Optimized Effective Potential Method Research Articles

Analytical Forces for the Optimized Effective Potential Calculations.

The optimized effective potential (OEP) equation is an ill-conditioned linear system when using finite basis sets. Without any special treatment, the obtained exchange-correlation (XC) potential may have unphysical oscillations. One way to alleviate this problem is to regularize the solutions; however, a regularized XC potential is not the exact solution to the OEP equation. As a result, the system's energy is no longer variational against the Kohn-Sham (KS) potential, and the analytical forces cannot be derived from the Hellmann-Feynman theorem. In this work, we develop a robust and nearly black-box OEP method to ensure that the system's energy is variational against the KS potential. The basic idea is to add a penalty function that regularizes the XC potential to the energy functional. Analytical forces can then be derived based on the Hellmann-Feynman theorem. Another key result is that the impact of the regularization can be much reduced by regularizing the difference between the XC potential and an approximate XC potential rather than regularizing the XC potential. Numerical tests show that forces and the energy differences between systems are not sensitive to the regularization coefficient, which indicates that in practice accurate structural and electronic properties can be obtained without extrapolating the regularization coefficient to zero. We expect this new method to be found useful for calculations that employ advanced, orbital-based functionals, especially for applications that require efficient force calculations.

Numerically stable optimized effective potential method with standard Gaussian basis sets.

We present a numerically stable optimized effective potential (OEP) method based on Gaussian basis sets. The key point of the approach is a sequence of preprocessing steps of the auxiliary basis set used to represent exchange or correlation potentials, the Kohn-Sham (KS) response function, and the right-hand side of the OEP equation in conjunction with a representation of exchange or correlation potentials via exchange or correlation charge densities whose electrostatic potentials generate the potentials. Due to the preprocessing, standard Gaussian basis sets from basis set libraries can be used in OEP calculations. As examples, we present numerical stable computational setups based on aux-cc-pwCVXZ basis sets with X = T, Q, 5 for the orbitals and aux-cc-pVDZ/mp2fit and aux-cc-pVTZ/mp2fit auxiliary basis sets and use them to calculate KS exchange potentials with the exact exchange-only KS method for various atoms and molecules. The resulting exchange potentials not only are numerically stable and physically reasonable but also show convergence with increasing quality of the orbital basis sets. The effect of incorporating exact conditions that the KS exchange potential has to obey is discussed. Moreover, it is briefly demonstrated that the presented approach not only works for KS exchange potentials but equally well for correlation potentials within the direct random phase approximation. Besides for OEP methods, the introduced preprocessing of auxiliary basis sets should also be beneficial in procedures to calculate back effective KS potentials from given electron densities.

One and two body densities for excited states of the helium confined atom

AbstractA study of the first excited states of the helium atom confined under impenetrable spherical walls is carried out. Both single particle and two body, intracule and extracule, densities are constructed. Crossing levels and Hund's rule are analyzed in terms of the contribution to the total energy from kinetic, electron–nucleus, and electron–electron energies. A study about the behavior of the single particle and two body densities is carried out. The Multiconfiguration Parameterized Optimized Effective Potential method is employed with a cut‐off factor to account for Dirichlet boundary conditions. Single particle density is analytically constructed whereas the Monte Carlo algorithm is used to calculate two body densities.

Optimized effective potential method and application to static RPA correlation

The optimized effective potential (OEP) method is a promising technique for calculating the ground state properties of a system within the density functional theory. However, it is not widely used as its computational cost is rather high and, also, some ambiguity remains in the theoretical framework. In order to overcome these problems, we first introduced a method that accelerates the OEP scheme in a static RPA-level correlation functional. Second, the Krieger–Li–Iafrate (KLI) approximation is exploited to solve the OEP equation. Although seemingly too crude, this approximation did not reduce the accuracy of the description of the magnetic transition metals (Fe, Co, and Ni) examined here, the magnetic properties of which are rather sensitive to correlation effects. Finally, we reformulated the OEP method to render it applicable to the direct RPA correlation functional and other, more precise, functionals. Emphasis is placed on the following three points of the discussion: (i) level-crossing at the Fermi surface is taken into account; (ii) eigenvalue variations in a Kohn–Sham functional are correctly treated; and (iii) the resultant OEP equation is different from those reported to date.

On orthogonality constrained multiple core-hole states and optimized effective potential method

An attempt to construct a multiple core-hole state within the optimized effective potential (OEP) methodology is presented. In contrast to the conventional Δ-self-consistent field method for hole states, the effects of removing an electron is achieved using some orthogonality constraints imposed on the orbitals so that a Slater determinant describing a hole state is constrained to be orthogonal to that of a neutral system. It is shown that single, double, and multiple core-hole states can be treated within a unified framework and can be easily implemented for atoms and molecules. For this purpose, a constrained OEP method proposed earlier for excited states (Glushkov and Levy, J. Chem. Phys. 2007, 126, 174106) is further developed to calculate single and double core ionization energies using a local effective potential expressed as a direct mapping of the external potential. The corresponding equations, determining core-hole orbitals from a one-particle Schrödinger equation with a local potential as well as correlation corrections derived from the second-order many-body perturbation theory are given. One of the advantages of the present direct mapping formulation is that the effective potential, which plays the role of the Kohn-Sham potential, has the symmetry of the external potential. Single and double core ionization potentials computed with the presented scheme were found to be in agreement with data available from experiment and other calculations. We also discuss core-hole state local potentials for the systems studied.

On the Coulson-Fischer wave function and a local static correlation potential

By using the optimized effective potential (OEP) method in conjunction with the Coulson–Fischer wave function, we develop a formalism, OEP-CF, for incorporating both exchange and static correlation effects. A local static correlation potential is presented which affords a qualitatively correct description of dissociative processes. The energies supported by the OEP-CF method are lower than the corresponding Hartree-Fock values and energies obtained from exchange–only OEP methods (x OEP) since, in the OEP-CF formalism, orbital orthogonality restrictions are relaxed. The OEP-CF equations, which determine a multiplicative correlation potential, are compared with those obtained with the more familiar x OEP method. A parametrized form of a local exchange-correlation Coulson–Fischer potential expressed in terms of the external potential is implemented for the diatomic systems H2 and HeH+. Such a representation allows the solution of both the one-electron Schrödinger equation and OEP-CF equations to be simplified. Applications to the dissociation of the H2 molecule and the HeH+ ion using the OEP-CF methodology demonstrate that the method is capable of describing the static correlation with an accuracy comparable with existing multiconfigurational-OEP-based methods. Potential energy curves computed with the OEP-CF method are compared with those obtained from the x OEP method. †

Investigation of Self-Interaction Corrections for an Exactly Solvable Model System: Orbital Dependence and Electron Localization.

A systematic investigation of two approximate self-interaction corrections (SICs), Perdew-Zunger SIC and Lundin-Eriksson SIC, and the local-density approximation (LDA) is performed for a model Hamiltonian whose exact many-body solution and exact LDA are known. Both SICs as well as LDA are applied in the calculation of ground-state energies, ground-state densities, energy gaps, and impurity densities of one-dimensional Hubbard chains differing in size, particle number, and interaction strength. The orbital-dependent potentials arising from either SIC are treated within the optimized-effective potential method, which we reformulate for the Hubbard model. The delocalization tendency of LDA is confronted with the localization tendency of SIC. A statistical analysis of the resulting data set sheds light on the role of SIC for weakly and strongly interacting particles and allows one to assess the performance of each methodology.

Scheme for calculating the orbital-dependent exchange-correlation potential using the virial theorem: Application to atomic systems

We present a density-functional scheme for calculating the orbital-dependent exchange-correlation potential using the virial theorem as a sum rule. In order to check the validity of this scheme, atomic-structure calculations only with the exchange potential are performed. The accuracy of this scheme is shown to be comparable to that of the optimized effective potential (OEP) method, while the computational workload is extremely reduced compared to the OEP method.

Numerically stable optimized effective potential method with balanced Gaussian basis sets

A solution to the long-standing problem of developing numerically stable optimized effective potential (OEP) methods based on Gaussian basis sets is presented by introducing an approach consisting of an exact exchange OEP method with an accompanying construction and balancing scheme for the involved auxiliary and orbital Gaussian basis sets that is numerically stable and that properly represents an exact exchange Kohn-Sham method. The method is a purely analytical method that does not require any numerical grid, scales like Hartree-Fock or B3LYP procedures, is straightforward to implement, and is easily generalized to take into account orbital-dependent density functionals other than the exact exchange considered in this work. Thus, the presented OEP approach opens the way to the development and application of novel orbital-dependent exchange-correlation functionals. It is shown that adequately taking into account the continuum part of the Kohn-Sham orbital spectrum is crucial for numerically stable Gaussian basis set OEP methods. Moreover, it is mandatory to employ orbital basis sets that are converged with respect to the used auxiliary basis representing the exchange potential. OEP calculations in the past often did not meet the latter requirement and therefore may have led to erroneously low total energies.

A Proposal for Calculating the Orbital-Dependent Exchange-Correlation Potential by Means of the Virial Theorem

We provide a density functional scheme for calculating the orbital-dependent exchange-correlation potential using the virial theorem as a sum rule. This is different from the optimized effective potential (OEP) method. To confirm the validity, atomic-structure calculations are performed for closed-shell atoms. The exchange energies are in good agreement with the previous values obtained by the OEP method, whereas the numerical speed has been considerably improved.

First-Principles Approach to Noncollinear Magnetism: Towards Spin Dynamics

A description of noncollinear magnetism in the framework of spin-density functional theory is presented for the exact exchange energy functional which depends explicitly on two-component spinor orbitals. The equations for the effective Kohn-Sham scalar potential and magnetic field are derived within the optimized effective potential (OEP) framework. With the example of a magnetically frustrated Cr monolayer it is shown that the resulting magnetization density exhibits much more noncollinear structure than standard calculations. Furthermore, a time-dependent generalization of the noncollinear OEP method is well suited for an ab initio description of spin dynamics. We also show that the magnetic moments of solids Fe, Co, and Ni are well reproduced.

Band gap calculations with Becke–Johnson exchange potential

Recently, a simple analytical form for the exchange potential was proposed by Becke andJohnson. This potential, which depends on the kinetic-energy density, was shown toreproduce very well the shape of the exact exchange potential (obtained with the optimizedeffective potential method) for atoms. Calculations on solids show that the Becke–Johnsonpotential leads to a better description of band gaps of semiconductors and insulators withrespect to the standard local density and Perdew–Burke–Ernzerhof approximations for theexchange–correlation potential. Comparison is also made with the values obtained with theEngel–Vosko exchange potential which was also developed using the exact exchangepotential.

Simple Iterative Procedure for Optimized Effective Potential Method in Density Functional Theory and in Current Density Functional Theory

The optimized effective potential (OEP) method in the density functional theory is reformulated on the basis of the variable coupling constant scheme. By this formulation, the origin of the so-called `orbital shift' in the OEP method is revealed. The method leads to a simple iterative procedure of the OEP method using the Thomas–Fermi model. This method is extended to the OEP method in the current density functional theory (CDFT) and in the time-dependent CDFT. As a numerical test, the procedure in the nonmagnetic time-independent case is applied to the exchange-only calculation of the Be atom. The present procedure converges more quickly than that of Kummel and Perdew [Phys. Rev. B 68 (2003) 035103].

Scaling analysis of the optimized effective potentials for the multiplet states of multivalent 3d ions

We apply the optimized effective potential method (OPM) to the multivalent 3dn (n = 2, …, 8) ions; Mν+ (ν = 2, …, 8). The total energy functional is approximated by the single-configuration Hartree–Fock. The exchange potential for the average energy configuration is decomposed into the potentials derived from F2(3d, 3d) and F4(3d, 3d) Slater integrals. To investigate properties of the density-functional potential, we have checked the scaling properties of several physical quantities such as the density, the 3d orbital and these potentials. We find that the potentials of the Slater integrals do not have the scaling property. Instead, the weighted potential of an ion i, which is the potential of the Slater integrals times the 3d-orbital density, satisfies the scaling property where qi3d is the occupation number of the 3d-orbital R3d(r) for ion i. Furthermore, the weighted potential can be approximated by the ion-independent functional of the 3d-orbital density ckR8/33d(r)/q3d where c2 = 0.366 and c4 = 0.223. This suggests that the weighted potential can be expressed as a functional of the 3d-orbital density.

Analytic energy gradients of the optimized effective potential method

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals.

Analysis of optimized effective potentials for multiplet states of 3d transition metal atoms

We apply the optimized effective potential method (OPM) to the multiplet energies of the 3dn transition metal atoms, where the orbital dependence of the energy functional with respect to the orbital wavefunction is the single-configuration HF form. We find that the calculated OPM exchange potential can be represented by the following two forms. Firstly, the difference between OPM exchange potentials of the multiplet states can be approximated by the linear combination of the potentials derived from the Slater integrals F2(3d, 3d) and F4(3d, 3d) for the average energy of the configuration. Secondly, the OPM exchange potential can be expressed as the linear combination of the OPM exchange potentials of the single determinants.

Optimized Effective Potential Method and KLI Approximation for Systems with Degenerate Orbitals in the LCAO Calculations

A new formulation of the optimized effective potential (OEP) method is presented. Our OEP method is applicable to the linear combination of atomic orbital (LCAO) calculation for systems which have degenerate orbitals, in contrast to the conventional OEP method. An approximation to our OEP equation for the exchange-correlation (xc) potential, similar to the KLI approximation, is derived.

Finite-basis-set optimized effective potential exchange-only method

The finite-basis-set optimized effective potential (OEP) method is presented from an integral equation point of view. It is shown that the projection method for solving the OEP integral equation provides a consistent and convenient approach for including orbital-dependent functionals and potentials in the finite-basis-set implementations of the Kohn–Sham theory. Different finite-basis-set realizations of the OEP method are introduced and tested within the exchange-only approximation. An exact condition involving the local multiplicative exchange potential and the nonlocal Hartree–Fock exchange potential built from Kohn–Sham orbitals is incorporated in our schemes. Numerical results are presented.

Can optimized effective potentials be determined uniquely?

Local (multiplicative) effective exchange potentials obtained from the linear-combination- of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior (although formally they should have the correct asymptotic behavior) and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals. Nonetheless, by making a judicious choice of the basis sets, we obtain accurate exchange potentials for atoms and molecules from an LCAO OEP procedure, which are significant improvements over local or gradient-corrected exchange functionals or the Slater potential. The Krieger–Li–Iafrate scheme offers better approximations to the OEP method.

Optimized effective potential method for polymers

The optimized effective potential (OEP) method allows for calculation of the local, effective single particle potential of density functional theory for explicitly orbital-dependent approximations to the exchange-correlation energy functional. In the present work the OEP method is used together with the approximation due to Krieger, Li, and Iafrate (KLI). We present the first application of this method to polymers. KLI calculations have been performed for the insulating polyethylene and the results have been compared to those from other orbital-dependent potentials. Various properties of the band structure are also calculated. The single-particle band gap strongly depends on the basis set with larger basis sets yielding narrow gaps. For certain physical quantities such as the total energy and the exchange energy, the various orbital-dependent Kohn–Sham exchange-only and Hartree–Fock results differ only slightly. For the highest occupied orbital energy the difference is more significant than expected. In order to get the right band gap in OEP the exchange contribution to the derivative discontinuity is calculated and added to the Kohn–Sham gap. The corrected gap obtained by the KLI approach is 12.8 eV compared with the Hartree–Fock and experimental values of 16.6 and 8.8 eV, respectively. We observe, however, the strong dependence of the derivative discontinuity on the basis set.