Generating Exchange-Correlation Functionals with a Simplified, Self-Consistent Correlation Factor Model.

We focus on the spherically averaged exchange-correlation hole ρXC(r, u) of density functional theory, which describes the reduction in the electron density at a distance u due to the reference electron localized at position r. The correlation factor (CF) approach, where the model exchange hole ρXmodel(r, u) is multiplied by a CF (fC(r, u)) to yield an approximation to the exchange-correlation hole ρXC(r, u) = fC(r, u) ρXmodel(r, u), has proven to be a powerful tool for the development of new approximations. One of the remaining challenges within the CF approach is the self-consistent implementation of the resulting functionals. To address this issue, here we propose a simplification of the previously developed CFs such that self-consistent implementations become feasible. As an illustration of the simplified CF model, we develop a new meta-GGA functional, and using only a minimum of empiricism, we provide an easy derivation of an approximation that is of an accuracy similar to more involved meta-GGA functionals.

The electron-centric approach to the exchange-correlation energy.

The Kohn-Sham theory addresses the challenge of representing the kinetic energy by re-quantizing density functional theory at a level of non-interacting electrons. It transforms the many-electron problem into a fictitious non-interacting electron problem, with the many-electron effects concealed within the exchange-correlation (XC) energy, which is expressed in terms of the electron density ρ(r). Unlike the wave function, ρ(r) can be viewed as a classical quantity, and expressing the XC energy in terms of it circumvents the need for correlated wave functions. In this work, we once again employ the re-quantization strategy and determine the XC energy using a local one-particle Schrödinger equation. The ground-state eigenfunction of the corresponding Hamiltonian is a reference point (r) dependent orbital φr,σ(u, σ') which is subsequently used to generate the XC hole and the XC energy. The spin coordinate is denoted by σ and u is the electron-electron separation. The one-particle equation for φr,σ(u, σ') includes a local potential vr,σ(u, σ') that we approximate using two simple physical constraints. We assess the approximation by applying it to the helium iso-electronic series, the homogeneous electron gas, and the dissociation of the hydrogen molecule.

The factorization ansatz for non-local approximations to the exchange-correlation hole.

Among the various types of approximations to the exchange-correlation energy (EXC), the completely non-local approach is one of the lesser explored approximation schemes. It has not yet reached the predictive power of the widely used generalized gradient approximations, meta-generalized gradient approximations, hybrids, etc. In non-local functionals pursued here, the electron density at every point in space is employed to express the exchange-correlation energy per particle ϵXC(r) at a given position r. Here, we use the non-local, spherical-averaged density ρ(r,u)=∫dΩu4πρ(r+u) as a starting point to construct approximate exchange-correlation holes through the factorization ansatz ρXC(r, u) = f(r, u)ρ(r, u). We present upper and lower bounds to the exchange energy per particle ϵX(r) in terms of ρ(r, u). The factor f(r, u) is then designed to satisfy various conditions that represent important exchange and correlation effects. We assess the resulting approximations and find that the complex, oscillatory structure of ρ(r, u) makes the construction of a corresponding f(r, u) very challenging. This conclusion, identifying the main issue of the non-local approximation, is supported by a detailed analysis of the resulting exchange-correlation holes.

The fourth-order expansion of the exchange hole and neural networks to construct exchange-correlation functionals.

The curvature Qσ of spherically averaged exchange (X) holes ρX,σ(r, u) is one of the crucial variables for the construction of approximations to the exchange-correlation energy of Kohn-Sham theory, the most prominent example being the Becke-Roussel model [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)]. Here, we consider the next higher nonzero derivative of the spherically averaged X hole, the fourth-order term Tσ. This variable contains information about the nonlocality of the X hole and we employ it to approximate hybrid functionals, eliminating the sometimes demanding calculation of the exact X energy. The new functional is constructed using machine learning; having identified a physical correlation between Tσ and the nonlocality of the X hole, we employ a neural network to express this relation. While we only modify the X functional of the Perdew-Burke-Ernzerhof functional [Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], a significant improvement over this method is achieved.

Extending the source-sink potential method to include electron-nucleus coupling.

The source-sink potential (SSP) method provides a simple tool for the qualitative analysis of the conductance of molecular electronic devices, and often analytical expressions for the conductance can be obtained. Here, we extend the SSP approach to account for decoherent, inelastic electron transport by including the non-adiabatic coupling between the electrons and the nuclei in the molecule. This coupling results in contributions to electron transport that can modify the qualitative structure-conductance relationships that we unraveled previously with SSP. In the approach proposed, electron-nucleus interactions are treated starting from the harmonic approximation for the nuclei, using a non-perturbative approach to account for the non-adiabatic coupling. Our method qualitatively describes experimentally observed phenomena and allows for a simple analysis that often provides analytical formulas in terms of the physical parameters of the junction, e.g., vibrational energies, non-adiabatic coupling, and molecule-contact coupling.

Constructing and representing exchange-correlation holes through artificial neural networks.

One strategy to construct approximations to the exchange-correlation (XC) energy EXC of Kohn-Sham density functional theory relies on physical constraints satisfied by the XC hole ρXC(r, u). In the XC hole, the reference charge is located at r and u is the electron-electron separation. With mathematical intuition, a given set of physical constraints can be expressed in a formula, yielding an approximation to ρXC(r, u) and the corresponding EXC. Here, we adapt machine learning algorithms to partially automate the construction of X and XC holes. While machine learning usually relies on finding patterns in datasets and does not require physical insight, we focus entirely on the latter and develop a tool (ExMachina), consisting of the basic equations and their implementation, for the machine generation of approximations. To illustrate ExMachina, we apply it to calculate various model holes and show how to go beyond existing approximations.

The correlation factor approach: Combining density functional and wave function theory.

Several of the limitations of approximate exchange-correlation functionals within Kohn-Sham density functional theory can be eliminated by extending the single-determinant reference system to a multi-determinant one. Here, we employ the correlation factor ansatz to combine multi-configurational, self-consistent field (MCSCF) with approximate density functionals. In the proposed correlation factor approach, the exchange-correlation hole ρXC(r, u), a function of the reference point r and the electron-electron separation u, is written as a product of the correlation factor fC(r, u) and an exchange plus static-correlation hole ρXS(r, u), i.e., ρXC CFXS(r, u) = fC(r, u)ρXS(r, u). ρXS(r, u) is constructed to reproduce the exchange-correlation energy of an MCSCF reference wave function. The correlation factor fC(r, u) is designed to account for dynamic correlation effects that are absent in ρXS(r, u). The resulting approximation to the exchange-correlation energy, which we refer to as CFXStatic, is free of empirical parameters, and it combines the qualitatively correct description of the electronic structure obtainable with MCSCF with the advantages of approximate density functionals in accounting for dynamic correlation.

Non-Hermitian quantum mechanics and exceptional points in molecular electronics.

In non-Hermitian (NH) quantum mechanics, Hamiltonians are studied whose eigenvalues are not necessarily real since the condition of hermiticity is not imposed. Certain symmetries of NH operators can ensure that some or all of the eigenvalues are real and thus suitable for the description of physical systems whose energies are always real. While the mathematics of NH quantum mechanics is well developed, applications of the theory to real quantum systems are scarce, and no closed system is known whose Hamiltonian is NH. Here, we consider the elementary textbook example of a NH Hamiltonian matrix, and we show how it naturally emerges as a simplifying concept in the modeling of molecular electronic devices. We analyze the consequences of non-Hermiticity and exceptional points in the spectrum of NH operators for the molecular conductance and the spectral density of simple models for molecules on surfaces.

Construction of self-interaction-corrected exchange-correlation functionals within the correlation factor approach.

To model the exchange-correlation hole ρXC(r, u) of Kohn-Sham theory, we employ the correlation factor ansatz, which has recently been developed in our group. In this ansatz, ρXC(r, u) is written as a product of the correlation factor fC(r, u) and an exchange-hole model ρX(r, u), i.e., ρXC(r, u) = fC(r, u)ρX(r, u). In particular, we address the one-electron, self-interaction error and introduce a modified correlation factor model where fC(r, u) is constructed such that it reduces identically to one in localized one-electron regions of a many-electron system. This self-interaction corrected exchange-correlation hole is then used to generate the corresponding exchange-correlation energy functional. The new functional is implemented into a Kohn-Sham program and assessed by calculating various molecular properties. We find that, overall, a significant improvement is obtained compared to previous versions of the correlation factor model.

The correlation factor model for the exchange-correlation energy and its application to transition metal compounds.

In the recently developed correlation factor (CF) model [Precechtelova et al., J. Chem. Phys. 143, 144102 (2015)], the exchange-correlation (XC) hole is approximated. Since various constraints satisfied by the XC-hole are known, approximations to it can be designed which largely avoid empirical adjustments. In the CF approach, the XC-hole is written as a product of an exchange hole times a CF. An important constraint satisfied by the CF model is that it correctly reproduces the exact exchange energy in the high density limit. This is achieved by employing the exact exchange-energy per particle (ϵXr) as an input variable, i.e., the CF model builds on exact exchange. Variations of the initial CF model are proposed which ensure that the exact answer is obtained in the homogeneous limit. Furthermore, we apply a correction to the depth of the XC-hole that is designed to capture strong correlation. EC functionals that build on exact exchange, such as hybrids, often fail for systems that exhibit sizeable electron correlation. Despite this fact and despite the reduction of empiricism to a single parameter within CF, accurate atomization energies are obtained for strongly-correlated transition metal compounds. The CF model significantly improves upon widely used functionals such as Perdew-Burke-Ernzerhof (PBE), PBE hybrid, and Tao-Perdew-Staroverov-Scuseria (TPSS).

Impact of offset energies on the yield of interfacial charge separation in molecular photocells.

We display that the process of charge carriers' separation at molecular photocells is a complex phenomenon that is controlled by the cumulative action of Coulomb interaction for electron-hole pairs and LUMO-LUMO offset energies. By applying quantum scattering theory and the Lippmann-Schwinger equation, we provide a comprehensive framework of the device operation in which the operating molecular photocell is described by a wave function. We find that depending on the magnitude of offset energies, the electron-hole interaction can normally decrease or abnormally increase the charge separation yield. To rationalize the results, we benefit from the spectral information provided in the model. Our analysis helps to optimize molecular photocells operation.

Synthesis of oxazolidinones: rhodium-catalyzed C-H amination of N-mesyloxycarbamates.

N-Mesyloxycarbamates undergo intramolecular C-H amination reactions to afford oxazolidinones in good to excellent yields in the presence of rhodium(ii) carboxylate catalysts. The reaction is performed under green conditions and potassium carbonate is used, forming biodegradable potassium mesylate as a reaction by-product. This method enables the production of electron-rich, electron-deficient, aromatic and heteroaromatic oxazolidinones in good to excellent yields. Conformationally restricted cyclic secondary N-mesyloxycarbamates furnish cis-oxazolidinones in high yields and selectivity; DFT calculations are provided to account for the observed selectivity. trans-Oxazolidinones were prepared from acyclic secondary N-mesyloxycarbamates using Rh2(oct)4. The selectivity was reverted with a cytoxazone N-mesyloxycarbamate precursor using large chiral rhodium(ii) carboxylate complexes, affording the corresponding cis-oxazolidinone. This orthogonal selectivity was used to achieve the formal synthesis of (-)-cytoxazone.

The impact of long-range electron-hole interaction on the charge separation yield of molecular photocells.

We discuss the effects of charge carrier interaction and recombination on the operation of molecular photocells. Molecular photocells are devices where the energy conversion process takes place in a single molecular donor-acceptor complex attached to electrodes. Our investigation is based on the quantum scattering theory, in particular on the Lippmann-Schwinger equation; this minimizes the complexity of the problem while providing useful and non-trivial insight into the mechanism governing photocell operation. In this study, both exciton pair creation and dissociation are treated in the energy domain, and therefore there is access to detailed spectral information, which can be used as a framework to interpret the charge separation yield. We demonstrate that the charge carrier separation is a complex process that is affected by different parameters, such as the strength of the electron-hole interaction and the non-radiative recombination rate. Our analysis helps to optimize the charge separation process and the energy transfer in organic solar cells and in molecular photocells.

The shell model for the exchange-correlation hole in the strong-correlation limit.

We present a model for the exchange-correlation hole and the exchange-correlation energy in the strong-correlation (SC) limit of density functional theory. The SC limit is useful in the construction of exchange-correlation functionals through interpolation of the adiabatic connection. The new approximation (referred to as shell model) is an improvement of the non-local radius (NLR) model recently proposed by Wagner and Gori-Giorgi [Phys. Rev. A 90, 052512 (2014)]. The NLR model does not correctly reproduce the limit of the strongly correlated homogeneous electron gas and this shortcoming is remedied by the shell model. As in the case of the NLR model, the spherically averaged electron density ρ(r,u)=∫dΩu4πρ(r+u) is the starting point for the construction of the shell model and it is also its computational bottleneck. We show how ρ(r, u), the NLR, and the shell model can be implemented efficiently. For this purpose, analytical integrals for the normalization and the energy density of the underlying holes are provided. Employing the shell model, we illustrate how improved adiabatic connection interpolations can be constructed.

Simple model of a coherent molecular photocell.

Electron transport in molecular electronic devices is often dominated by a coherent mechanism in which the wave function extends from the left contact over the molecule to the right contact. If the device is exposed to light, photon absorption in the molecule might occur, turning the device into a molecular photocell. The photon absorption promotes an electron to higher energy levels and thus modifies the electron transmission probability through the device. A model for such a molecular photocell is presented that minimizes the complexity of the problem while providing a non-trivial description of the device mechanism. In particular, the role of the molecule in the photocell is investigated. It is described within the Hückel method and the source-sink potential approach [F. Goyer, M. Ernzerhof, and M. Zhuang, J. Chem. Phys. 126, 144104 (2007)] is used to eliminate the contacts in favor of complex-valued potentials. Furthermore, the photons are explicitly incorporated into the model through a second-quantized field. This facilitates the description of the photon absorption process with a stationary state calculation, where eigenvalues and eigenvectors are determined. The model developed is applied to various generic molecular photocells.

Modeling of molecular photocells: Application to two-level photovoltaic system with electron-hole interaction.

We present a novel simple model to describe molecular photocells where the energy conversion process takes place by a single molecular donor-acceptor complex attached to electrodes. By applying quantum scattering theory, an open quantum system method, the coherent molecular photocell is described by a wave function. We analyze photon absorption, energy conversion, and quantum yield of a molecular photocell by considering the effects of electron-hole interaction and non-radiative recombination. We model the exciton creation, dissociation, and subsequent effects on quantum yield in the energy domain. We find that depending on the photocell structure, the electron-hole interaction can normally decrease or abnormally increase the cell efficiency. The proposed model helps to understand the mechanisms of molecular photocells, and it can be used to optimize their yield.

Construction of exchange-correlation functionals through interpolation between the non-interacting and the strong-correlation limit.

Drawing on the adiabatic connection of density functional theory, exchange-correlation functionals of Kohn-Sham density functional theory are constructed which interpolate between the extreme limits of the electron-electron interaction strength. The first limit is the non-interacting one, where there is only exchange. The second limit is the strong correlated one, characterized as the minimum of the electron-electron repulsion energy. The exchange-correlation energy in the strong-correlation limit is approximated through a model for the exchange-correlation hole that is referred to as nonlocal-radius model [L. O. Wagner and P. Gori-Giorgi, Phys. Rev. A 90, 052512 (2014)]. Using the non-interacting and strong-correlated extremes, various interpolation schemes are presented that yield new approximations to the adiabatic connection and thus to the exchange-correlation energy. Some of them rely on empiricism while others do not. Several of the proposed approximations yield the exact exchange-correlation energy for one-electron systems where local and semi-local approximations often fail badly. Other proposed approximations generalize existing global hybrids by using a fraction of the exchange-correlation energy in the strong-correlation limit to replace an equal fraction of the semi-local approximation to the exchange-correlation energy in the strong-correlation limit. The performance of the proposed approximations is evaluated for molecular atomization energies, total atomic energies, and ionization potentials.

Design of exchange-correlation functionals through the correlation factor approach.

The correlation factor model is developed in which the spherically averaged exchange-correlation hole of Kohn-Sham theory is factorized into an exchange hole model and a correlation factor. The exchange hole model reproduces the exact exchange energy per particle. The correlation factor is constructed in such a manner that the exchange-correlation energy correctly reduces to exact exchange in the high density and rapidly varying limits. Four different correlation factor models are presented which satisfy varying sets of physical constraints. Three models are free from empirical adjustments to experimental data, while one correlation factor model draws on one empirical parameter. The correlation factor models are derived in detail and the resulting exchange-correlation holes are analyzed. Furthermore, the exchange-correlation energies obtained from the correlation factor models are employed to calculate total energies, atomization energies, and barrier heights. It is shown that accurate, non-empirical functionals can be constructed building on exact exchange. Avenues for further improvements are outlined as well.

Coherent molecular transistor: control through variation of the gate wave function.

In quantum interference transistors (QUITs), the current through the device is controlled by variation of the gate component of the wave function that interferes with the wave function component joining the source and the sink. Initially, mesoscopic QUITs have been studied and more recently, QUITs at the molecular scale have been proposed and implemented. Typically, in these devices the gate lead is subjected to externally adjustable physical parameters that permit interference control through modifications of the gate wave function. Here, we present an alternative model of a molecular QUIT in which the gate wave function is directly considered as a variable and the transistor operation is discussed in terms of this variable. This implies that we specify the gate current as well as the phase of the gate wave function component and calculate the resulting current through the source-sink channel. Thus, we extend on prior works that focus on the phase of the gate wave function component as a control parameter while having zero or certain discrete values of the current. We address a large class of systems, including finite graphene flakes, and obtain analytic solutions for how the gate wave function controls the transistor.

Communication: A non-empirical correlation factor model for the exchange-correlation energy.

A persistent challenge in density functional theory is the construction of a nonempirical correlation functional that is compatible with the exact exchange energy. To solve this problem, we develop a correlation factor approach in which an exchange hole model, yielding the exact exchange energy, is multiplied by a correlation factor that turns the exchange hole into an exchange-correlation hole. This results in an accurate correlation energy functional that is determined solely through physical constraints. Subject to the properties of the employed exchange hole model, the proposed correlation factor model to the exchange-correlation energy becomes exact in the high-density limit. In this limit, the exchange-correlation energy is dominated by exchange.