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.     

Basis set generation for the SCF calculation

A method for basis set generation for SCF calculations is proposed. Using SCF orbitals and orbital energies obtained in the extended basis set the Fock operator can be expressed as its spectral resolution. The sum of differences between occupied orbital energies and corresponding eigenvalues obtained by the diagonalization of this operator in the new smaller basis set is a criterion of the quality of this new set. The present method consists of the minimization of this sum by changing the parameters that determine the new basis functions. An example of the optimization of the different Gaussian basis sets for the LiH molecule is described.     

Use of exchange maximization to generate starting vectors for self-consistent field calculations on metal cluster/adsorbate systems

Localized molecular orbitals (LMOs) derived from exchange maximization with respect to all atom-centered basis functions in the basis set are shown to generate a good starting electronic field for self-consistent field calculations on extended systems such as metal clusters, for which well-defined chemical bonds are not present. Examples studied are a cluster of 20 Ni atoms and the Pt(97)CO, Ag(43)/H(3)CNON, Ag(91)/H(2)CO, and vinylidene/Ni metal cluster plus adsorbate systems. It is also shown that improved starting vectors can be obtained by remixing a subset of the LMOs with the largest exchange eigenvalues through diagonalization of the Fock matrix computed with a null electronic field. Employing only a subset of the exchange-maximized LMOs in the first iterations, and then gradually expanding the space in which the diagonalizations are carried out in succeeding cycles, is shown to be an effective means of guiding the SCF procedure to the converged full-basis solution.     

Convergence optimization of restricted open-shell self-consistent field calculations

Different self-consistent field (SCF) iteration schemes for open-shell systems are discussed. After a brief summary of the well-known level shifting and damping procedure, we describe the quadratically convergent SCF (QCSCF) approach based on the gradient and the Hessian matrix in a space of orbital rotation parameters. An analytical expression for the latter is derived for the general many-shell case. Starting from the expression for the energy change obtained by the QCSCF method, we then present a simplified direct procedure avoiding matrix diagonalization but also the difficulties of the QCSCF method in handling the Hessian matrix. Numerical calculations on some open-shell systems involving transition-metal complexes show that this method leads to rapid and reliable convergence of the iteration process in cases where the usual SCF procedure of iterative diagonalization tends to diverge. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 617–637, 1997     

The trust-region self-consistent field method: towards a black-box optimization in Hartree-Fock and Kohn-Sham theories

The trust-region self-consistent field (TRSCF) method is presented for optimizing the total energy E(SCF) of Hartree-Fock theory and Kohn-Sham density-functional theory. In the TRSCF method, both the Fock/Kohn-Sham matrix diagonalization step to obtain a new density matrix and the step to determine the optimal density matrix in the subspace of the density matrices of the preceding diagonalization steps have been improved. The improvements follow from the recognition that local models to E(SCF) may be introduced by carrying out a Taylor expansion of the energy about the current density matrix. At the point of expansion, the local models have the same gradient as E(SCF) but only an approximate Hessian. The local models are therefore valid only in a restricted region-the trust region-and steps can only be taken with confidence within this region. By restricting the steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the total energy is ensured in each iteration of the TRSCF method. Examples are given where the TRSCF method converges monotonically and smoothly, but where the standard DIIS method diverges.     

Variational principle for transition matrix

A variational principle for the transition matrix is considered. Conception of the MC SCF (in particular, CASSCF ) approach for the transition matrix is proposed as a solution of the variational problem on the optimal evolution of a packet of two stationary states with some additional conditions. An ordinary MC SCF method for a single state is a special case of the proposed approach. Some aspects of the solving of the equations for the optimal transition matrix are treated. The method can be used in atomic and molecular calculations of transition energies, oscillator strengths, and other properties. © 1993 John Wiley & Sons, Inc.     

Projection between basis sets: A means of defining the electronic structure of functional subunits

The general problem of the transfer of the representation of a set of orthonormal functions from one basis to a different, nonequivalent one is considered and specialized to a matrix formulation convenient for use in molecular electronic structure calculations. A procedure is suggested for treating problems where the transfer of representation breaks into a subset of most interest and one of less interest as for example the occupied and virtual orbitals of a Hartree-Fock SCF calculation. These techniques are then applied to obtain a representation of a methyl group from an SCF wave function for methane.     

A technique for orbital exponent optimization in ab initio HF?SCF?LCAO?MO calculations

A technique for Slater orbital exponent optimization in an HF? SCF? LCAO? MO calculation is proposed in which orbital exponent variation is incorporated into the SCF scheme. This is accomplished by rewriting Slater's rules so that the shielding terms depend on the molecular charge distribution through the elements of the population matrix. The SCF scheme then includes a calculation of a new set of orbital exponents from the coefficients of self-consistent molecular orbitals obtained from the previous set of exponents. The process is iterated until the energy attains its lowest value. The technique is illustrated by minimal basis calculations on LiH, BH, and HF. Near optimization is obtained with considerably less effort than is necessary for other reported techniques. Aside from interesting properties, the technique can be important for extended basis calculations where exponent optimization is a difficult task.     

Methods for parallel computation of SCF NMR chemical shifts by GIAO method: Efficient integral calculation,multi-Fock algorithm,and pseudodiagonalization

We implemented our gauge-including atomic orbital (GIAO) NMR chemical shielding program on a workstation cluster, using the parallel virtual machine (PVM) message-passing system. On a modest number of nodes, we achieved close to linear speedup. This program is characterized by several novel features. It uses the new integral program of Wolinski that calculates integrals in vectorized batches, increases efficiency, and simplifies parallelization. The self-consistent field (SCF) step includes a multi-Fock algorithm, i.e., the simultaneous calculation of several Fock matrices with the same integral set, increasing the efficiency of the direct SCF procedure. The SCF diagonalization step, which is difficult to parallelize, has been replaced by pseudodiagonalization. The latter, widely used in semiempirical programs, becomes important in ab initio type calculations above a certain size, because the ultimate scaling of the diagonalization step is steeper than that of integral computation. Examples of the calculation of the NMR shieldings in large systems at the SCF level are shown. Parallelization of the density functional code is underway. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 816–825, 1997     

One electron orbitals intrinsic to the reduced hamiltonian

One electron orbitals are determined from the reduced hamiltonian by a simple one-step diagonalization. These reduced hamiltonian orbitals (RHO's) are uniquely determined and virtual orbitals obtained in this procedure are on a par with filled orbitals. These RHO's appear well suited for CI calculations. Minimum basis set calculations are presented for H₂O and compared with similar SCF studies.     

KSSOLV-GPU: an Efficient GPU-Enabled MATLAB Toolbox for Solving the Kohn-Sham Equations within Density Functional Theory in Plane-Wave Basis Set?

KSSOLV (Kohn-Sham Solver) is a MATLAB (Matrix Laboratory) toolbox for solving the Kohn-Sham density functional theory (KS-DFT) with the plane-wave basis set. In the KS-DFT calculations, the most expensive part is commonly the diagonalization of Kohn-Sham Hamiltonian in the self-consistent field (SCF) scheme. To enable a personal computer to perform medium-sized KS-DFT calculations that contain hundreds of atoms, we present a hybrid CPU-GPU implementation to accelerate the iterative diagonalization algorithms implemented in KSSOLV by using the MATLAB built-in Parallel Computing Toolbox. We compare the performance of KSSOLV-GPU on three types of GPU, including RTX3090, V100, and A100, with conventional CPU implementation of KSSOLV respectively and numerical results demonstrate that hybrid CPU-GPU implementation can achieve a speedup of about 10 times compared with sequential CPU calculations for bulk silicon systems containing up to 128 atoms.     

Linear-scaling implementation of molecular electronic self-consistent field theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.     

Some analytical properties of the dilated SCF Equations

The self-consistent field (SCF ) equations for many-electron systems, suitable within the complex-coordinate method, are derived. The formulation is based on a general bivariational theorem for non-Hermitian operators, with an emphasis on the analytic structure invoked by the complex dilation of the total Hamiltonian. The dilation structure of the resulting SCF equations is stressed and the concomitant analytical properties are discussed. The solutions are classified with respect to these properties, and interpreted in terms of a general form of the symmetry dilemma. The role of the dilated SCF equations for resonance calculations is discussed.     

SCF theory of intermolecular interactions without basis set superposition error

Special SCF LCAO MO type equations are derived, permitting “supermolecule” calculations for intermolecular interactions, excluding basis set superposition error (BSSE) from the beginning on the basis of the “chemical Hamiltonian approach”. (No additional “monomer” calculations are necessary to correct for BSSE.) The formalism excluding the BSSE results in a non-Hermitean Fock matrix; an algorithm is proposed to obtain the required molecular orbitals, in which no integral transformation is needed.     

On the use of symmetry in the ab initio quantum mechanical simulation of nanotubes and related materials

Nanotubes can be characterized by a very high point symmetry, comparable or even larger than the one of the most symmetric crystalline systems (cubic, 48 point symmetry operators). For example, N = 2n rototranslation symmetry operators connect the atoms of the (n,0) nanotubes. This symmetry is fully exploited in the CRYSTAL code. As a result, ab initio quantum mechanical large basis set calculations of carbon nanotubes containing more than 150 atoms in the unit cell become very cheap, because the irreducible part of the unit cell reduces to two atoms only. The nanotube symmetry is exploited at three levels in the present implementation. First, for the automatic generation of the nanotube structure (and then of the input file for the SCF calculation) starting from a two‐dimensional structure (in the specific case, graphene). Second, the nanotube symmetry is used for the calculation of the mono‐ and bi‐electronic integrals that enter into the Fock (Kohn‐Sham) matrix definition. Only the irreducible wedge of the Fock matrix is computed, with a saving factor close to N. Finally, the symmetry is exploited for the diagonalization, where each irreducible representation is separately treated. When M atomic orbitals per carbon atom are used, the diagonalization computing time is close to Nt, where t is the time required for the diagonalization of each 2M × 2M matrix. The efficiency and accuracy of the computational scheme is documented. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Development of parallel density functional program using distributed matrix to calculate all-electron canonical wavefunction of large molecules

We developed a new parallel density-functional canonical molecular-orbital program for large molecules based on the resolution of the identity method. In this study, all huge matrices were decomposed and saved to the distributed local memory. The routines of the analytical molecular integrals and numerical integrals of the exchange-correlation terms were parallelized using the single program multiple data method. A conventional linear algebra matrix library, ScaLAPACK, was used for matrix operations, such as diagonalization, multiplication, and inversion. Anderson's mixing method was adopted to accelerate the self-consistent field (SCF) convergence. Using this program, we calculated the canonical wavefunctions of a 306-residue protein, insulin hexamer (26,790 orbitals), and a 133-residue protein, interleukin (11,909 orbitals) by the direct-SCF method. In regard to insulin hexamer, the total parallelization efficiency of the first SCF iteration was estimated to be 82% using 64 Itanium 2 processors connected at 3.2 GB/s (SGI Altix3700), and the calculation successfully converged at the 17-th SCF iteration. By adopting the update method, the computational time of the first and the final SCF loops was 229 min and 156 min, respectively. The whole computational time including the calculation before the SCF loop was 2 days and 17 h. This study put the calculations of the canonical wavefunction of 30,000 orbitals to practical use.     

Approximate molecular orbital theory: the ese MO formalism. I. General theory and notation

A new formalism for simplified molecular orbital calculations is elucidated. The focus of the formalism is the production of a good approximation to the LCAO SCF F matrix of Roothaan's equations, and preoccupation with approximations to individual molecular integrals is avoided. The great majority of multicentre two-electron integrals of the exact formalism are found to be largely inconsequential to the attainment of a good approximation to the F matrix.     

Exploitation of symmetry in periodic Self-Consistent-Field ab initio calculations: application to large three-dimensional compounds

Symmetry can dramatically reduce the computational cost (running time and memory allocation) of Self-Consistent-Field ab initio calculations for crystalline systems. Crucial for running time is use of symmetry in the evaluation of one- and two-electron integrals, diagonalization of the Fock matrix at selected points in reciprocal space, reconstruction of the density matrix. As regards memory allocation, full square matrices (overlap, Fock and density) in the Atomic Orbital (AO) basis are avoided and a direct transformation from the packed AO to the SACO (Symmetry Adapted Crystalline Orbital) basis is performed, so that the largest matrix to be handled has the size of the largest sub-block in the latter basis. We here illustrate the effectiveness of this scheme, following recent advancements in the CRYSTAL code, concerning memory allocation and direct basis set transformation. Quantitative examples are given for large unit cell systems, such as zeolites (all-silica faujasite and silicalite MFI) and garnets (pyrope). It is shown that the full SCF of 3D systems containing up to 576 atoms and 11136 Atomic Orbitals in the cell can be run with a hybrid functional on a single core PC with 500 MB RAM in about 8 h.     

Perturbative virtual SCF CI treatment for energy levels of coupled oscillator systems

A perturbative SCF CI treatment to obtain energy levels of coupled oscillator systems is proposed. The method uses the virtual SCF basis set, and the SCF equations are solved by means of a perturbative treatment that provides the diagonal matrix elements involved in the CI calculation. The off-diagonal matrix elements are calculated using a commutation relationship derived from exact quantum theorems. Numerical results for several systems are obtained and compared with those from others SCF, SCF CI , and variational treatments.     

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.