Ground-state Electron Density Research Articles

New applications of the jellium model for the study of atomic clusters

Application of the jellium model for investigation of the electronic structure and photoionization of metal clusters and fullerenes is discussed. The valence electrons are considered either within the Hartree-Fock and the local density approximations. The random phase approximation is utilized to account for the many-electron correlations in the response of a system to an external field. It is shown that the photodetachment cross section and photoelectron angular distribution in metal cluster anions are described reasonably well within the jellium model. Its application to fullerenes requires the use of corrections for a better description of the ground state electron density.

Weak harmonic confinement of the quintet solution of a Moshinsky atom with 4 electrons

Bruch, in early work, treated a spatially free Moshinsky atom with four parallel-spin electrons interacting harmonically. Here we add a harmonic external potential, having an unrelated spring constant \(k\), and use the variational method, with a one-parameter trial wave function, to examine the quintet ground state energy and electron density, particularly in the weak confinement limit \(k\rightarrow 0\). The results are compared with Bruch’s and modest contact is made with the early work of Post.

Interpretation of Crystal Effects on NMR Chemical Shift Tensors: Electron and Shielding Deformation Densities

The relationship between the NMR observables and the supramolecular structure of any system is not straightforward. In this work we examine the influence of the crystal packing for three purine derivatives (hypoxanthine, theobromine, and 6-(2-methoxy)benzylaminopurine) on the principal components of the NMR chemical shift tensors (CSTs). We employ density functional calculations to obtain various molecular properties (the ground-state electron density, the magnitudes and orientations of the components of NMR chemical shift tensor, and the spatial distribution of the isotropic magnetic shielding) for the isolated molecules and for the molecules embedded in supramolecular clusters modeling the crystal environment and evaluate their differences. The concept has enabled us to rationalize the effect of the crystal packing on the NMR CSTs in terms of the redistribution of the ground-state electron density induced by intermolecular interactions in the solid state.

Use of Harriman’s construction in determining molecular equilibrium states

Information-theoretic description of the electron probabilities and currents in molecules is extended to cover the complex amplitudes (wave functions) of quantum mechanics. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. The previous one-electron development in such an entropic perspective on the molecular electronic structure is extended to cover N-electron states by adopting the Harriman-type framework of equidensity orbitals. This analysis emphasizes the phase part of electronic states, which generates the probability-current density and the associated non-classical entropy contributions, which allow one to distinguish the information content of states generating the same electron density and differing in their current composition. A complementary character of the Fisher and Shannon information measures is explored in the associated vertical (density-constrained) information principles, for determining the equilibrium state corresponding to the fixed ground-state electron density. It is argued that the lowest “thermodynamic” state generally differs from the true ground state of the system, by exhibiting the space-dependent phase and hence also the non-vanishing probability current, linked to the system electron distribution.

Modelling X-ray scattering factors from fluids of some fluorinated molecules and related compounds

There is current interest, both for basic science and technological applications, in fluorocarbons such as CF4 and the related molecules. Here we first report a Hartree–Fock calculation of the ground-state electron density in the octahedral molecule SF6 using the experimentally determined bond length. From this density, the number of electrons lying inside a sphere of radius r, denoted by Q(r), centred on the S nucleus has been extracted. The X-ray scattering factor f(k) is then modelled using Q(r) and compared with f(k) for the isoelectronic united atom ytterbium. The molecule CF4, together with its isoelectronic partners PF3, CrF2 and AsF, is again compared with regard to f(k) with the united atom Mo. Finally, interest in modelling PbF is pointed out, in relation to the quantum-chemical theory already available.

Can orbital-free density functional theory simulate molecules?

Orbital-free density functional theory (OFDFT), with its attractive linearly scaling computation cost and low prefactor, is one of the most powerful first principles methods for simulating large systems (~10(4)-10(6) atoms). However, approximating the electron kinetic energy with density functionals limits the accuracy and generality of OFDFT compared to Kohn-Sham density functional theory (KSDFT). In this work, we test whether the Huang-Carter (HC) kinetic energy density functional (KEDF), which contains the physics to properly describe covalently bonded semiconductor materials, can also be used to describe covalent bonds in molecules. In particular, we calculate a variety of homonuclear diatomic molecules with the HC functional within OFDFT. The OFDFT bond dissociation energy, equilibrium bond length, and vibrational frequency of these dimers are in remarkably good agreement with benchmark KSDFT results, given the lack of orbitals in the calculation. We vary the two parameters λ (controlling the reduced density gradient contribution to the nonlocal kernel) and β (the exponent of the density in the nonlocal term) present in the HC KEDF and find that the optimal λ correlates with the magnitude of the highest occupied molecular orbital-lowest unoccupied molecular orbital energy gap. Although the HC KEDF represents a significant improvement over previous KEDFs in describing covalent systems, deficiencies still exist. Despite the similar overall shape of the KSDFT and OFDFT ground state electron densities, the electron density within the bonding region is still quite different. Furthermore, OFDFT is not yet able to give reasonable description of magnetic states. The energy orderings of the triplet and singlet states of Si(2) and Al family dimers are not consistent with KSDFT or experimental results and the spin polarization distributions also differ widely between the two theories.

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r, r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V x (r) determined solely by the HF ground state density for the Be atom.

Finite element solution of the Kohn‐Sham equations

AbstractMany properties of condensed matter as for example electric conductivity, magnetism as well as the mechanical response upon external excitations are determined by the electronic structure of the material. Its evaluation represents a coupled, quantum mechanical many body problem, consisting of the positively charged atomic cores and the negatively charged, fermionic electrons. By the Hohenberg‐Kohn theorem a method was established to calculate the ground state electron density through a density functional. Then the Kohn‐Sham equations resemble a nonlinear, single electron problem with an effective potential, accounting for the Coulomb interactions between the particles as well as for quantum mechanical effects. They can be solved within a self consistent field procedure. In this work a real space formulation for the calculation of the electronic structure in the context of density functional theory is shown. In a first step a finite element based solution algorithm for the Kohn‐Sham equations is developed, based upon which the electron density is obtained in a non‐periodic setting. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

In this article, we provide ap riorierror estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsacker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. We prove that, for any local minimizer Φ 0 of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of Φ 0 for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal ap riori error estimates for the spectral discretization method.

Local Defects are Always Neutral in the Thomas–Fermi–von Weiszäcker Theory of Crystals

The aim of this article is to propose a mathematical model describing the electronic structure of crystals with local defects in the framework of the Thomas-Fermi-von Weizsacker (TFW) theory. The approach follows the same lines as that used in {\it E. Cances, A. Deleurence and M. Lewin, Commun. Math. Phys., 281 (2008), pp. 129--177} for the reduced Hartree-Fock model, and is based on thermodynamic limit arguments. We prove in particular that it is not possible to model charged defects within the TFW theory of crystals. We finally derive some additional properties of the TFW ground state electronic density of a crystal with a local defect, in the special case when the host crystal is modelled by a homogeneous medium.

Theorems on ground-state phase transitions in Kohn–Sham models given by the Coulomb density functional

Some theorems on derivatives of the Coulomb density functional with respect to the coupling constant λ are given. A model Fermion system with the reduced coupling constant, λ < 1, is defined to reproduce a ground-state electron density nGS(r) and the ground state energy. Fixing the charge density, possible phase transitions as level crossings occur only at discrete points along the λ axis. We prove that Dini's λ derivatives of the reduced energy density functional are given by expectation values of the Coulomb interaction with respect to two-state vectors at the crossing point. If the density is v-representable also for λ < 1, the accumulation of phase transition points is forbidden when λ → 1. Our theorems ensure the existence of a converging series of models defined by the multi-reference density functional theory.

A form of the single-particle kinetic energy density of an inhomogeneous electron liquid from a combination of one-body potential and ground-state electron density

Gál and March have recently proposed a form of the single-particle kinetic energy density in density functional theory in terms of the one-body potential V(r) and the ground-state electron density n(r) generated thereby. Here, with a minor modification of the GM form, examples are given for (a) harmonic trapping and (b) a bare Coulomb potential. The case of the He atom is also considered, via the Chandrasekhar variational wave function. Finally, the use of the semiempirical fine-tuned Hartree–Fock n(r) for spherical atoms due to Cordero et al. is briefly referred to.

Acetyl group orientation modulates the electronic ground-state asymmetry of the special pair in purple bacterial reaction centers

Recent experimental data point to an asymmetric ground-state electronic distribution in the special pair (P) of purple bacterial reaction centers, which acts as the primary electron donor in photosynthesis. We have performed a density functional theory investigation on an extended model including the bacteriochlorophyll dimer and a few relevant surrounding residues to explore the origin of this asymmetry. We find strong evidence that the ground-state electron density in P is intrinsically asymmetric due to protein-induced distortions of the porphyrin rings, with excess electron charge on the P(M) bacteriochlorophyll cofactor. Moreover, the electron charge asymmetry is strongly modulated by the specific orientation of the C3(1) acetyl group, which is hydrogen bonded to His168. The electronic excitation has a significant charge transfer character inducing a displacement of electron charge from P(L) to P(M), in agreement with experimental data in the excited state. These results are relevant for the understanding of the unidirectional electron transfer path in photosynthesis.

Proposed approximate relation in inhomogeneous electron liquids between exchange-only potential and its Slater counterpart

We consider here exchange-only density functional theory. The search for the exchange-only potential V x (r) involves either the optimal effective potential, or as an approximate alternative, calculating corrections to the Slater (Sl) potential = 2e x (r)/n(r), e x (r) and n(r) being, respectively, the exchange energy density due to Dirac and the ground-state electron density. The Levy–Perdew virial-like integral relation between the total exchange energy and the gradient of V x (r) is an important element in the present proposal.

One-body potential theory of molecules and solids modified semiempirically for electron correlation

The study of Cordero, March and Alonso (CMA) for four spherical atoms, Be, Ne, Mg and Ar, semiempirically fine-tunes the Hartree–Fock (HF) ground-state electron density by inserting the experimentally determined ionization potentials. The present Letter, first of all, relates this approach to the very recent work of Bartlett ‘towards an exact correlated orbital theory for electrons’. Both methods relax the requirement of standard DFT that a one-body potential shall generate the exact ground-state density, though both work with high quality approximations. Unlike DFT, the CMA theory uses a modified HF non-local potential. It is finally stressed that this potential generates also an idempotent Dirac density matrix. The CMA approach is thereby demonstrated to relate, albeit approximately, to the DFT exchange-correlation potential.

Heuristic exchange gradient-type functional for the hydrogen atom in two dimensions

Following the idea of Gill and Pople [Phys. Rev. A 47, 2383 (1993)] in three dimensions, we have constructed a gradient-corrected functional for the hydrogen atom for the two-dimensional case. The corresponding potential is constrained by the requirement that it cancel precisely the Coulomb potential of the normalized ground-state electronic charge density distribution which characterizes the localized single-electron state around a proton. Within this condition for a gradient-corrected functional, the dimensionality dependence is quantified and found to be relatively small.

Introducing PROFESS 2.0: A parallelized, fully linear scaling program for orbital-free density functional theory calculations

Orbital-free density functional theory (OFDFT) is a first principles quantum mechanics method to find the ground-state energy of a system by variationally minimizing with respect to the electron density. No orbitals are used in the evaluation of the kinetic energy (unlike Kohn–Sham DFT), and the method scales nearly linearly with the size of the system. The PRinceton Orbital-Free Electronic Structure Software (PROFESS) uses OFDFT to model materials from the atomic scale to the mesoscale. This new version of PROFESS allows the study of larger systems with two significant changes: PROFESS is now parallelized, and the ion–electron and ion–ion terms scale quasilinearly, instead of quadratically as in PROFESS v1 (L. Hung and E.A. Carter, Chem. Phys. Lett. 475 (2009) 163). At the start of a run, PROFESS reads the various input files that describe the geometry of the system (ion positions and cell dimensions), the type of elements (defined by electron–ion pseudopotentials), the actions you want it to perform (minimize with respect to electron density and/or ion positions and/or cell lattice vectors), and the various options for the computation (such as which functionals you want it to use). Based on these inputs, PROFESS sets up a computation and performs the appropriate optimizations. Energies, forces, stresses, material geometries, and electron density configurations are some of the values that can be output throughout the optimization.

Differential equation for the Dirac single-particle first-order density matrix in terms of the ground-state electron density

The March-Suhai (MS) partial differential equation for the Dirac density matrix {gamma}{sub s}(r,r'), proved for one- and two-level occupancies, involves both the ground-state density n(r), with its low-order derivatives, and the positive definite kinetic energy density t{sub s}(r). Here, we examine the relation between the equation of motion for {gamma}{sub s}(r,r'), with input now being the one-body potential of density-functional theory, and the MS equation. The important link is the differential virial theorem, which can be used to remove t{sub s}(r) from the MS differential equation. For multiple occupancy, the Pauli potential enters in an important manner. In one dimension, however, the appearance of the Pauli potential can be avoided, obtaining a necessary condition for {gamma}{sub s}(x,x{sup '}) to satisfy for arbitrary level occupancy, in the form of a MS-type differential equation.

Mixed quantum-classical dynamics with time-dependent external fields: A time-dependent density-functional-theory approach

A mixed quantum-classical method aimed at the study of nonadiabatic dynamics in the presence of external electromagnetic fields is developed within the framework of time-dependent density functional theory. To this end, we use a trajectory-based description of the quantum nature of the nuclear degrees of freedom according to Tully's fewest switches trajectories surface hopping, where both the nonadiabatic coupling elements between the different potential energy surfaces, and the coupling with the external field are given as functionals of the ground-state electron density or, equivalently, of the corresponding Kohn-Sham orbitals. The method is applied to the study of the photodissociation dynamics of some simple molecules in gas phase.

Relation between single-particle kinetic energy and exchange energy in DFT for the inhomogeneous electron liquid in the Be atom

In a recent study, the authors have used the semi empirical fine-tuned Hartree–Fock ground-state electron density n(r) of Cordero et al. [Phys. Rev. A 75, 052502 (2007)] for the Be atom to calculate the phase θ(r) from a non-linear pendulum-like equation. Since the density amplitude n(r)1/2 plus θ(r) determine, in turn, the idempotent Dirac density matrix γ(r, r′), we use n(r) and θ(r) first of all to calculate the exchange energy density e X (r) of the density functional theory (DFT). This enables us to obtain the Slater (Sl) approximation to the exchange-only potential. A comparison can then be made, by integrating the earlier predicted exchange-correlation force −∂V XC (r)/∂r, of V XC (r) with . Relationship to the Becke semiempirical density gradient approximation for exchange is also established. Some brief discussion of the Perdew–Burke–Ernzerhof density functional is added.