Self-consistent Field Equations Research Articles

High-order finite element method for atomic structure calculations

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrödinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the κ=±1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10−8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrödinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program summaryProgram Title: featomCPC Library link to program files:https://doi.org/10.17632/962fzmm7f7.1Licensing provisions: MITProgramming language: Fortran with command-line interfacesNature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of density functional theory for isolated atoms.Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also. Self-consistent field equations are solved using Pulay mixing.Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for κ=±1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes.Restrictions: Spherical symmetry.

Gravitation with modified fluid Lagrangian: Variational principle and an early dark energy model

Variational principle is the main approach to obtain complete and self-consistent field equations in gravitational theories. This method works well in pure field cases such as $f(R)$ and Horndeski gravities. However, debates exist in the literature over the modification of perfect fluid. This paper aims to clarify this issue. For a wide class of modified fluid Lagrangian, we show that the variational principle is unable to give complete field equations. One additional equation is required for completeness. Adopting the local energy conservation equation gives the modified fluid a good thermodynamic interpretation. Our result is the first modified fluid theory that can incorporate energy conservation. As an application of this framework, we propose a specific modified fluid model to realize early dark energy triggered by cosmic radiation-matter transition. This model naturally explains why early dark energy occurs around matter-radiation equality and is useful in erasing the Hubble tension.

Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids.

In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn-Hilliard-like model, Bazant-Storey-Kornyshev model, and Maggs-Podgornik-Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations.

Directed walk in probability space that locates mean field solutions to spin models.

Despite their formal simplicity, most lattice spin models cannot be easily solved, even under the simplifying assumptions of mean field theory. In this paper, we present a method for generating mean field solutions to classical continuous spins. We focus our attention on systems with nonlocal interactions and nonperiodic boundaries, which require careful handling with existing approaches, such as Monte Carlo sampling. Our approach utilizes functional optimization to derive a closed-form optimality condition and arrive at self-consistent mean field equations. We show that this approach significantly outperforms conventional Monte Carlo sampling in convergence speed and accuracy. To convey the general concept behind the approach, we first demonstrate its application to a simple system: a finite one-dimensional dipolar chain in an external electric field. We then describe how the approach naturally extends to more complicated spin systems and to continuum field theories. Furthermore, we numerically illustrate the efficacy of our approach by highlighting its utility on nonperiodic spin models of various dimensionality.

Electric double layer of spherical pH-responsive polyelectrolyte brushes in an electrolyte solution: A strong stretching theory accounting for excluded volume interaction and mass action law

In this paper, we study the electrostatics of pH-responsive polyelectrolyte-grafted spherical particles by using a strong stretching theory that takes into account the excluded volume interaction and the density of chargeable sites on the polyelectrolyte molecules. Based on free energy formalism, we obtain self-consistent field equations for determining the structure and electrostatics of spherical polyelectrolyte brushes. First, we find that the smaller the radius of the inner core, the longer the height of the polyelectrolyte brush. Then, we also prove that an increase in the excluded volume interaction yields a swelling of the polyelectrolyte brush height. In addition, we demonstrate how the effect of pH, bulk ionic concentration, and lateral separation between adjacent polyelectrolyte chains on the electrostatic properties of a spherical polyelectrolyte brush is affected by the radius of the inner core, the excluded volume interaction, and the chargeable site density.

Structural and electrostatic properties between pH-responsive polyelectrolyte brushes studied by augmented strong stretching theory.

In this paper, we study electrostatic and structural properties between pH-responsive polyelectrolyte brushes by using a strong stretching theory accounting for excluded volume interactions, the density of polyelectrolyte chargeable sites, and the Born energy difference between the inside and outside of the brush layer. In a free energy framework, we obtain self-consistent field equations to determine electrostatic properties between two pH-responsive polyelectrolyte brushes. We elucidate that in the region between two pH-responsive polyelectrolyte brushes, electrostatic potential at the centerline and osmotic pressure increase not only with excluded volume interaction but also with the density of chargeable sites on a polyelectrolyte molecule. Importantly, we clarify that when two pH-responsive polyelectrolyte brushes approach each other, the brush thickness becomes short and that a large excluded volume interaction and a large density of chargeable sites yield the enhanced contract of polyelectrolyte brushes. In addition, we also demonstrate how the influence of such quantities as pH, the number of Kuhn monomers, the density of charged sites, the lateral separation between adjacent polyelectrolyte brushes, and Kuhn length on the electrostatic and structural properties between the two polyelectrolyte brushes is affected by the exclusion volume interaction. Finally, we investigate the influence of the Born energy difference on the thickness of polyelectrolyte brushes and the osmotic pressure between two pH-responsive polyelectrolyte brushes.

Highly accurate and constrained density functional obtained with differentiable programming

Using an end-to-end differentiable implementation of the Kohn-Sham self-consistent field equations, we obtain a highly accurate neural network--based exchange and correlation (XC) functional of the electronic density. The functional is optimized using information on both energy and density while exact constraints are enforced through an appropriate neural network architecture. We evaluate our model against different families of XC approximations and show that at the meta-GGA level our functional exhibits unprecedented accuracy for both energy and density predictions. For nonempirical functionals, there is a strong linear correlation between energy and density errors. We use this correlation to define an XC functional quality metric that includes both energy and density errors, leading to an improved way to rank different approximations.

Exploring the antibacterial activity of 1, 2 diaminoethane hexanedionic acid by spectroscopic, electronic, ELF, LOL, RDG analysis and molecular docking studies using DFT method

Diaminoethane hexanedionic acid, selected for its antibacterial property is extensively used for spectroscopic elucidations and computational aspects using density functional methods. Self-consistent field equation has been solved iteratively to reach the equilibrium geometry corresponding to the saddle point on the potential energy surface (PES). Assignments of vibrational spectra have been carried out with the aid of Normal coordinate analysis following the SQMFF methodology. NLO analysis through first order hyperpolarizability parameter helps in finding the potential of DAHA as a good NLO candidate whereas Charge delocalization and the stability of the compound was deliberated using natural bond orbital (NBO) analysis. An explicit surface analysis was carried out using the wave function analyser Multiwfn software, in order to study the reactivity of the compound. TD-DFT approach was applied to assign the electronic transition perceived in UV–visible spectrum measured experimentally. Frontier molecular orbital energy gap confirms the bioactivity of the molecule. NCI analysis gives information about inter and intra non covalent interactions and ESP is used to recognise the nucleophilic and electrophilic regions of molecule. Molecular docking was conceded with four antibacterial proteins confirm the bioactivity of molecule and drug likeness factors were calculated to comprehend the biological assets of DAHA.

Vibronic mean-field and perturbation theory for Jahn-Teller and pseudo-Jahn-Teller molecules

A generalisation of vibrational mean-field theory is developed for non-adiabatic Jahn-Teller and pseudo-Jahn-Teller molecules with coupled vibrational and electronic degrees of freedom treated within a quasi-diabatic framework. Solutions to the vibronic self-consistent field equations of motion generate qualitatively correct zeroth-order vibronic wavefunctions even in the strong coupling limit. These serve as an efficient basis for further perturbative or variational corrections. Particular attention is paid to complications arising in multi-mode Jahn-Teller systems, where the degenerate ground state exhibits artificial symmetry-breaking in the mean-field approximation. This effect is shown to behave like a second-order phase transition, with the symmetry of finite-temperature mean-field solutions restored above a critical temperature.

A trust-region augmented Hessian implementation for restricted and unrestricted Hartree-Fock and Kohn-Sham methods.

We present a new implementation of a trust-region augmented Hessian approach (TRAH-SCF) for restricted and unrestricted Hartree-Fock and Kohn-Sham methods. With TRAH-SCF, convergence can always be achieved even with tight convergence thresholds, which requires just a modest number of iterations. Our convergence benchmark study and our illustrative applications focus on open-shell molecules, also antiferromagnetically coupled systems, for which it is notoriously complicated to converge the Roothaan-Hall self-consistent field (SCF) equations. We compare the number of TRAH iterations to reach convergence with those of Pulay's original and Kollmar's (K) variants of the direct inversion of the iterative subspace (DIIS) method and also analyze the obtained SCF solutions. Often, TRAH-SCF finds a symmetry-broken solution with a lower energy than DIIS and KDIIS. For unrestricted calculations, this is accompanied by a larger spin contamination, i.e., larger deviation from the desired spin-restricted ⟨S2⟩ expectation value. However, there are also rare cases in which DIIS finds a solution with a lower energy than KDIIS and TRAH. In rare cases, both TRAH-SCF and KDIIS may also converge to a non-aufbau solution. For those calculations, standard DIIS always diverges. For cases that converge smoothly with either method, TRAH usually needs more iterations to converge than DIIS and KDIIS because for every new set of orbitals, the level-shifted Newton-Raphson equations are solved approximately and iteratively. In such cases, the total runtime of TRAH-SCF is still competitive with the DIIS-based approaches even if extended basis sets are employed, which is illustrated for a large hemocyanin model complex.

Pressure Effects on Self-Assembly in Mixtures Containing Zwitterionic Amphiphiles.

To understand the responses of self-assembly in mixtures containing zwitterionic amphiphilic chains to high pressure, we introduce a self-consistent field theory in combination with a molecular equation-of-state model for them in a primitive way. The free energy density for those in the bulk state is first formulated. Its locally equilibrated excess part is then incorporated into Edwards Hamiltonian along with the electrostatic energy contributions to elicit the saddle point approximation to the partition function with proper self-consistent field equations. It is shown that charge-charge correlations enhance self-assembling tendency for the amphiphiles with the opposite charges on one component side, as the medium dielectric constant εr decreases. Those with the opposite charges at the two chain ends respond in a more complicated way to εr. Densification by applied pressure strengthens the self-assembly for both at a moderate εr, similar to typical phospholipids, but pressure effects are strongly dependent on the position of charges along the chains at a lower εr. It is argued that the manipulation of the dielectric environment and disparity in component dispersion interactions can yield useful materials exhibiting various types of baroresponsivity or thermoresponsivity with re-entrant self-assembly.

The Sternheimer approach to all-electron real-space density-functional perturbation theory with atomic basis set

We present an efficient perturbative method to get the response density matrix using localized non-orthogonal basis sets. This scheme is based on the solution of the coupled perturbed self-consistent field equation with the Sternheimer approach, which only requires the occupied states and avoids a sum over unoccupied states. We present a complete derivation of the Sternheimer approach to perturbation theory within the framework of the linear combination of atomic orbitals. To demonstrate the capabilities of this method, we have implemented it in the all-electron Fritz Haber Institute ab initio molecular simulation package and applied it to benchmark molecules. For the response properties with respect to the atomic displacement and to the homogeneous electric field, the results are in excellent agreement with those of the previous traditional method and fully validate this Sternheimer approach.

Polarizable Density Embedding for Large Biomolecular Systems.

We present an efficient and robust fragment-based quantum-classical embedding model capable of accurately capturing effects from complex environments such as proteins and nucleic acids. This is realized by combining the molecular fractionation with conjugate caps (MFCC) procedure with the polarizable density embedding (PDE) model at the level of Fock matrix construction. The PDE contributions to the Fock matrix of the core region are constructed using the local molecular basis of the individual fragments rather than the supermolecular basis of the entire system. Thereby, we avoid complications associated with the application of the MFCC procedure on environment quantities such as electronic densities and molecular-orbital energies. Moreover, the computational cost associated with solving self-consistent field (SCF) equations of the core region remains unchanged from that of purely classical polarized embedding models. We analyze the performance of the resulting model in terms of the reproduction of the electrostatic potential of an insulin monomer protein and further in the context of solving problems related to electron spill-out. Finally, we showcase the model for the calculation of one- and two-photon properties of the Nile red molecule in a protein environment. Based on our analyses, we find that the combination of the MFCC approach with the PDE model is an efficient, yet accurate approach for calculating molecular properties of molecules embedded in structured biomolecular environments.

State-Targeted Energy Projection: A Simple and Robust Approach to Orbital Relaxation of Non-Aufbau Self-Consistent Field Solutions.

Orbital optimization is crucial when using a non-Aufbau Slater determinant that involves promotion of an electron from a (nominally) occupied molecular orbital to an unoccupied one, or else ionization from a molecular orbital that lies below the highest occupied frontier molecular orbital. However, orbital relaxation of a non-Aufbau determinant risks "variational collapse" back to the Aufbau solution of the self-consistent field (SCF) equations. Algorithms such as the maximum overlap method (MOM) that are designed to avoid this collapse are not guaranteed to work, and more robust alternatives increase the cost per SCF iteration. Here, we introduce an alternative procedure called state-targeted energy projection (STEP) that is based on level shifting and is identical in cost to a normal SCF procedure, yet converges in numerous cases where MOM suffers variational collapse. Benchmark calculations on small-molecule reference data suggest that ΔSCF calculations based on STEP are an accurate way to compute both ionization and excitation energies, including core-level ionization and excited states with significant double-excitation character. For the molecule 2,4,6-trifluoroborazine, ΔSCF calculations based on STEP afford excellent agreement with experiment for both vertical and adiabatic ionization energies, the latter requiring geometry optimization of a non-Aufbau valence hole. Semiquantitative agreement with experiment is obtained for the absorption spectrum of chlorophyll a. Finally, the importance of asymptotic exchange and correlation is illustrated by application to Rydberg states using spin-scaled Møller-Plesset perturbation theory with a non-Aufbau reference determinant. Together, these results suggest that STEP offers a reliable and affordable alternative to the MOM procedure for determining non-Aufbau solutions of the SCF equations.

An Overview of Self-Consistent Field Calculations Within Finite Basis Sets.

A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree–Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn–Sham–Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn–Sham–Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.

Modal optimisation within the time-independent eigenstate-free Raman wavefunction formalism

The time-independent eigenstate-free Raman wavefunction approach, which is based on the vibrational inhomogeneous Schrödinger equation (ISE), is a promising alternative to the well-known sum-over-states and time-dependent methods for the calculation of optical bandshapes and resonance Raman intensities. The solution of the ISE, the Raman wavefunction (RWF), can be obtained within a vibrational configuration interaction (VCI) ansatz. In this method, the optimisation of vibrational modals can provide a better convergence of the calculated RWF towards the full VCI limit. For this purpose, the variational principle for the ISE problem is formulated. The variational condition for the modal functions leads to the complex inhomogeneous vibrational self-consistent field equations. The key aspects of the implementation of this approach are discussed. The performance of this method in comparison with the conventional scheme involving VCI bases of different quality is illustrated for the X ~ 2 A 2 ⟵ X ~ 1 A 1 transition in furan. The quantitative effects due to the modal optimisation are found to be relatively small, as compared to the impact of moderate extension of the configuration basis.

Nonlocal statistical field theory of dipolar particles forming chain-like clusters

We present a nonlocal statistical field theory of a diluted solution of dipolar particles which are capable of forming chain-like clusters in accordance with the ’head-to-tail’ mechanism. As in our previous study [Yu.A. Budkov 2018 J. Phys.: Condens. Matter 30 344001], we model dipolar particles as dimers comprised of oppositely charged point-like groups, separated by fluctuating distance. For the special case of the Yukawa-type distribution function of distance between the charged groups of dipolar particles we obtain an analytical expression for the electrostatic free energy of solution within the random phase approximation. We show that an increase in the association constant leads to a decrease in the absolute value of the electrostatic free energy of solution, preventing its phase separation which is in agreement with the former computer simulations and theoretical results. We obtain a non-linear integro-differential equation for the self-consistent field potential created by the fixed external charges in a solution medium, taking into account the association of dipolar particles. As a consequence of the derived self-consistent field equation, in regime of weak electrostatic interactions, we obtain an analytical expression for the electrostatic potential of the point-like test ion, surrounded by the chain-like clusters of the dipolar particles. We show that in the mean-field approximation the association does not change the bulk dielectric permittivity of the solution, but increases the solvation radius of the point-like charge, relative to the theory of non-associating dipolar particles.

Self-consistent field theory of density correlations in classical fluids

More than half of a century has passed since the free energy of classical fluids defined by second Legendre transform was derived as a functional of density-density correlation function. It is now becoming an increasingly significant issue to develop the correlation functional theory that encompasses the liquid state theory, especially for glassy systems where out of equilibrium correlation fields are to be investigated. Here we have formulated a field theoretic perturbation theory that incorporates two-body fields (both of density-density correlation field and its dual field playing the role of two-body interaction potential) into a density functional integral representation of the Helmholtz free energy. Quadratic density fluctuations are only considered in the saddle-point approximation of two-body fields as well as density field. We have obtained a set of self-consistent field equations with respect to these fields, which simply reads a modified mean-field equation of density field where the bare interaction potential in the thermal energy unit is replaced by minus the direct correlation function given in the mean spherical approximation. Such replacement of the interaction potential in the mean-field equation belongs to the same category as the local molecular field theory proposed by Weeks and co-workers. Notably, it has been shown that even the mean-field part of the free energy functional given by the self-consistent field theory includes information on short-range correlations between fluid particles, similarly to the formulation of the local molecular field theory. The advantage of our field theoretic approach is not only that the modified mean-field equation can be improved systematically, but also that fluctuations of two-body fields in nonuniform fluids may be considered, which would be relevant especially for glass-forming liquids.

Global elucidation of broken symmetry solutions to the independent particle model through a Lie algebraic approach.

Broken symmetry solutions-solutions to the independent particle model that do not obey all symmetries required by the Hamiltonian-have attracted significant interest for capturing multireference properties with mean-field scaling. However, identification and optimization of broken-symmetry solutions is difficult owing to the non-linear nature of the self-consistent field (SCF) equations, particularly for solutions belonging to low-symmetry subgroups and where multiple broken symmetry solutions are sought. Linearization of SCF solution space results in the Lie algebra, which this work utilizes as a framework for elucidation of the set of solutions that exist at the desired symmetry. To demonstrate that searches constructed in the Lie algebra yield the set of broken symmetry solutions, a grid-based search of real-restricted, real-unrestricted, complex-restricted, paired-unrestricted, and real-general solutions of the (nearly D 4h ) H4 molecule is performed.

A Hybrid Monte Carlo Self-Consistent Field Model of Physical Gels of Telechelic Polymers.

We developed a hybrid Monte Carlo self-consistent field technique to model physical gels composed of ABA triblock copolymers and gain insight into the structure and interactions in such gels. The associative A blocks of the polymers are confined to small volumes called nodes, while the B block can move freely as long as it is connected to the A blocks. A Monte Carlo algorithm is used to sample the node configurations on a lattice, and Scheutjens–Fleer self-consistent field (SF-SCF) equations are used to determine the change in free energy. The advantage of this approach over more coarse grained methods is that we do not need to predefine an interaction potential between the nodes. Using this MC-SCF hybrid simulation, we determined the radial distribution functions of the nodes and structure factors and osmotic compressibilities of the gels. For a high number of polymers per node and a solvent-B Flory–Huggins interaction parameter of 0.5, phase separation is predicted. Because of limitations in the simulation volume, we did however not establish the full phase diagram. For comparison, we performed some coarse-grained MC simulations in which the nodes are modeled as single particles with pair potentials extracted from SF-SCF calculations. At intermediate concentrations, these simulations gave qualitatively similar results as the MC-SCF hybrid. However, at relatively low and high polymer volume fractions, the structure of the coarse-grained gels is significantly different because higher-order interactions between the nodes are not accounted for. Finally, we compare the predictions of the MC-SCF simulations with experimental and modeling data on telechelic polymer networks from literature.