Self-consistent effective local potentials

An effective local potential (ELP) is a multiplicative operator whose deviation from a given nonlocal potential has the smallest variance evaluated with a prescribed single-determinant wave function. ELPs are useful in density functional theory as alternatives to optimized effective potentials (OEPs) because they do not require special treatment in finite basis set calculations as OEPs do. We generalize the idea of variance-minimizing potentials by introducing the concept of a self-consistent ELP (SCELP), a local potential whose deviation from its nonlocal counterpart has the smallest variance in terms of its own Kohn-Sham orbitals. A semi-analytical method for computing SCELPs is presented. The OEP, ELP, and SCELP techniques are applied to the exact-exchange-only Kohn-Sham problem and are found to produce similar results for many-electron atoms.     

Effective local potentials for orbital-dependent density functionals

Practicality of the Kohn-Sham density functional scheme for orbital-dependent functionals hinges on the availability of an efficient procedure for constructing local exchange-correlation potentials in finite basis sets. We have shown recently that the optimized effective potential (OEP) method, commonly used for this purpose, is not free from difficulties. Here we propose a robust alternative to OEPs, termed effective local potentials (ELPs), based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart. The ELP method is applied to the exact-exchange-only problem and shown to be promising for overcoming troubles with OEPs.     

Optimized effective potential method for individual low-lying excited states

This paper presents an optimized effective potential (OEP) approach based on density functional theory (DFT) for individual excited states that implements a simple method of taking the necessary orthogonality constraints into account. The amended Kohn-Sham (KS) equations for orbitals of excited states having the same symmetry as the ground one are proposed. Using a variational principle with some orthogonality constraints, the OEP equations determining a local exchange potential for excited states are derived. Specifically, local potentials are derived whose KS determinants minimize the total energies and are simultaneously orthogonal to the determinants for states of lower energies. The parametrized form of an effective DFT potential expressed as a direct mapping of the external potential is used to simplify the OEP integral equations. A performance of the presented method is examined by exchange-only calculations of excited state energies for simple atoms and molecules.     

Uniqueness of the iterative solution of the optimized effective potential equation

The optimized effective potential (OEP) equation can be used in a numerically efficient self-consistent form to solve for the density functional exchange and correlation potentials, as shown in a recent paper of Kummel and Perdew [Phys. Rev. Lett. 90, 43004 (2003)]. The uniqueness of an iterative solution of the OEP equation has not yet been adequately addressed. In this paper, it is shown that no nonconstant multiplicative potentials that can contaminate an iterative solution of the OEP equation exist and, hence, that formally the exact exchange-correlation potential determined form of the OEP equation is unique to within a constant.     

Density-functional theory with effective potential expressed as a mapping of the external potential: applications to open-shell molecules

In this paper we apply the direct-mapping density-functional theory (DFT) to open-shell systems, in order to get many-electron wave functions having the same transformation properties as the eigenstates of the exact Hamiltonians. Such a case is that of spin, where in order to get the magnetic properties, the many-particle states must be eigenstates not only of S(z) but also of S2. In this theory the Kohn and Sham [Phys. Rev. A 140, 1133 (1965)] potential is expressed directly as a mapping of the external potential. The total energies of the molecules calculated were satisfactory as their relative deviations (deltaEE) from the exact Hartree-Fock ones were of the order of 10(-4). This accuracy is much higher than that of the standard DFT in its local exchange potential approximation. This method does not need an approximate density as input, as the effective potential is derived directly from the external potential.     

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. © 2012 Wiley Periodicals, Inc.     

Local effective potential theory: nonuniqueness of potential and wave function

In local effective potential energy theories such as the Hohenberg-Kohn-Sham density functional theory (HKS-DFT) and quantal density functional theory (Q-DFT), electronic systems in their ground or excited states are mapped to model systems of noninteracting fermions with equivalent density. From these models, the equivalent total energy and ionization potential are also obtained. This paper concerns (i) the nonuniqueness of the local effective potential energy function of the model system in the mapping from a nondegenerate ground state, (ii) the nonuniqueness of the local effective potential energy function in the mapping from a nondegenerate excited state, and (iii) in the mapping to a model system in an excited state, the nonuniqueness of the model system wave function. According to nondegenerate ground state HKS-DFT, there exists only one local effective potential energy function, obtained as the functional derivative of the unique ground state energy functional, that can generate the ground state density. Since the theorems of ground state HKS-DFT cannot be generalized to nondegenerate excited states, there could exist different local potential energy functions that generate the excited state density. The constrained-search version of HKS-DFT selects one of these functions as the functional derivative of a bidensity energy functional. In this paper, the authors show via Q-DFT that there exist an infinite number of local potential energy functions that can generate both the nondegenerate ground and excited state densities of an interacting system. This is accomplished by constructing model systems in configurations different from those of the interacting system. Further, they prove that the difference between the various potential energy functions lies solely in their correlation-kinetic contributions. The component of these functions due to the Pauli exclusion principle and Coulomb repulsion remains the same. The existence of the different potential energy functions as viewed from the perspective of Q-DFT reaffirms that there can be no equivalent to the ground state HKS-DFT theorems for excited states. Additionally, the lack of such theorems for excited states is attributable to correlation-kinetic effects. Finally, they show that in the mapping to a model system in an excited state, there is a nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. Examples of the nonuniqueness of the potential energy functions for the mapping from both ground and excited states and the nonuniqueness of the wave function are provided for the exactly solvable Hooke's atom. The work of others is also discussed.     

Temperature-responsive protein pores

We describe temperature-responsive protein pores containing single elastin-like polypeptide (ELP) loops. The ELP loops were placed within the cavity of the lumen of the alpha-hemolysin (alphaHL) pore, a heptamer of known crystal structure. The cavity is roughly spherical with a molecular surface volume of about 39,500 A3. In an applied potential, the wild-type alphaHL pore remained open for long periods. In contrast, the ELP loop-containing alphaHL pores exhibited transient current blockades, the nature of which depended on the length and sequence of the inserted loop. Together with similar results obtained with poly(ethylene glycols) covalently attached within the cavity, the data suggest that the transient current blockades are caused by excursions of ELP into the transmembrane beta-barrel domain of the pore. Below its transition temperature, the ELP loop is fully expanded and blocks the pore completely, but reversibly. Above its transition temperature, the ELP is dehydrated and the structure collapses, enabling a substantial flow of ions. Potential applications of temperature-responsive protein pores in medical biotechnology are discussed.     

Deriving effective mesoscale potentials from atomistic simulations

We demonstrate how an iterative method for potential inversion from distribution functions developed for simple liquid systems can be generalized to polymer systems. It uses the differences in the potentials of mean force between the distribution functions generated from a guessed potential and the true distribution functions to improve the effective potential successively. The optimization algorithm is very powerful: convergence is reached for every trial function in few iterations. As an extensive test case we coarse-grained an atomistic all-atom model of polyisoprene (PI) using a 13:1 reduction of the degrees of freedom. This procedure was performed for PI solutions as well as for a PI melt. Comparisons of the obtained force fields are drawn. They prove that it is not possible to use a single force field for different concentration regimes.     

Size extensivity of the direct optimized effective potential method

We investigate the size extensivity of the direct optimized effective potential procedure of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)]. The choice of reference potential within the finite basis construction of the local Kohn-Sham potential can lead to a method that is not size extensive. Such a situation is encountered when one employs the Fermi-Amaldi potential, which is often used to enforce the correct asymptotic behavior of the exact exchange-correlation potential. The size extensivity error with the Fermi-Amaldi reference potential is shown to behave linearly with the number of electrons in the limit of an infinite number of well separated monomers. In practice, the error tends to be rather small and rapidly approaches the limiting linear behavior. Moreover, with a flexible enough potential basis set, the error can be decreased significantly. We also consider one possible reference potential, constructed from the van Leeuwen-Baerends potential, which provides a size extensive implementation while also enforcing the correct asymptotic behavior.     

Transiting the molecular potential energy surface along low energy pathways: The TRREAT algorithm

The Transition Rapidly exploring Random Eigenvector Assisted Tree (TRREAT) algorithm is introduced to perform searches along low curvature pathways on a potential energy surface (PES). The method combines local curvature information about the PES with an iterative Rapidly exploring Random Tree algorithm (LaValle, Computer Science Department, Iowa State University, 1998, TR98–11) that quickly searches high‐dimensional spaces for feasible pathways between local minima. Herein, the method is applied to identifying conformational changes of molecular systems using Cartesian coordinates while avoiding a priori definition of collective variables. We analyze the pathway identification problem for alanine dipeptide, cyclohexane and glycine using nonreactive and reactive forcefields. We show how TRREAT‐identified pathways can be used as valuable input guesses for double‐ended methods such as the Nudged Elastic Band when ascertaining transition state energies. This method can be utilized to improve/extend the reaction databases that lie at the core of automatic chemical reaction mechanism generator software currently developed to build kinetic models of chemical reactions. © 2013 Wiley Periodicals, Inc.     

Optimized effective potential from a correlated wave function: optimized effective potential-generalized valence bond (OEP-GVB)

The optimized effective potential (OEP) method allows orbital-dependent functionals to be used in density functional theory. Traditionally the orbital-dependent functional of interest has been the Hartree-Fock energy, leading to exact exchange density functional theory. Here we present results that use a generalized valence-bond (GVB) wave function, a multiconfigurational wave function that includes static correlation and dissociates to the proper limits. We demonstrate the effectiveness of the OEP-GVB method by showing the dissociation of H2 and the excitation spectrum of He.     

A simple effective potential for exchange

The optimized effective potential (OEP) for exchange was introduced some time ago by Sharp and Horton and by Talman and Shadwick. The integral equation for the OEP is difficult to solve, however, and a variety of approximations have therefore been proposed. These are explicitly orbital dependent and require the same two-electron integrals as Hartree-Fock theory. We have found a remarkably simple approximate effective potential that closely resembles the Talman-Shadwick potential in atoms. It depends only on total densities and requires no two-electron integrals.     

On finite basis set implementation of the exchange-only optimized effective potential method

In this paper, we analyze a structure of the basis set optimized effective potential (OEP) equations from the Fredholm alternative point of view and present one of possible numerical schemes to solve the OEP equation in a stable manner. The solution is constructed as a sum of a unique solution on the subspace of eigenfunctions of the response matrix with non-zero eigenvalues and a non-unique solution on a counterpart subspace with singular eigenvalues. Non-uniqueness of a solution is exploited to obtain a local effective potential that satisfies the condition for the highest occupied molecular orbital (HOMO) without restricting the variational freedom of the optimization procedure. Unlike the existing methods we implement the HOMO condition using the functions of the null-subspace. Numerical results for the total and orbital energies based on the proposed scheme are close to the corresponding literature data.     

Ab initio density functional theory: the best of both worlds?

Density functional theory (DFT), in its current local, gradient corrected, and hybrid implementations and their extensions, is approaching an impasse. To continue to progress toward the quality of results demanded by today's ab initio quantum chemistry encourages a new direction. We believe ab initio DFT is a promising route to pursue. Whereas conventional DFT cannot describe weak interactions, photoelectron spectra, or many potential energy surfaces, ab initio DFT, even in its initial, optimized effective potential, second-order many-body perturbation theory form [OEP (2)-semi canonical], is shown to do all well. In fact, we obtain accuracy that frequently exceeds MP2, being competitive with coupled-cluster theory in some cases. Furthermore, this is accomplished within a relatively fast computational procedure that scales like iterative second order. We illustrate our results with several molecular examples including Ne2,Be2,F2, and benzene.     

The effective fragment potential: small clusters and radial distribution functions

The effective fragment potential (EFP) method for treating solvent effects provides relative energies and structures that are in excellent agreement with the analogous fully quantum [i.e., Hartree-Fock (HF), density functional theory (DFT), and second order perturbation theory (MP2)] results for small water clusters. The ability of the method to predict bulk water properties with a comparable accuracy is assessed by performing EFP molecular dynamics simulations. The resulting radial distribution functions (RDF) suggest that as the underlying quantum method is improved from HF to DFT to MP2, the agreement with the experimental RDF also improves. The MP2-based EFP method yields a RDF that is in excellent agreement with experiment.     

Charge transfer interaction in the effective fragment potential method

An approximate formula is derived and implemented in the general effective fragment potential (EFP2) method to model the intermolecular charge transfer interaction. This formula is based on second order intermolecular perturbation theory and utilizes canonical molecular orbitals and Fock matrices obtained with preparative self-consistent field calculations. It predicts charge transfer energies that are in reasonable agreement with the reduced variational space energy decomposition analysis. The formulas for the charge transfer gradients with respect to EFP translational and rotational displacements are also derived and implemented.     

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.     

The role of orbital products in the optimized effective potential method

The orbital products of occupied and virtual orbitals are employed as an expansion basis for the charge density generating the local potential in the optimized effective potential method thus avoiding the use of auxiliary basis sets. The high computational cost arising from the quadratic increase of the dimension of this product basis with system size can be greatly reduced by elimination of the linearly dependent products according to a procedure suggested by Beebe and Linderberg [Int. J. Quantum Chem. 12, 683 (1977)]. Numerical results from this approach show a very good agreement with those obtained from balancing the auxiliary basis for the expansion of the local potential with the orbital basis set.     

Basis set modeling for molecular calculations using effective core potential

A new approach for developing of basis sets to be used along with effective core potential is systematically studied. The behavior of the LCAO coefficients versus the ln(α) of the respective primitives can provide simple guidelines to establish the range over which the basis set should be developed or modified, especially when using effective core potential. Double-zeta basis sets were modeled for SBK pseudopotential from all-electron basis sets for a series of compounds containing elements of the second period of the periodic table. Application of the modeled basis sets at the Hartree–Fock and MP2 levels of theory shows that the new method provides molecular properties as accurate as those calculated by all-electron calculations. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1918–1929, 1997