Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+G?), where G? is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+G?) rather than (1+G?). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.     

Density functional theory with fractional orbital occupations

In contrast to the original Kohn-Sham (KS) formalism, we propose a density functional theory (DFT) with fractional orbital occupations for the study of ground states of many-electron systems, wherein strong static correlation is shown to be described. Even at the simplest level represented by the local density approximation (LDA), our resulting DFT-LDA is shown to improve upon KS-LDA for multi-reference systems, such as dissociation of H(2) and N(2), and twisted ethylene, while performing similar to KS-LDA for single-reference systems, such as reaction energies and equilibrium geometries. Because of its computational efficiency (similar to KS-LDA), this DFT-LDA is applied to the study of the singlet-triplet energy gaps (ST gaps) of acenes, which are "challenging problems" for conventional electronic structure methods due to the presence of strong static correlation effects. Our calculated ST gaps are in good agreement with the existing experimental and high-level ab initio data. The ST gaps are shown to decrease monotonically with the increase of chain length, and become vanishingly small (within 0.1 kcal/mol) in the limit of an infinitely large polyacene. In addition, based on our calculated active orbital occupation numbers, the ground states for large acenes are shown to be polyradical singlets.     

Dyson-corrected orbital energies for the perturbative treatment of electron correlation

The effect of replacing the Hartree–Fock one-particle energies with ionization potentials obtained from inverse Dyson equation when calculating electron correlation energies perturbatively is investigated. Though the energy shifts vary from system to system, the slight decrease of the resulting excitation energies at around equilibrium geometries leads to a slight increase of the correlation energies in most cases. In the dissociation limit the inverse Dyson equation opens the gap, thus nondiverging potential curves emerge even at the restricted Hartree–Fock (RHF)+RS2 level. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 713–719, 1998     

Analysis of electron-positron wavefunctions in the nuclear-electronic orbital framework

The nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) approach is extended and applied to the positronic systems PsH, LiPs, and e(+)LiH. In this implementation, all electrons and positrons are treated quantum mechanically, and all nuclei are treated classically. This approach utilizes molecular orbital techniques with Gaussian basis sets for the electrons and positrons and includes electron-positron correlation with explicitly correlated Gaussian-type geminal functions. An efficient strategy is developed to reduce the number of variational parameters in the NEO-XCHF calculations. The annihilation rates, electron and positron densities, and electron-positron contact densities are compared to available results from higher-level calculations. Our analysis illustrates that the NEO-XCHF method produces qualitative to semi-quantitative results for these properties at a relatively low computational cost by treating only the essential electron-positron correlation explicitly. The NEO-HF method, which does not include explicit correlation and therefore is extremely efficient, is found to provide qualitatively accurate electron-positron contact densities for the e(+)LiH system but not for the LiPs system. Thus, the utility of the NEO-HF method for determining where annihilation occurs is system dependent and not generally reliable. The NEO-XCHF method, however, provides a computationally practical and reliable approach for determining where annihilation will occur in positronic systems.     

Density functional Gaussian-type orbital approach in theoretical study of S2F2 isomerization

The structures of two isomers, difluorodisulfane (FSSF) and thiothionylfluoride (SSF₂), and the corresponding transition structure were generated with density functional theory (DFT) methods. Three groups of DFT methods were used: local (Local Spin Density Approximation, LSDA), nonlocal (local with gradient corrections; BLYP and BP86), and hybrid methods that include a mixture of Hartree-Fock (HF) exchange with nonlocal correlation (Becke3BLYP, Becke3P86). An extended basis set [6-311 + + G(3df)] was used for all calculations, although satisfactory results can be obtained with the 6-311G(d) basis set. The geometries obtained were compared with both restricted Hartree-Fock (RHF) calculated and experimentally obtained values. The energy outcome and the activation barrier for the isomerization were evaluated. It was determined that excellent geometries can be obtained with the Becke3B86 hybrid method, whereas for reasonable energies MP2 single-point calculations on these geometries are necessary. © 1996 by John Wiley & Sons, Inc.     

Modeling positrons in molecular electronic structure calculations with the nuclear-electronic orbital method

The nuclear-electronic orbital (NEO) method was modified and extended to positron systems for studying mixed positronic-electronic wavefunctions, replacing the mass of the proton with the mass of the positron. Within the modified NEO framework, the NEO-HF (Hartree-Fock) method provides the energy corresponding to the single-configuration mixed positronic-electronic wavefunction, minimized with respect to the molecular orbitals expressed as linear combinations of Gaussian basis functions. The electron-electron and electron-positron correlation can be treated in the NEO framework with second-order perturbation theory (NEO-MP2) or multiconfigurational methods such as the full configuration interaction (NEO-FCI) and complete active space self-consistent-field (NEO-CASSCF) methods. In addition to implementing these methods for positronic systems, strategies for calculating electron-positron annihilation rates using NEO-HF, NEO-MP2, and NEO-FCI wavefunctions were also developed. To apply the NEO method to the positronium hydride (PsH) system, positronic and electronic basis sets were optimized at the NEO-FCI level and used to compute NEO-MP2 and NEO-FCI energies and annihilation rates. The effects of basis set size on NEO-MP2 and NEO-FCI correlation energies and annihilation rates were compared. Even-tempered electronic and positronic basis sets were also optimized for the e+LiH molecule at the NEO-MP2 level and used to compute the equilibrium bond length and vibrational energy.     

Density functional perturbational orbital theory of spin polarization in electronic systems. I. Formalism

A perturbational approach is presented for the general analysis of spin-polarization effect on electronic structures and energies within spin-density functional formalism. Explicit expressions for the changes in Kohn-Sham [Phys. Rev. 140, 1133 (1965)] orbital energies and coefficients as well as for the change in total electronic energy are derived upon using the local spin density and self-interaction-corrected exchange-correlation functionals. The application of the method for atoms provides analytical expressions for the exchange splitting energy and spin-polarization energy. The atomic exchange parameters are obtained from the expressions for the elements with Z=1-92 and they match well with Stoner exchange parameters for 3d metal elements.     

Localized orbital corrections for the calculation of ionization potentials and electron affinities in density functional theory

This paper describes the extension of a previously reported empirical localized orbital correction model to the correction of ionization potential energies (IP) and electron affinities (EA) for atoms and molecules of first and second row elements. The B3LYP localized orbital correction version of the model (B3LYP-LOC) uses 22 heuristically determined parameters that improve B3LYP DFT IP and EA energy calculations on the G2 data set of 134 molecules from a mean absolute deviation (MAD) from experiment of 0.137 to 0.039 eV. The method significantly reduces the number of outliers and overall MAD to error levels below that achieved with G2 wave function based theory; furthermore, the new model has zero additional computational cost beyond standard DFT calculations. Although the model is heuristic and is based on a multiple linear regression to experimental errors, each of the parameters is justified on physical grounds, and each provides insight into the fundamental limitations of DFT, most importantly the failure of current DFT methods to accurately account for nondynamical electron correlation.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

Iterative diagonalization for orbital optimization in natural orbital functional theory

A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F . The off‐diagonal elements of the latter are determined explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be determined from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calculated with the Hartree‐Fock orbitals after the occupation numbers have been optimized. The method has been tested on the G2/97 set of molecules for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off‐diagonal elements of F . The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Colle-Salvetti-type correction for electron-nucleus correlation in the nuclear orbital plus molecular orbital theory

A Colle-Salvetti (CS)-type electron-nucleus correction in the nuclear orbital plus molecular orbital theory is proposed. The CS-type correction is designed to satisfy the cusp condition for the electron-nucleus interaction. Since the CS-type correction is expressed in terms of the electron and nucleus densities, its evaluation is computationally feasible. Numerical assessment confirms that the CS-type correction performs well for the small G2 set.