Noninteracting Reference Research Articles

Leveling the Mountain Range of Excited-State Benchmarking through Multistate Density Functional Theory.

The performance of multistate density functional theory (MSDFT) with nonorthogonal state interaction (NOSI) is assessed for 100 vertical excitation energies against the theoretical best estimates extracted to the full configuration interaction accuracy on the database developed by Loos et al. in 2018 (Loos2018). Two optimization techniques, namely, block-localized excitation and target state optimization, are examined along with two ways of estimating the transition density functional (TDF) for the correlation energy of the Hamiltonian matrix density functional. The results from the two optimization methods are similar. It was found that MSDFT-NOSI using the spin-multiplet degeneracy constraint for the TDF of spin-coupling interaction, along with the M06-2X functional, yields a root-mean-square error (RMSE) of 0.22 eV, which performs noticeably better than time-dependent density functional theory (DFT) at an RMSE of 0.43 eV using the same functional and basis set on the Loos2018 database. In comparison with wave function theory, NOSI has smaller errors than CIS(D∞), LR-CC2, and ADC(3) all of which have an RMSE of 0.28 eV, but somewhat greater than STEOM-CCSD (RMSE of 0.14 eV) and LR-CCSD (RMSE of 0.11 eV) wave function methods. In comparison with Kohn-Sham (KS) DFT calculations, the multistate DFT approach has little double counting of correlation. Importantly, there is no noticeable difference in the performance of MSDFT-NOSI on the valence, Rydberg, singlet, triplet, and double-excitation states. Although the use of another hybrid functional PBE0 leads to a greater RMSE of 0.36 eV, the deviation is systematic with a linear regression slope of 0.994 against the results with M06-2X. The present benchmark reveals that density functional approximations developed for KS-DFT for the ground state with a noninteracting reference may be adopted in MSDFT calculations in which the state interaction is key.

Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory.

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb-exchange energy EHx plus an external potential energy ∫dr v(r)D(r), and a correlation matrix density functional . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of . We introduce the multistate correlation potential, as the functional derivative of , which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Using gradient-based algorithms to determine ground-state energies on a quantum computer

Variational algorithms are promising candidates to be implemented on near-term quantum computers. In our work, we investigate the variational Hamiltonian ansatz (VHA), where a parametrized trial state of the quantum mechanical wave function is optimized to obtain the ground-state energy. In the VHA, the trial state is given by a noninteracting reference state modified by unitary rotations using generators that are part of the Hamiltonian describing the system. The lowest energy is obtained by optimizing the angles of those unitary rotations. A standard procedure to optimize the variational parameters is to use gradient-based algorithms. However, shot noise and the intrinsic noise of the quantum device affect the evaluation of the required gradients. We study how different methods for obtaining the gradient, specifically the finite-difference and the parameter-shift rule, are affected by shot noise and the noise of the quantum computer. To this end, we simulate a simple quantum circuit, as well as the two-site and six-site Hubbard models.

Ab initio effective one‐electron potential operators: Applications for charge‐transfer energy in effective fragment potentials

The concept of effective one-electron potentials (EOPs) has proven to be extremely useful in efficient description of electronic structure of chemical systems, especially extended molecular aggregates such as interacting molecules in condensed phases. Here, a general method for EOP-based elimination of electron repulsion integrals is presented, that is tuned toward the fragment-based calculation methodologies such as the second generation of the effective fragment potentials (EFP2) method. Two general types of the EOP operator matrix elements are distinguished and treated either via the distributed multipole expansion or the extended density fitting (DF) schemes developed in this work. The EOP technique is then applied to reduce the high computational costs of the effective fragment charge-transfer (CT) terms being the bottleneck of EFP2 potentials. The alternative EOP-based CT energy model is proposed, derived within the framework of intermolecular perturbation theory with Hartree-Fock noninteracting reference wavefunctions, compatible with the original EFP2 formulation. It is found that the computational cost of the EFP2 total interaction energy calculation can be reduced by up to 38 times when using the EOP-based formulation of CT energy, as compared to the original EFP2 scheme, without compromising the accuracy for a wide range of weakly interacting neutral and ionic molecular fragments. The proposed model can thus be used routinely within the EFP2 framework.

Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory.

A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.

Multipronged ESI-MS Approach for Studying Glycan-Binding Protein Interactions with Glycoproteins.

A multipronged electrospray ionization mass spectrometry (ESI-MS) approach for investigating glycan-mediated interactions between water-soluble glycan-binding proteins (GBPs) and glycoproteins (GPs) is described. First, the catch-and-release (CaR)-ESI-MS assay, carried with ion mobility separation prior to GBP "release" (i.e., CaRIMS-ESI-MS), is employed to rapidly identify GBP-GP binding in solution. The apparent affinity ( Ka) of the GBP for the GP is then measured using the competitive proxy ligand-ESI-MS binding assay. Finally, N-glycans, enzymatically released as free oligosaccharides from the GP, are screened against the GBP using ESI-MS to identify the glycans that are recognized by the GBP. Measurements performed at multiple GBP concentrations allow for the affinities of released N-glycans (grouped as compositional isomers) to be ranked. Implementation of the approach is illustrated using the known interactions between a C-terminal domain fragment of human galectin-3 (hGal-3C) and three human serum GPs, α-1-acid glycoprotein (AGP), haptoglobin phenotype 1-1 (Hp1-1) and α-2-macroglobulin (α2M). Specific binding of hGal-3C to each GP was successfully detected by CaRIMS-ESI-MS; no binding was detected for a noninteracting reference protein, which served as a negative control. The Ka measured by proxy ligand-ESI-MS for AGP, Hp1-1 and α2M (4 × 105 M-1, 2 × 105 M-1 and 3 × 105 M-1, respectively) are consistent with the reported apparent affinities of hGal-3 for serum GPs and their N-glycans. Screening the N-glycan libraries of each of the GPs against hGal-3C identified ligands corresponding to a total of 20 different saccharide compositions with sialylated bi-, tri-, and tetra-antennary structures. The results of binding measurements performed at two different hGal-3C concentrations revealed that all of the N-glycan ligands exhibit affinities ≥104 M-1.

Exponential-six potential scaling for the calculation of free energies in molecular simulations

An adjustable, scaled form of the exponential-six (exp-6) potential is presented. The potential form allows stable scaling from a fully interacting exp-6 system to a non-interacting reference system for the direct computation of free energy differences or efficient particle growth simulations, particularly for high-density systems. Additional scaling parameters were introduced to overcome known endpoint effects, whereby reducing the potential to an ideal gas state can produce singularities in simulation averages or prohibit the sampling of close particle distances. The scaled potential is validated in several ways, using Hamiltonian thermodynamic integration, by comparison to vapour–liquid and solid–liquid coexistence free energies reported in the literature, and by the application of the Gibbs–Helmholtz equation. Forms of the scaled exp-6 potential for its implementation into molecular simulations and the thermodynamic integration methods are also developed.

Practical auxiliary basis implementation of Rung 3.5 functionals.

Approximate exchange-correlation functionals for Kohn-Sham density functional theory often benefit from incorporating exact exchange. Exact exchange is constructed from the noninteracting reference system's nonlocal one-particle density matrix γ(r(->), r(->)'). Rung 3.5 functionals attempt to balance the strengths and limitations of exact exchange using a new ingredient, a projection of γ(r(->), r(->)') onto a semilocal model density matrix γ(SL)(ρ(r(->)), ∇ρ(r(->)), r(->) - r(->)'). γSL depends on the electron density ρ(r(->) at reference point r(->), and is closely related to semilocal model exchange holes. We present a practical implementation of Rung 3.5 functionals, expanding the r(->) - r(->)' dependence of γSL in an auxiliary basis set. Energies and energy derivatives are obtained from 3D numerical integration as in standard semilocal functionals. We also present numerical tests of a range of properties, including molecular thermochemistry and kinetics, geometries and vibrational frequencies, and bandgaps and excitation energies. Rung 3.5 functionals typically provide accuracy intermediate between semilocal and hybrid approximations. Nonlocal potential contributions from γSL yield interesting successes and failures for band structures and excitation energies. The results enable and motivate continued exploration of Rung 3.5 functional forms.

Extension of many-body theory and approximate density functionals to fractional charges and fractional spins

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 G(0), 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, G(0). 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.

Compression of High Grafting Density Opposing Polymer Brushes Using Molecular Dynamics Simulations in Explicit Solvent

Opposing polymer brush layers at high grafting density were examined under confinement and characterized with respect to structure and interaction forces using molecular dynamics simulations in an explicit solvent. The brush system underwent a static compression, where the system is simulated at several discrete separation distances. These simulations are all at the same solvent chemical potential as a non-interacting reference state to produce a realistic compression. Normal pressure-distance profiles were generated and compared to density profiles at each separation distance to determine structure-property relationships. Significant interpenetration of brush layers occurred at high compression, to the extent that each brush reached the opposing surface. Higher interpenetration corresponded to a sharp increase in the pressure-distance curve, suggesting a correlation between interpenetration and interaction forces. We find clear differences from literature values using implicit solvent techniques.

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.

Metric measures of interparticle interaction in an exactly solvable two-electron model atom

The exact ground-state solutions for the model of two particles in a confining harmonic oscillator potential interacting through a repulsive harmonic oscillator force are used from the standpoint of geometric distances [Phys. Rev. Lett. 106, 050401 (2011).] between wave functions and densities. The distances from the noninteracting reference state are calculated at a specified confinement by increasing the coupling of the interparticle interaction. Based on the analytic expressions for coupling-dependent geometric measures, a discussion of the Hohenberg-Kohn mapping is given.

Orbital-dependent Representation of Correlation Energy Functional

Abstract Exchange-correlation energy functionals depending on the Kohn-Sham orbitals and eigenvalues resolve some of the well-known deficiencies of the local density and generalized gradient approximations. Such functionals can be derived in first-principles fashion by use of standard many-body techniques, if the Kohn-Sham Hamiltonian is utilized as non-interacting reference Hamiltonian. In this way one can establish an exact relation for the exchange-correlation functional, which opens several routes for the derivation of approximate functionals. Straightforward expansion in powers of the electron-electron coupling constant gives, to first order, the exact exchange of density functional theory and, to second order, a correlation functional which has the same structure as the second order Møller-Plesset term. This simplest first-principles correlation functional reproduces both the shell structure in the exact correlation potential and dispersion forces. On the other hand, it overestimates all correlation effects and is variationally instable for systems with a very small HOMO-LUMO gap. Both deficiences can be resolved by partial resummation of the Kohn-Sham perturbation expansion. In this contribution two such resummations are discussed, a minimum form designed to address the variational instability at no computational cost (based on the Epstein-Nesbet diagrams) and a more systematic variant (based on the ring diagrams).

Fractional occupation in Kohn-Sham density-functional theory and the treatment of non-pure-statev-representable densities

In the framework of Kohn-Sham density-functional theory, systems with ground-state densities that are not pure-state $v$-representable (PSVR) in the noninteracting reference system occur frequently. In the present contribution, an algorithm, which allows the solution of such systems, is proposed. It is shown that the use of densities which do not correspond to a ground state of their noninteracting reference system is forbidden. As a consequence, the proposed algorithm considers only noninteracting ensemble $v$-representable densities. The Fe atom, a well-known non-PSVR system, is used as an illustration. Finally, the problem is analyzed within finite-temperature density-functional theory, where the physical significance of fractional occupations is exposed and the question of why degenerate states can be unequally occupied is resolved.

Quantized liquid density-functional theory for hydrogen adsorption in nanoporous materials

We develop a finite-temperature quantized version of density-functional theory of atomic and molecular liquids (QLDFT). Following the Kohn-Sham partitioning of the free energy, we introduce a noninteracting reference fluid of particles obeying the Maxwell-Boltzmann statistics. The kinetic and potential energy of the reference fluid are evaluated exactly. All remaining contributions, including interactions between fluid particles and corrections due to the appropriate quantum statistics are subsumed by an excess (in electronic DFT called exchange-correlation) functional. Two variants to approximate the excess functional are presented: the simplest local-interaction expression (LIE-0) avoids the direct calculation of interparticle interactions and includes them in the excess functional, which is parametrized to reproduce experimental equation of state of normal hydrogen. The more sophisticated LIE-1 approximation is based on the weighted local-density approximation and includes the explicit interparticle interaction potential as well as the local approximation of the excess functional, the latter being weighted by the average over a spherical environment to include nonlocal effects in an approximate way. We apply LIE-0 and LIE-1 to two benchmark systems, bulk fluid hydrogen and hydrogen in a slit pore, and compare it with classical molecular-dynamics simulations employing the same potential. Both functionals produce similar results for direct quantum effects in adsorption free energy. At the same time, LIE-1 also yields a reasonable description of the fluid structure and classical packing effects, which are not reproduced by LIE-0. The source code of our implementation of the LIE-QLDFT is distributed under the GNU public license and is included as a supporting material.

Calculation of Proteins’ Total Side-Chain Torsional Entropy and Its Influence on Protein–Ligand Interactions

Despite the high density within a typical protein fold, the ensemble of sterically permissible side-chain repackings is vast. Here, we examine the extent of this variability that survives energetic biases due to van der Waals interactions, hydrogen bonding, salt bridges, and solvation. Monte Carlo simulations of an atomistic model exhibit thermal fluctuations among a diverse set of side-chain arrangements, even with the peptide backbone fixed in its crystallographic conformation. We have quantified the torsional entropy of this native-state ensemble, relative to that of a noninteracting reference system, for 12 small proteins. The reduction in entropy per rotatable bond due to each kind of interaction is remarkably consistent across this set of molecules. To assess the biophysical importance of these fluctuations, we have estimated side-chain entropy contributions to the binding affinity of several peptide ligands with calmodulin. Calculations for our fixed-backbone model correlate very well with experimentally determined binding entropies over a range spanning more than 80 kJ/(mol·308 K).

Time‐dependent density functional calculations on the electronic spectra of the neutral nickel complex [Ni(LISQ)2] (LISQ = 3,5‐di‐tert‐butyl‐o‐diiminobenzosemiquinonate(1−)) and its monoanion and dication

The electronic spectrum of the neutral nickel complex [Ni(LISQ)2] (LISQ = 3,5-di-tert-butyl-o-diiminobenzosemiquinonate(1-)) and the spectra of its anion and dication have been calculated by means of time-dependent density functional theory. The electronic ground state of the neutral complex exhibits an open shell singlet diradical character. The mandatory multireference problem for this electronic ground state has been treated approximately by using the unrestricted and spin symmetry broken Kohn-Sham Slater determinant as the wave function for the noninteracting reference system in the time-dependent density functional calculations. A reasonable agreement with observed transition energies and band intensities has been achieved. This holds also for the long wavelength transitions that are shown to be of charge transfer type. The charge distributions in the electronic ground state and the corresponding low lying excited states, however, are rather similar. Thus, the known failure of standard time-dependent density functional theory to describe improperly long range charge transfer transitions is absent in this work. The applied computational scheme might be adequate for calculating electronic spectra of transition metal complexes with noninnocent ligands.

Pair density functional theory utilizing the noninteracting reference system: An effective initial theory

We present a pair density (PD) functional scheme utilizing the noninteracting reference system. In order to check to what extent this scheme can express the correlation effects, actual calculations are performed for the neutral neon atom. It is shown that this scheme reproduces about 20% of the correlation energy through use of noninteracting single determinants with an approximate correlating kinetic energy functional. Thus, since it obviously provides the $N$-representable and correlated PD, this scheme can be positioned as an effective initial theory in the field of the PD functional theory, just like the position of the Hartree-Fock approximation in the field of the wave function theory.

Quadratic response functions in the relativistic four-component Kohn-Sham approximation

A formulation and implementation of the quadratic response function in the adiabatic four-component Kohn-Sham approximation is presented. The noninteracting reference state is time-reversal symmetric and formed from Kramers pair spinors, and the energy density is gradient corrected. Example calculations are presented for the optical properties of disubstituted halobenzenes in their meta and ortho conformations. It is demonstrated that correlation and relativistic effects are not additive, and it is shown that relativity alone reduces the mubeta-response signal by 62% and 75% for meta- and ortho-bromobenzene, respectively, and enhances the same response by 17% and 21% for meta- and ortho-iodobenzene, respectively. Of the employed functionals, CAM-B3LYP shows the best performance and gives hyperpolarizabilities beta distinctly different from B3LYP.

Spin−Spin Contributions to the Zero-Field Splitting Tensor in Organic Triplets, Carbenes and BiradicalsA Density Functional and Ab Initio Study

An evaluation study for the direct dipolar electron spin-spin (SS) contribution to the zero-field splitting (ZFS) tensor in electron paramagnetic resonance (EPR) spectroscopy is presented. Calculations were performed on a wide variety of organic systems where the SS contribution to the ZFS dominates over the second-order spin-orbit coupling (SOC) contribution. Calculations were performed using (hybrid) density functional theory (DFT), as well as complete active space self-consistent field (CASSCF) wave functions. In the former case, our implementation is an approximation, because we use the two-particle reduced spin-density matrix of the noninteracting reference system. In the latter case, the SS contribution is approximated by a mean-field method which, nevertheless, gives accurate results, compared to the approximation free computation of the SS part in a CASSCF framework. For the case of the triplet dioxygen molecule, it was shown that restricted open-shell density functional theory (RODFT), as well as CASSCF, can provide accurate spin-spin couplings while spin-unrestricted DFT leads to much larger errors. Furthermore, 15 organic radicals, including several 1,3 and 1,5 diradicals, dinitroxide biradicals, and even a chlorophyll a model system, were examined as test cases to demonstrate the accuracy and efficiency of our approach within a DFT framework. Accurate D values with root-mean-square deviations of 0.0035 cm(-1) were obtained. Furthermore, all trends, including those due to substituent effects, were correctly reproduced. In a different set of calculations, the polyacenes benzene, naphthalene, anthracene, and tetracene were studied. Applying DFT, the absolute D values were noticeably underestimated, but it was possible to correctly reproduce the trend to smaller D values with larger size of the systems. Finally, it was demonstrated that our approach is also well-suited for the study of carbenes. The smaller organic radicals of this work were also studied, through the use of CASSCF wave functions. This was a special advantage in the case of the triplet polyacenes, where the CASSCF approach gave better results than the DFT method. In comparing spin-restricted and spin-unrestricted results, it was shown through a natural orbital analysis and comparison to high-level ab initio calculations that even small amounts of spin polarization introduced by the unrestricted calculations lead to large deviations between the unrestricted Kohn-Sham (UKS) and restricted open-shell Kohn-Sham (ROKS) approaches. It is challenging to understand why the ROKS results show much better correlation with the experimental data.