Density Functional Theory Procedures Research Articles

Post-CCSD(T) Thermochemistry of Chlorine Fluorides as a Challenging Test Case for Evaluating Density Functional Theory and Composite Ab Initio Methods.

Quantum chemistry plays a key role in exploring the chemical properties of highly reactive chlorine polyfluorides compounds (ClFn). Here, we investigate the thermochemical properties of ClFn species (n = 2-6) by means of high-level thermochemical procedures approximating the CCSDT(Q) and CCSDTQ5 energies at the complete basis set limit. We consider total atomization energies (TAEs), Cl-F bond dissociation energies (BDEs), F2 elimination energies (F2 elim.), ionization potentials (IPs), and electron affinities (EAs). The TAEs have significant contributions from post-CCSD(T) correlation effects. The higher-order triple excitations, CCSDT-CCSD(T), are negative and amount to -0.338 (ClF2), -0.727 (ClF3), -0.903 (ClF4), -1.335 (ClF5), and -1.946 (ClF6) kcal/mol. However, the contributions from quadruple (and, where available, also quintuple) excitations are much larger and positive and amount to +1.335 (ClF2), +1.387 (ClF3), +2.367 (ClF4), +2.399 (ClF5), and +3.432 (ClF6) kcal/mol. Thus, the contributions from post-CCSD(T) excitations exceed the threshold of chemical accuracy in nearly all cases. Due to their increasing hyper-valency and multireference character, the ClFn series provides an interesting and challenging test case for both density functional theory and low-level composite ab initio procedures. Here, we highlight the limitations in achieving overall chemical accuracy across all DFT and most composite ab initio procedures.

Optical and morphological properties of DCM thin films co-doped of Znq2 by PVD: Theoretical and experimental investigations

In this article, new experimental results of the morphological and optical properties of thin films of 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran co-doped by 8-hydroxyquinoline zinc (Znq2) for different concentrations were presented. The physical vapor deposition in a high vacuum was used to create the thin films on glass substrates with a thickness of 100 nm. The morphological properties of the samples were identified using the Atomic Force Microscopy (AFM) apparatus in tapping mode. TEM images show amorphous structures. The phase composition of the samples was assessed using FTIR methods. The Tauc plot approach was used to examine the measured transmittance spectra in order to estimate the energy gap. Analysis of the results showed that the co-doping of DCM thin film by Znq2 influenced the structural and optical properties of the samples. To evaluate the experimental results found, theoretical calculations based on the maximum one-photon absorption (OPA) wavelengths of DCM and Znq2 have been first measured through the UV–Vis spectral technique. Then, the experimental conclusions on values for DCM and Znq2 have been compared with their corresponding simulation data procured from the time-dependent self-consistent-field (TD-SCF) computation utilizing the density functional theory (DFT) at B3LYP/6-311G(d,p) level. Besides, we have explored the first and second frontier molecular orbitals (MOs) for title compounds and their energy gaps via the DFT procedure. The calculations of maximum OPA wavelengths and the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) contributions on both molecules have also been crosschecked with the results of already published data in the literature.

Design Rules, Accurate Enthalpy Prediction, and Synthesis of Stoichiometric Eu3+ Quantum Memory Candidates.

Stoichiometric Eu3+ compounds have recently shown promise for building dense, optically addressable quantum memory as the cations' long nuclear spin coherence times and shielded 4f electron optical transitions provide reliable memory platforms. Implementing such a system, though, requires ultranarrow, inhomogeneous linewidth compounds. Finding this rare linewidth behavior within a wide range of potential chemical spaces remains difficult, and while exploratory synthesis is often guided by density functional theory (DFT) calculations, lanthanides' 4f electrons pose unique challenges for stability predictions. Here, we report DFT procedures that reliably reproduce known phase diagrams and correctly predict two experimentally realized quantum memory candidates. We are the first to synthesize the double perovskite halide Cs2NaEuF6. It is an air-stable compound with a calculated band gap of 5.0 eV that surrounds Eu3+ with mononuclidic elements, which are desirable for avoiding inhomogeneous linewidth broadening. We also analyze computational database entries to identify phosphates and iodates as the next generation of chemical spaces for stoichiometric quantum memory system studies. This work identifies new candidate platforms for exploring chemical effects on quantum memory candidates' inhomogeneous linewidth while also providing a framework for screening Eu3+ compound stability with DFT.

Theoretical survey on the electronic, linear and nonlinear optical properties of substituted benzenes and polycondensed π-systems. A density functional theory study

Herein dipole moments, static and dynamic (hyper)polarizabilities, delocalization energies, distortions of the phenyl rings of the substituted benzenes are computed using density functional theory procedure. It is revealed that the biphenyl molecule exhibits the largest frequency-dependent second order hyperpolarizabilities. Furthermore, the application of an external electric field along the direction parallel to the molecular plane induces a considerable decrease in the energy of the unoccupied orbitals of biphenyl, resulting in more red-shifted virtual orbitals compared with benzene and the highest static second-order hyperpolarizability for the molecule. It is also found that when the two adjacent rings share only one carbon atom, high σ and π-electronic contributions to the second-order hyperpolarizabilities are expected. Interactions of the type π* → π* with high delocalization energies are predicted for pyridine, nitrobenzene, chlorobenzene, bromobenzene, phenol, and benzoic acid. The quantum–mechanical calculations of the present work may provide theoretical insights into designing high (hyper)polarizability materials.

Density function theory calculation to study the oxidation potential of electron-donating compounds; affirming the oxidation mechanism by NICS calculations

The manuscript describes a method for understanding the correlation of structural features and first oxidation potentials [Formula: see text] of electron-donating compounds (EDCs) with tetrathiafulvalene (TTF), dithiadiazafulvalenes (DTDAF), and tetraazafulvalene (TAF) frameworks. The density functional theory (DFT) procedure at B3LYP (6-31 + g(d)) was used for geometric optimization, given the large dimensions of the molecules studied, and their high structural similarity. First of all, the correlation between the oxidation potential and the highest occupied molecular orbital (HOMO) energy level as an effective quantum chemical descriptor was examined. Then, nucleus-independent chemical shifts (NICSs) calculation was applied to affirm the oxidation mechanism and interpret the effect of replacing the sulfur atoms by nitrogen, on the oxidation process. Finally, a more comprehensive investigation of structural features that affect the oxidation potential, topological, geometrical, constitutional, as well as, electrostatic, charged partial surface area, quantum-chemical, molecular orbital, and thermodynamic descriptors was calculated. A predictive model was developed based on the genetic algorithm multivariate linear regression (GA-MLR). There was an outstanding agreement between the theoretical and the experimental values obtained for the first oxidation potentials of the test set (Q2Ext = 0.981).

Spin-Dependent First-Principles Study on Optoelectronic Properties of Neodymium Zirconates Pyrochlores Nd2Zr2O7 in Fd-3m and Pmma Phases.

Rare-earth zirconate pyrochlores (RE2Zr2O7) are of much fundamental and technological interest as optoelectronic, scintillator and thermal barrier coating materials. For the first time, we report the detailed optoelectronic properties of rare-earth zirconates Nd2Zr2O7 in both, i.e., for spin up and spin down states, via the use of first-principles density functional theory (DFT) procedure. To obtain the desired optoelectronic properties, we used a highly accurate method called full-potential linearized augmented plane wave (FPLAPW) within the generalized gradient approximation (GGA), parametrized with Hubbard potential U as an exchange-correlation function. The band gaps predicted for Nd2Zr2O7 were of the order 2.4 eV and 2.5 eV in Fd-3m and Pmma symmetrical phases, respectively. For both the phases, our research involved a complete examination of the optical properties of Nd2Zr2O7, including extinction coefficient, absorption coefficient, energy loss, function, reflectivity, refractive index, and real optical conductivity, analyzed in the spectral range from 0.0 eV to 14 eV. The calculated optical properties in both phases showed a considerable spin-dependent effect. The electronic bonding characteristics of different species in Nd2Zr2O7 within the two crystal symmetries were explored via the density distribution mapping of charge.

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.

Difficulty of the evaluation of the barrier height of an open-shell transition state between closed shell minima: The case of small C4n rings.

C4n cyclacenes exhibit strong bond-alternation in their equilibrium geometry. In the two equivalent geometries, the system keeps an essentially closed-shell character. The two energy minima are separated by a transition state suppressing the bond-alternation, where the wave function is strongly diradical. This paper discusses the physical factors involved in this energy difference and possible evaluations of the barrier height. The barrier given as the energy difference between the restricted density functional theory (DFT)/B3LYP for the equilibrium and the broken symmetry DFT/B3LYP of the transition state is either negative or small, in contradiction with the most reliable Wave Function Theory calculations. The minimal (two electrons in two molecular orbitals) Complete Active Space self-consistent field (CASSCF) overestimates the barrier, and the subsequent second-order perturbation cancels it. Due to the collective character of the spin-polarization effect, it is necessary to perform a full π CASSCF + second-order perturbation to reach a reasonable value of the barrier, but this type of treatment cannot be applied to large molecules. DFT procedures treating on an equal foot the closed-shell and open-shell geometries have been explored, such as Mixed-Reference Spin-Flip Time-dependent-DFT and a new spin-decontamination proposal, namely, DFT-dressed configuration interaction, but the results still depend on the density functional. M06-2X without or with spin-decontamination gives the best agreement with the accurate wave function results.

The Density Functional Theory Account of Interplaying Long-Range Exchange and Dispersion Effects in Supramolecular Assemblies of Aromatic Hydrocarbons with Spin

Aromatic hydrocarbons with fused benzene rings and regular triangular shapes, called n-triangulenes according to the number of rings on one edge, form groundstates with n-1 unpaired spins because of topological reasons. Here, we focus on methodological aspects emerging from the density functional theory (DFT) treatments of dimer models of the n = 2 triangulene (called also phenalenyl), observing that it poses interesting new problems to the issue of long-range corrections. Namely, the interaction comprises simultaneous spincoupling and van der Waals effects, i.e., a technical conjuncture not considered explicitly in the benchmarks calibrating long-range corrections for the DFT account of supramolecular systems. The academic side of considering dimer models for calculations and related analysis is well mirrored in experimental aspects, and synthetic literature revealed many compounds consisting of stacked phenalenyl cores, with intriguing properties, assignable to their long-range spin coupling. Thus, one may speculate that a thorough study assessing the performance of state-of-the-art DFT procedures has relevance for potential applications in spintronics based on organic compounds.

Interaction properties of phenol and styrene from plastic fumes on β-Arsenene sheets: A first-principles study

The recognizing efficiency of β-Arsenene nanosheet (β-AsNS) towards the plastic fumes, such as phenol and styrene is assessed by adopting the DFT (Density Functional Theory) procedure. The cohesive formation energy of the commanding material, β-AsNS is assessed to be −1.538 eV per atom, which verifies the configurational durability of the base component. The electronic fingerprints of the isolated β-AsNS and carcinogenic toxicants’ (phenol and styrene) adsorbed β-AsNS like the electron difference density and energy band gap are ciphered. Furthermore, the interaction properties of the toxicants adsorbed β-AsNS, namely the Bader charge transfer, binding energy, and average energy gap variation are evaluated to ascertain the chemi-detecting capacity of β-AsNS towards the plastic fumes, phenol, and styrene.

Mechanical and thermodynamic stability, structural, electronics and magnetic properties of new ternary thorium-phosphide silicides ThSixP1-x: First-principles investigation and prospects for clean nuclear energy applications

Thorium compounds have attracted immense scientific and technological attention with regard to both fundamental and practical implications, owing to unique chemical and physical properties like high melting point, high density and thermal conductivity. Hereby, we investigate the mechanical and thermodynamic stability and report on the structural, electronic and magnetic properties of new silicon-doped cubic ternary thorium phosphides ThSixP1-x (x = 0, 0.25, 0.5, 0.75 and 1). The first-principles density functional theory procedure was adopted within full–potential linearized augmented plane wave (FP-LAPW) method. The exchange and correlation potential terms were treated within Generalized-Gradient-Approximation functional modified by Perdew-Burke-Ernzerrhof parameterizations. The proposed compounds showed mechanical and thermodynamic stable structure and hence can be synthesized experimentally. The calculated lattice parameters, bulk modulus, total energy, density of states, electronic band structure and spin magnetic moments of the compounds revealed considerable correlation to the Si substitution for P and the relative Si/P doping concentration. The electronic and magnetic properties of the doped compounds rendered them non-magnetic but metallic in nature. The main orbital contribution to the Fermi level arises from the hybridization of Th(6d+5f) and (Si+P)3p states. Reported results may have potential implications with regard to both fundamental point of view and technological prospects such as fuel materials for clean nuclear energy.

Computational modeling of the kinetics and mechanism of the new generation of glutathione peroxidase nanomimic: selenosubtilisin and tellurosubtilisin

The catalytic cycles of a new generation of the GPx mimic, selenosubtilisin and tellurosubtilisin were illuminated via the density functional theory and solvent-assisted proton exchange procedure to show the indirect proton exchange through a hydrogen-bonded transfer network. The reactivity of selenosubtilisin and tellurosubtilisin toward methanethiol as a reduction step and hydrogen peroxide as an oxidation step was investigated. The results show that all steps of the proposed catalytic cycles are spontaneous and kinetically favorable. From the kinetics viewpoint, the oxidation of selenic acid (ΔG≠ = 23.86 kcal mol−1) and tellurenic acid (ΔG≠ = 20.66 kcal mol−1) is considered as the rate-determining steps. The comparison of the barriers for oxidation/reduction of the chalcogens shows a decay from the selenosubtilisin to tellurosubtilisin. On the basis of the turnover frequency calculations, the catalytic cycle of tellurosubtilisin is faster than that of selenosubtilisin, because of the lower span energy. To confirm the nature of bond cleavage/formation, during the catalytic reactions, topological analyses were applied by using the quantum theory of atoms in molecules procedure. The high values of the electron localization function and localized orbital locator at the transition state regions, calculated by the MultiWFN program, confirm the critical bond formation.

Benchmark study of DFT and composite methods for bond dissociation energies in argon compounds

We introduce a database of 14 accurate bond dissociation energies (BDEs) of noble gas compounds. Reference CCSD(T)/CBS BDEs are obtained by means of W1 theory. We evaluate the performance of contemporary density functional theory (DFT), double-hybrid DFT (DHDFT), and composite ab initio procedures. A general improvement in performance is observed along the rungs of Jacob’s Ladder; however, only a handful of functionals give good performance for predicting the bond dissociation energies in the NGC14 database. Thus, this database represents a challenging test for DFT methods. Most of the conventional DFT functionals (71%) result in root-mean-square deviations (RMSDs) between 10.0 and 82.1 kJ mol−1. The rest of the DFT functionals attain RMSDs between 2.5 and 8.9 kJ mol−1. The best performing functionals from each rung of Jacob’s Ladder are (RMSD given in parenthesis): HCTH407 (30.9); M06-L (5.4); PBE0 (2.8); B1B95, M06, and PW6B95 (2.7–2.9); CAM-B3LYP-D3 (5.4); and B2T-PLYP (2.5 kJ mol−1).

Calculation of the Jahn-Teller parameters with DFT

In this review, ? density functional theory (DFT) procedure is presented to calculate the Jahn?Teller (JT) parameters in a non-empirical way, which does not depend on the system at hand. Moreover, the intrinsic distortion path (IDP) model that gives further insight into the mechanism of the distortion is presented. The summarized results and their comparison to experimentally estimated values and high-level ab initio calculations, not only prove the good ability of the used approach, but also provide many answers to the intriguing behavior of JT active molecules.

Synthesis, experimental and theoretical characterizations of a 1,2,4-triazole Schiff base and its nickel(II) complex

A new Schiff base, HL, derived from condensation reaction of 2,4-dihydroxybenzaldehyde and 4-amino-3-methyl-1H-1,2,4-triazole-5(4H)-thione together with its Ni(II) complex have been synthesized and characterized by several spectroscopies methods. The optimized geometries, assignment of the IR bands and NMR chemical shifts as well as exploring of the frontier orbitals of two synthesized compounds have been calculated using the density functional theory (DFT) procedures. The most stable tautomer of the HL Schiff base and kinetics of its tautomerization have been explored theoretically. In the square planar structure of the [Ni(L)Cl] complex, the anionic L- acts as a NOS tridentate ligand. The chloro ligand is positioned trans to the azomethine nitrogen. The DFT-computed IR frequencies and NMR chemical shifts are good in agreement with their corresponding experimental values, approving validity of the optimized geometries for two investigated compounds. The [Ni(L)Cl] complex exhibits high catalytic activities in a three-component cyclocondensation for synthesis of the tetrahydrobenzo [b]pyrans.

Highly Sensitive Ratiometric Chemosensor and Biomarker for Cyanide Ions in the Aqueous Medium

A newly designed cyanide-selective chemosensor based on chromone containing benzothiazole groups [3-(2,3-dihydro-benzothiazol-2-yl)-chromen-4-one (DBTC)] was synthesized and structurally characterized by physico-chemical, spectroscopic, and single-crystal X-ray diffraction analyses. The compound DBTC can detect cyanide anions based on nucleophilic addition as low as 5.76 nM in dimethyl sulfoxide–N-(2-hydroxyethyl)piperazine-N′-ethanesulfonic acid buffer (20 mM, pH 7.4) (v/v = 1:3). The binding mode between receptor DBTC and cyanide nucleophile has also been demonstrated by experimental studies using various spectroscopic tools and theoretical studies, and the experimental work has also been verified by characterizing one supporting compound of similar probable structure of the final product formed between DBTC and cyanide ion (DBTC–CN compound) by single-crystal X-ray analysis for detailed structural analyses. In theoretical study, density functional theory procedures have been used to calculate the molecular structure and the calculation of the Fukui function for evaluation of the electrophilic properties of each individual acceptor atom. Furthermore, the efficacy of the probe (DBTC) to detect the distribution of CN– ions in living cells has been checked by acquiring the fluorescence image using a confocal microscope. Notably, the paper strips with DBTC were prepared, and these could serve as efficient and suitable CN– test kits successfully.

DFT study of nano zinc/copper voltaic cells.

To facilitate the development of new materials for use in batteries, it is necessary to develop ab initio full-electron computational techniques for modeling potential new battery materials. Here, we tested density functional theory procedures that are accurate enough to obtain the energetics of a zinc/copper voltaic cell. We found the magnitude of the zero-point energy correction to be 0.01-0.2kcal/mol per atom or molecule and the magnitude of the dispersion correction to be 0.1-0.6kcal/mol per atom or molecule for Zn n , (H2O) n , [Formula: see text], [Formula: see text], and Cu n . Counterpoise correction significantly affected the values of ∆[Formula: see text], ∆[Formula: see text], and ∆Esolv by 1.0-3.1kcal/mol per atom or molecule at the B3PW91/6-31G(d) level of theory, but by only 0.04-0.4kcal/mol per atom or molecule at the B3PW91/cc-pVTZ level of theory. The application of B3PW91/6-31G(d) yielded results that differed from macroscopic experimental values by 0.1-7.1kcal/mol per atom or molecule, whereas applying B3PW91/cc-pVTZ produced results that differed from macroscopic experimental values by 0.1-4.8kcal/mol per atom or molecule, with the smallest differences occurring for reactions with a small macroscopic experimental ∆E and the largest differences occurring for reactions with a large macroscopic experimental ∆E, implying size consistency.

Computational Kinetic Modeling of the Catalytic Cycle of Glutathione Peroxidase Nanomimic: Effect of Nucleophilicity of Thiols on the Catalytic Activity.

The catalytic cycle of a new derivative of ebselen, 1, was elucidated via three steps by the density functional theory and solvent-assisted proton exchange procedure involving indirect proton exchange through a hydrogen-bonded transfer network. Different behaviors of the aromatic and aliphatic thiols were investigated in the reduction of selenoxide (step 2 → 3) and selenurane (step 3 → 1) based on their nucleophilicity. The reduction of selenoxide in the presence of thiophenol (ΔG‡ = 15.9 kcal·mol-1) is faster than that of methanethiol (ΔG‡ = 29.3 kcal·mol-1), and methanethiol makes the reduction of selenoxide unspontaneous and kinetically unfavorable (ΔG = 2.8 kcal·mol-1). The nucleophilic attack may be enhanced by using the thiophenol backbone at the selenium center to lower the energy barrier of the selenoxide reduction (ΔG‡ = 15.9 kcal·mol-1). On the basis of the turnover frequency calculations, during the catalytic cycle, the rate of the reaction was analyzed and discussed. Low values of the electron density and Laplacian at the transition states are the evidence of the covalent O-H and O-O bonds rupture in the presence of methanethiol and thiophenol. The nature of the critical bond points was characterized, using the quantum theory of atoms in molecules, based on the electron location function and localized orbital locator values. Finally, the charge transfer process at the rate-determining step was investigated based on the natural bond orbital analysis.

Vacuum UV Polarization Spectroscopy of p-Terphenyl.

p-Terphenyl is used as a component in a variety of optical devices. In this investigation, the electronic transitions of p-terphenyl are investigated by synchrotron radiation linear dichroism (SRLD) spectroscopy in the range 30000-58000 cm-1 (330-170 nm) on molecular samples aligned in stretched polyethylene, thereby extending the region investigated by polarization spectroscopy into the vacuum UV. The resulting partial absorbance curves reveal that the vacuum UV band system with a maximum at 55000 cm-1 (180 nm) is predominantly short axis-polarized. This result is of interest in the optical applications of p-terphenyl, for example as a wavelength shifter. The observed polarization spectra are compared with the results of quantum chemical model calculations. Convoluted versions of the transitions predicted with the semiempirical ZINDO method and with the long-range-corrected time-dependent density functional theory (TD-DFT) procedures TD-CAM-B3LYP, TD-LC-ωPBE, and TD-ωB97XD are in similar qualitative agreement with the observed partial absorbance curves throughout the investigated spectral regions, while TD-B3LYP fails to predict qualitatively the spectrum of p-terphenyl in the region above 40000 cm-1 (250 nm).

Symmetry properties of the electron density and following from it limits on the KS-DFT applications

ABSTRACTAt present, the Density Functional Theory (DFT) approach elaborated by Kohn with co-authors more than 50 years ago became the most widely used method for study molecules and solids. Using modern computation facilities, it can be applied to systems with million atoms. In the atmosphere of such great popularity, it is particularly important to know the limits of the applicability of DFT methods. In this report, I will discuss two cases when the conventional DFT approaches, using only electron density ρ and its gradients, cannot be applied (I will not consider the Ψ-versions of DFT). The first case is quite evident. In the degenerated states, the electron density may not be defined, since electronic and nuclear motions cannot be separated, the vibronic interaction mixed them. The second case is related to the spin of the state. As it was rigorously proved by group theoretical methods at the theorem level, the electron density does not depend on the total spin S of the arbitrary N-electron state. It means that the Kohn–Sham equations have the same form for states with different S. The critical survey of elaborated DFT procedures, taking into account spin, shows that they modified only exchange functionals, the correlation functionals do not correspond to the spin of the state. The point is that the conception of spin cannot be defined in the framework of the electron density formalism, which corresponds to the one-particle reduced density matrix. This is the main reason of the problems arising in the study by DFT of magnetic properties of the transition metals. The possible way of resolving these problems can be found in the two-particle reduced density matrix formulation of DFT.