Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification

Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.     

Systematic sparse matrix error control for linear scaling electronic structure calculations

Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.     

液态RbCl的态密度

给出了在分子动力学模拟基础上Fumi-Tosi势离子液体的正则模式分析方法,用Fumi-Tosi势(包括长程势)代替Lennard-Jones势,并且用等效Coulomb势处理长程Coulomb作用.讨论了Hessian矩阵元的计算方法和Hessian矩阵特征值的计算方法.计算实践表明，取用余误差函数形式的等效库仑势,可以合理地得到Hessian矩阵和态密度.液态RbCl中构型平均态密度的数值结果表明，液态RbCl的态密度表现出与Lennard-Jones液体的态密度相仿的特点.     

Transforms for idempotency purification of density matrices in linear-scaling electronic-structure calculations

A family of purification transforms (of the nth-order convergence, n>2) is proposed, which is a generalization of the McWeeny transform (of the second-order convergence). These transforms can be applied in calculational schemes where the number of basis functions exceeds the number of occupied orbitals in the density matrix. Another family of purification transforms, proposed recently by Kryachko [Chem. Phys. Lett. 318 (2000) 210], is shown to be applicable in such schemes only where these numbers are equal. Various properties of these two families are discussed.     

Bringing about matrix sparsity in linear-scaling electronic structure calculations

The performance of linear-scaling electronic structure calculations depends critically on matrix sparsity. This article gives an overview of different strategies for removal of small matrix elements, with emphasis on schemes that allow for rigorous control of errors. In particular, a novel scheme is proposed that has significantly smaller computational overhead compared with the Euclidean norm-based truncation scheme of Rubensson et al. (J Comput Chem 2009, 30, 974) while still achieving the desired asymptotic behavior required for linear scaling. Small matrix elements are removed while ensuring that the Euclidean norm of the error matrix stays below a desired value, so that the resulting error in the occupied subspace can be controlled. The efficiency of the new scheme is investigated in benchmark calculations for water clusters including up to 6523 water molecules. Furthermore, the foundation of matrix sparsity is investigated. This includes a study of the decay of matrix element magnitude with distance between basis function centers for different molecular systems and different methods. The studied methods include Hartree–Fock and density functional theory using both pure and hybrid functionals. The relation between band gap and decay properties of the density matrix is also discussed.     

A perspective on the localizability of Hartree–Fock orbitals

A common perception about molecular systems with a nonlocal electronic structure (as manifested by a nonlocal Hartree–Fock (HF) density matrix), such as conjugated π-systems, is that they can only be described in terms of nonlocal molecular orbitals. This view is mostly founded on chemical intuition, and further, this view is strengthened by traditional approaches for obtaining local occupied and virtual orbital spaces, such as the occupied Pipek–Mezey orbitals, and projected atomic orbitals. In this article, we discuss the limitations for localizability of HF orbitals in terms of restrictions posed by the delocalized character of the underlying density matrix for the molecular system and by the orthogonality constraint on the molecular orbitals. We show that the locality of the orbitals, in terms of nonvanishing charge distributions of orbitals centered far apart, is much more strongly affected by the orthogonality constraint than by the physical requirement that the occupied orbitals must represent the electron density. Thus, the freedom of carrying out unitary transformations among the orbitals provides the flexibility to obtain highly local occupied and virtual molecular orbitals, even for molecular systems with a nonlocal density matrix, provided that a proper localization function is used. As an additional consideration, we clear up the common misconception that projected atomic orbitals in general are more local than localized orthogonal virtual orbitals.     

The Fukui matrix: a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues.     

Analyzing the electric response of molecular conductors using “electron deformation” orbitals and occupied‐virtual electron transfer

The concept of “electron deformation orbitals” (EDOs) is used to investigate the electric response of conducting metals and oligophenyl chains. These orbitals and their eigenvalues are obtained by diagonalization of the deformation density matrix (difference between the density matrices of the perturbed and unperturbed systems) and can be constructed as linear combinations of the unperturbed molecular orbitals within “frozen geometry” conditions. This form of the EDOs allows calculating the part of the electron deformation density associated to an effective electron transfer from occupied to virtual orbitals (valence to conduction band electron transfer in the band model of conductivity). It is found that the “electron deformation” orbitals pair off, displaying the same eigenvalue but opposite sign. Each pair represents an amount of accumulation/depletion of electron charge at different molecular regions. In the oligophenyl systems investigated only one pair contributes effectively to the charge flow between molecular ends, resulting from the promotion of electrons from occupied orbitals to close in energy virtual orbitals of appropriate symmetry and overlapping. Analysis of this pair along explains the differences in conductance of olygophenyl chains based on phenyl units. © 2014 Wiley Periodicals, Inc.     

Calculating the density of states for large‐scale molecular systems

We present a new method for calculating the density of states of large‐scale molecular systems. The new algorithm does not require calculation of all eigenvalues of the force constant matrix. The histogram for the eigenvalue distribution is constructed by performing a sequence of sparse LDL^T matrix factorizations. Moreover, the entire calculation can be fully parallelized on a multiprocessor computer system.     

Estimating bounds on the highest and lowest eigenvalues of any matrix

A method for estimating the bounds for the highest and lowest eigenvalues of a finite-dimensional matrix is presented. The method is tested for the Hamiltonian matrix for several particle-in-a-box-like systems. We also provide results for the well-studied benchmark case of the ro-vibrational states of H₃ ⁺, and consider bounds obtained for a completely random non symmetric matrix. Finally, we discuss how the error in a Chebychev expansion solution of quantum scattering depends on the error made in estimating the highest eigenvalue of the Hamiltonian matrix. Received: 7 May 1999 / Accepted: 30 July 1999 / Published online: 15 December 1999     

Open-shell reduced density matrix functional theory

Open-shell reduced density matrix functional theory is established by investigating the domain of the exact functional. For spin states that are the ground state, a particularly simple set is found to be the domain. It cannot be generalized to other spin states. A number of conditions satisfied by the exact density matrix functional is formulated and tested for approximate functionals. The exact functional does not suffer from fractional spin error, which is the source of the static correlation error in dissociated molecules. We prove that a simple approximation (called the Buijse-Baerends functional, Mu?ller or square root functional) has a non-positive fractional spin error. In the case of the H atom the error is zero. Numerical results for a few atoms are given for approximate density and density matrix functionals as well as a recently developed range-separated combination of both.     

The AGP model for fermion systems

A mathematically rigorous treatment of the antisymmetrized geminal power (AGP) model is given. The model is advocated as an extremely flexible tool for studying a wide variety of fermion systems, characterized by a single Slater determinant, on the one hand, or Yang's most highly correlated wave function, on the other. When the first-order reduced density matrix of the actual system has eigenvalues which are evenly degenerate, the model can treat one-particle properties exactly. A formula is obtained for the precise energy of an AGP state in the Born-Oppenheimer approximation. An estimate is given of the error which occurs when the second-order density matrix of an AGP state is approximated by the widely used expression involving the so-called anomalous Green's function. © 1997 John Wiley & Sons, Inc.     

Response calculations based on an independent particle system with the exact one-particle density matrix: excitation energies

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H(2) and HeH(+) using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ("virtual") natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional.     

Block deformation analysis: Density matrix blocks as intramolecular deformation density

Block deformation analysis as deformation density of atomic orbitals is introduced to analyze intramolecular interactions. In this respect, density matrix blocks in terms of natural atomic orbitals are employed to find interacting and noninteracting multicenter subsystem and extract the corresponding deformation density. Eigenanalysis of this deformation density is performed to result eigenvalues and eigenorbitals as displaced charge due to the intramolecular interaction and orbital space responsible for charge reorganization, respectively, that possesses advantages of other methods, simultaneously. It is applied to several small molecules, different types of carbon allotropes including zero-, one-, and two-dimensional nanostructures, and challenging systems such as ortho-hydrogen atoms in planar biphenyl. Results highly correlate with delocalization and Wiberg bond indices and show that eigenvalues of block deformation analysis deserved to be considered as bonding index.     

On finite basis set implementation of the exchange-only optimized effective potential method

In this paper, we analyze a structure of the basis set optimized effective potential (OEP) equations from the Fredholm alternative point of view and present one of possible numerical schemes to solve the OEP equation in a stable manner. The solution is constructed as a sum of a unique solution on the subspace of eigenfunctions of the response matrix with non-zero eigenvalues and a non-unique solution on a counterpart subspace with singular eigenvalues. Non-uniqueness of a solution is exploited to obtain a local effective potential that satisfies the condition for the highest occupied molecular orbital (HOMO) without restricting the variational freedom of the optimization procedure. Unlike the existing methods we implement the HOMO condition using the functions of the null-subspace. Numerical results for the total and orbital energies based on the proposed scheme are close to the corresponding literature data.     

Comparison of DFT methods for molecular orbital eigenvalue calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.     

Truncation of small matrix elements based on the Euclidean norm for blocked data structures

Methods for the removal of small symmetric matrix elements based on the Euclidean norm of the error matrix are presented in this article. In large scale Hartree-Fock and Kohn-Sham calculations it is important to be able to enforce matrix sparsity while keeping errors under control. Truncation based on some unitary-invariant norm allows for control of errors in the occupied subspace as described in (Rubensson et al. J Math Phys 49, 032103). The Euclidean norm is unitary-invariant and does not grow intrinsically with system size and is thus suitable for error control in large scale calculations. The presented truncation schemes repetitively use the Lanczos method to compute the Euclidean norms of the error matrix candidates. Ritz value convergence patterns are utilized to reduce the total number of Lanczos iterations.     

Fast noniterative orbital localization for large molecules

We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ("Cholesky molecular orbitals") demonstrates their localized character inherited from the sparsity of the density matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.     

Decomposition of deformation density into orbital components

In this research, deformation density matrix has been introduced as matrix representation of the density difference between the complex and fragments. The deformation density matrix is then diagonalized to obtain the magnitude of displaced charges as eigenvalues. Correspondingly, the eigenvectors reveal the spaces responsible for reorganization of the electrons because of the complex formation. The formalism has been applied on some CO₂ planar clusters, and the results showed that how the deformation density can be successfully separated into in‐plane and out‐of‐plane contributions. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008