Conduction of molecular electronic devices: Qualitative insights through atom-atom polarizabilities

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.

External fields in conceptual density functional theory

The necessity of the recent incorporation of new external variables in the context of conceptual DFT (CDFT) is discussed based on the ever-increasing portfolio of experimental reaction conditions in the endeavor of experimentalists to synthesize new molecules with unprecedented properties. Electric and magnetic fields (ε and B), mechanical forces (F), and confinement are proposed as valuable new variables, extending conventional CDFT and its associated response functions. A finite field approach is used to calculate the evolution of both global and local descriptors in a selected series of atomic and molecular applications, and from it derive new response function involving, with one exception, the first derivative to the field considered. The electric field results, displaying, for example, a case of a field-induced enantioselectivity in the Fukui function, may be instrumental in the recent upsurge of chemistry in oriented external electric fields. The study of atomic electronegativity and hardness in magnetic fields displays a piecewise behavior, associated to configurational jumps upon increasing field strength and reveals an overall compression of their ranges for stronger fields, which may be guiding upon investigating chemistry in extremely high fields like in white dwarfs. The evolution of the electronegativity and hardness of diatomics under mechanical force can elegantly be traced back to differences in their equilibrium distance in the neutral, cationic, and anionic state. The well-known reduction of the polarizability under confinement can be seen as a fore-runner of the increasing hardness of atoms under pressure, presently under investigation. Periodicity showing up in a spontaneous way in the variety of properties is a leitmotiv in this study, as well as the interconnections/analogies between the different response functions.

From Density Functional Theory to Conceptual Density Functional Theory and Biosystems

The position of conceptual density functional theory (CDFT) in the history of density functional theory (DFT) is sketched followed by a chronological report on the introduction of the various DFT descriptors such as the electronegativity, hardness, softness, Fukui function, local version of softness and hardness, dual descriptor, linear response function, and softness kernel. Through a perturbational approach they can all be characterized as response functions, reflecting the intrinsic reactivity of an atom or molecule upon perturbation by a different system, including recent extensions by external fields. Derived descriptors such as the electrophilicity or generalized philicity, derived from the nature of the energy vs. N behavior, complete this picture. These descriptors can be used as such or in the context of principles such as Sanderson’s electronegativity equalization principle, Pearson’s hard and soft acids and bases principle, the maximum hardness, and more recently, the minimum electrophilicity principle. CDFT has known an ever-growing use in various subdisciplines of chemistry: from organic to inorganic chemistry, from polymer to materials chemistry, and from catalysis to nanotechnology. The increasing size of the systems under study has been coped with thanks to methodological evolutions but also through the impressive evolution in software and hardware. In this flow, biosystems entered the application portfolio in the past twenty years with studies varying (among others) from enzymatic catalysis to biological activity and/or the toxicity of organic molecules and to computational peptidology. On the basis of this evolution, one can expect that “the best is yet to come”.

Quantum Circuit Rules for Molecular Electronic Systems: Where Are We Headed Based on the Current Understanding of Quantum Interference, Thermoelectric, and Molecular Spintronics Phenomena?

In this minireview, we discuss important aspects of the various quantum phenomena (such as quantum interference, spin-dependent charge transport, and thermoelectric effects) relevant in single-molecule charge transport and list some of the basic circuit rules devised for different molecular systems. These quantum phenomena, in conjunction with the existing empirical circuit rules, can help in predicting some of the structure-property relationships in molecular circuits. However, a universal circuit law that predicts the charge transport properties of a molecular circuit has not been derived yet. Having such law(s) will help to design and build a complex molecular device leading to exciting unique applications that are not possible with the traditional silicon-based technologies. Based on the existing knowledge in the literature, here we open the discussion on the possible future research directions for deriving unified circuit law(s) to predict the charge transport in complex single-molecule circuits.

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.

Captodative Substitution Enhances the Diradical Character of Compounds, Reduces Aromaticity, and Controls Single-Molecule Conductivity Patterns: A Valence Bond Study

The present contribution uses a valence bond (VB) perspective to consider the captodative substitution strategy, a method to enhance the diradical character of (potentially aromatic) compounds. We confirm the qualitative reasoning that has generally been used to rationalize the diradical-character-enhancing effect of captodative substitution: this type of substitution scheme disproportionally stabilizes specific Dewar/diradical(oid) VB structures, thus increasing their weight in the full ground-state wave function. Furthermore, we assess the effect of captodative substitution on the aromaticity of the considered compound. We observe a clear trade-off between diradical character and aromaticity for our model systems: as one of these properties increases, the other decreases. This finding is especially significant within the field of single-molecule electronics because it enables unification of the previously observed inverse proportionality between the aromaticity of a compound and the magnitude of conductance through that molecule, with the observed proportionality between diradical character and the magnitude of conductance associated with a compound. To some extent, both properties, i.e., aromaticity and diradical character, appear to be the flip-sides of the same coin.

Influence of long-range interactions on quantum interference in molecular conduction. A tight-binding (Hückel) approach

When certain pairs of atoms in a π-conjugated molecule are connected with nanometer-scale source and drain electrodes, the remarkable quantum interference (QI) effect may arise. In this case, the electron transmission probability is significantly suppressed due to the QI effect. Tight-binding approaches, such as the Hückel molecular orbital (HMO) model, have revealed important features of this quantum phenomenon. However, important deviations from experiments and from more sophisticated calculations are known for a variety of cases. Here, we propose an extension of the HMO method to include non-nearest-neighbor interactions. Such long-range interactions (LRIs) are implemented in the HMO model in the form of a damping function that decays as the topological distance-the number of bonds separating two atoms-gets larger. The proposed model is further developed so that a geometric modification, i.e., the rotation around a single bond, can be taken into account. Our results show that LRI affects both the location of the antiresonance peak due to QI and the intensity of QI, even suppressing it in some cases. These results agree well with what was observed in a Density Functional based Tight-Binding (DFTB) study reported in the literature. These properties can be interpreted on the basis of a graph-theoretic path-counting model as well as the molecular orbital theory. In addition, the geometric LRI model is shown to reproduce the change of transmission as a function of rotation around the single bond separating two benzene rings in biphenyl, in agreement with what was observed in both experiment and DFTB calculation.

Exploring chemical space with alchemical derivatives: alchemical transformations of H through Ar and their ions as a proof of concept

Alchemical derivatives have been used previously to obtain information about transformations in which the number of electrons is unchanged. Here an approach for combining changes in both the number of electrons and the nuclear charge is presented.

Exploring Chemical Space with Alchemical Derivatives: BN-Simultaneous Substitution Patterns in C60

With the idea of using alchemical derivatives to explore in an efficient, computer- and cost-effective way Chemical Space was launched several years ago. In the context of Conceptual DFT response functions, these energies vs nuclear charge derivatives permit the estimatation of the energy of transmutants of a given starting or reference molecule showing different nuclear compositions. After an explorative study on small and planar molecules ( Balawender et al. J. Chem. Theory Comput. 2013 , 9 , 5327 ) by the present authors of this paper, the present study fully exploits the computational advantages of the alchemical derivatives in larger three-dimensional systems. Starting from a single reference calculation on C₆₀, the complete BN substitution pattern, from single substituted C₅₈BN via the belt (C₂₀(BN)₂₀ and the ball C₁₂(BN)₂₄ structures to the fully substituted (BN)₃₀, is explored. Successive and simultaneous substitution strategies are followed and compared, indicating that both techniques yield identical results up to 13 substitutions but that for higher substitutions the simultaneous approach needs to be taken. Due to the cost-efficiency of the algorithm this path can indeed be followed as opposed to earlier work in the literature where for each step a full SCF calculation was at stake leading to prohibitively large computational demands for adopting the simultaneous approach. Previously formulated rules governing the substitution pattern by Kar and co-workers are scrutinized in this context and reformulated giving chemical insight in the gradual substitution process and the relative energies of the isomers. In its present form the method offers an interesting venue to study BN substitution patterns in higher fullerenes and graphene and in general paves the way for more efficient exploration of the Chemical Space.

Close relation between quantum interference in molecular conductance and diradical existence

An empirical observation of a relationship between a striking feature of electronic transmission through a π-system, destructive quantum interference (QI), on one hand, and the stability of diradicals on the other, leads to the proof of a general theorem that relates the two. Subject to a number of simplifying assumptions, in a π-electron system, QI occurs when electrodes are attached to those positions of an N-carbon atom N-electron closed-shell hydrocarbon where the matrix elements of the Green's function vanish. These zeros come in two types, which are called easy and hard. Suppose an N+2 atom, N+2 electron hydrocarbon is formed by substituting 2 CH2 groups at two atoms, where the electrodes were. Then, if a QI feature is associated with electrode attachment to the two atoms of the original N atom system, the resulting augmented N+2 molecule will be a diradical. If there is no QI feature, i.e., transmission of current is normal if electrodes are attached to the two atoms, the resulting hydrocarbon will not be a diradical but will have a classical closed-shell electronic structure. Moreover, where a diradical exists, the easy zero is associated with a nondisjoint diradical, and the hard zero is associated with a disjoint one. A related theorem is proven for deletion of two sites from a hydrocarbon.