Auxiliary Density Functional Theory Research Articles

Bimetallic ions pre-intercalated hydrated vanadium oxides for high-performance aqueous zinc-ion batteries

A new generation of rechargeable batteries known as Aqueous Zinc-Ion Batteries (AZIB) offers benefits including affordability, eco-friendliness, dependability, and safety. Hydrated vanadium oxides (V2O5·nH2O) show potential as cathodes for AZIB due to their layered structure and multivalent properties. However, the zinc storage performance is limited by the structural instability and slow zinc ion mobility. In this work, TiK-VOH (the bimetallic ions system of Ti4+ and K+ co-pre-intercalated in V2O5·nH2O) with lamellar fish scale-like structure is synthesized using a simple sol-gel method. The high positive charge density of the tetravalent Ti4+ is more beneficial to attract the electrons in the V-O layer and regulate the layer spacing (12.1 Å), which not only improves the diffusion kinetics of zinc ions, but also acts as a pillar to stabilize the layer structure. K+ mainly plays the role of improving electrical conductivity, and then accelerates the charge transfer in the electrode reaction process. Auxiliary density functional theory simulation further confirms that the diffusion energy barrier and Fermi level are optimized by pre-intercalating bimetallic ions. Based on the synergistic effect of Ti4+ and K+, the TiK-VOH material shows excellent zinc storage performance, the highest specific capacity of TiK-VOH reaches 393.4 mAh g−1 at a current density of 0.2 A g−1, even at 10 A g−1 (increased by 20 times), it still maintains a discharge specific capacity of 202 mAh g−1 with a capacity retention rate of 94 % after 2000 cycles, the overall performance is much better than the Ti-VOH and K-VOH. This work offers an alternative perspective for the modification of vanadium-based cathode of AZIBs.

Exchange-correlation kernel for perturbation dependent auxiliary functions in auxiliary density perturbation theory

ContextAnalytic exchange-correlation kernel formulations are of the outermost importance for density functional theory (DFT) perturbation calculations. In this paper, the working equation for the exchange-correlation kernel of the generalized gradient approximation (GGA) for perturbation dependent auxiliary functions is derived and discussed in the framework of auxiliary density functional theory (ADFT). The presented new formulation is extended to the unrestricted approach, too. A comprehensive discussion of the implementation of the GGA ADFT kernel, using either the native exchange-correlation functional implementations in deMon2k or the ones from the LibXC library, is given. Calculations with analytic exchange-correlation kernels are compared to their finite difference counterparts. The obtained results are in quantitative agreement. Nevertheless, analytic GGA ADFT kernel implementations show substantial improvement in the computational performance. Similar results are reported for analytic second derivatives of effective core potential (ECP) and model core potential (MCP) matrix elements when compared to their finite difference counterparts in molecular frequency analyses.MethodAll calculations are performed in the framework of ADFT as implemented in deMon2k. In the ADFT analytic frequency calculations, auxiliary density perturbation theory was used. The underlying two-center exchange-correlation kernel matrix elements are calculated by numerical integration either with analytic or finite difference kernel expressions. Validation calculations are performed with the VWN and PBE functionals employing DFT-optimized DZVP basis sets in conjunction with automatically generated GEN-A2 auxiliary density function sets. In the (Pt3Cu)n cluster benchmark calculations, the RPBE functional was used. For Pt atoms, the quasi-relativistic LANL2DZ effective core potential with the corresponding valence basis set was employed, whereas for Cu atoms, the all-electron DFT-optimized TZVP basis was applied. The auxiliary density was expanded by the automatically generated GEN-A2* auxiliary function set. We run all benchmark calculations in parallel on 24 cores.

Theoretical study of PdNi and PdCu clusters embedded on graphene modified by monovacancy and nitrogen doping.

The stability and reactivity of Pd4Ni4 and Pd4Cu4 clusters embedded on graphene modified by monovacancy and nitrogen doping were investigated using auxiliary density functional theory (ADFT) calculations. The most stable structure of the Pd4Ni4 cluster is found in high spin multiplicity, whereas the lowest stable energy structure of the Pd4Cu4 cluster is a close shell system. The interaction energies between the bimetallic clusters and the defective graphene systems are significantly higher than those reported in the literature for the Pd-based clusters deposited on pristine graphene. It is observed that the composites studied present a HOMO-LUMO gap less than 1 eV, which suggests that they may present a good chemical reactivity. Therefore, from the results obtained in this work it can be inferred that the single vacancy graphene and pyridinic N-doped graphene are potentially good support materials for Pd-based clusters.

Cycloaddition reactions via "on water" protocol reactions: A density functional theory study.

In this work, the reactions of quadricyclane with dimethyl azodicarboxylate (DMAD) and of quadricyclane with diethyl azodicarboxylate (DEAD) in gas phase and in water environments were studied by a first-principles investigation within the framework of auxiliary density functional theory (ADFT). For these type of organic reactions is known that water is required to accelerate them. Since the reason of why this occur is still unknown, this work aims to gain insight into this reaction mechanism. For this investigation, the generalized gradient approximation as well as a hybrid functional were employed. The obtained optimized structures for the reactants, of the products and of the transition states are reported, together with the corresponding frequency analysis results and the reaction profiles. Along the proposed concerted reaction mechanism, a critical points search of the electron density and a charge analysis were performed. The calculated potential energy barriers of these reactions in gas phase and in water environments are compared. In agreement with experiment, the obtained results indicate that both reactions occur faster in water than in gas phase. This study shows that there is a change in the polarity of the two most important carbon atoms of the formed compounds along the reactions and that the decrease of the activation energy barrier which occurs in liquid phase in these reactions is because the structures of the main transition states are stabilized by the water environment. Therefore, the here obtained results demonstrate the important role played by the water-molecule framework into the activation energy barrier and structures of the molecules that participate in the DMAD and DEAD cycloaddition reactions.

A multi-GPU implementation of Real-Time Time-Dependent Auxiliary Density Functional Theory for the investigation of nanosystems irradiations

We report a new Multi-GPU (Graphical Processor Unit) implementation of real-time time-dependent Auxiliary Density Functional Theory (DFT) for simulations of attosecond electronic dynamics in molecular systems subjected to strong perturbations. Our code relies on the Kohn-Sham formalism of DFT and has been implemented in the deMon2k Fortran code. We expand single-particle wave functions (i.e molecular orbitals) as linear combinations of Gaussian-type-orbitals centered on atoms. The density matrix propagation is carried out on GPU while the Kohn-Sham potential is operated on CPUs (Central Processor Unit) with the help of variationally fitted densities. We propose a parallelization strategy using the MAGMA/CUDA libraries to calculate the exponential of dense Hermitian matrices entering the mathematical definition of the propagator, here using Taylor expansions. We report performance benchmarks on water droplets and on fullerenes (C50 to C540). They show a clear advantage of GPU over CPU (using the Scalapack library). The benchmarks also show the benefit of using more than one GPU for systems comprised of up to more than 10,000 basis functions. There, a speed-up of almost 40 between pure 40 CPU and four 4 GPU is obtained. Attosecond electron dynamics simulation in molecular systems comprised of several thousands of electrons becomes amenable to routine simulations in our code. We assess the accuracy of the GPU implementation considering various applications, namely, the calculation of extreme UV absorption spectra with non-Hermitian dynamics, the response of C180 to an electric perturbation, and finally the irradiation of a DNA/protein complex by a 0.4 MeV proton. The results demonstrate the robustness of the implementation. This work also paves the way for future even more efficient implementations. Program SummaryProgram title: deMon2k-RT-TDDFTCPC Library link to program files: https://doi.org/10.17632/whwc5zkst5.1Developer's repository link: DOI: 10.5281/zenodo.8301468Licensing provisions: CeCILL Free Software License Agreement v2.1Programming language: FortranExternal routines/libraries: MPI, BLAS/LAPACK, CUDA, MAGMASupplementary material: User manual, installation instructions, test cases. Nature of problem: Electron dynamics within molecular systems is amenable to simulation with Real-Time Time-Dependent Density Functional Theory (RT-TD-DFT). The methodology consists in propagating on the attosecond time scale the single particle electronic wave functions on the underlying Kohn-Sham potential [1,2]. In deMon2k-RT-TDDFT, the propagation is coupled to the Auxiliary DFT framework for fast evaluation of the potential [3]. A remaining computational bottleneck is the matrix exponential calculation of the Hermitian Hamiltonian matrix. This task is achieved in deMon2k via diagolanization or expansions formulas (Taylor, Chebychev or Baker-Campbell-Hausdorff). The operation is cumbersome on central-processor-unit machines for large matrices (e.g. > 5000 × 5000), which corresponds to molecular systems comprising around 2,000 electrons. Alternatives are needed to overcome this bottleneck.Solution method: We propose to carry out the matrix exponentiation on graphical-processor-units thanks to the MAGMA[4]/CUDA libraries. The code permits a drastic reduction of the computational cost, opening the attosecond dynamic simulations of large molecular systems comprised thousands of electrons with controlled accuracy. The method is tested here for Taylor expansions. Additional comments including restrictions and unusual features: The code described in this article opens the realization of RT-TD-ADFT simulations. The code is linked to the Master version 6.1.6 of deMon2k (http://www.demon-software.com), which must be pre-installed by the user.

Cartesian constraints in QM/MM optimizations.

With the rise of quantum mechanical/molecular mechanical (QM/MM) methods, the interest in the calculation of molecular assemblies has increased considerably. The structures and dynamics of such assemblies are usually governed to a large extend by intermolecular interactions. As a result, the corresponding potential energy surfaces are topological rich and possess many shallow minima. Therefore, local structure optimizations of QM/MM molecular assemblies can be challenging, in particular if optimization constraints are imposed. To overcome this problem, structure optimization in normal coordinate space is advocated. To do so, the external degrees of freedom of a molecule are separated from the internal ones by a projector matrix in the space of the Cartesian coordinates. Here we extend this approach to Cartesian constraints. To this end, we devise an algorithm that adds the Cartesian constraints directly to the projector matrix and in this way eliminates them from the reduced coordinate space in which the molecule is optimized. To analyze the performance and stability of the constrained optimization algorithm in normal coordinate space, we present constrained minimizations of small molecular systems and amino acids in gas phase as well as water employing QM/MM constrained optimizations. All calculations are performed in the framework of auxiliary density functional theory as implemented in the program deMon2k.

Irradiation of Plutonium Tributyl Phosphate Complexes by Ionizing Alpha Particles: A Computational Study.

The PUREX solvent extraction process, widely used for recovering uranium and plutonium from spent nuclear fuel, utilizes an organic solvent composed of tributyl phosphate (TBP). The emission of ionizing particles such as alpha particles, resulting from the decay of plutonium, makes the organic solvent vulnerable to degradation. Here, we study the ultrashort time alpha irradiation of tributylphosphate (TBP) and Pu(NO3)4(TBP)2 complex formed in the PUREX process. Electron dynamics is propagated by Real-Time-Dependent Auxiliary Density Functional Theory (RT-TD-ADFT). We investigate the use of previously proposed absorption boundary conditions (ABC) in the molecular orbital space to treat secondary electron emission. Basis set and exchange correlation functional effects with ABC are reported as well as a detailed analysis of the ABC parametrization. Preliminary results on the water molecule and then on TBP show that the phenomenological nature of the ABC parameters necessitates selecting appropriate values for each system under study. Irradiation of free and complexed TBP shows an influence of the ligands on the variation of atomic charges on the femtosecond time scale. An accumulation of atomic charges in the alkyl chains of TBP is observed in the case where the nitrate groups are predominantly irradiated. In addition, we find that the Pu atom regains its electric charge very rapidly after being hit by the projectile, with the coordination sphere serving as an electron reservoir to preserve its formal redox state. This study paves the road toward a full understanding of the degradation of organic extracants employed in the nuclear industry.

Organic Molecular Intercalated V3O7·H2O with High Operating Voltage for Long Cycle Life Aqueous Zn‐Ion Batteries

AbstractV3O7·H2O (VO) is an attractive cathode material for high‐capacity aqueous Zn‐ion batteries (AZIBs), but it is limited by slow ion mobility and low working platform voltage. Here, a 1,3‐propane diamine (DP)‐intercalated VO with nanoribbon‐assembled thorn flower‐like structure is fabricated by a facile hydrothermal method, noted as VO‐DP. The study shows that the zinc ion diffusion coefficient in VO‐DP (3.1 × 10−8 cm−2 s−1) is five orders of magnitude higher than that of a pure VO counterpart. Auxiliary density functional theory simulation shows that the embedded energy of zinc ions in VO‐DP significantly decreases from 0.24 to −2.5 eV, thus leading to excellent diffusion kinetics and superior rate performance. Benefiting from these unique properties, AZIBs composed of VO‐DP cathodes exhibit high operating voltage (0.89 V), remarkable capacities of 473 mA h g−1 at 0.05 A g−1, excellent rate capability (144 mA h g−1 at 10 A g−1) and long‐term cycling performance (73% capacity retention over 15 000 cycles at 10 A g−1).

Stability and activity of PdCu clusters embedded on pyridinic N-doped graphene: a density functional theory investigation

The stability and reactivity of Pd6-nCun (n = 0−3) clusters supported on pyridinic N-doped graphene (PNG) were investigated using the auxiliary density functional theory method. First, the stability of Pd6-nCun (n = 0−3) clusters supported on PNG was analyzed using charge transfer, interaction energy, and electron density critical point calculations. Then, the reactivity of these clusters supported on PNG was computed using frontier molecular orbitals and oxygen adsorption energies. According to the calculated interaction energy, as the Cu content increased, the interaction energy between Pd6-nCun (n = 0−3) clusters and PNG increased. The charge transfer indicated that the clusters donate charge to the PNG. Based on the energy of the frontier orbitals, the Pd5Cu cluster supported on PNG exhibits the smallest HOMO–LUMO gap inferring that this system is the most reactive, whereas the less reactive system corresponds to the Pd4Cu2 cluster supported on PNG. Finally, the oxygen adsorption energies on Pd6 (−3.86 eV) and Pd5Cu (−3.87 eV) clusters supported on PNG were similar. However, the Pd5Cu cluster has a lower Pd content than the Pd6 cluster. Therefore, this work reveals that PdCu systems supported on PNG could be cheap and efficient catalysts for the oxygen reduction reaction.

Efficient implementation of time-dependent auxiliary density functional theory.

The random phase approximation of time-dependent auxiliary density functional theory (TDADFT) is rederived from auxiliary density perturbation theory. Our exhaustive validation of TDADFT reveals an upshift of the excitation energies by ∼0.1eV with respect to standard time-dependent density functional theory. For the computationally efficient implementation of TDADFT, floating point operation optimized three-center electron repulsion integral recurrence relations and their double asymptotic expansions are implemented into the Davidson solver. The computational efficiency of TDADFT is benchmarked with four sets of molecules comprising alkanes, fullerenes, DNA fragments, and zeolites. The results show that TDADFT has a computational scaling between 1.3 and 1.9 with respect to the number of basis functions, which is lower than the scaling of standard time-dependent density functional theory. Due to its computational simplifications, TDADFT is particularly well suited for Born-Oppenheimer molecular dynamics simulations. As illustrative examples, we present the temperature effects on the gas-phase absorption spectra of benzene, naphthalene, and anthracene.

Stability and catalytic properties of Pt–Ni clusters supported on pyridinic N-doped graphene nanoflakes: an auxiliary density functional theory study

Stability and catalytic properties of Pt–Ni clusters supported on pyridinic N-doped graphene nanoflakes: an auxiliary density functional theory study

Structures and properties of Co13−xCux (x=0–13) nanoclusters and their interaction with pyridinic N3-doped graphene nanoflake

The structures and properties of icosahedral Co13−xCux (x = 0–13) nanoclusters and their interaction with pyridinic N3-doped graphene (PNG) nanoflake were studied using auxiliary density functional theory. First, the spin multiplicity, harmonic frequency, spin magnetic moment per atom, average bond length (ABL), vertical ionization potential (VIP), vertical electron affinity (VEA), and chemical hardness (η) of the icosahedral Co13−xCux (x = 0–13) nanoclusters were investigated. Subsequently, the Co13−xCux (x = 0–13) nanoclusters were supported on PNG nanoflake and their interaction energies (Eint) and charge transfers were computed. The spin multiplicity and spin magnetic moment per atom decrease as the quantity of Co atoms diminishes in the Co13−xCux (x = 0–13) nanoclusters revealing that the magnetic behavior is dependent on the quantity of Co atoms in the Co13−xCux nanoclusters, whereas the ABL values raise as the number of Cu atoms increases in Co13−xCux (x = 0–13) nanoclusters. The Co9Cu4 and Co8Cu5 nanoclusters present the smallest η confirming the minor electronic stabilities for these systems. On the interaction between Co13−xCux (x = 0–13) nanoclusters and PNG nanoflake, charge transfer occurs from the nanoclusters to the PNG nanoflake. This study demonstrated that PNG nanoflake is an adequate material to stabilize the Co13−xCux (x = 0–13) nanoclusters.

Reliability and performances of real-time time-dependent auxiliary density functional theory

We recently adapted the Auxiliary DFT framework as implemented in deMon2k to the simulation of time-dependent problems via the Runge and Gross equations. Our implementation of the so-called Real-Time-Time-Dependent ADFT (RT-TD-ADFT) fully benefits from the algorithms available in deMon2k to carry out variational density fitting, notably the MINRES algorithm recently proposed for self-consistent-field calculations. We test here MINRES for the first time in the context of RT-TD-ADFT. We report extensive benchmarks calculations to assess the reliability of the ADFT framework. These encompass the construction of absorption spectra in the gas phase and in solvent, the calculation of electronic stopping power curves, the irradiation of zeolites by swift ions and the investigation of charge migrations with attosecond time resolution. All our results are very encouraging. We show that even small auxiliary basis sets are sufficient to obtain results almost undisguisable from those obtained with large and flexible auxiliary bases. Overall, we establish the reliability of RT-TD-ADFT to simulate electronics dynamics in large or very large molecular systems.

Femtosecond responses of hydrated DNA irradiated by ionizing rays focus on the sugar-phosphate part

In this article, we investigate the mechanisms of DNA ionization upon irradiation by 0.5 meV alpha particles. We focus on the sugar-phosphate group and its hydration shell. In radiation chemistry, the term quasi-direct effect refers the physical and chemical responses taking place after irradiation of solvent molecules pertaining to the solvation shells of solutes. The molecular mechanisms accounting for the quasi-direct effect are actually largely elusive, especially for those prevailing in the early timescales (< 10–12 s). We report Real-Time Time-Dependent Auxiliary Density Functional Theory simulations carried out within the framework of hybrid QM/MM scheme (Quantum Mechanics/Molecular Mechanics) with polarizable and non-polarizable embedding. Ten water molecules from the solvation shell of DNA backbone are independently irradiated. We find that during the first femtoseconds after irradiation, the holes formed on the irradiated water remain at their sites of formation. Electrostatic induction within the environment does not significantly impact charge migrations. We address the hypothesis that charge migration driven by electron correlation is responsible for an ultrafast H2O+ to DNA charge transfers, which would account for a quasi-direct effect. We find that pure charge migration at fixed nuclear positions is not responsible for the quasi-direct effect when considering sugar-phosphate solvation shells.

Cubic aromaticity in ligand-stabilized doped Au superatoms.

The magnetic response of valence electrons in doped gold-based M@Au8L8 q superatoms (M = Pd, Pt, Ag, Au, Cd, Hg, Ir, and Rh; L = PPh3; and q = 0, +1, +2) is studied by calculating the gauge including magnetically induced currents (GIMIC) in the framework of the auxiliary density functional theory. The studied systems include 24 different combinations of the dopant, total cluster charge, and cluster structure (cubic-like or oblate). The magnetically induced currents (both diatropic and paratropic) are shown to be sensitive to the atomic structure of clusters, the number of superatomic electrons, and the chemical nature of the dopant metal. Among the cubic-like structures, the strongest aromaticity is observed in Pd- and Pt-doped M@Au8L8 0 clusters. Interestingly, Pd- and Pt-doping increases the aromaticity as compared to a similar all-gold eight-electron system Au9L8 +1. With the recent implementation of the GIMIC in the deMon2k code, we investigated the aromaticity in the cubic and butterfly-like M@Au8 core structures, doped with a single M atom from periods 5 and 6 of groups IX-XII. Surprisingly, the doping with Pd and Pt in the cubic structure increases the aromaticity compared to the pure Au case not only near the central atom but encompassing the whole metallic core, following the aromatic trend Pd > Pt > Au. These doped (Pd, Pt)@Au8 nanoclusters show a closed shell 1S21P6 superatom electronic structure corresponding to the cubic aromaticity rule 6n + 2.

Photoabsorption spectra of helicenes

In this work, photoabsorption spectra of some polycyclic aromatic hydrocarbons (PAHs) from time-dependent auxiliary density functional theory (TD-ADFT) are presented. In particular photoabsorption spectra of oligoacenes from three up to six fused rings, phenanthrene, benzo[c]phenanthrene, pentahelicene and hexahelicene are given. These spectra are presented to see semi-quantitative differences between isomers. The energy interval, from 1 up to 30 eV, of these spectra fill the gap left by the scarce spectroscopic data for helicenes.

Magnetically induced currents and aromaticity in ligand-stabilized Au and AuPt superatoms

Understanding magnetically induced currents (MICs) in aromatic or metallic nanostructures is crucial for interpreting local magnetic shielding and NMR data. Direct measurements of the induced currents have been successful only in a few planar molecules but their indirect effects are seen in NMR shifts of probe nuclei. Here, we have implemented a numerically efficient method to calculate gauge-including MICs in the formalism of auxiliary density functional theory. We analyze the currents in two experimentally synthesized gold-based, hydrogen-containing ligand-stabilized nanoclusters [HAu9(PPh3)8]2+ and [PtHAu8(PPh3)8]+. Both clusters have a similar octet configuration of Au(6s)-derived delocalized “superatomic” electrons. Surprisingly, Pt-doping in gold increases the diatropic response of the superatomic electrons to an external magnetic field and enhances the aromaticity of [PtHAu8(PPh3)8]+. This is manifested by a stronger shielding of the hydrogen proton in the metal core of the cluster as compared to [HAu9(PPh3)8]2+, causing a significant upfield shift in agreement with experimental proton NMR data measured for these two clusters. Our method allows the determination of local magnetic shielding properties for any component in large 3D nanostructures, opening the door for detailed interpretation of complex NMR spectra.

On the growth behavior, structures, energy, and magnetic properties of bimetallic $$\hbox {M}_{{n}}\hbox {Pd}_{{n}}$$ (M = Co, Ni; n = 1–10) clusters

In this work, the growth behavior, structures, energetic and magnetic properties of $$\hbox {Co}_{{n}}\hbox {Pd}_{{n}}$$ and $$\hbox {Ni}_{{n}}\hbox {Pd}_{{n}}$$ (n = 1–10) clusters were investigated employing auxiliary density functional theory (ADFT). Initial geometries for successive optimization were extracted from Born–Oppenheimer molecular dynamics (BOMD) trajectories. It is demonstrated that when the cluster size increases, the Co and Ni atoms became shrouded by Pd atoms, leading to the initial formation of M@Pd (M = Co and Ni) core–shell structures. The spin multiplicities of the $$\hbox {Co}_{{n}}\hbox {Pd}_{{n}}$$ and $$\hbox {Ni}_{{n}}\hbox {Pd}_{{n}}$$ (n = 1–10) clusters increase with cluster size. The CoPd clusters exhibit higher spin multiplicity and are characterized by higher spin magnetic moments than the NiPd counterparts. This study reveals that the spin density distributions are located on the Co and Ni atoms in the respective clusters. As the cluster size increases both systems tend to donate more easily electrons and the binding energies per atom grows monotonically.

First-principle study of the structures, growth pattern, and properties of (Pt3Cu)n, n = 1-9, clusters.

In this work, a first-principles systematic study of (Pt3Cu)n, n = 1-9, clusters was performed employing the linear combination of Gaussian-type orbital auxiliary density functional theory approach. The growth of the clusters has been achieved by increasing the previous cluster by one Pt3Cu unit at a time. To explore in detail the potential energy surface of these clusters, initial structures were obtained from Born-Oppenheimer molecular dynamics trajectories generated at different temperatures and spin multiplicities. For each cluster size, several dozens of structures were optimized without any constraints. The most stable structures were characterized by frequency analysis calculations. This study demonstrates that the obtained most stable structures prefer low spin multiplicities. To gain insight into the growing pattern of these systems, average bond lengths were calculated for the lowest stable structures. This work reveals that the Cu atoms prefer to be together and to localize inside the cluster structures. Moreover, these systems tend to form octahedra moieties in the size range of n going from 4 to 9 Pt3Cu units. Magnetic moment per atom and spin density plots were obtained for the neutral, cationic, and anionic ground state structures. Dissociation energies, ionization potential, and electron affinity were calculated, too. The dissociation energy and the electron affinity increase as the number of Pt3Cu units grows, whereas the ionization potential decreases.

Variational properties of auxiliary density functionals

The evolution of variational Coulomb fitting from a purely practical scheme to reduce computational burden to a formal variant of Hohenberg–Kohn–Sham density functional theory (auxiliary density functional theory, ADFT) is discussed. After a summary of the historical evolution, an analysis of its connection with the Hohenberg–Kohn theorem is given, some implications for the Euler equation and for time-dependent DFT are given and some implications for the deMon2k code delineated.