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In this work, we investigate the performance of a recently proposed transcorrelated (TC) approach based on a single-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)] for systems involving more than two electrons. The benefit of such an approach relies on its simplicity as efficient numerical–analytical schemes can be set up to compute the two- and three-body integrals occurring in the effective TC Hamiltonian. To obtain accurate ground state energies within a given basis set, the present TC scheme is coupled to the recently proposed TC–full configuration interaction quantum Monte Carlo method [Cohen et al., J. Chem. Phys. 151, 061101 (2019)]. We report ground state total energies on the Li–Ne series, together with their first cations, computed with increasingly large basis sets and compare to more elaborate correlation factors involving electron–electron–nucleus coordinates. Numerical results on the Li–Ne ionization potentials show that the use of the single-parameter correlation factor brings on average only a slightly lower accuracy (1.2 mH) in a triple-zeta quality basis set with respect to a more sophisticated correlation factor. However, already using a quadruple-zeta quality basis set yields results within chemical accuracy to complete basis set limit results when using this novel single-parameter correlation factor. Calculations on the H2O, CH2, and FH molecules show that a similar precision can be obtained within a triple-zeta quality basis set for the atomization energies of molecular systems. [ABSTRACT FROM AUTHOR]

We demonstrate how similarity-transformed full configuration interaction quantum Monte Carlo (FCIQMC) based on the transcorrelated Hamiltonian can be applied to make highly accurate predictions for the binding curve of the beryllium dimer, marking the first case study of a molecular system with this method. In this context, the non-Hermitian transcorrelated Hamiltonian, resulting from a similarity transformation with a Jastrow factor, serves the purpose to effectively address dynamic correlation beyond the used basis set and thus allows for obtaining energies close to the complete basis set limit from FCIQMC already with moderate basis sets and computational effort. Building on results from other explicitly correlated methods, we discuss the role of the Jastrow factor and its functional form, as well as potential sources for size consistency errors, and arrive at Jastrow forms that allow for high accuracy calculations of the vibrational spectrum of the beryllium dimer. [ABSTRACT FROM AUTHOR]

The best practice in computational methods for determining vertical ionization energies (VIEs) is assessed, via reference to experimentally determined VIEs that are corroborated by highly accurate coupled-cluster calculations. These reference values are used to benchmark the performance of density functional theory (DFT) and wave function methods: Hartree-Fock theory, second-order M0ller-Plesset perturbation theory, and Electron Propagator Theory (EPT). The core test set consists of 147 small molecules. An extended set of six larger molecules, from benzene to hexacene, is also considered to investigate the dependence of the results on molecule size. The closest agreement with experiment is found for ionization energies obtained from total energy difference calculations. In particular, DFT calculations using exchange-correlation functionals with either a large amount of exact exchange or long-range correction perform best. The results from these functionals are also the least sensitive to an increase in molecule size. In general, ionization energies calculated directly from the orbital energies of the neutral species are less accurate and more sensitive to an increase in molecule size. For the single-calculation approach, the EPT calculations are in closest agreement for both sets of molecules. For the orbital energies from DFT functionals, only those with long-range correction give quantitative agreement with dramatic failing for all other functionals considered. The results offer a practical hierarchy of approximations for the calculation of vertical ionization energies. In addition, the experimental and computational reference values can be used as a standardized set of benchmarks, against which other approximate methods can be compared. [ABSTRACT FROM AUTHOR]

Discontinuous changes in the electronic structure upon infinitesimal changes to the Hamiltonian are demonstrated. These are revealed in one and two electron molecular systems by full configuration interaction (FCI) calculations when the realm of the nuclear charge is extended to be fractional. FCI electron densities in these systems show dramatic changes in real space and illustrate the transfer, hopping, and removal of electrons. This is due to the particle nature of electrons seen in stretched systems and is a manifestation of an energy derivative discontinuity at constant number of electrons. Dramatic errors of density functional theory densities are seen in real space as this physics is missing from currently used approximations. The movements of electrons in these simple systems encapsulate those in real physical processes, from chemical reactions to electron transport and pose a great challenge for the development of new electronic structure methods. [ABSTRACT FROM AUTHOR]

The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are designed for physical systems with integer charges and spins, not in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of G0, the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, G0. We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the local density approximation and generalized gradient approximations; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional (or Hartree-Fock theory) and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental bandgaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge. It should play an important role in developing accurate approximate density functionals and many-body theory. [ABSTRACT FROM AUTHOR]

The conventional analysis of Perdew and Levy, and Sham and Schlüter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((∂ρs(r′,r))/∂N)vs, at integers. (3) Consistent with the Perdew-Levy-Sham-Schlüter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons. [ABSTRACT FROM AUTHOR]

Many useful concepts developed within density functional theory provide much insight for the understanding and prediction of chemical reactivity, one of the main aims in the field of conceptual density functional theory. While approximate evaluations of such concepts exist, the analytical and efficient evaluation is, however, challenging, because such concepts are usually expressed in terms of functional derivatives with respect to the electron density, or partial derivatives with respect to the number of electrons, complicating the connection to the computational variables of the Kohn-Sham one-electron orbitals. Only recently, the analytical expressions for the chemical potential, one of the key concepts, have been derived by Cohen, Mori-Sánchez, and Yang, based on the potential functional theory formalism. In the present work, we obtain the analytical expressions for the real-space linear response function using the coupled perturbed Kohn-Sham and generalized Kohn-Sham equations, and the Fukui functions using the previous analytical expressions for chemical potentials of Cohen, Mori-Sánchez, and Yang. The analytical expressions are exact within the given exchange-correlation functional. They are applicable to all commonly used approximate functionals, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals. The analytical expressions obtained here for Fukui function and linear response functions, along with that for the chemical potential by Cohen, Mori-Sánchez, and Yang, provide the rigorous and efficient evaluation of the key quantities in conceptual density functional theory within the computational framework of the Kohn-Sham and generalized Kohn-Sham approaches. Furthermore, the obtained analytical expressions for Fukui functions, in conjunction with the linearity condition of the ground state energy as a function of the fractional charges, also lead to new local conditions on the exact functionals, expressed in terms of the second-order functional derivatives. We implemented the expressions and demonstrate the efficacy with some atomic and molecular calculations, highlighting the importance of relaxation effects. [ABSTRACT FROM AUTHOR]

The difficulty of approximate density functionals in describing the energetics of Diels–Alder reactions and dimerization of aluminum complexes is analyzed. Both of these reaction classes involve formation of cyclic or bicyclic products, which are found to be underbound by the majority of functionals considered. We present a consistent view of these results from the perspective of delocalization error. This error causes approximate functionals to give too low energy for delocalized densities or too high energy for localized densities, as in the cyclic and bicyclic reaction products. This interpretation allows us to understand better a wide range of errors in main-group thermochemistry obtained with popular density functionals. In general, functionals with minimal delocalization error should be used for theoretical studies of reactions where there is a loss of extended conjugation or formation of highly branched, cyclic, and cagelike molecules. [ABSTRACT FROM AUTHOR]

The pseudobond method is used in quantum mechanical/molecular mechanical (QM/MM) simulations in which a covalent bond connects the quantum mechanical and classical subsystems. In this method, the molecular mechanical boundary atom is replaced by a special quantum mechanical atom with one free valence that forms a bond with the rest of the quantum mechanical subsystem. This boundary atom is modified through the use of a parametrized effective core potential and basis set. The pseudobond is designed to reproduce the properties of the covalent bond that it has replaced, while invoking as small a perturbation as possible on the system. Following the work of Zhang [J. Chem. Phys. 122, 024114 (2005)], we have developed new pseudobond parameters for use in the simulation of enzymatic systems. Our parameters yield improved electrostatics and deprotonation energies, while at the same time maintaining accurate geometries. We provide parameters for Cps(sp3)–C(sp3), Cps(sp3)–C(sp2,carbonyl), and Cps(sp3)–N(sp3) pseudobonds, which allow the interface between the quantum mechanical and molecular mechanical subsystems to be constructed at either the Cα–Cβ bond of a given amino acid residue or along the peptide backbone. In addition, we demonstrate the efficiency of our parametrization method by generating residue-specific pseudobond parameters for a single amino acid. Such an approach may enable higher accuracy than general purpose parameters for specific QM/MM applications. [ABSTRACT FROM AUTHOR]

Electron transfer (ET) reactions are one of the most important processes in chemistry and biology. Because of the quantum nature of the processes and the complicated roles of the solvent, theoretical study of ET processes is challenging. To simulate ET processes at the electronic level, we have developed an efficient density functional theory (DFT) quantum mechanical (QM)/molecular mechanical (MM) approach that uses the fractional number of electrons as the order parameter to calculate the redox free energy of ET reactions in solution. We applied this method to study the ET reactions of the aqueous metal complexes Fe(H2O)62+/3+ and Ru(H2O)62+/3+. The calculated oxidation potentials, 5.82 eV for Fe(II/III) and 5.14 eV for Ru(II/III), agree well with the experimental data, 5.50 and 4.96 eV, for iron and ruthenium, respectively. Furthermore, we have constructed the diabatic free energy surfaces from histogram analysis based on the molecular dynamics trajectories. The resulting reorganization energy and the diabatic activation energy also show good agreement with experimental data. Our calculations show that using the fractional number of electrons (FNE) as the order parameter in the thermodynamic integration process leads to efficient sampling and validate the ab initio QM/MM approach in the calculation of redox free energies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Wu-Yang method for determining the optimized effective potential (OEP) and implicit density functionals from a given electron density is revisited to account for its ill-posed nature, as recently done for the direct minimization method for OEP’s from a given orbital functional [T. Heaton-Burgess, F. A. Bulat, and W. Yang, Phys. Rev. Lett. 98, 256401 (2007)]. To address the issues on the general validity and practical applicability of methods that determine the Kohn-Sham (local) multiplicative potential in a finite basis expansion, a new functional is introduced as a regularized version of the original work of Wu and Yang. It is shown that the unphysical, highly oscillatory potentials that can be obtained when unbalanced basis sets are used are the controllable manifestation of the ill-posed nature of the problem. The new method ensures that well behaved potentials are obtained for arbitrary basis sets. [ABSTRACT FROM AUTHOR]

We examine the development and investigate the performance of exchange-correlation functionals constructed from the adiabatic connection. Our method is based on a direct modeling of the adiabatic connection curve in the coupling-constant space and is very flexible in the models. Several different models are investigated in the construction of new families of exchange-correlation functionals. Also the performance of two of these models (MCY1 and MCY2) is investigated over a wider range of systems and properties, with comparison made to the performance of established functionals. Overall, the adiabatic functionals improve upon widely used hybrid and generalized gradient approximation functionals, particularly in correctly describing one-electron systems and reaction energy barriers. [ABSTRACT FROM AUTHOR]

The evaluation of ≤S⁁2> in density functional theory (DFT) is considered. Wang et al. [J. Chem. Phys. 102, 3477 (1995)] have derived an approximate, local density expression for <<⁁2> and in the present study their formula is evaluated using densities from unrestricted Hartree-Fock (UHF) and a range of DFT exchange-correlation functionals. The results are compared with those obtained by evaluating the conventional UHF expression using the Kohn-Sham orbitals, which is appropriate for the noninteracting system. A generalized gradient approximation for ≤S⁁2> is then proposed and investigated [ABSTRACT FROM AUTHOR]

New exchange-correlation functionals that address the important issue of many-electron self-interaction are developed. This is carried out by considering the performance of the functional on systems with fractional numbers of electrons at the same time as more standard thermochemical tests. The inclusion of Coulomb-attenuated exchange in the functional is facilitated by use of the adiabatic connection coupled with a short-range and long-range splittings. The new functionals have a good performance on thermochemistry and a much improved description of the total energy versus number of electrons and henceforth a much smaller many-electron self-interaction error. [ABSTRACT FROM AUTHOR]

Metal shielding constants and chemical shifts are determined for nine transition metal complexes using an uncoupled formalism with orbitals and eigenvalues determined using the Yang-Wu implementation [W. Yang and Q. Wu, Phys. Rev. Lett. 89, 143002 (2002)] of the optimized effective potential approach in density functional theory. Preliminary calculations using generalized gradient approximation functionals quantify the influence of the variables in the optimized effective potential implementation. In particular, a flexible potential expansion is necessary for a precise calculation of these quantities. Hybrid functionals are then considered. Expanding the potential in the primary orbital basis yields chemical shifts that are a notable improvement over conventional hybrid values, and which are a marginal improvement over those obtained using a high-quality generalized gradient approximation. Similar shifts are obtained using a more flexible potential expansion, although care is required to avoid unphysical structure in the exchange-correlation potential. [ABSTRACT FROM AUTHOR]

One of the most important challenges in density functional theory (DFT) is the proper description of fractional charge systems relating to the self-interaction error (SIE). Traditionally, the SIE has been formulated as a one-electron problem, which has been addressed in several recent functionals. However, these recent one-electron SIE-free functionals, while greatly improving the description of thermochemistry and reaction barriers in general, still exhibit many of the difficulties associated with SIE. Thus we emphasize the need to surpass this limit and shed light on the many-electron SIE. After identifying the sufficient condition for functionals to be free from SIE, we focus on the symptoms and investigate the performance of most popular functionals. We show that these functionals suffer from many-electron SIE. Finally, we give a SIE classification of density functionals. [ABSTRACT FROM AUTHOR]

We develop a self-interaction-free exchange-correlation functional which is very accurate for thermochemistry and kinetics. This is achieved by theoretical construction of the functional form and nonlinear fitting. We define a simple interpolation of the adiabatic connection that uses exact exchange, generalized gradient approximation (GGA) and meta-GGA functionals. The performance is optimized by fitting a small number of empirical parameters. Overall the new functional improves significantly upon hybrids and meta-GGAs while correctly describing one-electron systems. The mean absolute error on a large set of reaction barriers is reduced to 1.99 kcal/mol. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we extend our density-functional theory calculations, with generalized gradient approximation and hybrid functionals, using Slater-type orbitals (STOs), to the determination of second-order molecular properties. The key to the entire methodology involves the fitting of all STO basis function products to an auxiliary STO basis, through the minimization of electron-repulsion integrals. The selected properties are (i) dipole polarizabilities, (ii) nuclear magnetic shielding constants, and (iii) nuclear spin–spin coupling constants. In all cases the one-electron integrals involving STOs were evaluated by quadrature. The implementation for (ii) involved some complexity because we used gauge-including atomic orbitals. The presence of two-electron integrals on the right-hand side of the coupled equations meant that the fitting procedure had to be implemented. For (iii) in the hybrid case, fitting procedures were again required for the exchange contributions. For each property we studied a number of small molecules. We first obtained an estimate of the basis set limit using Gaussian-type orbitals (GTOs). We then showed how it is possible to reproduce these values using a STO basis set. For (ii) a regular TZ2P quality STO basis was adequate; for (i) the addition of one set of diffuse functions (determined by Slater’s rules) gave the required accuracy; for (iii) it was necessary to add a set of 1s functions, including one very tight function, to give the desired result. In summary, we show that it is possible to predict second-order molecular properties using STO basis sets with an accuracy comparable with large GTO basis sets. We did not encounter any major difficulties with either the selection of the bases or the implementation of the procedures. Although the energy code (especially in the hybrid case) may not be competitive with a regular GTO code, for properties we find that STOs are more attractive. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We report the extension of our Slater-type orbital (STO) basis code for density functional theory (DFT) calculations to include “exact exchange,” so that Hartree–Fock and hybrid DFT calculations may be performed. Exchange integrals are evaluated through the insertion of a resolution of the identity, using the same auxiliary STO basis set which was used in our original generalized gradient approximation DFT studies. Three-center two-electron integrals are evaluated by quadrature. A scheme for introducing diatomic auxiliary STO basis sets is introduced, which enables much larger calculations to be performed. We also report an extension to the evaluation of second-order molecular properties using these functionals. Calculations for excitation energies to valence and Rydberg states are included. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The most common form of density functional calculations on molecular systems used generalized gradient approximation exchange-correlation functionals (such calculations can be applied to larger systems because no exact exchange is included). The most efficient and fastest such codes use an auxiliary basis set to fit the density so that only three-center integrals need to be evaluated. The codes DGAUSS and TURBOMOL use Gaussian basis sets, whereas the long-established ADF code uses Slater basis sets. We here examine the use of Slater basis sets. Our new code evaluates all required integrals numerically by quadrature. We report calculations on the G2 molecular set, contrasting them with similar calculations using Gaussian basis sets. Our conclusion, as far as energetics and structure are concerned, is that very similar predictions may be obtained from basis sets of the same size, and at approximately the same cost. [ABSTRACT FROM AUTHOR]

The purpose of this work is to search for a justifiable form for a molecular dynamic correlation functional. A detailed examination of Colle and Salvetti's derivation of the LYP functional is presented. It is argued that the leading term is all important, and furthermore that it should account for αβ correlation. This term only depends upon the densities, and it has a truncation factor which is obtained from the size of the correlation hole. It is -c∫ρ αρβ/(ρ( 1 + dρ[sup ⅓))dr. It reproduces the αβ correlation energies of (He-Ar) to a very high accuracy. The correlation functional which represents σσ correlation is more complex, because the two particle Hartree-Fock density matrix is zero at electron coalescence. The functional must therefore depend upon (∇p)². Using these and related arguments we have found a four parameter generalized gradient functional which appears to perform nearly as well as the LYP functional. However unlike the LYP functional, it has two identifiable terms for αβ correlation, and two identifiable terms for σσ correlation. Together with our previously derived exchange functional, we have therefore obtained an exchange-correlation functional for molecular studies, the form for which can be more understandably justified. The performance of this new Generalized Gradient Approximation functional for molecular predictions is reported. It is a considerable improvement on the BLYP functional, and is in the category of an optimum Generalized Gradient functional. Finally the present status of the science of searching for exchange-correlation functions is reviewed. It is suggested that it may not be possible to find a local functional which is significantly more accurate for chemistry than the presently used Generalized Gradient Approximation functionals. [ABSTRACT FROM AUTHOR]