Thermodynamic responses of electronic systems

Spin-Polarized Conceptual Density Functional Theory from the Convex Hull

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., {N, S} vs {Nα, Nβ}) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions {P+(r), P-(r)} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Grand Canonical Ensemble Modeling of Electrochemical Interfaces Made Simple

Grand canonical ensemble (GCE) modeling of electrochemical interfaces, in which the electrochemical potential is converged to a preset constant, is essential for understanding electrochemistry and electrocatalysis at the electrodes. However, it requires developing efficient and robust algorithms to perform practical and effective GCE modeling with density functional theory (DFT) calculations. Herein, we developed an efficient and robust fully converged constant-potential (FCP) algorithm based on Newton's method and a polynomial fitting to calculate the necessary derivative for DFT calculations. We demonstrated with the constant-potential geometry optimization and Born-Oppenheimer molecular dynamics (BOMD) calculations that our FCP algorithm is resistant to the numerical instability that plagues other algorithms, and it delivers efficient convergence to the preset electrochemical potential and renders accurate forces for updating the nuclear positions of an electronically open system, outperforming other algorithms. The implementation of our FCP algorithm enables flexibility in using various computational codes and versatility in performing advanced tasks including the constant-potential enhanced-sampling BOMD simulations that we showcased with the modeling of the electrochemical hydrogenation of CO, and it is thus expected to find a wide spectrum of applications in the modeling of chemistry at electrochemical interfaces.

Approximating constant potential DFT with canonical DFT and electrostatic corrections

The complexity of electrochemical interfaces has led to the development of several approximate density functional theory (DFT)-based schemes to study reaction thermodynamics and kinetics as a function of electrode potential. While fixed electrode potential conditions can be simulated with grand canonical ensemble DFT (GCE-DFT), various electrostatic corrections on canonical, constant charge DFT are often applied instead. In this work, we present a systematic derivation and analysis of the different electrostatic corrections on canonical DFT to understand their physical validity, implicit assumptions, and scope of applicability. Our work highlights the need to carefully address the suitability of a given model for the problem under study, especially if physical or chemical insight in addition to reaction energetics is sought. In particular, we analytically show that the different corrections cannot differentiate between electrostatic interactions and covalent or charge-transfer interactions. By numerically testing different models for CO₂ adsorption on a single-atom catalyst as a function of the electrode potential, we further show that computed capacitances, dipole moments, and the obtained physical insight depend sensitively on the chosen approximation. These features limit the scope, generality, and physical insight of these corrective schemes despite their proven practicality for specific systems and energetics. Finally, we suggest guidelines for choosing different electrostatic corrections and propose the use of conceptual DFT to develop more general approximations for electrochemical interfaces and reactions using canonical DFT.

Properties of the density functional response kernels and its implications on chemistry

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.

Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the “|dμ| big is good” rule or the maximum hardness principle. Expressions for the change in chemical potential (μ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign.

Molecular Interactions From the Density Functional Theory for Chemical Reactivity: The Interaction Energy Between Two-Reagents

Reactivity descriptors indicate where a reagent is most reactive and how it is most likely to react. However, a reaction will only occur when the reagent encounters a suitable reaction partner. Determining whether a pair of reagents is well-matched requires developing reactivity rules that depend on both reagents. This can be achieved using the expression for the minimum-interaction-energy obtained from the density functional reactivity theory. Different terms in this expression will be dominant in different circumstances; depending on which terms control the reactivity, different reactivity indicators will be preferred.

The electronic temperature and the effective chemical potential parameters of an atom in a molecule. A Fermi-Dirac semi-local variational approach

We developed a numerical procedure to compute the electronic temperature and the effective (local) chemical potential undergone by electrons belonging to a particular molecular species. Our strategy relies on consider atomic basins as open quantum (sub)systems within the context of the quantum theory of atoms in molecules. Each basin is represented by the two parameters, the electronic temperature and the effective chemical potential, which are determined by distributing electrons (fermions) imbedded in each atomic region, through a Fermi-Dirac semi-local variational procedure. The results obtained for 40 different chemical species show that the effective chemical potential is a useful tool to reveal the most acidic/basic atoms in a molecule while the electronic temperature is closely related to the concept of chemical hardness at the local level. Our numerical data also indicate that the electronic temperature values undergone by electrons imbedded in atomic basins are way beyond the room temperature condition, allowing to fractionally occupy several of the one-particle quantum states. In this context, we developed two new indexes useful to reveal outstanding orbitals involved in the chemical reactivity of atoms in molecules.

Proposal of a Fermi-Dirac-Derived Reactivity Descriptor: Beyond the Frontier MO Model

In this paper, we derive a reactivity descriptor stemming from the Fermi-Dirac population scheme, applied to density functional calculations on molecular systems. Assuming that molecular orbitals only marginally change when temperature is slightly increased from 0 K, we study the response of electron density to a change in temperature. Connection with usual conceptual density functional theory descriptors is made, and the T-variation of electron density for some representative examples is given and discussed.

Conceptual density functional theory based electronic structure principles

In this review article, we intend to highlight the basic electronic structure principles and various reactivity descriptors as defined within the premise of conceptual density functional theory (CDFT). Over the past several decades, CDFT has proven its worth in providing valuable insights into various static as well as time-dependent physicochemical problems. Herein, having briefly outlined the basics of CDFT, we describe various situations where CDFT based reactivity theory could be employed in order to gain insights into the underlying mechanism of several chemical processes.

Extending conceptual DFT to include additional variables: oriented external electric field

The extension of the E = E[N, v] functional for exploring chemical reactivity in a conceptual DFT context to include external electric fields is discussed. Concentrating on the case of a homogeneous field the corresponding response functions are identified and integrated, together with the conventional response functions such as permanent dipole moment and polarizability, in an extended response function tree associated with the E = E[N, v, ε] functional. In a case study on the dihalogens F2, Cl2, Br2, I2 the sensitivity of condensed atomic charges (∂q/∂ε) is linked to the polarizability of the halogen atoms. The non-integrated (∂ρ(r)/∂ε) response function, directly related to the field induced density change, is at the basis of these features. It reveals symmetry breaking for a perpendicular field, not detectable in its atom condensed counterpart, and accounts for the induced dipole moment directly related to the molecular polarizability. The much higher sensitivity of the electronic chemical potential/electronegativity as compared to the chemical hardness is highlighted. The response of the condensed Fukui functions to a parallel electric field increases when going down in the periodic table and is interpreted in terms of the extension of the outer contours in the non-condensed Fukui function. In the case of a perpendicular field the (∂f(r)/∂ε) response function hints at stereoselectivity with a preferential side of attack which is not retrieved in its condensed form. In an application the nucleophilic attack on the carbonyl group in H2CO is discussed. Similar to the dihalogens, stereoselectivity is displayed in the Fukui function for nucleophilic attack (f+) in the case of a perpendicular electric field, and opposite to the one that would arise based on the induced density. Disentangling the expression for the evolution of the Fukui function in the presence of an electric field reveals that this difference can be traced back to local differences in the polarization or induced density between the anionic and the neutral system. This difference may be exploited, e.g. for an appropriately substituted H2CO, to generate enantioselectivity.

Temperature-Dependent Approach to Electronic Charge Transfer

A charge transfer model is developed within the framework of the grand canonical ensemble through the analysis of the behavior of the fractional charge as a function of the chemical potential of the bath when the temperature and the external chemical potential are kept fixed. Departing from the fact that, before the interaction between two species, each one has a zero fractional charge, one can identify two situations after the interaction occurs where the fractional charge of at least one of the species is different from zero, indicating that there has been charge transference. One of them corresponds to the case when one of the species is immersed in a bath conformed by the other one, while the other is related to the case in which both species are present in equal amounts (stoichiometric proportion). Correlations between the fractional charges and average energies, thus obtained with experimental equilibrium constants, kinetic rate constants, hydration constants, and bond enthalpies, indicate that, although at the experimental temperatures, they are very small quantities, they have chemically meaningful information. Additionally, in the stoichiometric case, one also finds a rather good correlation between the equalized chemical potential and the one obtained from experimental information for a test set of diatomic and triatomic molecules.