Conventional Self-consistent Field Research Articles

Efficient and Robust Ab Initio Self-Consistent Field Acceleration Algorithm Based on a Semiempirical Model Hamiltonian.

A novel doubly iterative self-consistent field (SCF) approach using a semiempirical model Hamiltonian (denoted as the SMH algorithm) is proposed to accelerate the Hartree-Fock (HF) and density functional theory (DFT) calculations. This method first constructs the Fock matrix exactly in each SCF macroiteration, followed by a few SCF microiterations, where the Fock matrix is incrementally updated using an inexpensive semiempirical approximation. This leads to an improved wave function in each SCF macroiteration with minimal additional cost, and therefore a reduced number of exact Fock builds is required for SCF convergence. The SMH algorithm can be combined with conventional SCF convergence techniques such as level shifting, direct inversion in the iterative subspace (DIIS), and energy-DIIS (EDIIS). When integrated with DIIS, SMH enhances the convergence of simple organic molecules by approximately 10% compared to plain DIIS, with speedups of up to 60% for the more challenging transition metal systems compared to EDIIS + DIIS. Our results show that SMH is a reliable SCF accelerator that seldom deteriorates convergence and is highly robust.

The Grassmann interpolation method for spin-unrestricted open-shell systems.

The recently reported Grassmann interpolation (G-Int) method [J. A. Tan and K. U. Lao, J. Chem. Phys. 158, 051101 (2023)] has been extended to spin-unrestricted open-shell systems. In contrast to closed-shell systems, where G-Int has to be performed only once since the α and β density matrices are the same, spin-unrestricted open-shell systems require G-Int to be performed twice-one for the α spin and another for the β spin density matrix. In this work, we tested the performance of G-Int to the carbon monoxide radical cation CO●+ and nickelocene complex, which have the doublet and triple ground states, respectively. We found that the Frobenius normerrors associated with the interpolations for the α and β spin density matrices are comparable for a given molecular geometry. These G-Int density matrices, when used as an initial guess for a self-consistent field (SCF) calculation, outperform the conventional SCF guess schemes, such as the superposition of atomic densities, purified superposition of atomic densities, core Hamiltonian, and generalized Wolfsberg-Helmholtz approximation. Depending on the desired accuracy, these G-Int density matrices can be used to directly evaluate the SCF energy without performing SCF iterations. In addition, the spin-unrestricted G-Int density matrices have been used for the first time to directly calculate the atomic charges using the Mulliken and ChElPG population analysis.

Automatic Differentiation for the Direct Minimization Approach to the Hartree-Fock Method.

Automatic differentiation (AD) has become an important tool for optimization problems in computational science, and it has been applied to the Hartree-Fock method. Although the reverse-mode AD is more efficient than the forward-mode, eigenvalue calculation in the self-consistent field (SCF) method has impeded the use of the reverse-mode AD. Here, we propose a method to directly minimize Hartree-Fock energy under the orthonormality constraint of the molecular orbitals using reverse-mode AD by avoiding eigenvalue calculation. According to our validation, the proposed method was more stable than the conventional SCF method and achieved comparable accuracy.

Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane-Wave Basis Set.

A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane-wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density-dependent (ODD) functionals including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the excited state; (2) an outer loop for minimizing the energy in a tangential direction in the space of the orbitals. For ODD functionals, a third loop is used to find the unitary transformation that minimizes the energy functional among occupied orbitals only. Combined with the maximum overlap method, the algorithm converges in challenging cases where conventional self-consistent field algorithms tend to fail. The benchmark tests presented include two charge-transfer excitations in nitrobenzene and an excitation of CO to degenerate π* orbitals where the importance of complex orbitals is illustrated. The application of this method to several metal-to-ligand charge-transfer and metal-centered excited states of an FeII photosensitizer complex is described, and the results are compared to reported experimental estimates. This method is also used to study the effect of the Perdew-Zunger self-interaction correction on valence and Rydberg excited states of several molecules, both singlet and triplet states, and the performance compared to semilocal and hybrid functionals.

Vibrational optical activity of chiral carbon nanoclusters treated by a generalized π-electron method.

Cross sections of inelastic light scattering accompanied by vibronic excitation in large conjugated carbon structures is assessed at the π-electron level. Intensities of Raman and vibrational Raman optical activity (VROA) spectra of fullerenes are computed, relying on a single electron per atom. When considering only first neighbor terms in the Hamiltonian (a tight-binding (TB) type or Hückel-model), Raman intensities are captured remarkably well, based on comparison with frequency-dependent linear response of the self-consistent field (SCF) method. Resorting to π-electron levels when computing spectral intensities brings a beneficial reduction in computational cost as compared to linear response SCF. At difference with total intensities, the first neighbor TB model is found inadequate for giving the left and right circularly polarized components of the scattered light, especially when the molecular surface is highly curved. To step beyond first neighbor approximation, an effective π-electron Hamiltonian, including interaction of all sites is derived from the all-electron Fockian, in the spirit of the Bloch-equation. Chiroptical cross-sections computed by this novel π-electron method improve upon first-neighbor TB considerably, with no increase in computational cost. Computed VROA spectra of chiral fullerenes, such as C76 and C28, are reported for the first time, both by conventional linear response SCF and effective π-electron models.

Boundary effects in confined copolymer system and compressible SCFT model

The conventionally used Modified Self-consistent Field Theory (MSCFT) for confined system is known to have two major difficulties: numerical instability and dependence on artificial constraints. Here we use Compressible Model based Self-consistent Field Theory (CMSCFT), which has no artificial constraints, and furthermore overcomes numerical instability problems. CMSCFT asymptotically converges to the conventional Self-consistent Field Theory (SCFT) as the compressibility tends to zero. Satisfactory simulations on some hard-surface confinements, focusing on one-dimensional situation, are also given. We suggest that CMSCFT will be a promising tool for copolymer systems with hard surface confinements.

Effective preconditioning for ab initio ground state energy minimization with non-orthogonal localized molecular orbitals

The non-orthogonal localized molecular orbital (NOLMO) is the most localized representation of electronic degrees of freedom. As such, NOLMOs are thus potentially the most efficient for linear-scaling calculations of electronic structures for large systems. However, direct ab initio calculations with NOLMO have not been fully implemented and widely used, partly because of the slow convergence issue in the optimization of NOLMO. Towards realizing the potential of NOLMO for large systems, we applied an energy minimum variational principle for carrying out ab initio self-consistent-field (SCF) calculations with NOLMOs. We developed an effective preconditioning approach using the diagonal part of the second order derivatives and show that the convergence of the energy optimization is significantly improved. The speed of convergence of the energy and density are comparable with that of the conventional SCF approach, thus paving the way for the optimization of NOLMO in linear scaling calculations for large systems.

Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: Exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.

Large scale Hartree-Fock calculations with conventional SCF algorithm. Influence of integral and index compression on Fock matrix construction

It is shown that a compression of two-electron integrals and their indices significantly improves efficiency of the conventional self-consistent field (SCF) algorithm for a solution of the Hartree-Fock equation by decrease the Fock matrix calculation time. The improvement is reached not only due to a reduction of the integral file size, but mainly because data compression reduces or even can eliminate a cache conflict in data transfer from the hard drive to the main computer memory. Thus, the conventional SCF algorithm with the data compression becomes very efficient and permits to carry out large-scale Hartree-Fock calculations. The largest Hartree-Fock calculations have been performed for RNA 433D structure from the PDB data bank with 6080 basis functions formed from 6-31G basis on a workstation with 1 GHz Alpha processor.

Localized orbitals in the coupled cluster singles and doubles model

In the vast majority of many-body perturbation theory and coupled-cluster method (MBPT/CCM) calculations, the self-consistent field (SCF) wavefunction is employed as the zeroth-order wave-function, and its associated canonical orbitals as the molecular orbital basis. The recent development of the coupled-cluster singles and doubles (CCSD) formalism provides an opportunity to explore alternative reference function and orbital choices, on a consistent basis. Here we examine, specifically, sets of orbitals that are more “localized” than the SCF orbitals. In particular, the bonding orbitals are selected to resemble traditional chemical bonds. CCSD calculations in the localized basis on methane employing a better than double-zeta basis set are performed and compared with conventional SCF basis results.

An efficient self-consistent field method for large systems of weakly interacting components

An efficient method for removing the self-consistent field (SCF) diagonalization bottleneck is proposed for systems of weakly interacting components. The method is based on the equations of the locally projected SCF for molecular interactions (SCF MI) which utilize absolutely localized nonorthogonal molecular orbitals expanded in local subsets of the atomic basis set. A generalization of direct inversion in the iterative subspace for nonorthogonal molecular orbitals is formulated to increase the rate of convergence of the SCF MI equations. Single Roothaan step perturbative corrections are developed to improve the accuracy of the SCF MI energies. The resulting energies closely reproduce the conventional SCF energy. Extensive test calculations are performed on water clusters up to several hundred molecules. Compared to conventional SCF, speedups of the order of (N/O)2 have been achieved for the diagonalization step, where N is the size of the atomic orbital basis, and O is the number of occupied molecular orbitals.

Binary SCF: GAMESS improvements for energy evaluation based on SCF methods

This paper describes a new algorithm that improves computational efficiency for energy evaluation in GAMESS, named Binary SCF (BitSCF). This algorithm is based on the Conventional Self Consistent Field (CSCF) method and it presents two main innovations. The first one is the use of a bit map for the control of the 2e − integrals evaluated and previously stored in disk. The bit map reduces disk demand significantly when storing 2e − integrals. The second innovation is the use of i-loop balancing, a new proposal for static workload distribution through the nodes. The i-loop balancing optimizes parallelism in GAMESS, reducing an important sequential portion of its source code. The results obtained confirm a reduction of up to 49% in the disk demand for the 2e − integrals storage, considering different simulated molecules on a 13-node Beowulf cluster and 6-31G and STO-6G basis set. The total and CPU times stay below the times obtained with the GAMESS CSCF method for larger molecules. The total execution and CPU times were equivalent at the CSCF times for smaller molecules. The results show an improvement in the GAMESS scalability as well as a better cost vs. benefit relationship when executing the program.

Application of density functional theory to tethered polymer chains: Effect of intermolecular attractions

Density functional theory (DFT) was used to study polymer chains, tethered to a surface and in the presence of a solvent. For reasons of computational practicality, it is common practice to remove the explicit solvent molecules from the problem. Contact was made with two such models, which we call the “implicit-solvent” and “continuum-solvent” approximations. First, DFT was applied to tethered chains in an implicit solvent. Using the equation of state of bead–spring chains as input, we found excellent agreement of the theory with density profiles obtained in molecular dynamics simulations on the same model as a function of temperature. Next, DFT was applied to tethered chains in an incompressible, continuum solvent. Using the Flory–Huggins theory as input, our DFT equations reduced to conventional self-consistent field theory. From our DFT formalism, we demonstrated that the implicit-solvent problem, at a given temperature, is equivalent to the continuum-solvent problem, provided the χ parameter and total density are interpreted appropriately.

Linear and sublinear scaling formation of Hartree–Fock-type exchange matrices

We present a new method (LinK) to form the exact exchange matrix, as needed in Hartree–Fock and hybrid density functional theory calculations, with an effort capable of scaling only linearly with molecular size. It preserves the highly optimized structure of conventional direct self-consistent field (SCF) methods with only negligible prescreening overhead and does not impose predefined decay properties. Our LinK method leads to very early advantages as compared to conventional methods for systems with larger band gaps. Due to negligible screening overhead it is also competitive with conventional SCF schemes both for small molecules and systems with small band gaps. For the formation of an exchange-type matrix in coupled perturbed SCF theory our LinK method can exhibit sublinear scaling, or more precisely, independence of the computational effort from molecular size.

Semiempirical Fragment Self-Consistent Field Monte Carlo simulations for liquid chlorosilanes

We applied the recently developed Neglect of Diatomic Differential Overlap Fragment Self-Consistent Field Monte Carlo method to the simulation of the liquid state of chlorinated monosilanes. This semiempirical technique divides the periodic simulation cell into subsystem, where the random move of an atom takes place, and environment exerting only secondary effects on the former. Expanding the electronic wave function on the basis of atomic hybrid orbitals, that form strictly localised molecular orbitals corresponding to chemical bonds in classical molecules, the wave function of the environment is determined from coupled 2 × 2 secular equations. For the subsystem the conventional Self-Consistent Field equations, with a perturbation term in the Fockian and a much lower dimensionality than for the whole system, have to be solved. Thus the computational efforts drastically decrease as the dependence on the number of atoms in the environment reduces from quartic or cubic, as in conventional ab initio or semiempirical methods, to quadratic. Our simulation for chlorosilanes predicts that in the liquid state the preferred orientation of two neighbouring molecules is the one with the maximum number of SiH…ClSi hydrogen bonds. We found that gas phase SiCl bond lengths increase by 6 to 16 pm in the liquid state as a result of that association. HSiCl and ClSiCl bond angles change much less, by 2–4 degrees, as compared to the gas phase geometry. Since, to our knowledge, there are no experimental data published for these systems, our results may serve as preliminary information on liquid chlorosilanes.

Novel semiempirical method for quantum Monte Carlo simulation: Application to amorphous silicon

We propose a novel bulk phase Monte Carlo simulation technique, in which the energy is calculated by quantum mechanical methods. The semiempirical fragment self-consistent field technique applied divides the periodic simulation cell into two parts. The first is the subsystem where the important change (the random movement of an atom or molecule) takes place and the second is the environment exerting only secondary effects on the former. Expanding the electronic wave function on the basis of strictly localized molecular orbitals and/or atomic hybrid orbitals the wave function of the environment is obtained from simple coupled 2×2 secular equations. The conventional self-consistent field equations, with a perturbation term in the Fockian, have to be solved only for the subsystem. In this way the computational efforts are decreased drastically, as the dependence on the number of atoms in the environment reduces to quadratic instead of cubic or quartic as in conventional semiempirical or ab initio methods, respectively. We wrote a computer code and applied our method to amorphous silicon. Starting from a distorted tetrahedrally bonded random network model we performed Monte Carlo simulations using the fragment self-consistent field energy calculation. After equilibration we obtained distribution functions almost identical to the ones corresponding to the distortion free tetrahedrally bonded network. This finding confirms the adequacy of our method for this specific case.

Unrestricted Hartree–Fock method for infinite systems with antiferromagnetic array: Analysis of antiferromagnetic state of trans‐polyacetylene

AbstractAn improved unrestricted Hartree–Fock (UHF) crystal orbital method for infinite one‐dimensional systems having antiferromagnetic spin array is presented. This variational version of the UHF method is composed of two steps of self‐consistent‐field (SCF) calculations. The second SCF equation is to be solved using the canonical transformation between specific crystal orbitals around the Fermi level of the first conventional SCF solution. The energy gain and the forbidden band gap (correlation gap) associated with the hypothetical antiferromagnetic state of trans‐polyacetylene are examined as an example. © 1993 John Wiley & Sons, Inc.

A VARIATIONAL APPROACH TO THE STUDY OF SURFACE ELECTRONIC PROPERTIES

A variational approach to the study of surface electronic properties is proposed. The method is based on a direct variation of the density matrix and bypasses the intermediate steps of searching for eigenvectors in the conventional self-consistent field theory. The method developed is applicable to clean surfaces as well as to surfaces with adsorbed layers.

Limit cycle solutions of a scalar model nonlinear field theory and the Hartree quantum levels

A scalar model nonlinear field theory is constructed from an action principle so that the inherent "limit cycle solutions" of the theory correspond to the eigensolutions of the conventional Hartree equation for helium. The properties of the "limit cycle transition" solutions are discussed and offer an alternative method, for solving the nonlinear Hartree eigenvalue equation, which has the distinct advantage of not requiring the spherical averaging approximation inherent in the conventional "self-consistent field" calculations. The elements which control the speed of convergence of a limit cycle transition solution into a limit cycle solution are analyzed; a method of speeding up the process in order to make it competitive, in terms of computer time, with the conventional self-consistent field calculation is discussed.