exemplified for a toy model of a quartic oscillator, where with only five states the relative error in the lower bound for the ground state is reduced to 6 · 10-6, which is the same as the relative error of the least upper bound obtained with the same basis functions. The lower bound method presented in this paper suggests that lower bounds may become a staple of eigenvalue computations
Lower Bounds for Nonrelativistic Atomic Energies
No full textA recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H-) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory
Sztuczna inteligencja przemysłowa, czyli kolejny przełom w branży
No full textWyniki projektu pt.
„Opracowanie na drodze prac B+R platformy optymalizacji produkcji Nazca 4.0”Artykuł opisuje, w jaki sposób sztuczna inteligencja (SI) wkracza do przemyśle i stanowi kolejny przełom w rozwoju tej branży. Zwraca uwagę na kluczowe obszary, w których SI może znacząco poprawić wydajność, takie jak prognozowanie awarii maszyn, optymalizacja procesów czy analiza ogromnych zbiorów danych w czasie rzeczywistym. Autor podkreśla, że sukces wdrożeń zależy nie tylko od zaawansowanych algorytmów czy odpowiedniej infrastruktury IT, lecz przede wszystkim od zmiany podejścia w organizacji: od kultury innowacji po gotowość pracowników do uczenia się nowych umiejętności. W treści wskazuje się również na rosnące znaczenie bezpieczeństwa i etyki w kontekście zastosowań SI w przemyśle, zwłaszcza gdy mowa o przetwarzaniu wrażliwych danych. Artykuł pokazuje ponadto przykłady konkretnych zastosowań, takich jak systemy wizyjne kontrolujące jakość produktów czy inteligentne układy sterowania liniami produkcyjnymi. Z całości wnioskować można, że rozsądne wykorzystanie sztucznej inteligencji stanowi szansę na przewagę konkurencyjną, szybsze reagowanie na zmiany rynkowe oraz utrzymanie wysokich standardów jakości i bezpieczeństwa w zakładach przemysłowych.Ze środków EFRR w ramach RPO WSL na lata 2014-2020 (umowa nr UDA-RPSL.01.02.00-24-047G/19-00
Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues
No full textRitz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bound
Joachim Pollak (1798-1879): On the Role of the Rabbi in the Jewish Community of Trebitsch in 19th Century
No full textThis B.A. thesis is a case study, contributing toward the understanding to the nineteenth-century rabbinate in Moravia. It is based on a unique convolute of Hebrew correspondence of R. Joachim Pollak (1798-1879) who served as rabbi of Trebitsch since 1828 till his death. The convolute, now in the archive of the Jewish Museum in Prague, consists of over four hundred letters written by Pollak himself or adressed to him. The author analyzed the Hebrew manuscript, provided its paleographical analysis and placed it in the context of the epistolographic genre. The chronological list contains the names of Pollakʼs correspondents, many of whom were identified by the author. The letters, interpreted in the light of other archival and secondary sources, provide a unique insight into the competences, duties and activities of a local Moravian rabbi. Powered by TCPDF (www.tcpdf.org
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