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Inhomogeneous Electron Gas

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Abstract

This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.

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... In a con form, it is necessary to emphasize that part of the general functional that correspond the exchange correlation interaction. The exchange correlation energy has been expres with density of the particles-n(r) [58,59]: ...
... In a concise form, it is necessary to emphasize that part of the general functional that corresponds to the exchange correlation interaction. The exchange correlation energy has been expressed with density of the particles-n(r) [58,59]: ...
... Equation (3) represents the single-particle Schrödinger equation with a form identical to that of the Schrödinger equation for non-interacting particles in an external potential. The DFT formalism [58,59], proposed by Hohenberg, Kohn and Sham, is suitable for the interaction between electrons and positrons. The positron wave function ϕ + (r) is calculated as follows [57]: The positron annihilation rate λ in an inhomogeneous electron gas is proportional to the overlap of electron n−(r) and positron n + (r) densities in the LDA. ...
... In a con form, it is necessary to emphasize that part of the general functional that correspond the exchange correlation interaction. The exchange correlation energy has been expres with density of the particles-n(r) [58,59]: ...
... In a concise form, it is necessary to emphasize that part of the general functional that corresponds to the exchange correlation interaction. The exchange correlation energy has been expressed with density of the particles-n(r) [58,59]: ...
... Equation (3) represents the single-particle Schrödinger equation with a form identical to that of the Schrödinger equation for non-interacting particles in an external potential. The DFT formalism [58,59], proposed by Hohenberg, Kohn and Sham, is suitable for the interaction between electrons and positrons. The positron wave function ϕ + (r) is calculated as follows [57]: The positron annihilation rate λ in an inhomogeneous electron gas is proportional to the overlap of electron n−(r) and positron n + (r) densities in the LDA. ...
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... Still, PIMC simulations are feasible over substantial parts of the relevant parameter regime. In addition, we investigate the impact of temperature and density on the real space density, which is the essential quantity governing the celebrated Hohenberg-Kohn theorems [44,45]. The theorems state that the ground-state electronic density of a system n 0 (r) uniquely determines its properties and even constructs a functional E[n] with a global minimum at n 0 (r). ...
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... First-principles calculations were performed with the Vienna Ab initio Simulation Package (VASP) 41,42 on the basis of density functional theory (DFT). 43,44 The Perdew−Burke−Ernzerhof (PBE) 45 form of generalized gradient approximation (GGA) exchange−correlation functional was implemented. The projector augmented wave (PAW) pseudopotential method 46,47 was used for the plane-wave expansion with an energy cutoff of 500 eV. ...
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... The calculations in this paper have been carried out at zero temperature. Plane wave-based density functional theory (DFT) calculations were applied using the Vienna Ab initio Simulation Package (VASP) [45][46][47]. The exchange-correlation energy functional were approximated using projected augmented wave (PAW) [48] based on generalized gradient approximation (GGA) [49] with Perdew-Burke-Ernzerhof (PBE) functional [49,50]. ...
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... The Vienna ab initio simulation package (VASP) [21] program was used for all the calculation, in the projector augmented wave (PAW) pseudopotential, which is based on density functional theory (DFT) [22]. A spin-polarized generalized gradient approximation (GGA) was used to solve the Kohn-Sham equations with the Perdew-Burke-Ernzerhof functional (PBE) exchange-correlation functional, with plane-wave pseudopotential [23]. ...
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... Our first-principles calculations are based on the generalized-gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [69][70][71][72][73] within the framework of the density-functional theory (DFT) using projector-augmented-wave (PAW) [74] wave functions as implemented in the Vienna Ab-Initio Simulation Package (VASP) [75,76]. The effect of Van-der-Waals (VdW) interactions was taken into account by using the empirical correction scheme of Grimme (DFT-D2) [77]. ...
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... We performed first-principles quantum mechanical calculations based on the density functional theory (DFT) (Hohenberg and Kohn 1964;Kohn and Sham 1965) using Vienna Ab initio Simulation Package (VASP) (Kresse and Furthmüller 1996a, b;Kresse and Hafner 1993). The local density approximation (LDA) was selected to treat the exchange and correlation (XC) functional (Perdew and Zunger 1981) (Table S1). ...
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Dispersion correction in theoretical determination of cyclopeptide conformations is emphasized. Whether in gas approximation or in solvation simulation, the density functional theory with London dispersion correction (DFT-D3) demonstrates that only 2∼3 conformers can stably coexist for cycloaspeptides (A, D, G) at B3LYP-D3 and CAM-B3LYP-D3. Conformational rationality is confirmed by electronic circular dichroism (ECD). Whether for Cotton effect or for excitation energy, TD-B3LYP-D3 has better performances than TD-CAM-B3LYP-D3 because the former can better reproduce the experiment. A molecular orbital analysis is used to interpret ECD, where two energy bands observed in experiment originates from the ππ* transitions other than the σπ* transitions. Long-range correction and solvent effect make H-bonds shorten, and dispersion correction makes them further shorten.
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MXenes are two-dimensional layered materials that have attracted increased attention for technological applications, e.g., electrodes, support for single atom catalysts, etc. Several experimental and computational studies have been reported, however, our atomistic understanding is still far from good, in particular, for MXene-ionic liquid interfaces, which is crucial to predict and control the charge-discharge rate of those systems. Here, we have assessed the adsorption of ionic liquids forming species, on surfaces of Ti3C2Tx (where T = F, O or OH) MXenes by means of abinitio calculations at density functional theory level. The F and O terminated MXenes showed to be stables upon the adsorption, whereas the OH terminated one suffered important structural deformations when interacting with cationic species. The adsorption of molecular species are stronger, with values ranging in between -1.10 and -7.36eV, indicating the contribution of Coulombic and induction interactions as a result of the charge transference among MXenes and adsorbates. The adsorption of ionic pairs on the Ti3C2F2 and Ti3C2O2 monolayers were driven by dispersion interactions, that represent at least 84% of the total adsorption energy. Additionally, the work function of the stable MXenes structures were barely affected by the adsorption of the ionic pairs, and hence, the adsorption process did not affect the electronic properties of the single Ti3C2Tx monolayers.
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Density functional theory (DFT) is a widely adopted methodology that gives quantum-level understanding of matter and guides materials discovery. In parallel, artificial intelligence (AI) for materials is an emerging interdisciplinary research direction whose major purpose is to accelerate the process of materials discovery. However, the shortage of informative data—which are crucial for model training—makes accurate prediction, design, synthesis, and characterization a challenge for materials. Experimental data are regarded as expensive, while the relatively less expensive data obtained from DFT calculations suffer from systematic errors rooted in approximated density functionals. Recently, Kirkpatrick et al. constructed the density functional DeepMind 21 (DM21) via deep learning, which solved fractional electron problems and outperformed hand-designed functionals. The foundation of DM21 reveals that data calculated from accurate AI-designed functionals can be supplementary to experimental data in prediction models training and improve models’ performance, thus accelerating material discovery processes.
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Electrochromic materials can tune the illumination and heat exchange of a building with the environment and thereby save energy in lighting, heating, and air conditioning in a cost‐effective way, which is vital in realizing carbon neutrality. 2D frameworks such as coordination nanosheets (CONASHs) that are widely explored for a wide range of applications in energy storage and conversion can be a cluster of novel electrochromic materials. In this work, a series of transition metal benzenehexathiol (TM‐BHT) CONASHs are theoretically investigated via first‐principles simulations. During ion intercalation and deintercalation in TM‐BHTs, changes in lattice structures, lithium diffusion barriers, atomic charges, bond strength, and electronic properties are explored in‐depth. The incurred changes are then correlated with critical electrochromic properties, including the transmittance adjustment ranges in the visible light, near‐infrared, solar spectrum, and mid‐infrared. Among the various TM‐BHT systems, Cu‐BHT and Ag‐BHT are the most promising broadband electrochromic materials for optical and thermal management in the wavelength range from visible to mid‐infrared. The theoretical guidance from this work paves a new path toward electrochromic applications of CONASHs that exploit the versatility of these 2D materials.
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Here, the optical, dielectric and electronic properties of two-dimensional (2D) isomers of NbS2 monolayer, namely planar isomer (P–NbS2 isomer) and hexagonal isomer (H–NbS2 isomer), are being explored. It is found that the P–NbS2 isomer is an indirect band gap semiconducting, and it has a band gap of approximately 0.36 eV between the valence and conduction bands. On the other hand, the H–NbS2 isomer exhibit metallic characteristics, which is evident from the band structure and also from the density of states. The optical properties such as absorption coefficient, dielectric function, and optical conductivity, along with the refractive index, reflection, and extinction coefficients in both the isomers, have also been calculated. These isomers of NbS2 exhibit excellent optical response and high dielectric behavior in the low-energy region. The dielectric study suggests that both the isomers are highly anisotropic in nature. These monolayers of NbS2 isomer may be a potential candidate for futuristic metal-based electronic applications such as nanoelectronics, optoelectronics, and photonics.
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The knowledge about the mechanisms of the morphological control of nanoparticles (NPs) is directly correlated with the atomic configurations of their exposed surfaces, which can facilitate materials functionalization according to the surface-dependent properties. In this context, this study focused on modeling via the density functional theory (DFT) the (001), (100), (101), (103), (110), (111), and (112) surfaces of the CaXO4 (X = Mo or W) (scheelite phase) to offer a comprehensive study of their structural and electronic properties. Additionally, a systematic mapping of the NPs morphology was elaborated as a function of the modulation of the surface energies. For the surfaces of both systems, a stability order of (001) > (112) > (111) > (101) > (110) > (103) > (100) was observed. Differences were observed in both systems concerning the outermost polyhedral distortion and their atomic charges. The analysis of the energy band alignment of the surfaces revealed the potential use of both materials in photocatalytic environmental remediation. The methodology and results presented herein can be useful for targeting the synthesis and functionalization of CaXO4 and related materials.
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To study the structural properties of Co2FeZ (Z = Al, Si, Ga) (CFZ) alloys, we will use the approximation method GGA-PBE based on the method of plane waves increased by linear waves at full potential using the theory of functional density in both the Hg2CuTi and Cu2MnAl-type structures. From the most stable state we determine the other properties such as the magnetic, elastic and thermoelectric properties. The band structure calculation reveals indirect band gap in spin down channel and zero band gap in spin up channel of valence and conduction bands confirming the spin gapless semiconducting nature of these compounds. Calculated Seebeck coefficient in spin up and spin down channel reveals that the CFZ behaves as both n and p type thermoelectric materials with better output efficiency. The transport properties of these materials are discussed on the basis of Seebeck coefficient, electrical conductivity coefficient, thermal conductivity and figure-of-merit coefficient. By analyzing the nature of the bonding between the different atoms that form CFZ that each of them has a strong covalent character.
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Koopmans spectral functionals aim to describe simultaneously ground-state properties and charged excitations of atoms, molecules, nanostructures, and periodic crystals. This is achieved by augmenting standard density functionals with simple but physically motivated orbital-density-dependent corrections. These corrections act on a set of localized orbitals that, in periodic systems, resemble maximally localized Wannier functions. At variance with the original, direct supercell implementation (Phys. Rev. X 2018, 8, 021051), we discuss here (i) the complex but efficient formalism required for a periodic boundary code using explicit Brillouin zone sampling and (ii) the calculation of the screened Koopmans corrections with density functional perturbation theory. In addition to delivering improved scaling with system size, the present development makes the calculation of band structures with Koopmans functionals straightforward. The implementation in the open-source Quantum ESPRESSO distribution and the application to prototypical insulating and semiconducting systems are presented and discussed.
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Peptides and foldamers have recently gained increasing attention as chiral catalysts to achieve challenging (asymmetric) transformations. We previously reported that short helically folded aliphatic oligoureas in combination with achiral Brønsted bases are effective H-bonding catalysts for C-C bond-forming reactions─i.e., the conjugate addition of 1,3-dicarbonyl pronucleophiles to nitroalkenes─with high reactivity and selectivity and at remarkably low chiral catalyst/substrate molar ratios. This theoretical investigation at the density functional theory level of theory, aims to both analyze how the substrates of the reaction interact with the foldamer catalyst and rationalize a chain-length dependence effect on the catalytic properties. We confirm that the first two ureas are the only H-bond donors available to interact with external molecules. Moreover, each urea site interacts with one of the two reactants allowing a short distance between the two reacting carbons, thus facilitating the conjugated addition. Additionally, it was observed that the molecular recognition and catalyst-substrate interactions are mainly governed by electrostatic interactions but not orbital interactions (see from NBO if this is finally true). On these grounds, an electrostatic potential (ESP) analysis showed an important internal charge separation in the catalyst, the positive ESP region being concentrated around the first two ureas, with its area extending as the number of residues increases.
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The energetics of the regioselective mononitration of 9,10-BN-naphthalene with acetyl nitrate (H3C2NO4) were modeled with ab initio simulations in the gas phase and an acetonitrile solvent. The single-electron-transfer (SET) nitration mechanism leading to a σ-complex and a single-step nitration mechanism were modeled. The energy barrier for the single-step mechanism was lower than that for the SET mechanism in the gas phase. However, the two are much more energetically competitive in the solvent. The σ-complex was found to be unstable in the gas phase owing to the interaction with the counterion. Using the single-step mechanism, the carbon site 1 nearest boron had the lowest activation energy for nitration of 22.6 kcal/mol, while site 3 had the second lowest barrier of 24.6 kcal/mol. Details on the molecular structures at intermediate and transition states as well as charges in different configurations are discussed.
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The quasi-classical limit of the many-particle perturbation theory is considered for electrons moving in the Coulomb potential of the nucleus. The problem is analyzed using the Green's function formulation of many-particle physics. When corrections involving the gradient of the self-consistent potential are neglected, the results agree to all orders in the interaction strength with the procedure of Lewis for including correlation corrections in the Thomas-Fermi model. The disagreement between the work of Lewis and the recent work of Baraff is shown to be due to the latter's neglect of anomalous contributions to the perturbation theory which arise, because the local Fermi momentum is a function of the interaction strength. The calculation of inhomogeneity corrections is also considered.
Article
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of ℏ. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.
Article
The Fermi-Thomas-Dirac equation is modified tc include correlations ; between electrons. An application is made to the equation of state. No ; numerical work with the modified equation is reported. (anth);
Article
The limits of validity of the correlation-energy calculations in the regions of high density, low density, and actual metallic electron densities are discussed. Simple physical arguments are given which show that the high-density calculation of Gell-Mann and Brueckner is valid for rs1 while the low-density calculation of Wigner is valid for rs20. For actual metallic densities it is shown that the contribution to the correlation energy from long-wavelength momentum transfers (k<βk0<0.47rs12k0) may be accurately calculated in the random phase approximation. This contribution is calculated using the Bohm-Pines extended Hamiltonian, and is shown to be E(β)=-0.458β2rs+0.866β3rs32-0.98β4rs2+0.019β4rs+0. 706β5rs52+ry. An identical result is obtained by a suitable expansion of the result of Gell-Mann and Brueckner; the validity of the Bohm-Pines neglect of subsidiary conditions in the calculation of the ground-state energy is thereby explicitly established. The contribution to the correlation energy from sufficiently high momentum transfers (kk0) will arise only from the interaction between electrons of antiparallel spin, and may be estimated using second-order perturbation theory. The contribution arising from intermediate momentum transfers (0.47rs12k0kk0) cannot be calculated analytically; the interpolation procedures for this domain proposed by Pines and Hubbard are shown to be nearly identical, and their accuracy is estimated as ∼15%. The result for the over-all correlation energy using the interpolation procedure of Pines is Ec(-0.115+0.031 lnrs)ry.
Article
A new method is described for computing the effect of correlation, inhomogeneity, and exchange on the Thomas-Fermi model of the atom. The method makes use of the many-body point of view, rather than an independent-particle point of view, by considering the hierarchy equation linking the n-particle Green's functions. The hierarchy is truncated by a prescription equivalent to the Gell-Mann and Brueckner theory of the high-density electron gas, resulting in a description of the atom in which the exchange interaction is replaced by the effective interaction. The physical significance of this replacement is noted. The Green's function for this model is then expanded as a series in powers of &planck;. The lowest order term is found to describe the Thomas-Fermi model of the atom. The equation for the next higher term contributing to this expansion is manipulated so as to yield an ordinary differential equation for the corresponding correction to the potential. This equation contains a term which expresses the effect of inhomogeneity and another which arises from the correlation of the electrons and from exchange. The inhomogeneity term is one which has been found previously. Study of the correlation term shows that it depends on the separation energy of an electron from an infinite electron gas, which suggests a generalization by which the method might be made applicable to those outer regions of the atom for which the electron density is below that to which the Gell-Mann and Brueckner theory would apply.
Article
A theoretical investigation is made of the distribution of electrons round a positively charged impurity dissolved in a monovalent metal. Applications are made to dissolved hydrogen, where the impurity is a proton, and to atoms such as zinc, gallium, etc., which are usually considered to contribute their electrons to the conduction electrons. In all cases the positive charge must be screened; and in many cases this is shown to occur through the formation of bound states below the level of the Fermi distribution. The relation of these results to the Hume-Rothery rule is discussed. The ideas introduced are used to calculate the heats of solution of hydrogen and of polyvalent metals in the noble metals, and to discuss the magnetic properties of these alloys. A detailed discussion is given of x-ray emission and absorption, the vacancy left in the x-ray shell being here treated as the positive impurity. In certain cases quantitative predictions are made about the energies of x-ray absorption edges.Discussions along the same lines are given of the optical absorption of the noble metals, and of the x-ray emission spectra of certain alloys.
Article
The shielding of a small fixed charge in a high-density electron gas is calculated by means of a technique similar to that used by Gell-Mann and Brueckner for calculating the correlation energy. A closed expression for the density of displaced electrons is derived and evaluated numerically for several densities of the electron gas. An asymptotic form is also given.
Article
The energy of interaction between free electrons in an electron gas is considered. The interaction energy of electrons with parallel spin is known to be that of the space charges plus the exchange integrals, and these terms modify the shape of the wave functions but slightly. The interaction of the electrons with antiparallel spin, contains, in addition to the interaction of uniformly distributed space charges, another term. This term is due to the fact that the electrons repell each other and try to keep as far apart as possible. The total energy of the system will be decreased through the corresponding modification of the wave function. In the present paper it is attempted to calculate this "correlation energy" by an approximation method which is, essentially, a development of the energy by means of the Rayleigh-Schr\"odinger perturbation theory in a power series of ${e}^{2}$.
Article
The quantity εc is defined as the correlation energy per particle of an electron gas expressed in rydbergs. It is a function of the conventional dimensionless parameter rs, where rs-3 is proportional to the electron density. Here εc is computed for small values of rs (high density) and found to be given by εc=Alnrs+C+O(rs). The value of A is found to be 0.0622, a result that could be deduced from previous work of Wigner, Macke, and Pines. An exact formula for the constant C is given here for the first time; earlier workers had made only approximate calculations of C. Further, it is shown how the next correction in rs can be computed. The method is based on summing the most highly divergent terms of the perturbation series under the integral sign to give a convergent result. The summation is performed by a technique similar to Feynman's methods in field theory.
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J. Friedel, Dr. R. Balian, and Dr. C. De Dominicis for valuable discussions. PHYSICAL REVIEW VOLUME 136, NUM BER 3 B 9 iVOVEM B ER 1964
Enrico Fermi International School of Physics, Varenna, 1963 (unpublished). second area of concentration is the question of proper description of the electromagnetic radiation emanating from a laser; i.e. , questions of coherence and correlation
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Lamb, Jr., Lecture Notes, Enrico Fermi International School of Physics, Varenna, 1963 (unpublished). second area of concentration is the question of proper description of the electromagnetic radiation emanating from a laser; i.e., questions of coherence and correlation. ' And finally, the problem of interaction of laser light with matter has attracted considerable interest. ' It is this latter question to which we are devoting ourselves in this paper. a ' R. Glauber, Phys. Rev. 130, 2529 (1963);
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E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963);
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E. Wolf, Proc. Phys. Soc. (London) 80, 1269 (1962).
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Z. Fried s.nd W. M. Frank, Nuovo Cimento 27, 218 (1963).