## Final Report Summary - MODENADYNA (MODeling Electron Non-Adiabatic DYNAmics)

The goals pursued in MODENADYNA were mainly concerned with the development of theoretical and computational methods suited to support the search of innovative materials and the engineering of nano devices with targeted electronic properties. Success in these activities relies, among other factors, on our understanding of the physics at the nanoscale and on the flexibility of the tools to simulate the systems under investigation. Theoretical simulations can allow us to test hypotheses, to elucidate fundamental mechanisms, and to deal virtually with what cannot be yet realized experimentally in laboratories. In the long term, this can help to improve a large variety of technologies and even to develop new technologies. Notable instances are the technologies related to the reduction of energy consumption or new devices to produce energy more efficiently. Reaching these goals can imply substantial socioeconomical benefits related to public-health and economic-growth issues. In the short term, the results obtained within MODENADYNA are expected to be relevant to researchers working on the development and application of first-principles computational methods.

In simulating electronic properties on the nanoscale realistically, the crux of the problem is due to the challenge of dealing with many degrees of freedoms which are related to each other by an intricate network of interactions. Materials of technological interest can be made of elementary units composed of a few up to thousands of atoms (and even more, in case of systems of biological relevance). The need of schematizations and reduced descriptions is indispensable. Yet, this should be done in such a way not to rule out the possibility of extracting useful spatial- and/or time-resolved information.

The state of electrons can be described quantum mechanically by means of the solution of a Time-Dependent (TD) Schrödinger equation: a wave function which refers to many interacting electrons moving within the considered system. Density Functional Theory (DFT) and its TD extension (TDDFT) reformulate the problem of describing equilibrium and non-equilibrium electronic properties in terms of the distribution of the electron density. The electron density can be expressed as a simple mathematical function which avoids the multidimensional character of the wave function and yet contains useful information on how electrons are distributed in space. The electron density, in turn, can be determined by mapping mathematically the “real” interacting electrons onto “virtual” non-interacting particles: which are known as the Kohn-Sham (KS) electrons. The task of dealing with systems composed of even up to several thousands of electrons becomes practical (with the help of modern computational infrastructures), if we can find proper approximate expressions for the quantities that connect the “virtual” non-interacting electrons to the “real” ones: these are the so-called exchange-correlation (xc) functionals.

Further schematizations permit to derive explicit and practical expressions for the xc functionals. These are the Local Density Approximation (LDA) and the Adiabatic Approximation (AA). In the LDA, one pretends that the electrons behave locally as in a homogenous gas with constant density. In the AA, one pretends that the electrons behave instantaneously as they were in equilibrium. On one hand, these approximations enable us to deal with nano structured systems with realistic implications; on the other hand, they also introduce limitations that make approaches often poorly predictive. Going beyond the LDA and AA is, arguably, the most changeling yet needed development in (TD)DFT.

The main objective of the project MODENADYNA was to make progress in overcoming the LDA and AA approximations. Overcoming the LDA means to take into account the effects of the spatial inhomogeneities in the systems. Spatial inhomogeneities are related to the way electrons localize because of intrinsic features such as the atomic shells and molecular bonds, or extrinsic features such as those introduced by man-made nano structured confinements or by additional electromagnetic fields. Overcoming the AA demands to take into account the dynamical manifestation of the aforementioned spatial inhomogeneities and the memory effects due to the fact that the xc functional depends on the whole evolution of the electron density and not just on its instantaneous distribution. Other non-adiabatic effects are due to the coupling to the motion of the atomic nuclei (which in reality are not fixed) and couplings to the environment (as, for example, the effects of liquid solvents on diluted molecular systems).

In two years of activities, the scientific outcomes of the project MODENADYNA can be summarized as follows (here listed approximately in chronological order of submission/preparation of the corresponding article):

1) Established the first two leading contributions of the asymptotic expansion for the correlation energy of large (non-relativist) atoms. This provides an exact condition to develop non-empirical correlation energy density.

2) Shown that for some one-dimension model systems, it is possible to devise an orbital-free approach as accurate as the exact exchange method. If this result could be extended to general three-dimensional systems, it would allow us to deal with very large systems more efficiently than ever done previously.

3) Tested a novel numerical implementation of the polarizable continuum model coupled with the TDDFT equations to account for solvation effects in real-space and real-time calculations. This is important to obtain closer agreement with experimental results dealing with molecules in liquid solvents. This implementation was released under GPL license to academic and non academic users.

4) Provided an estimation of the effects of same-spin dynamical correlations on the electron localization. This provides insights to design practical xc functionals able to deal with realistic inhomogeneities.

5) Supported an experimental investigation concerning the dynamics of polaron pair formation in a prototypical polymer thin film on a sub-20-fs time scale by means of TDDFT computational simulations. The elementary optical excitations of this polymer were characterized as hybridized exciton-polaron-pairs, strongly coupled to a dominant underdamped vibrational mode. The findings open up new perspectives for tailoring light-to-current conversion in organic materials.

6) Reported the characterization of the ultrafast lattice dynamics in P3HT:PC71BM and CH3NH3PbI3 to femtosecond optical excitation by means of ultrafast electron diffraction and TDDFT calculations. The temporal evolution of the atomic motions during charge transport and electron-hole separation was determined. The results can help to define a universally unifying process for organic and perovskite photovoltaics and could serve as a general designing strategy for further improving their performance.

7) Introduced a rigorous, physically appealing, and practical way to measure distances between exchange-only correlations of interacting many-electron systems, which works regardless of their size and inhomogeneity. This distance captures fundamental physical features such as the periodicity of atomic elements, and it can be used to effectively and efficiently analyze the performance of density functional approximations and can find useful applications in high-throughput materials design.

8) Demonstrated that the standard DFT electron localization function can be usefully complemented with the information extracted from opposite-spin correlated electron pairs by means of a procedure which, remarkably, only requires occupied Kohn-Sham orbitals. In addition to the improvements for the exemplary case of the Hydrogen molecule, the approach yields compelling trends in atoms and organic molecules.

9) Resolved the long-standing problem of finding practical, yet non-empirical, building blocks for the xc functionals useful to deal with spin currents which do not assume weak inhomogeneity. These building blocks may therefore have a wide range of applicability in atomic, molecular and condensed matter physics related to the phenomenology implied by magnetic fields and spin-orbit couplings.

10) Designed, implemented, and tested a novel functional approximations to determine the fundamental gaps of low-dimensional nano structured quantum dots systems.

11) Provided a description of the orbital-dependent expression of the 'Hartree-Exchange' functional in ensemble DFT in terms of a universal limit and a minimization procedure. This can be useful to enable calculations of specific excitation energies avoiding the complications introduced by a TD propagation.

The results have been disseminated at international and national conferences, workshops, meetings, and seminars. Publications in international peer-reviewed journals with high impact have appeared and several more have been recently submitted or are in preparation. International and national exchange scientific visits have been done (outgoing and incoming) with the twofold scope of disseminating the results and extending the network of collaborations. The Fellow has successfully consolidated his expertise in the field. Transfer of knowledge has also been performed in the form of teaching and/or training of Master/Ph.D. students and young researchers. The project and some of its activities have been described on the web site https://sites.google.com/site/pittalisstefano and some materials (such as presentations) were made available. Most of the publications are freely available on http://arxiv.org/. Outreach activities have included an interview of the Fellow by a Japanese radio station and a presentation for a broad audience made by the Scientist in Charge.

In simulating electronic properties on the nanoscale realistically, the crux of the problem is due to the challenge of dealing with many degrees of freedoms which are related to each other by an intricate network of interactions. Materials of technological interest can be made of elementary units composed of a few up to thousands of atoms (and even more, in case of systems of biological relevance). The need of schematizations and reduced descriptions is indispensable. Yet, this should be done in such a way not to rule out the possibility of extracting useful spatial- and/or time-resolved information.

The state of electrons can be described quantum mechanically by means of the solution of a Time-Dependent (TD) Schrödinger equation: a wave function which refers to many interacting electrons moving within the considered system. Density Functional Theory (DFT) and its TD extension (TDDFT) reformulate the problem of describing equilibrium and non-equilibrium electronic properties in terms of the distribution of the electron density. The electron density can be expressed as a simple mathematical function which avoids the multidimensional character of the wave function and yet contains useful information on how electrons are distributed in space. The electron density, in turn, can be determined by mapping mathematically the “real” interacting electrons onto “virtual” non-interacting particles: which are known as the Kohn-Sham (KS) electrons. The task of dealing with systems composed of even up to several thousands of electrons becomes practical (with the help of modern computational infrastructures), if we can find proper approximate expressions for the quantities that connect the “virtual” non-interacting electrons to the “real” ones: these are the so-called exchange-correlation (xc) functionals.

Further schematizations permit to derive explicit and practical expressions for the xc functionals. These are the Local Density Approximation (LDA) and the Adiabatic Approximation (AA). In the LDA, one pretends that the electrons behave locally as in a homogenous gas with constant density. In the AA, one pretends that the electrons behave instantaneously as they were in equilibrium. On one hand, these approximations enable us to deal with nano structured systems with realistic implications; on the other hand, they also introduce limitations that make approaches often poorly predictive. Going beyond the LDA and AA is, arguably, the most changeling yet needed development in (TD)DFT.

The main objective of the project MODENADYNA was to make progress in overcoming the LDA and AA approximations. Overcoming the LDA means to take into account the effects of the spatial inhomogeneities in the systems. Spatial inhomogeneities are related to the way electrons localize because of intrinsic features such as the atomic shells and molecular bonds, or extrinsic features such as those introduced by man-made nano structured confinements or by additional electromagnetic fields. Overcoming the AA demands to take into account the dynamical manifestation of the aforementioned spatial inhomogeneities and the memory effects due to the fact that the xc functional depends on the whole evolution of the electron density and not just on its instantaneous distribution. Other non-adiabatic effects are due to the coupling to the motion of the atomic nuclei (which in reality are not fixed) and couplings to the environment (as, for example, the effects of liquid solvents on diluted molecular systems).

In two years of activities, the scientific outcomes of the project MODENADYNA can be summarized as follows (here listed approximately in chronological order of submission/preparation of the corresponding article):

1) Established the first two leading contributions of the asymptotic expansion for the correlation energy of large (non-relativist) atoms. This provides an exact condition to develop non-empirical correlation energy density.

2) Shown that for some one-dimension model systems, it is possible to devise an orbital-free approach as accurate as the exact exchange method. If this result could be extended to general three-dimensional systems, it would allow us to deal with very large systems more efficiently than ever done previously.

3) Tested a novel numerical implementation of the polarizable continuum model coupled with the TDDFT equations to account for solvation effects in real-space and real-time calculations. This is important to obtain closer agreement with experimental results dealing with molecules in liquid solvents. This implementation was released under GPL license to academic and non academic users.

4) Provided an estimation of the effects of same-spin dynamical correlations on the electron localization. This provides insights to design practical xc functionals able to deal with realistic inhomogeneities.

5) Supported an experimental investigation concerning the dynamics of polaron pair formation in a prototypical polymer thin film on a sub-20-fs time scale by means of TDDFT computational simulations. The elementary optical excitations of this polymer were characterized as hybridized exciton-polaron-pairs, strongly coupled to a dominant underdamped vibrational mode. The findings open up new perspectives for tailoring light-to-current conversion in organic materials.

6) Reported the characterization of the ultrafast lattice dynamics in P3HT:PC71BM and CH3NH3PbI3 to femtosecond optical excitation by means of ultrafast electron diffraction and TDDFT calculations. The temporal evolution of the atomic motions during charge transport and electron-hole separation was determined. The results can help to define a universally unifying process for organic and perovskite photovoltaics and could serve as a general designing strategy for further improving their performance.

7) Introduced a rigorous, physically appealing, and practical way to measure distances between exchange-only correlations of interacting many-electron systems, which works regardless of their size and inhomogeneity. This distance captures fundamental physical features such as the periodicity of atomic elements, and it can be used to effectively and efficiently analyze the performance of density functional approximations and can find useful applications in high-throughput materials design.

8) Demonstrated that the standard DFT electron localization function can be usefully complemented with the information extracted from opposite-spin correlated electron pairs by means of a procedure which, remarkably, only requires occupied Kohn-Sham orbitals. In addition to the improvements for the exemplary case of the Hydrogen molecule, the approach yields compelling trends in atoms and organic molecules.

9) Resolved the long-standing problem of finding practical, yet non-empirical, building blocks for the xc functionals useful to deal with spin currents which do not assume weak inhomogeneity. These building blocks may therefore have a wide range of applicability in atomic, molecular and condensed matter physics related to the phenomenology implied by magnetic fields and spin-orbit couplings.

10) Designed, implemented, and tested a novel functional approximations to determine the fundamental gaps of low-dimensional nano structured quantum dots systems.

11) Provided a description of the orbital-dependent expression of the 'Hartree-Exchange' functional in ensemble DFT in terms of a universal limit and a minimization procedure. This can be useful to enable calculations of specific excitation energies avoiding the complications introduced by a TD propagation.

The results have been disseminated at international and national conferences, workshops, meetings, and seminars. Publications in international peer-reviewed journals with high impact have appeared and several more have been recently submitted or are in preparation. International and national exchange scientific visits have been done (outgoing and incoming) with the twofold scope of disseminating the results and extending the network of collaborations. The Fellow has successfully consolidated his expertise in the field. Transfer of knowledge has also been performed in the form of teaching and/or training of Master/Ph.D. students and young researchers. The project and some of its activities have been described on the web site https://sites.google.com/site/pittalisstefano and some materials (such as presentations) were made available. Most of the publications are freely available on http://arxiv.org/. Outreach activities have included an interview of the Fellow by a Japanese radio station and a presentation for a broad audience made by the Scientist in Charge.