Microsolvation of cobalt, nickel, and copper atoms with ammonia: a theoretical study of the solvated electron precursors

CONTEXT

The s-block metals dissolved in ammonia form metal-ammonia complexes with diffuse electrons which could be used for redox catalysis. In this theoretical paper, we investigated the possibility of the d-bloc transition metals (Mn, Fe, Co, Ni, and Cu) solvated by ammonia. It has been demonstrated that both Mn and Fe atoms undergo into an oxidative reaction with NH3 forming an inserted species, HMNH2. On the contrary, the Co, Ni, and Cu atoms can accommodate four NH3, via the coordination bond, to form the first solvation sphere within C2v, D2d, and Td point groups, respectively. Addition of a fifth NH3 constitute the second solvation shell by forming hydrogen bond with the other NH3s. Interestingly, M(NH3)4 (M = Co, Ni, and Cu) is a so-called solvated electron precursor and should be considered as a monocation M(NH3)4+ kernel in tight contact with one electron distributed over its periphery. This nearly free electron could be used to capture a CO2 molecule and engages in a reduction reaction.

METHODS

Geometry optimization of the stationary points on the potential energy surface was performed using density functional theory - CAM-B3LYP functional including the GD3BJ dispersion contribution - in combination with the 6-311 +  + G(2d, 2p) basis set for all the atoms. All first-principles calculations were performed using the Gaussian 09 quantum chemical packages. The natural electron configuration of transition atom engaged in the compounds has been found using the natural bond orbital (NBO) method. We used the EDR (electron delocalization range) approach to analyze the structure of solvated electrons in real space. We also used the electron localization function (ELF) to measure the degree of electronic localization within a chemical compound. The EDR and ELF analyses are done using the TopMod and Multiwfn packages, respectively.

Spin state and magnetic coupling in polynuclear Ni(II) complexes from density functional theory: is there an optimal amount of Fock exchange?

Reliable prediction of the ground-state spin and magnetic coupling constants in transition-metal complexes is a well-known challenge for density functional theory (DFT). One popular strategy for addressing this long-standing issue involves the modification of the fraction of Fock exchange in a hybrid functional. Here we explore the viability of this approach using three polynuclear metal-organic complexes based on a Ni4O4 cubane motif, having different ground state spin values (S = 0, 2, 4) owing to the use of different ligands. We systematically search for an optimum fraction of Fock exchange, across various global, range-separated, and double hybrid functionals. We find that for all functionals tested, at best there only exists a very narrow range of Fock exchange fractions which results in a correct prediction of the ground-state spin for all three complexes. The useful range is functional dependent, but general trends can be identified. Typically, at least two similar systems must be used in order to determine both an upper and lower limit of the optimal range. This is likely owing to conflicting demands of minimizing delocalization errors, which typically requires a higher percentage of Fock exchange, and addressing static correlation, which typically requires a lower one. Furthermore, we find that within the optimal range of Fock exchange, the sign and relative magnitude of Ni-Ni magnetic coupling constants are reasonably well reproduced, but there is still room for quantitative improvement in the prediction. Thus, the prediction of spin state and magnetic coupling in polynuclear complexes remains an ongoing challenge for DFT.

Koopmans' Theorem-Compliant Long-Range Corrected (KTLC) Density Functional Mediated by Black-Box Optimization and Data-Driven Prediction for Organic Molecules

Density functional theory (DFT) is a significant computational tool that has substantially influenced chemistry, physics, and materials science. DFT necessitates parametrized approximation for determining an expected value. Hence, to predict the properties of a given molecule using DFT, appropriate parameters of the functional should be set for each molecule. Herein, we optimize the parameters of range-separated functionals (LC-BLYP and CAM-B3LYP) via Bayesian optimization (BO) to satisfy Koopmans' theorem. Our results demonstrate the effectiveness of the BO in optimizing functional parameters. Particularly, Koopmans' theorem-compliant LC-BLYP (KTLC-BLYP) shows results comparable to the experimental UV-absorption values. Furthermore, we prepared an optimized parameter dataset of KTLC-BLYP for over 3000 molecules through BO for satisfying Koopmans' theorem. We have developed a machine learning model on this dataset to predict the parameters of the LC-BLYP functional for a given molecule. The prediction model automatically predicts the appropriate parameters for a given molecule and calculates the corresponding values. The approach in this paper would be useful to develop new functionals and to update the previously developed functionals.

On the self-consistency of DFT-1/2

Density functional theory (DFT)-1/2 is an efficient bandgap rectification method for DFT under local density approximation (LDA) or generalized gradient approximation. It was suggested that non-self-consistent DFT-1/2 should be used for highly ionic insulators like LiF, whereas self-consistent DFT-1/2 should still be used for other compounds. Nevertheless, there is no quantitative criterion prescribed for which implementation should work for an arbitrary insulator, which leads to severe ambiguity in this method. In this work, we analyze the impact of self-consistency in DFT-1/2 and shell DFT-1/2 calculations in insulators or semiconductors with ionic bonds, covalent bonds, and intermediate cases and show that self-consistency is required even for highly ionic insulators for globally better electronic structure details. The self-energy correction renders electrons more localized around the anions in self-consistent LDA-1/2. The well-known delocalization error of LDA is rectified, but with strong overcorrection, due to the presence of additional self-energy potential. However, in non-self-consistent LDA-1/2 calculations, the electron wave functions indicate that such localization is much more severe and beyond a reasonable range because the strong Coulomb repulsion is not counted in the Hamiltonian. Another common drawback of non-self-consistent LDA-1/2 is that the ionicity of the bonding gets substantially enhanced, and the bandgap can be enormously high in mixed ionic-covalent compounds like TiO₂.

Can range-separated functionals be optimally tuned to predict spectra and excited state dynamics in photoactive iron complexes?

Density functional theory is an efficient computational tool to investigate photophysical and photochemical processes in transition metal complexes, giving invaluable assistance in interpreting spectroscopic and catalytic experiments. Optimally tuned range-separated functionals are particularly promising, as they were created to address some of the fundamental deficiencies present in approximate exchange-correlation functionals. In this paper, we scrutinize the selection of optimally tuned parameters and its influence on the excited state dynamics, using the example of the iron complex [Fe(cpmp)₂]²⁺ with push-pull ligands. Various tuning strategies are contemplated based on pure self-consistent DFT protocols, as well as on the comparison with experimental spectra and multireference CASPT2 results. The two most promising sets of optimal parameters are then employed to carry out nonadiabatic surface-hopping dynamics simulations. Intriguingly, we find that the two sets lead to very different relaxation pathways and timescales. While the set of optimal parameters from one of the self-consistent DFT protocols predicts the formation of long-lived metal-to-ligand charge transfer triplet states, the set in better agreement with CASPT2 calculations leads to deactivation in the manifold of metal-centered states, in better agreement with the experimental reference data. These results showcase the complexity of iron-complex excited state landscapes and the difficulty of obtaining an unambiguous parametrization of long-range corrected functionals without experimental input.

One-electron self-interaction error and its relationship to geometry and higher orbital occupation

Density Functional Theory (DFT) sees prominent use in computational chemistry and physics; however, problems due to the self-interaction error (SIE) pose additional challenges to obtaining qualitatively correct results. As an unphysical energy an electron exerts on itself, the SIE impacts most practical DFT calculations. We conduct an in-depth analysis of the one-electron SIE in which we replicate delocalization effects for simple geometries. We present a simple visualization of such effects, which may help in future qualitative analysis of the one-electron SIE. By increasing the number of nuclei in a linear arrangement, the SIE increases dramatically. We also show how molecular shape impacts the SIE. Two- and three-dimensional shapes show an even greater SIE stemming mainly from the exchange functional with some error compensation from the one-electron error, which we previously defined [D. R. Lonsdale and L. Goerigk, Phys. Chem. Chem. Phys. 22, 15805 (2020)]. Most tested geometries are affected by the functional error, while some suffer from the density error. For the latter, we establish a potential connection with electrons being unequally delocalized by the DFT methods. We also show how the SIE increases if electrons occupy higher-lying atomic orbitals; seemingly one-electron SIE free methods in a ground are no longer SIE free in excited states, which is an important insight for some popular, non-empirical density functional approximations (DFAs). We conclude that the erratic behavior of the SIE in even the simplest geometries shows that robust DFAs are needed. Our test systems can be used as a future benchmark or contribute toward DFT development.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Many-Body Self-Interaction and Polarons

We address the many-body self-interaction in relation to polarons in density functional theory. Our study provides (i) a unified theoretical framework encompassing many-body and one-body forms of self-interaction and (ii) an efficient semilocal scheme for charge localization. Our theoretical formulation establishes a quantitative connection between the many-body and one-body forms of self-interaction in terms of electron screening, thereby conferring superiority to the concept of many-body self-interaction. Our semilocal methodology involves the use of a weak localized potential and applies equally to electron and hole polarons. We find that polarons free from many-body self-interaction have formation energies that are robust with respect to the functional adopted.

Screening the Band Shape of Molecules by Optimal Tuning of Range-Separated Hybrid Functional with TD-DFT: A Molecular Designing Approach

In the present article we have demonstrated the effectiveness of optimally tuned range-separated hybrid (RSH) functional to determine the electronic transitions from two fluorophore moieties (blue and yellow/orange) within a single white light emitter (WLE). First, the optimally tuned range separation parameter (ω) is calculated for two white emitting fluorophores (W1 and W2) already reported in the literature. The success of the optimally tuned RSH functional ω*B97XD, used in the TDDFT study, encouraged the authors to design eight new single organic white light emitters with frustrated energy transfer between the two individual fluorophore moieties (blue and yellow/orange). The simulated spectra (the band shapes, to be more specific) generated by TDDFT study and outcomes through natural transition orbital (NTO) and natural bond orbital (NBO) studies clearly demonstrate that all the designed eight organic molecules are potential white light emitters and can be synthesized in future.

Accurate Non-Adiabatic Couplings from Optimally-Tuned Range-Separated Hybrid Functionals

Precise theoretical calculations of non-adiabatic couplings, which describe the interaction between two Born-Oppenheimer surfaces, are important for the modeling of radiationless decay mechanisms in photochemical processes. Here, we demonstrate that accurate non-adiabatic couplings can be calculated in the framework of linear-response time-dependent density functional theory by using non-empirical, optimally-tuned range-separated hybrid (OT-RSH) functionals. We focus on molecular radicals, in which ultrafast non-radiative decay plays a crucial role, to find that the OT-RSH functional compares well to wave-function based reference data and competes with the accuracy of semi-empirical CAM-B3LYP calculations. Our findings show that the OT-RSH approach provides very accurate non-adiabatic couplings and, therefore, provides a computationally efficient alternative to wave-function based techniques.

Recent Advances in Cartesian-Grid DFT in Atoms and Molecules

In the past several decades, density functional theory (DFT) has evolved as a leading player across a dazzling variety of fields, from organic chemistry to condensed matter physics. The simple conceptual framework and computational elegance are the underlying driver for this. This article reviews some of the recent developments that have taken place in our laboratory in the past 5 years. Efforts are made to validate a viable alternative for DFT calculations for small to medium systems through a Cartesian coordinate grid- (CCG-) based pseudopotential Kohn–Sham (KS) DFT framework using LCAO-MO ansatz. In order to legitimize its suitability and efficacy, at first, electric response properties, such as dipole moment (μ), static dipole polarizability (α), and first hyperpolarizability (β), are calculated. Next, we present a purely numerical approach in CCG for proficient computation of exact exchange density contribution in certain types of orbital-dependent density functionals. A Fourier convolution theorem combined with a range-separated Coulomb interaction kernel is invoked. This takes motivation from a semi-numerical algorithm, where the rate-deciding factor is the evaluation of electrostatic potential. Its success further leads to a systematic self-consistent approach from first principles, which is desirable in the development of optimally tuned range-separated hybrid and hyper functionals. Next, we discuss a simple, alternative time-independent DFT procedure, for computation of single-particle excitation energies, by means of “adiabatic connection theorem” and virial theorem. Optical gaps in organic chromophores, dyes, linear/non-linear PAHs, and charge transfer complexes are faithfully reproduced. In short, CCG-DFT is shown to be a successful route for various practical applications in electronic systems.

Molecular orbital projectors in non-empirical jmDFT recover exact conditions in transition-metal chemistry

Low-cost, non-empirical corrections to semi-local density functional theory are essential for accurately modeling transition-metal chemistry. Here, we demonstrate the judiciously modified density functional theory (jmDFT) approach with non-empirical U and J parameters obtained directly from frontier orbital energetics on a series of transition-metal complexes. We curate a set of nine representative Ti(III) and V(IV) d1 transition-metal complexes and evaluate their flat-plane errors along the fractional spin and charge lines. We demonstrate that while jmDFT improves upon both DFT+U and semi-local DFT with the standard atomic orbital projectors (AOPs), it does so inefficiently. We rationalize these inefficiencies by quantifying hybridization in the relevant frontier orbitals. To overcome these limitations, we introduce a procedure for computing a molecular orbital projector (MOP) basis for use with jmDFT. We demonstrate this single set of d1 MOPs to be suitable for nearly eliminating all energetic delocalization error and static correlation error. In all cases, MOP jmDFT outperforms AOP jmDFT, and it eliminates most flat-plane errors at non-empirical values. Unlike DFT+U or hybrid functionals, jmDFT nearly eliminates energetic delocalization error and static correlation error within a non-empirical framework.

Optimal Tuning Perspective of Range-Separated Double Hybrid Functionals

We study the optimal tuning of the free parameters in range-separated double hybrid functionals, based on enforcing the exact conditions of piecewise linearity and spin constancy. We find that introducing the range separation in both the exchange and the correlation terms allows for the minimization of both fractional charge and fractional spin errors for singlet atoms. The optimal set of parameters is system specific, underlining the importance of the tuning procedure. We test the performance of the resulting optimally tuned functionals for the dissociation curves of diatomic molecules. We find that they recover the correct dissociation curve for the one-electron system, H2+, and improve the dissociation curves of many-electron molecules such as H2 and Li2, but they also yield a nonphysical maximum and only converge to the correct dissociation limit at very large distances.

One-Shot Approach for Enforcing Piecewise Linearity on Hybrid Functionals: Application to Band Gap Predictions

We present an efficient procedure for constructing nonempirical hybrid functionals to accurately predict band gaps of extended systems. We determine mixing parameters by enforcing the generalized Koopmans' condition on localized electron states, which are achieved by inserting an optimized potential probe. Application of this scheme to a large set of materials yields band gaps with a mean error of 0.30 eV with respect to experiment. Next, we consider a perturbative one-shot approach in which the single-particle eigenvalues are calculated with the wave functions obtained at the semilocal level. In this way, the computational cost is reduced by ∼85% without loss of accuracy. The scheme is found to be robust upon consideration of different defect species and functional forms.

Improving Results by Improving Densities: Density-Corrected Density Functional Theory

Density functional theory (DFT) calculations have become widespread in both chemistry and materials, because they usually provide useful accuracy at much lower computational cost than wavefunction-based methods. All practical DFT calculations require an approximation to the unknown exchange-correlation energy, which is then used self-consistently in the Kohn-Sham scheme to produce an approximate energy from an approximate density. Density-corrected DFT is simply the study of the relative contributions to the total energy error. In the vast majority of DFT calculations, the error due to the approximate density is negligible. But with certain classes of functionals applied to certain classes of problems, the density error is sufficiently large as to contribute to the energy noticeably, and its removal leads to much better results. These problems include reaction barriers, torsional barriers involving π-conjugation, halogen bonds, radicals and anions, most stretched bonds, etc. In all such cases, use of a more accurate density significantly improves performance, and often the simple expedient of using the Hartree-Fock density is enough. This Perspective explains what DC-DFT is, where it is likely to improve results, and how DC-DFT can produce more accurate functionals. We also outline challenges and prospects for the field.

Hybrid functionals with local range separation: Accurate atomization energies and reaction barrier heights

Range-separated hybrid approximations to the exchange-correlation density functional mix exact and semi-local exchange in a position-dependent manner. In their conventional form, the range separation is controlled by a constant parameter. Turning this constant into a density functional leads to a locally space-dependent range-separation function and thus a more powerful and flexible range-separation approach. In this work, we explore the self-consistent implementation of a local range-separated hybrid, taking into account a one-electron self-interaction correction and the behavior under uniform density scaling. We discuss different forms of the local range-separation function that depend on the electron density, its gradient, and the kinetic energy density. For test sets of atomization energies, reaction barrier heights, and total energies of atoms, we demonstrate that our best model is a clear improvement over common global range-separated hybrid functionals and can compete with density functionals that contain multiple empirical parameters. Promising results for equilibrium bond lengths, harmonic vibrational frequencies, and vertical ionization potentials further underline the potential and flexibility of our approach.

Breakdown of the ionization potential theorem of density functional theory in mesoscopic systems

The ionization potential (IP)-theorem of Kohn-Sham (KS) density functional theory (DFT) states that the energy of the highest occupied molecular orbital (HOMO) ϵ_HOMO equals the negative of the first IP, thus ascribing a physical meaning to one of the eigenvalues of the KS Hamiltonian. We scrutinize the fact that the validity of the IP-theorem relies critically on the electron density n(r), far from the system, to be determined by HOMO only, behaving as n(r)∼r→∞e^-2-2ϵ_HOMOr. While this behavior always holds for finite systems, it does not hold for mesoscopic ones, such as quasi-two-dimensional (Q2D) electron gas or Q2D crystals. We show that this leads to the violation of the IP-theorem for the latter class of systems. This finding has a strong bearing on the role of the KS valence band with respect to the work-function problem in the mesoscopic case. Based on our results, we introduce a concept of the IP band structure as an observable alternative to its unphysical KS counterpart. A practical method of the determination of the IP band structure in terms of DFT quantities is provided.

Electronic-structure methods for materials design

The accuracy and efficiency of electronic-structure methods to understand, predict and design the properties of materials has driven a new paradigm in research. Simulations can greatly accelerate the identification, characterization and optimization of materials, with this acceleration driven by continuous progress in theory, algorithms and hardware, and by adaptation of concepts and tools from computer science. Nevertheless, the capability to identify and characterize materials relies on the predictive accuracy of the underlying physical descriptions, and on the ability to capture the complexity of realistic systems. We provide here an overview of electronic-structure methods, of their application to the prediction of materials properties, and of the different strategies employed towards the broader goals of materials design and discovery.

Molecular DFT+U: A Transferable, Low-Cost Approach to Eliminate Delocalization Error

While density functional theory (DFT) is widely applied for its combination of cost and accuracy, corrections (e.g., DFT+U) that improve it are often needed to tackle correlated transition-metal chemistry. In principle, the functional form of DFT+U, consisting of a set of localized atomic orbitals (AOs) and a quadratic energy penalty for deviation from integer occupations of those AOs, enables the recovery of the exact conditions of piecewise linearity and the derivative discontinuity. Nevertheless, for practical transition-metal complexes, where both atomic states and ligand orbitals participate in bonding, standard DFT+U can fail to eliminate delocalization error (DE). Here, we show that by introducing an alternative valence-state (i.e., molecular orbital or MO) basis to the DFT+U approach, we recover exact conditions in cases for which standard DFT+U corrections have no error-reducing effect. This MO-based DFT+U also eliminates DE where standard AO-based DFT+U is already successful. We demonstrate the transferability of our approach on representative transition-metal complexes with a range of ligand field strengths, electron configurations (i.e., from Sc to Zn), and spin states.

Accurate Molecular Geometries in Complex Excited-State Potential Energy Surfaces from Time-Dependent Density Functional Theory

The interplay of electronic excitations and structural changes in molecules impacts nonradiative decay and charge transfer in the excited state, thus influencing excited-state lifetimes and photocatalytic reaction rates in optoelectronic and energy devices. To capture such effects requires computational methods providing an accurate description of excited-state potential energy surfaces and geometries. We suggest time-dependent density functional theory using optimally tuned range-separated hybrid (OT-RSH) functionals as an accurate approach to obtain excited-state molecular geometries. We show that OT-RSH provides accurate molecular geometries in excited-state potential energy surfaces that are complex and involve an interplay of local and charge-transfer excitations, for which conventional semilocal and hybrid functionals fail. At the same time, the nonempirical OT-RSH approach maintains the high accuracy of parametrized functionals (e.g., B3LYP) for predicting excited-state geometries of small organic molecules showing valence excited states.

Normal mode analysis of spectral density of FMO trimers: Intra- and intermonomer energy transfer

The intermolecular contribution to the spectral density of the exciton-vibrational coupling of the homotrimeric Fenna-Matthews-Olson (FMO) light-harvesting protein of green sulfur bacteria P. aestuarii is analyzed by combining a normal mode analysis of the protein with the charge density coupling method for the calculation of local transition energies of the pigments. Correlations in site energy fluctuations across the whole FMO trimer are found at low vibrational frequencies. Including, additionally, the high-frequency intrapigment part of the spectral density, extracted from line-narrowing spectra, we study intra- and intermonomer exciton transfer. Whereas the intrapigment part of the spectral density is important for fast intramonomer exciton relaxation, the intermolecular contributions (due to pigment-environment coupling) determine the intermonomer exciton transfer. Neither the variations of the local Huang-Rhys factors nor the correlations in site energy fluctuations have a critical influence on energy transfer. At room temperature, the intermonomer transfer in the FMO protein occurs on a 10 ps time scale, whereas intramonomer exciton equilibration is roughly two orders of magnitude faster. At cryogenic temperatures, intermonomer transfer limits the lifetimes of the lowest exciton band. The lifetimes are found to increase between 20 ps in the center of this band up to 100 ps toward lower energies, which is in very good agreement with the estimates from hole burning data. Interestingly, exciton delocalization in the FMO monomers is found to slow down intermonomer energy transfer, at both physiological and cryogenic temperatures.

Design Rules for Transparent Push-Pull Electron Acceptors: A Case Study on Perylenediimide Derivatives

Transparent photovoltaics are receiving increased attention for their wide range of applicability, but there have been few attempts to systematically tune organic materials to achieve transparency. In this Letter, we study the influence of simple structural modifications on the photoabsorption spectrum of the nonfullerene electron acceptor perylenediimide. Motivated by push-pull design strategies, we explore the effects of applying electron-donating functional groups with varying strengths using three design motifs. We demonstrate that relative Mulliken electronegativity, which can be easily computed using an optimally tuned screened range-separated hybrid functional, is a useful metric for finding suitable donor groups. We also demonstrate that donor groups that include a conjugated spacer are crucial to obtaining a planar molecule with uniform conjugation and low-energy charge-transfer excitations. These simple design rules can be used to design near-infrared absorbing transparent electron acceptors based on perylenediimide and other promising molecular platforms.