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The two-photon absorption (TPA) properties of four TPEB [tetrakis(phenylethynyl)benzene] derivatives (TD, para, ortho, and meta) with different donor/acceptor substitution patterns have been investigated experimentally by the femtosecond open-aperture Z-scan method and theoretically by the time-dependent density-functional theory (TDDFT) method. The four compounds show relatively large TPA cross sections, and the all-donor substituted species (TD) displays the largest TPA cross-section σ(2) = 520 ± 30 GM. On the basis of the calculated electronic structure, TD shows no TPA band in the lower energy region of the spectrum because the transition density is concentrated on particular transitions due to the high symmetry of the molecular structure. The centrosymmetric donor-acceptor TPEB para shows excitations resulting from transitions centered on D-π-D and A-π-A moieties, as well as transition between the D-π-D and A-π-A moieties; this accounts for the broad nature of the TPA bands for this compound. Calculations for two noncentrosymmetric TPEBs (ortho and meta) reveal that the diminished TPA intensities of higher-energy bands result from destructive interference between the dipolar and three-state terms. The molecular orbitals (MOs) of the TPEBs are derivable with linear combinations of the MOs of the two crossing BPEB [bis(phenylethynyl)benzene] derivatives. Overall, the characteristics of the experimental spectra are well-described based on the theoretical analysis.

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We present a complete linear scaling method for hybrid Kohn-Sham density functional theory electronic structure calculations and demonstrate its performance. Particular attention is given to the linear scaling computation of the Kohn-Sham exchange-correlation matrix directly in sparse form within the generalized gradient approximation. The described method makes efficient use of sparse data structures at all times and scales linearly with respect to both computational time and memory usage. Benchmark calculations at the BHandHLYP/3-21G level of theory are presented for polypeptide helix molecules with up to 53 250 atoms. Threshold values for computational approximations were chosen on the basis of their impact on the occupied subspace so that the different parts of the calculations were carried out at balanced levels of accuracy. The largest calculation used 307 204 Gaussian basis functions on a single computer with 72 GB of memory. Benchmarks for three-dimensional water clusters are also included, as well as results using the 6-31G** basis set.

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An efficient, linear-scaling implementation of Kohn-Sham density-functional theory for the calculation of molecular forces for systems containing hundreds of atoms is presented. The density-fitted Coulomb force contribution is calculated in linear time by combining atomic integral screening with the continuous fast multipole method. For higher efficiency and greater simplicity, the near-field Coulomb force contribution is calculated by expanding the solid-harmonic Gaussian basis functions in Hermite rather than Cartesian Gaussians. The efficiency and linear complexity of the molecular-force evaluation is demonstrated by sample calculations and applied to the geometry optimization of a few selected large systems.

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We study the formation of plasmon modes of small gold clusters by modeling the excitation spectra. The shape change of the longitudinal mode as a function of cluster size is studied using time-dependent Kohn-Sham theory and Gaussian basis sets. The presence of d electrons in gold atoms affect the plasmon formation process, resulting in a high excitation energy for transverse mode and a complicated spectra profile in general. The transverse mode can still be identified with the help of a frozen-orbital approximation.

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We present a method to compute Coulomb and exchange matrices with predetermined accuracy as measured by a matrix norm. The computation of these matrices is fundamental in Hartree-Fock and Kohn-Sham electronic structure calculations. We show numerically that, when modern algorithms for Coulomb and exchange matrix evaluation are applied, the Euclidean norm of the error matrix ε is related to the threshold value τ as ε = cτ(α). The presented extrapolation method automatically selects the integral thresholds so that the Euclidean norm of the error matrix is at the requested accuracy. This approach is demonstrated for a variety of systems, including protein-like systems, water clusters, and graphene sheets. The proposed method represents an important step toward complete error control throughout the self-consistent field calculation as described in [J. Math. Phys. 2008, 49, 032103].

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A Lagrangian approach has been used to derive gauge-origin independent expressions for two properties that rationalize magneto-optical activity, namely the Verdet constant V(ω) of the Faraday effect and the ℬ term of magnetic circular dichroism. The approach is expressed in terms of an atomic-orbital density-matrix based formulation of response theory and use London atomic orbitals to parametrize the magnetic field dependence. It yields a computational procedure which is both gauge-origin independent and suitable for linear-scaling at the level of time-dependent Hartree-Fock and density functional theory. The formulation includes a modified preconditioned conjugated gradient algorithm, which projects out the excited state component from the solution to the linear response equation. This is required when solving one of the response equations for the determination of the ℬ term and divergence is encountered if this component is not projected out. Illustrative results are reported for the Verdet constant of H2, HF, CO, N2O, and CH3CH2CH3 and for the ℬ term of pyrimidine, phosphabenzene, and pyridine. The results are benchmarked against gauge-origin independent CCSD values.

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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.

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Density fitting is an important method for speeding up quantum-chemical calculations. Linear-scaling developments in Hartree-Fock and density-functional theories have highlighted the need for linear-scaling density-fitting schemes. In this paper, we present a robust variational density-fitting scheme that allows for solving the fitting equations in local metrics instead of the traditional Coulomb metric, as required for linear scaling. Results of fitting four-center two-electron integrals in the overlap and the attenuated Gaussian damped Coulomb metric are presented, and we conclude that density fitting can be performed in local metrics at little loss of chemical accuracy. We further propose to use this theory in linear-scaling density-fitting developments.

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We present an implementation of a set of algorithms for performing Hartree-Fock calculations with resource requirements in terms of both time and memory directly proportional to the system size. In particular, a way of directly computing the Hartree-Fock exchange matrix in sparse form is described which gives only small addressing overhead. Linear scaling in both time and memory is demonstrated in benchmark calculations for system sizes up to 11 650 atoms and 67 204 Gaussian basis functions on a single computer with 32 Gbytes of memory. The sparsity of overlap, Fock, and density matrices as well as band gaps are also shown for a wide range of system sizes, for both linear and three-dimensional systems.

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Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification.

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We test the performance of four-component relativistic density functional theory by calculating the static and frequency-dependent electric dipole-dipole polarizabilities of all (ground-state) closed-shell atoms up to Ra. We consider 12 nonrelativistic functionals, including three asymptotically shape-corrected functionals, by using two smooth interpolation schemes introduced by the Baerends group: the gradient-regulated asymptotic connection (GRAC) procedure and the statistical averaging of (model) orbital potentials (SAOP). Basis sets of doubly augmented triple-zeta quality are used. The results are compared to experimental data or to accurate ab initio results. The reference static electric dipole polarizability of palladium has been obtained by finite-field calculations using the coupled-cluster singles, doubles, and perturbative triples method within this work. The best overall performance is obtained using hybrid functionals and their GRAC shape-corrected versions. The performance of SAOP is among the best for nonhybrid functionals for Group 18 atoms but its precision degrades when considering the full set of atoms. In general, we find that conclusions based on results obtained for the rare-gas atoms are not necessarily representative of the complete set of atoms. GRAC cannot be used with effective core potentials since the asymptotic correction is switched on in the core region.

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The bond length alternation (BLA), the highest-occupied-lowest-unoccupied (HO-LU) orbital energy gap, and the corresponding excitation energy are determined for trans-polyacetylene (PA) and polyyne (PY) using density functional theory. Results from the Coulomb-attenuated CAM-B3LYP functional are compared with those from the conventional BHHLYP and B3LYP hybrid functionals. BLA values and HO-LU gaps are determined using both finite oligomer and infinite chain calculations, subject to periodic boundary conditions. TDDFT excitation energies are determined for the oligomers. The oligomer excitation energies and HO-LU gaps are then used, in conjunction with the infinite chain HO-LU gap, to estimate the infinite chain excitation energy. Overall, BHHLYP and CAM-B3LYP give BLA values and excitation energies that are larger and more accurate than those obtained using B3LYP. The results highlight the degree to which excitation energies can be approximated using the HO-LU gaps-at the infinite limit, this approximation works well for B3LYP, but not for the other functionals, where the HO-LU gap is significantly larger. The study provides further evidence for the high-quality theoretical predictions that can be obtained from the CAM-B3LYP functional.

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Electronic circular dichroism (ECD) parameters of the disulphide chromophore have been calculated for dihydrogen disulphide, dimethyl disulphide, and cystine using density-functional theory, coupled-cluster theory, and multiconfigurational self-consistent field theory. The objective is twofold: first, to examine the performance of the Coulomb-attenuated CAM-B3LYP functional for the calculation of ECD spectra; second, to investigate the dependence of the ECD parameters on the conformation around the disulphide bridge. The CAM-B3LYP functional improves considerably on the B3LYP functional, giving results comparable to CCSD theory and to MCSCF theory in an extended active space. The conformational dependence of the ECD parameters does not change much upon substitution, which is promising for the application of ECD in structural investigations of proteins containing disulphide bridges.

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Solvent-solute interaction energies for three well-known drug molecules in water solution are computed at the Hartree-Fock and B3LYP density functional theory levels using a linear scaling technique, which allows one to explicitly include in the model water molecules up to 14 A away from the solute molecule. The dependence of calculated interaction energies on the amount of included solvent has been examined. It is found that it is necessary to account for water molecules within an 8 A radius around the drug molecule to reach the saturated solvent interaction level. Effects of electron correlation and basis set on solvent-solute interaction energies are discussed.

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Band gap studies of zigzag-edge graphene ribbons are presented. While earlier calculations at LDA level show that zigzag-edge graphene ribbons become half-metallic when cross-ribbon electric fields are applied, our calculations with hybrid density functional demonstrate that finite graphene ribbons behave as half-semiconductors. The spin-dependent band gap can be changed in a wide range, making possible many applications in spintronics.

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A major unresolved problem of density functional theory is the yet unknown exchange-correlation functional, which leads to a proliferation of its less or more successful approximations. A practical implementation of these numerous functionals can present a substantial challenge particularly if the higher order functional derivatives are required. We present a systematic method of functional implementation. The method allows a clean handling of a large number of functionals in a mutually independent way. We developed an extensive set of automatic test routines to facilitate functional and derivative testing with respect to the implementation correctness and numerical stability. An integral part of the presented solution is a program for automatic code generation from analytical formulas that uses only freely available tools. Code for evaluation of functionals and their first, second, third, and fourth derivatives can be generated, which accelerates the development, implementation, and testing of new functionals.

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A hierarchic sparse matrix data structure for Hartree-Fock/Kohn-Sham calculations is presented. The data structure makes the implementation of matrix manipulations needed for large systems faster, easier, and more maintainable without loss of performance. Algorithms for symmetric matrix square and inverse Cholesky decomposition within the hierarchic framework are also described. The presented data structure is general; in addition to its use in Hartree-Fock/Kohn-Sham calculations, it may also be used in other research areas where matrices with similar properties are encountered. The applicability of the data structure to ab initio calculations is shown with help of benchmarks on water droplets and graphene nanoribbons.

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A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field theories for the calculation of frequency-dependent molecular response properties and excitation energies is presented, based on a nonredundant exponential parametrization of the one-electron density matrix in the atomic-orbital basis, avoiding the use of canonical orbitals. The response equations are solved iteratively, by an atomic-orbital subspace method equivalent to that of molecular-orbital theory. Important features of the subspace method are the use of paired trial vectors (to preserve the algebraic structure of the response equations), a nondiagonal preconditioner (for rapid convergence), and the generation of good initial guesses (for robust solution). As a result, the performance of the iterative method is the same as in canonical molecular-orbital theory, with five to ten iterations needed for convergence. As in traditional direct Hartree-Fock and Kohn-Sham theories, the calculations are dominated by the construction of the effective Fock/Kohn-Sham matrix, once in each iteration. Linear complexity is achieved by using sparse-matrix algebra, as illustrated in calculations of excitation energies and frequency-dependent polarizabilities of polyalanine peptides containing up to 1400 atoms.

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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.

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A number of computational techniques are described that reduce the effort related to the continuous fast multipole method, used for the evaluation of Coulomb matrix elements as needed in Hartree-Fock and density functional theories. A new extent definition for Gaussian charge distributions is proposed, as well as a new way of dividing distributions into branches. Also, a new approach for estimating the error caused by truncation of multipole expansions is presented. It is found that the use of dynamically truncated multipole expansions gives a speedup of a factor of 10 in the work required for multipole interactions, compared to the case when all interactions are computed using a fixed multipole expansion order. Results of benchmark calculations on three-dimensional systems are reported, demonstrating the usefulness of our present implementation of the fast multipole method.