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A perturbation-method-based post-processing for the planewave discretization of Kohn–Sham models
Abstract
In this article, we propose a post-processing of the planewave solution of the Kohn–Sham LDA model with pseudopotentials. This post-processing is based upon the fact that the exact solution can be interpreted as a perturbation of the approximate solution, allowing us to compute corrections for both the eigenfunctions and the eigenvalues of the problem in order to increase the accuracy. Indeed, this post-processing only requires the computation of the residual of the solution on a finer grid so that the additional computational cost is negligible compared to the initial cost of the planewave-based method needed to compute the approximate solution. Theoretical estimates certify an increased convergence rate in the asymptotic convergence range. Numerical results confirm the low computational cost of the post-processing and show that this procedure improves the energy accuracy of the solution even in the pre-asymptotic regime which comprises the target accuracy of practitioners.
... many-particle interactions in a given approximation. More recently, a post-processing strategy has been proposed by some authors for planewave discretizations for non-linear eigenvalue problems [8,9,7,11], which considers the exact solution as a perturbation of the discrete (using the planewave basis) approximation. This is in spirit not so far from so-called two-grid methods, where a first problem is solved on a coarse basis, i.e. in a small discretization space, and a small problem is solved on a fine basis. ...
... These considerations motivate the numerical strategies to compute solutions to (9) in the following section. ...
... ). The limit value λ σ := lim k→∞ λ (k) σ then satisfies λ σ = ν σi (λ σ ) and thus (9). Strategy 2: For a given target value λ t ∈ R, consider the sequence of iterates λ ...
In this article, we propose a new numerical method and its analysis to solve eigen-value problems for self-adjoint Schr{\"o}dinger operators, by combining the Feshbach-Schur perturbation theory with planewave discretization. In order to analyze the method, we establish an abstract framework of Feshbach-Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new planewave discretization method. Finally, we present some numerical results that underline the theoretical findings.
... This correction improves the convergence rate by a factor 1 M . Other approaches dealing directly (see [7,8]) or indirectly (e.g. using pseudopotentials [3,5]) with the Coulomb singularities exist and are more efficient. ...
... Let ψ i be the corresponding eigenfunction such that (1.7) holds. Then for s < 5 2 and all ε > 0, we have a positive constant C s,ε independent of M such that ...
In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn–Sham equations. A priori estimates of plane-wave convergence for periodic Kohn–Sham calculations with pseudopotentials have been proved, however in most computations in practice, plane-wave cut-offs are not tight enough to target the desired accuracy. It is often advocated that the real quantity of interest is not the value of the energy but of energy differences for different configurations. The computed energy difference is believed to be much more accurate because of “discretization error cancellation”, since the sources of numerical errors are essentially the same for different configurations. In the present work, we focused on periodic linear Hamiltonians with Coulomb potentials where error cancellation can be explained by the universality of the Kato cusp condition. Using weighted Sobolev spaces, Taylor-type expansions of the eigenfunctions are available yielding a precise characterization of this singularity. This then gives an explicit formula of the first order term of the decay of the Fourier coefficients of the eigenfunctions. As a consequence, the error on the difference of discretized eigenvalues for different configurations is indeed reduced by an explicit factor. However, this error converges at the same rate as the error on the eigenvalue. Plane-wave methods for periodic Hamiltonians with Coulomb potentials are thus still inefficient to compute energy differences.
... [44] for the Hartree-Fock equations. For Kohn-Sham DFT, practical, not guaranteed, error bounds on the energy and the density matrix have been proposed for various discretization methods, notably planewave [8,22], and finite elements [19,41,46,60,61]. Let us also mention the recent work [7], which includes practical bounds on quantities of interest other than the energy and the density matrix, such as the interatomic forces. ...
In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.
... A second example, which is covered by this paper, is the Kohn-Sham model [23] and, in particular, the model based on the density functional theory [19]. This theory allows a reduction of the degrees of freedom, leading to a model which balances accuracy and computational cost, see also [11,12,36] for a more detailed introduction. ...
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energyadaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.
... [3,4,10]. In particular, [14] provides post-processing algorithms based on a perturbation method which is related to the discussion of this section. ...
We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn--Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.
... Despite this, significant progress has been made in the past decade towards rigorous error control, which makes this perspective realistic in the medium term. Let us mention in particular recent works on a priori and a posteriori discretization error bounds for DFT [7][8][9][10][11][12][13], including k-point sampling [14], and on the numerical analysis of SCF algorithms [15][16][17][18]. ...
We address the problem of bounding rigorously the errors in the numerical solution of the Kohn-Sham equations due to (i) the finiteness of the basis set, (ii) the convergence thresholds in iterative procedures, (iii) the propagation of rounding errors in floating-point arithmetic. In this contribution, we compute fully-guaranteed bounds on the solution of the non-self-consistent equations in the pseudopotential approximation in a plane-wave basis set. We demonstrate our methodology by providing band structure diagrams of silicon annotated with error bars indicating the combined error.
... A priori discretization error estimates have been constructed in 4 for planewave basis sets, and then in 5 for more general variational discretization methods. A posteriori error estimators of the discretization error have been proposed in [6][7][8] . A combined study of both the discretization and algorithmic errors was published in 9 (see also 10 ). ...
It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. The purpose of this paper is to provide a quantitative study of discretization error cancellation. The latter is the error component due to the fact that the model used in the calculation (e.g. Kohn-Sham LDA) must be discretized in a finite basis set to be solved by a computer. We first report comprehensive numerical simulations performed with Abinit on two simple chemical systems, the hydrogen molecule on the one hand, and a system consisting of two oxygen atoms and four hydrogen atoms on the other hand. We observe that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cut-off goes to infinity. We then analyze a simple one-dimensional periodic Schr\"odinger equation with Dirac potentials, for which analytic solutions are available. This allows us to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.
The purpose of this study was to evaluate the effects of particular development stakeholders on the livelihoods of pastoralists in a few districts in the West Hararge zone of Ethiopia's Oromia region. In order to conduct this investigation, 160 samples of primary data were surveyed, and the econometric model's logistic regression likelihood estimation was employed. Spearman's rank analysis was used to determine how development stakeholders ranked their engagement with government in the areas of education, productive safety net, cooperative Oromia promotion, Busa Gonofa government institution, pastoral development government sector, and water sector and care Ethiopia NGOs sectors, in order. The applied econometric logistic regression, shows from government and NGOs; were founded significantly a significant at a statistical significance level of five percent: Islamic Relief, Water sanitation and hygiene, Farm Africa, and rural road development from the government sector. Accordingly, the research findings indicate that the education sector for pastoralist development strength needs to be treated as such smoothly with some moderation to achieve its target with full rate, while government office responsible for the water sector needs to give priority to reaching its primary target at area than selected sectors also for Care Ethiopia NGOs sector needs urgent improvement.
In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schrödinger operators, by combining the Feshbach–Schur perturbation theory with the spectral Fourier discretization. In order to analyze the method, we establish an abstract framework of Feshbach–Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new spectral Fourier discretization method. Finally, we present some numerical results that underline the theoretical findings.
Most electronic structure models, such as Hartree-Fock, Kohn-Sham, coupled-cluster, multi-configuration self-consistent field (MCSCF), or GW, are originally set in infinite-dimensional function spaces, and need to be discretized in order to be solved on a computer. The errors on the physical quantities of interest resulting from this approximation are called discretization errors, and can be studied using modern tools of numerical analysis. This chapter is a pedagogical introduction to discretization error analysis. The fundamental concepts of variational approximation, a priori and a posteriori error estimators, post-processing, asymptotic expansions, and extrapolation are introduced in a general framework. To avoid unnecessary technicalities, they are illustrated on the example of the plane-wave discretization of the periodic Gross-Pitaevskii model for Bose-Einstein condensates. This model shares many common features with the Hartree-Fock and Kohn-Sham models, while being mathematically simpler. Extensions to Kohn-Sham models are discussed.
In this article, we provide a priori estimates for a perturbation-based post-processing method of the plane-wave approximation of nonlinear Kohn–Sham local density approximation (LDA) models with pseudopotentials, relying on Cancès et al. (2020, Post-processing of the plane-wave approximation of Schrödinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, draa044) for the proofs of such estimates in the case of linear Schrödinger equations. As in Cancès et al. (2016, A perturbation-method-based post-processing for the plane-wave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446–459), where these a priori results were announced and tested numerically, we use a periodic setting and the problem is discretized with plane waves (Fourier series). This post-processing method consists of performing a full computation in a coarse plane-wave basis and then to compute corrections based on the first-order perturbation theory in a fine basis, which numerically only requires the computation of the residuals of the ground-state orbitals in the fine basis. We show that this procedure asymptotically improves the accuracy of two quantities of interest: the ground-state density matrix, i.e. the orthogonal projector on the lowest N eigenvectors, and the ground-state energy.
In this article we prove a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works (Cancès, Dusson, Maday, Stamm and Vohralík, (2014), A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math., 352, 941--946; Cancès, Dusson, Maday, Stamm and Vohralík, (2016), A perturbation-method-based postprocessing for the planewave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446--459.) We consider here a Schrödinger operator on , where is a cubic box with periodic boundary conditions and where is a multiplicative operator by a regular-enough function . The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest N eigenvalues of , and, on the other hand, the ground-state density matrix that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn–Sham density functional theory. Interpreting the exact eigenpairs of as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e., Fourier) basis we compute first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This allows us to increase the accuracy by a factor proportional to the inverse of the kinetic energy cutoff of both the ground-state energy and the ground-state density matrix in Hilbert–Schmidt norm at a low computational extra cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger plane-wave basis and two fast Fourier transforms per eigenvalue.
We address the problem of bounding rigorously the errors in the numerical solution of the Kohn-Sham equations due to (i) the finiteness of the basis set, (ii) the convergence thresholds in iterative procedures, (iii) the propagation of rounding errors in floating-point arithmetic. In this contribution, we compute fully-guaranteed bounds on the solution of the non-self-consistent equations in the pseudopotential approximation in a plane-wave basis set. We demonstrate our methodology by providing band structure diagrams of silicon annotated with error bars indicating the combined error.
In this thesis, we study elliptic eigenvalue problems with singular potentials, motivated by several models in physics and quantum chemistry, and we propose a discontinuous Galerkin hp finite element method for their solution. In these models, singular potentials occur naturally (associated with the interaction between nuclei and electrons). Our analysis starts from elliptic regularity in non homogeneous weighted Sobolev spaces. We show that elliptic operators with singular potential are isomorphisms in those spaces and that we can derive weighted analytic type estimates on the solutions to the linear eigenvalue problems. The isotropically graded hp method provides therefore approximations that converge with exponential rate to the solution of those eigenproblems. We then consider a wide class of nonlinear eigenvalue problems, and prove the convergence of numerical solutions obtained with the symmetric interior penalty discontinuous Galerkin method. Furthermore, when the non linearity is polynomial, we show that we can obtain the same analytic type estimates as in the linear case, thus the numerical approximation converges exponentially. We also analyze under what conditions the eigenvalue converges at an increased rate compared to the eigenfunctions.For both the linear and nonlinear case, we perform numerical tests whose objective is both to validate the theoretical results, but also evaluate the role of sources of errors not considered previously in the analysis, and to help in the design of hp/dG graded methods for more complex problems.
In this article, we provide a mathematical analysis of the perturbation
method for extended Kohn-Sham models, in which fractional occupation numbers
are allowed. All our results are established in the framework of the reduced
Hartree-Fock (rHF) model, but our approach can be used to study other kinds of
extended Kohn-Sham models, under some assumptions on the mathematical structure
of the exchange-correlation functional. The classical results of Density
Functional Perturbation Theory in the non-degenerate case (that is when the
Fermi level is not a degenerate eigenvalue of the mean-field Hamiltonian) are
formalized, and a proof of Wigner's (2n + 1) rule is provided. We then focus on
the situation when the Fermi level is a degenerate eigenvalue of the rHF
Hamiltonian, which had not been considered so far.
Density functional theory (DFT) is an incredible success story. The low
computational cost, combined with useful (but not yet chemical) accuracy, has
made DFT a standard technique in most branches of chemistry and materials
science. Electronic structure problems in a dazzling variety of fields are
currently being tackled. However, DFT has many limitations in its present form:
Too many approximations, failures for strongly correlated systems, too slow for
liquids, etc. This perspective reviews some recent progress and ongoing
challenges.
We describe the design and implementation of KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations. These types of problems arise in electronic structure calculations, which are nowadays essential for studying the microscopic quantum mechanical properties of molecules, solids, and other nanoscale materials. KSSOLV is well suited for developing new algorithms for solving the Kohn-Sham equations and is designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. The toolbox makes use of the object-oriented programming features available in MATLAB so that the process of setting up a physical system is straightforward and the amount of coding effort required to prototype, test, and compare new algorithms is significantly reduced. All of these features should also make this package attractive to other computational scientists and students who wish to study small- to medium-size systems.
The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and s- interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
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, F[n(r)], independent of v(r), such that the expression E≡∫v(r)n(r)dr+F[n(r)] has as its minimum value the correct ground-state energy associated with v(r). The functional F[n(r)] is then discussed for two situations: (1) n(r)=n0+ñ(r), ñn0≪1, and (2) n(r)=ϕ(rr0) with ϕ 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.
We present a broad overview of the models in use in Computational quantum chemistry (typically N-body Schrödinger, Hartree-Fock type, density functional type models), the hierarchy of the different approximations (ab initio, semi-empirical,...), of their physical foundations. We shall emphasize the mathematical difficulties these models rise, and the way they are treated (or not...). We explain the most standard discretization schemes (the most advanced aspects being treated in other contributions below) such as the LCAO approximation etc., the algorithms in use for their resolution (SCF,...), and the related numerical analysis. At the end of the chapter, it is expected that the reader has a good knowledge of the basic techniques, and of the mathematical and numerical analysis needed to make rigorously founded computations. Chapter headings are: I: The Basic Modelling; II: Ground State Calculations; III: The Condensed Phases; IV: Quantum Chemistry and its Relation to Experiment; V: The Time Dependent Models and their Discretization.
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.
The paper selects listed companies which were merged or acquired in 2005 as the empirical samples, chooses 24 specific indexes related to financial condition and governance structure of firms. Then the logistic regression analysis is used to establish one forecast model with 73.6 percent general accuracy basing on success rate and another with 75.3 percent general accuracy basing on M&A performance. The study shows that the company with good governance structure, favorable government efficiency, but mired in short-term operation difficulties is apt to become M&A target; furthermore, the operation revenue growth ability of target firm before M&A has positive effect on M&A performance.
Levy's 1-matrix energy functional (Ref. 4) is modified and extended to
include all ensemble-representable l-matrices in its domain. This
constitutes both a generalization and a simplification of earlier
observations by Gilbert. The generalization negates some criticisms of
the Donnelly-Parr analysis of a 1-matrix energy functional (Ref. 3)
since it was assumed to be defined only for pure-state representable
1-matrices. Further study of this analysis suggests that the taking of
arbitrary variations in the 1-matrix at a certain point in their study
may result in an invalid Euler equation for the 1-matrix energy
functional, although this does not alter the main results of that work.
The redefinition of Levy's functional to accommodate the larger domain
allows direct application of Harriman's analysis of the geometry of
density matrices in finite dimensional situations. A decomposition of
the equivalent of the Vee expectation value into two terms, one with
explicit and another with implicit occupation number dependence,
illustrates the role of N-representability. The Vee expectation value is
approximated by means of the Schwartz inequality.
An overview of relativistic density functional theory covering its foundations, the construction of explicit functionals and applications to spherical atoms is given. After a brief summary of the relevant field theoretical background we discuss the Hohenberg-Kohn theorem for quantum electrodynamical systems as well as the corresponding Kohn-Sham equations, emphasising in particular the renormalisation of ground state energies and currents required. We then outline the transition from the full quantum electrodynamical Kohn-Sham equations to the more practical variants which are actually used in applications. As an extension of the Kohn-Sham equations we also summarise the relativistic optimised-potential-method (OPM) which, in addition to the kinetic energy, also treats the exchange energy on the basis of Kohn-Sham orbitals. As far as the construction of explicit functionals is concerned, we review the local density approximation (LDA) and the weighted density approximation (WDA) for the exchange-correlation energy as well as the gradient expansion of the kinetic energy, again addressing in detail questions of renormalisation. The relativistic corrections to the ground state, single particle and exchange energies as well as exchange potentials of atoms are then examined within the exchange-only limit of the no-sea approximation to the full relativistic Kohn-Sham equations, comparing the LDA and the WDA with the results obtained by the relativistic OPM. In addition, we investigate transverse exchange and correlation contributions within the LDA by comparison with quantum chemical data.
We present a simple procedure for generating first-principles norm-conserving pseudopotentials which are very soft and therefore have good convergence properties when used in plane-wave calculations of the band structure and total-energy of materials. Using diamond as a test case, we show that our pseudopotential is more efficient for plane wave calculations than the most commonly used pseudopotentials.
This paper has three aims: (i) To discuss some of the mathematical connections between N-particle wave functions ψ and their single-particle densities ρ (x). (ii) To establish some of the mathematical underpinnings of “universal density functional” theory for the ground state energy as begun by Hohenberg and Kohn. We show that the HK functional is not defined for all ρ and we present several ways around this difficulty. Several less obvious problems remain, however. (iii) Since the functional mentioned above is not computable, we review examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy.
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
We provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizs\"{a}cker (TFW) model and for the spectral discretization of the Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the ground state energy and density of molecular systems in the condensed phase. The TFW model is stricly 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. Under a coercivity assumption on the second order optimality condition, we prove that for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of any Kohn-Sham ground state, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method. Comment: 50 pages
The treatment of adiabatic perturbations within density-functional theory is examined, at arbitrary order of the perturbation expansion. Due to the extremal property of the energy functional, standard variation-perturbation theorems can be used. The different methods (Sternheimer equation, extremal principle, Green’s function, and sum over state) for obtaining the perturbation expansion of the wave functions are presented. The invariance of the Hilbert space of occupied wave functions with respect to a unitary transformation leads to the definition of a ‘‘parallel-transport-gauge’’ and a ‘‘diagonal-gauge’’ perturbation expansion. Then, the general expressions are specialized for the second, third, and fourth derivative of the energy, with an example of application of the method up to third order.
When perturbation theory is applied to a quantity for which a variational principle holds (eigenenergies of Hamiltonian, Hartree-Fock or density-functional-theory, etc.), different variation-perturbation theorems can be derived. A general demonstration of the existence of variational principles for an even order of perturbation, when constraints are present, is provided here. Explicit formulas for these variational principles for even orders of perturbation, as well as for the ‘‘2n+1 theorem,’’ to any order of perturbation, with or without constraints, are also exhibited. This approach is applied to the case of eigenenergies of quantum-mechanical Hamiltonians, studied previously by other methods.
A method for calculating approximately the coupling energy of weakly interacting fragments is presented. The method is a simplified version of the density-functional scheme of Kohn and Sham and is applicable whenever the electron density of the coupled fragments does not deviate too markedly from a sum of isolated fragment densities. The coupling energy is expressed directly in terms of properties of the isolated fragments and the only nontrivial computational step is the determination of an eigenvalue sum for the coupled system with a fixed potential. Neither self-consistency cycling nor a solution of Poisson’s equation for the coupled fragments is required. The method is therefore particularly appropriate when full density-functional calculations are tractable for the isolated fragments but difficult for the coupled system, e.g., a molecule interacting with a surface. Explicit calculations for dimers illustrate that the approach is very accurate for weakly interacting systems, and that reasonable results can be obtained even for strong covalent bonds.
Universal variational functionals of densities, first-order density matrices, and natural spin-orbitals are explicitly displayed for variational calculations of ground states of interacting electrons in atoms, molecules, and solids. In all cases, the functionals search for constrained minima. In particular, following Percus [Formula: see text] is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron-electron repulsion energies of an N-representable trial electron density rho. Q[rho] searches all antisymmetric wavefunctions Psi(rho) which yield the fixed. rho. Q[rho] then delivers that expectation value which is a minimum. Similarly, [Formula: see text] is shown to be the universal functional for the electron-electron repulsion energy of an N-representable trial first-order density matrix gamma, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus, the v-representability problem, which is especially severe for trial first-order density matrices, has been solved. Universal variational functionals in Hartree-Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in density functional theory.
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