References:

[1] https://doi.org/10.1021/acs.jpclett.1c02461

[2] https://doi.org/10.1103/PhysRevLett.121.183401

[3] https://www.science.org/doi/10.1126/sciadv.aax0024

[4] https://link.aps.org/doi/10.1103/PhysRevResearch.4.013011

[5] https://arxiv.org/pdf/2010.11809.pdf, Phys. Rev. Lett. (2022).

[6] Tkatchenko and Fedorov, submitted.

First, we will introduce a particular way to profit from results of model systems [1,2]. We will show that model results can be used in an in principle exact way, which we term “Connector Theory”, in order to describe materials properties. Within this approach, a quantity of interest is calculated for a model system as a function of a parameter once and forever, and the results are stored and shared. Under certain conditions, the model result for an appropriate choice of parameter (called connector) can then be used to replace the quantity of interest in the real material. We will discuss the principles and general properties of the connector approach, and show that it leads to interesting approximations.

Finally, we will look at recent results obtained in the framework of Green’s functions. We will in particular concentrate on excitons, with results that go beyond standard textbook expectations and illustrate the rich playground of many-body effects in electronic spectra.

References

[1] M. Vanzini, A. Aouina, M. Panholzer, M. Gatti, and L. Reining, arXiv:1903.07930v4

[2] A. Aouina, M. Gatti, and L. Reining, Faraday Discussions 224, 27 (2020)

In this talk I will explain why Temple’s lower bound and its later improvement by Lehmann could not provide lower bounds with “chemical accuracy”. In recent work, we have shown that one may use a very different approach, based on a matrix in which the Ritz eigenstates are coupled via variances to an exact energy eigenstate to obtain lower bounds whose accuracy is competitive with that of the Ritz upper bound. The methodology will be demonstrated for some toy model as well as the He, Li and Be atoms. Upper and lower bounds are obtained with nano-Hartree accuracy. Perhaps the most difficult part of the new algorithm is that like the older Temple based method one needs to compute variances and this is quite challenging when considering Coulomb potentials. A method will be suggested by which one may accurately estimate variances from matrix elements of the Hamiltonian, without having to expressly compute matrix elements of the Hamiltonian squared operator.

Struggling with the concept of transition path times distributions for proteins naturally led to the question of what is the quantum analog and what can we learn from the same in the quantum mechanical context. This presented us with the challenge of understanding quantum mechanical transition times, and more specifically the time scale of quantum tunneling, a question of some practical interest in view of attosecond experiments on ionization of the He and H atoms as well as the so called Larmor tunneling times of Rb atoms. The highlights of these studies are: a. Theoretical resolution of the tunneling flight as the well known Wigner phase time [3]; b. Verification that the tunneling flight time may be superluminal, even when considering Dirac electrons (in one spatial dimension) [4]; Understanding that this superluminal property does not contradict special relativity, in the sense that it does not lead to superluminal signal transition [4]. Our recent study of electronic transition path time distributions will be presented, demonstrating resonance scattering and the inability of the fewest switches surface hopping algorithm to correctly account for such quantum mechanical phenomena [5]. Finally, transition path time distributions will be presented for fermions, bosons and distinguishable particles [6].

The study of time in quantum mechanics led also to the derivation of a weak value time energy uncertainty relation [7], which in turn led to a renewed interest in lower bound theory, a topic which has remained stagnant for many years, due to the fact that the quality of lower bounds was far inferior to the Ritz upper bounds invented by Ritz in 1908 and 1909. Twenty years later, Temple presented a method for obtaining convergent lower bounds. Ninety years later, the Ritz method is a staple of university courses while the Temple lower bound remains relatively hidden, due to its slow convergence. Finding “good” lower bounds remained a challenge. The Temple class of lower bounds is based on a Cauchy-Schwartz inequality. In recent work, we have shown that one may use a very different approach, based on a matrix in which the Ritz eigenstates are coupled via variances to an exact energy eigenstate to obtain lower bounds [8] whose accuracy is competitive with that of the Ritz upper bound [8,9]. The methodology will be demonstrated for some toy models as well as the He, Li and Be atoms. Using correlated Gaussian basis sets, upper and lower bounds may be obtained with nano-Hartree accuracy [9,10]. Perhaps the most difficult part of the new algorithm is that like the older Temple based method one needs to compute variances and this is quite challenging when considering Coulomb potentials. A method will be suggested by which one may accurately estimate variances from matrix elements of the Hamiltonian, without having to expressly compute matrix elements of the Hamiltonian squared operator.

References

[1] E. Pollak, The transition path time distribution and the transition path free energy barrier, Phys. Chem. Chem. Phys. 18, 28872 – 28882 (2016). DOI: 10.1039/C6CP05052B

[2] R. Dutta and E. Pollak, What can we learn from transition path time distributions for protein folding and unfolding?, Phys. Chem. Chem. Phys. 23, 23787-23795 (2021). DOI: 10.1039/D1CP03296H

[3] T. Rivlin, E. Pollak, and R. S. Dumont, Determination of the tunneling flight time as the reflected phase time, Phys. Rev. A 103, 012225 (2021).

[4] R.S. Dumont, T. Rivlin and E. Pollak, The relativistic tunneling flight time may be superluminal, but it does not imply superluminal signaling, New J. Phys. 22, 093060 (2020).

[5] X. He, B. Wu, J. Liu, T. Rivlin and E. Pollak, Transition path flight times and nonadiabatic electronic transitions, preprint, to be published.

[6] R. Ianconescu and E. Pollak, A comparison of transition path flight times for fermions, bosons and distinguishable particles, Preprint, to be published.

[7] E. Pollak and S. Miret-Artés, Uncertainty relations for time-averaged weak values, Phys. Rev. A 99, 012108 (2019).

[8] E Pollak and R Martinazzo, Lower bounds for Coulombic systems, J. Chem. Th. Comp. 17, 1535-1547 (2021).

[9] R.T. Ireland, P. Jeszenszki, E. Mátyus, R. Martinazzo, M. Ronto, E. Pollak, Lower Bounds for Nonrelativistic Atomic Energies, ACS Phys. Chem. Au 2, 23–37 (2022).

[10] M. Ronto, P. Jeszenszki, E. Mátyus and E. Pollak, Eigenvalue lower bounds as a practical tool for numerical electronic energy calculations, preprint, to be published.