Energy, Momentum, and Angular Momentum Transfer between Electrons and Nuclei.

The recently developed exact factorization approach condenses all electronic effects on the nuclear subsystem into scalar and vector potentials that appear in an effective time dependent Schrödinger equation. Starting from this equation, we derive subsystem Ehrenfest identities characterizing the energy, momentum, and angular momentum transfer between electrons and nuclei. An effective electromagnetic force operator induced by the electromagnetic field corresponding to the effective scalar and vector potentials appears in all three identities. The effective magnetic field has two components that can be identified with the Berry curvature calculated with (a) different Cartesian coordinates of the same nucleus and (b) arbitrary Cartesian coordinates of two different nuclei. (a) has a classical interpretation as the induced magnetic field felt by the nucleus, while (b) has no classical analog. Subsystem Ehrenfest identities are ideally suited for quantifying energy transfer in electron-phonon systems. With two explicit examples we demonstrate the usefulness of the new identities.

Fock-Space Embedding Theory: Application to Strongly Correlated Topological Phases.

A many-body wave function can be factorized in Fock space into a marginal amplitude describing a set of strongly correlated orbitals and a conditional amplitude for the remaining weakly correlated part. The marginal amplitude is the solution of a Schrödinger equation with an effective Hamiltonian that can be viewed as embedding the marginal wave function in the environment of weakly correlated electrons. Here, the complementary equation for the conditional amplitude is replaced by a generalized Kohn-Sham equation, for which an orbital-dependent functional approximation is shown to reproduce the topological phase diagram of a multiband Hubbard model as a function of crystal field and Hubbard parameters. The roles of band filling and interband fluctuations are elucidated.

Combining Eliashberg Theory with Density Functional Theory for the Accurate Prediction of Superconducting Transition Temperatures and Gap Functions.

We propose a practical alternative to Eliashberg equations for the ab initio calculation of superconducting transition temperatures and gap functions. Within the recent density functional theory for superconductors, we develop an exchange-correlation functional that retains the accuracy of Migdal's approximation to the many-body electron-phonon self-energy, while having a simple analytic form. Our functional is based on a parametrization of the Eliashberg self-energy for a superconductor with a single Einstein frequency, and enables density functional calculations of experimental excitation gaps. By merging electronic structure methods and Eliashberg theory, the present approach sets a new standard in quality and computational feasibility for the prediction of superconducting properties.

Extending Solid-State Calculations to Ultra-Long-Range Length Scales.

We present a method that enables solid-state density functional theory calculations to be applied to systems of almost unlimited size. Computations of physical effects up to the micron length scale but which nevertheless depend on the microscopic details of the electronic structure, are made possible. Our approach is based on a generalization of the Bloch state, which involves an additional sum over a finer grid in reciprocal space around each k point. We show that this allows for modulations in the density and magnetization of arbitrary length on top of a lattice-periodic solution. Based on this, we derive a set of ultra-long-range Kohn-Sham equations. We demonstrate our method with a sample calculation of bulk LiF subjected to an arbitrary external potential containing nearly 3500 atoms. We also confirm the accuracy of the method by comparing the spin density wave state of bcc Cr against a direct supercell calculation starting from a random magnetization density. Furthermore, the spin spiral state of γ-Fe is correctly reproduced.

Competing Spin Transfer and Dissipation at Co/Cu(001) Interfaces on Femtosecond Timescales.

By combining interface-sensitive nonlinear magneto-optical experiments with femtosecond time resolution and ab initio time-dependent density functional theory, we show that optically excited spin dynamics at Co/Cu(001) interfaces proceeds via spin-dependent charge transfer and back transfer between Co and Cu. This ultrafast spin transfer competes with dissipation of spin angular momentum mediated by spin-orbit coupling already on sub 100 fs timescales. We thereby identify the fundamental microscopic processes during laser-induced spin transfer at a model interface for technologically relevant ferromagnetic heterostructures.

Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF.

We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length R = Rc, where an avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the R-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical Rc by 0.5 bohr. This effect is very accurately captured by a simple and rigorously derived correction, with an M-1 prefactor, to the exchange-correlation potential in density functional theory, M = reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wave function and conditional electronic density, ∇Rχ(R) and ∇Rn(r, R), it couples the Kohn-Sham equations at neighboring R points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation-an approximation that reduces the search for nonadiabatic density functionals to the search for a single function y(n).

Exact Single-Electron Approach to the Dynamics of Molecules in Strong Laser Fields.

We present an exact single-electron picture that describes the correlated electron dynamics in strong laser fields. Our approach is based on the factorization of the electronic wave function as a product of a marginal and a conditional amplitude. The marginal amplitude, which depends only on one electronic coordinate and yields the exact one-electron density and current density, obeys a time-dependent Schrödinger equation with an effective time-dependent potential. The exact equations are used to derive an approximation that is a step towards general and feasible ab initio single-electron calculations for molecules. The derivation also sheds new light on the usual interpretation of the single-active electron approximation. From the study of model systems, we find that the exact and approximate single-electron potentials for processes with negligible two-electron ionization lead to qualitatively similar dynamics, but that the ionization barrier in the exact single-electron potential may be explicitly time dependent.

Spin Flips versus Spin Transport in Nonthermal Electrons Excited by Ultrashort Optical Pulses in Transition Metals.

A joint theoretical and experimental investigation is performed to understand the underlying physics of laser-induced demagnetization in Ni and Co films with varying thicknesses excited by 10 fs optical pulses. Experimentally, the dynamics of spins is studied by determining the time-dependent amplitude of the Voigt vector, retrieved from a full set of magnetic and nonmagnetic quantities performed on both sides of films, with absolute time reference. Theoretically, ab initio calculations are performed using time-dependent density functional theory. Overall, we demonstrate that spin-orbit induced spin flips are the most significant contributors with superdiffusive spin transport, which assumes only that the transport of majority spins without spin flips induced by scattering does not apply in Ni. In Co it plays a significant role during the first ∼20 fs only. Our study highlights the material dependent nature of the demagnetization during the process of thermalization of nonequilibrium spins.

Electron-nuclear wave-packet dynamics through a conical intersection.

We investigate the coupled electron-nuclear dynamics in a model system showing a conical intersection (CoIn) between two excited state potential energy surfaces. Within the model, a single electron and nucleus move in two dimensions in an external static field. It is demonstrated that the nuclear density conserves its initial Gaussian shape when directly passing the CoIn, whereas the electronic density remains approximately constant. This is in sharp contrast to the picture which evolves from an analysis within the basis of adiabatic electronic states. There, dramatic changes are seen in the dynamics of the different nuclear components of the total wave function. It is thus documented that, in the case of a highly efficient population transfer between the respective adiabatic states, neither the nuclear nor the electronic density is influenced by the existence of a CoIn. This is the case because the nuclear-electronic wave packet moves on the complete potential energy surface which changes its topology smoothly as a function of all particle coordinates.

Exact Factorization-Based Density Functional Theory of Electrons and Nuclei.

The ground state energy of a system of electrons (r=r_{1},r_{2},…) and nuclei (R=R_{1},R_{2},…) is proven to be a variational functional of the electronic density n(r,R) and paramagnetic current density j_{p}(r,R) conditional on R, the nuclear wave function χ(R), an induced vector potential A_{µ}(R) and a quantum geometric tensor T_{µν}(R). n, j_{p}, A_{µ} and T_{µν} are defined in terms of the conditional electronic wave function Φ_{R}(r). The ground state (n,j_{p},χ,A_{µ},T_{µν}) can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing effective scalar and vector potentials v_{s}(r) and A_{xc}(r) that depend parametrically on R, (ii) the Schrödinger equation for χ(R), and (iii) Euler-Lagrange equations that determine T_{µν}. The theory is applied to the E⊗e Jahn-Teller model.

Electronic Flux Density beyond the Born-Oppenheimer Approximation.

In the Born-Oppenheimer approximation, the electronic wave function is typically real-valued and hence the electronic flux density (current density) seems to vanish. This is unfortunate for chemistry, because it precludes the possibility to monitor the electronic motion associated with the nuclear motion during chemical rearrangements from a Born-Oppenheimer simulation of the process. We study an electronic flux density obtained from a correction to the electronic wave function. This correction is derived via nuclear velocity perturbation theory applied in the framework of the exact factorization of electrons and nuclei. To compute the correction, only the ground state potential energy surface and the electronic wave function are needed. For a model system, we demonstrate that this electronic flux density approximates the true one very well, for coherent tunneling dynamics as well as for over-the-barrier scattering, and already for mass ratios between electrons and nuclei that are much larger than the true mass ratios.

An exact factorization perspective on quantum interferences in nonadiabatic dynamics.

Nonadiabatic quantum interferences emerge whenever nuclear wavefunctions in different electronic states meet and interact in a nonadiabatic region. In this work, we analyze how nonadiabatic quantum interferences translate in the context of the exact factorization of the molecular wavefunction. In particular, we focus our attention on the shape of the time-dependent potential energy surface-the exact surface on which the nuclear dynamics takes place. We use a one-dimensional exactly solvable model to reproduce different conditions for quantum interferences, whose characteristic features already appear in one-dimension. The time-dependent potential energy surface develops complex features when strong interferences are present, in clear contrast to the observed behavior in simple nonadiabatic crossing cases. Nevertheless, independent classical trajectories propagated on the exact time-dependent potential energy surface reasonably conserve a distribution in configuration space that mimics one of the exact nuclear probability densities.

First-Principles Calculation of the Real-Space Order Parameter and Condensation Energy Density in Phonon-Mediated Superconductors.

We show that the superconducting order parameter and condensation energy density of phonon-mediated superconductors can be calculated in real space from first principles density functional theory for superconductors. This method highlights the connection between the chemical bonding structure and the superconducting condensation and reveals new and interesting properties of superconducting materials. Understanding this connection is essential to describe nanostructured superconducting systems where the usual reciprocal space analysis hides the basic physical mechanism. In a first application we present results for MgB2, CaC6 and hole-doped graphane.

Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes.

We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.

Laser-induced electron localization in H2⁺: mixed quantum-classical dynamics based on the exact time-dependent potential energy surface.

We study the exact nuclear time-dependent potential energy surface (TDPES) for laser-induced electron localization with a view to eventually developing a mixed quantum-classical dynamics method for strong-field processes. The TDPES is defined within the framework of the exact factorization [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett., 2010, 105, 123002] and contains the exact effect of the couplings to the electronic subsystem and to any external fields within a scalar potential. We compare its features with those of the quasistatic potential energy surfaces (QSPES) often used to analyse strong-field processes. We show that the gauge-independent component of the TDPES has a mean-field-like character very close to the density-weighted average of the QSPESs. Oscillations in this component are smoothened out by the gauge-dependent component, and both components are needed to yield the correct force on the nuclei. Once the localization begins to set in, the gradient of the exact TDPES tracks one QSPES and then switches to the other, similar to the description provided by surface-hopping between QSPESs. We show that evolving an ensemble of classical nuclear trajectories on the exact TDPES accurately reproduces the exact dynamics. This study suggests that the mixed quantum-classical dynamics scheme based on evolving multiple classical nuclear trajectories on the exact TDPES will be a novel and useful method to simulate strong field processes.

Nuclear velocity perturbation theory for vibrational circular dichroism: An approach based on the exact factorization of the electron-nuclear wave function.

The nuclear velocity perturbation theory (NVPT) for vibrational circular dichroism (VCD) is derived from the exact factorization of the electron-nuclear wave function. This new formalism offers an exact starting point to include correction terms to the Born-Oppenheimer (BO) form of the molecular wave function, similar to the complete-adiabatic approximation. The corrections depend on a small parameter that, in a classical treatment of the nuclei, is identified as the nuclear velocity. Apart from proposing a rigorous basis for the NVPT, we show that the rotational strengths, related to the intensity of the VCD signal, contain a new contribution beyond-BO that can be evaluated with the NVPT and that only arises when the exact factorization approach is employed. Numerical results are presented for chiral and non-chiral systems to test the validity of the approach.

The exact forces on classical nuclei in non-adiabatic charge transfer.

The decomposition of electronic and nuclear motion presented in Abedi et al. [Phys. Rev. Lett. 105, 123002 (2010)] yields a time-dependent potential that drives the nuclear motion and fully accounts for the coupling to the electronic subsystem. Here, we show that propagation of an ensemble of independent classical nuclear trajectories on this exact potential yields dynamics that are essentially indistinguishable from the exact quantum dynamics for a model non-adiabatic charge transfer problem. We point out the importance of step and bump features in the exact potential that are critical in obtaining the correct splitting of the quasiclassical nuclear wave packet in space after it passes through an avoided crossing between two Born-Oppenheimer surfaces and analyze their structure. Finally, an analysis of the exact potentials in the context of trajectory surface hopping is presented, including preliminary investigations of velocity-adjustment and the force-induced decoherence effect.

Optical response of extended systems using time-dependent density functional theory.

In this chapter, time-dependent density functional theory is introduced and an outline of the Runge-Gross theorem is presented. An equation for linear response within time-dependent density functional theory is derived. A key ingredient of this equation is the exchange-correlation kernel for which several modern-day approximation exist. These approximations are investigated for their ability to capture the excitonic physics in absorption and electron loss spectra. To this end, results for medium (Si and diamond) to large (LiF and Ar) band-gap insulators are presented, which exhibit excitonic physics to varying degrees.

Is the molecular Berry phase an artifact of the Born-Oppenheimer approximation?

We demonstrate that the molecular Berry phase and the corresponding nonanalyticity in the electronic Born-Oppenheimer wave function is, in general, not a true topological feature of the exact solution of the full electron-nuclear Schrödinger equation. For a numerically exactly solvable model we show that a nonanalyticity, and the associated geometric phase, only appear in the limit of infinite nuclear mass, while a perfectly smooth behavior is found for any finite nuclear mass.

Classical nuclear motion coupled to electronic non-adiabatic transitions.

Based on the exact factorization of the electron-nuclear wave function, we have recently proposed a mixed quantum-classical scheme [A. Abedi, F. Agostini, and E. K. U. Gross, Europhys. Lett. 106, 33001 (2014)] to deal with non-adiabatic processes. Here we present a comprehensive description of the formalism, including the full derivation of the equations of motion. Numerical results are presented for a model system for non-adiabatic charge transfer in order to test the performance of the method and to validate the underlying approximations.