Molecular fragment approach to the study of pyrrole oligomers and polypyrrole

A molecular fragment approach is used to compute ionization potentials, transition energies and electron affinities of pyrrole oligomers. The calculations of these quantities include correlation energy contributions evaluated by integrating a functional of the two-particle Hartree-Fock density matrix. Pyrrole oligomers with chains of up to 16 rings are explicitly treated and the calculated quantities extrapolated to the limit of an infinitely long chain, to predict polymer properties. The theoretical results compare favorably with data on gas-phase ionization potentials deduced from experimental oxidation potentials, and with optical absorption peaks recorded in solution or on solid films. The large discrepancy between electron affinities obtained from the eigenvalues of an independent-particle frozen-orbital calculation and those obtained from separate, correlated calculations on the neutral system and the negative ion in shown. Received: 24 March 1998 / Accepted: 2 September 1998 / Published online: 7 December 1998     

Convergence of the density functional one-centre expansion for the molecular continuum: N2 and (CH3)3N

A large-scale one-centre expansion with a radial B-spline basis set is implemented for bound and continuum states. A Kohn-Sham hamiltonian is employed with Hartree and exchange-correlation potentials calculated from the SCF electron density taken from a previous LCAO calculation. An inverse iteration method is used to obtain the continuum wavefunction, from which the cross section and asymmetry parameter are calculated. The convergence with respect to angular momentum and cut-off radius is analysed for N₂. The relevance of multipolar contributions even at large distances is shown and suggestions for further improvements are given. In order to show that the present method is suitable to treat systems of moderate size, the (CH₃)₃N molecule has also been calculated and the results are compared with experiment. Received: 19 May 1998 / Accepted: 20 August 1998 / Published online: 7 December 1998     

Multireference perturbation CI III. Fast evaluation of the one-particle density matrix

Within the frame of multireference perturbation configuration interaction we have developed a fast algorithm, based on diagrammatic techniques, for the calculation of the first-order correction to the one-particle density matrix. As an example of an application we have chosen the evaluation of the dipole moment of the CO molecule, where utilization of the first-order density is shown to corroborate the variational calculation. Received: 4 August 1998 / Accepted: 21 September 1998 / Published online: 16 November 1998     

Global geometry optimization of small silicon clusters at the level of density functional theory

By an application to small silicon clusters Si _N (with N = 4,5,7,10) it is shown that truly global geometry optimization on an ab initio or density functional theory level can be achieved, at a computational cost of approximately 1–5 traditional local optimization runs (depending on cluster size). This extends global optimization from the limited area of empirical potentials into the realm of ab initio quantum chemistry. Received: 24 February 1998 / Accepted: 6 March 1998 / Published online: 17 June 1998     

Ab initio correlation corrections to the Hartree-Fock quasi band-structure of periodic systems employing Wannier-type orbitals

A size-consistent ab initio formalism to calculate correlation corrections to ionization potentials as well as electron affinities of periodic systems is presented. Our approach is based on a Hartree-Fock scheme which directly yields local orbitals without any a posteriori localization step. The use of local orbitals implies non-zero off-diagonal matrix elements of the Fock operator, which are treated as an additional perturbation and give rise to localization diagrams. Based on the obtained local orbitals, an effective Bloch Hamiltonian is constructed to second order of perturbation theory with all third-order localization diagrams included. In addition, the summation of certain classes of diagrams up to infinite order in the off-diagonal Fock elements as well as the Epstein-Nesbet partitioning of the full Hamiltonian are discussed. The problem of intruder states, frequently encountered in many-body perturbation theory, is dealt with by employing the theory of intermediate Hamiltonians. As model systems we have chosen cyclic periodic structures up to an oligoethylene ring in double-zeta basis; however, the theory presented here straightforwardly carries over to infinite periodic systems. Received: 30 April 1998 / Accepted: 27 July 1998 /  Published online: 7 October 1998     

Real-time time-dependent density functional theory using density fitting and the continuous fast multipole method

An implementation of real-time time-dependent density functional theory (RT-TDDFT) within the TURBOMOLE program package is reported using Gaussian-type orbitals as basis functions, second and fourth order Magnus propagator, and the self-consistent field as well as the predictor–corrector time integration schemes. The Coulomb contribution to the Kohn–Sham matrix is calculated combining density fitting approximation and the continuous fast multipole method. Performance of the implementation is benchmarked for molecular systems with different sizes and dimensionalities. For linear alkane chains, the wall time for density matrix time propagation step is comparable to the Kohn-Sham (KS) matrix construction. However, for larger two- and three-dimensional molecules, with up to about 5,000 basis functions, the computational effort of RT-TDDFT calculations is dominated by the KS matrix evaluation. In addition, the maximum time step is evaluated using a set of small molecules of different polarities. The photoabsorption spectra of several molecular systems calculated using RT-TDDFT are compared to those obtained using linear response time-dependent density functional theory and coupled cluster methods.     

The generalized hybrid orbital method for combined quantum mechanical/molecular mechanical calculations: formulation and tests of the analytical derivatives

 Hybrid quantum mechanical (QM) and molecular mechanical (MM) potentials are becoming increasingly important for studying condensed-phase systems but one of the outstanding problems in the field has been how to treat covalent bonds between atoms of the QM and MM regions. Recently, we presented a generalized hybrid orbital (GHO) method that was designed to tackle this problem for hybrid potentials using semiempirical QM methods [Gao et al. (1998) J Phys Chem A 102: 4714–4721]. We tested the method on some small molecules and showed that it performed well when compared to the purely QM or MM potentials. In this article, we describe the formalism for the determination of the GHO energy derivatives and then present the results of more tests aimed at validating the model. These tests, involving the calculation of the proton affinities of some model compounds and a molecular dynamics simulation of a protein, indicate that the GHO method will prove useful for the application of hybrid potentials to solution-phase macromolecular systems. Received: 4 October 1999 / Accepted: 18 December 1999 / Published online: 5 June 2000     

Grid-free DFT implementation of local and gradient-corrected XC functionals

Following an approach to density functional theory calculations based on the matrix representation of operators, we implemented a scheme as an alternative to traditional grid-based methods. These techniques allow integrals over exchange-correlation operators to be evaluated through matrix manipulations. Both local and gradient-corrected functionals can be treated in a similar way. After deriving all the required expressions, selected examples with various functionals are given. Received: 7 March 1998 / Accepted: 21 May 1998 / Published on line: 6 August 1998     

Reparameterization of hybrid functionals based on energy differences of states of different multiplicity

 Low-spin/high-spin energy splittings for Fe(II) transition-metal complexes – particularly in weak ligand fields – cannot be well described by density functional methods. Different density functionals yield results which differ by up to 1 eV in transition-metal complexes with sulfur-rich first coordination spheres. We attribute this failure to the fact that the high-spin state is systematically favoured in Hartree–Fock-type theories, because Fermi correlation is included in the exact exchange, while Coulomb correlation is not. We thus expect that the admixture of exact exchange to a given density functional will heavily influence the energy splitting between states of different multiplicity. We demonstrate that the energy splitting depends linearly on the coefficient of exact exchange admixture. This remarkable result is found for all the Fe(II)–S complexes studied. From this observation we conclude in connection with experimental results that Becke's 20% admixture should be reduced to about 15% if meaningful energetics are sought for transition-metal compounds. We rationalize that this reduction by 5% will not affect the quality of the hybrid functional since we arrive at a slightly modified functional, which lies between the pure density functional and the hybrid density functional, which both give good results for “standard” systems. Received: 13 July 2001 / Accepted: 31 August 2001 /  Published online: 16 November 2001     

Gas-phase properties and Fukui indices of sulfine (CH2SO). Potential energy surface and maximum hardness principle for its protonation process. A density functional study

Gas-phase thermochemical properties of sulfine (CH₂SO) and the potential energy surface of its protonation process were studied by the density functional method employing different exchange-correlation potentials. All calculations showed that the most stable protonated isomer is planar with the proton bonded to the oxygen atom in a trans arrangement of the skeleton. Three transition states were located that allow interconversion between the different isomers. Hardnesses and Fukui indices were calculated to follow the reactivity trend along the protonation path and to explain the preference for a particular protonation site on neutral sulfine. Proton affinity, gas-phase basicity and heat of formation values, obtained for the first time fully quantum mechanically, agree well with those derived by a recent mass spectrometry experimental study. Good agreement between density functional theory and previous high-level theoretical and experimental data was also found for the heat of formation of sulfine and its most stable protonated form. Received: 12 October 1998 / Accepted: 24 November 1998 / Published online: 16 March 1999     

A theoretical study of substituent effects on the structure of isolated and condensed three-membered rings. A comparison between radicals and parent hydrocarbons

Substituent effects on the structure of radicals and parent hydrocarbons formed by isolated or condensed three-membered rings have been investigated by Hartree-Fock, post-Hartree-Fock and density functional methods. The trends of structural parameters computed for the hydrocarbon systems are in agreement with available experimental data. Substituent effects can be rationalized in terms of interactions between localized orbitals obtained by natural bond analysis. The effects are even larger in free radicals and can be analyzed using the same model. Received: 13 March 1998 / Accepted: 13 July 1998 / Published online: 9 October 1998     

Linear scaling computation of the Fock matrix. III. Formation of the exchange matrix with permutational symmetry

 A direct comparison is made between two recently proposed methods for linear scaling computation of the Hartree–Fock exchange matrix to investigate the importance of exploiting two-electron integral permutational symmetry. Calculations on three-dimensional water clusters and graphitic sheets with different basis sets and levels of accuracy are presented to identify specific cases where permutational symmetry may or may not be useful. We conclude that a reduction in integrals via permutational symmetry does not necessarily translate into a reduction in computation times. For large insulating systems and weakly contracted basis sets the advantage of permutational symmetry is found to be negligible, while for noninsulating systems and highly contracted basis sets a fourfold speedup is approached. Received: 8 October 1999 / Accepted: 3 January 2000 / Published online: 21 June 2000     

Equation for the direct determination of the density matrix: Time-dependent density equation and perturbation theory

The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time-dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schr?dinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time-independent equation by replacing the energy E with . The perturbation theory formulas for the density equation have also been given for both the time-dependent and time-independent cases. Received: 16 June 1998 / Accepted: 2 September 1998 / Published online: 8 February 1999     

Solution conformations of structured peptides: continuum electrostatics versus distance-dependent dielectric functions

To compare different implicit solvent potentials, the folding thermodynamics of the helical peptide RN24 and the β-hairpin peptide BH8 are studied by molecular dynamics simulation with adaptive umbrella sampling. As the potential energy functions, the analytical continuum solvent (ACS) potential and three simplified variants, termed EPSR₁, EPSR₄, and EPSR₁₀, are used. The ACS potential is a combination of the standard CHARMM force field for the internal energy (bonds, angles, dihedrals) and the van der Waals energy with the analytical continuum electrostatic (ACE) potential and a non-polar solvation potential. The EPSR potentials differ from the ACS potential by the use of Coulomb's law with a distance-dependent dielectric function to calculate the electrostatic energy. With the ACS potential, quantitative agreement with experiment is obtained for the helix propensity (RN24: 62% calculated vs 50–60% experiment) and the β-hairpin propensity (BH8: 33% calculated vs 19–37% experiment) of the peptides. During the simulations with the EPSR potentials, no significant formation of secondary structure is observed. It is shown that the preference for coil conformations over conformations with secondary structure by the EPSR potentials is due to an overestimation of the energy of salt bridge formation, independent of the magnitude of the Coulomb energy relative to the other energy terms. Possible improvements of the distance-dependent dielectric functions which may permit their application to the simulation of peptide folding, are discussed. Received: 11 July 1998 / Accepted: 22 September 1998 / Published online: 17 December 1998     

Fast and accurate Coulomb calculation with Gaussian functions

Coulomb interaction is one of the major time-consuming components in a density functional theory (DFT) calculation. In the last decade, dramatic progresses have been made to improve the efficiency of Coulomb calculation, including continuous fast multipole method (CFMM) and J-engine method, all developed first inside Q-Chem. The most recent development is the advent of Fourier transform Coulomb method developed by Fusti-Molnar and Pulay, and an improved version of the method has been recently implemented in Q-Chem. It replaces the least efficient part of the previous Coulomb methods with an accurate numerical integration scheme that scales in O(N2) instead of O(N4) with the basis size. The result is a much smaller slope in the linear scaling with respect to the molecular size and we will demonstrate through a series of benchmark calculations that it speeds up the calculation of Coulomb energy by several folds over the efficient existing code, i.e., the combination of CFMM and J-engine, without loss of accuracy. Furthermore, we will show that it is complementary to the latter and together the three methods offer the best performance for Coulomb part of DFT calculations, making the DFT calculations affordable for very large systems involving thousands of basis functions.     

Linear scaling computation of the Fock matrix. VII. Parallel computation of the Coulomb matrix

We present parallelization of a quantum-chemical tree-code for linear scaling computation of the Coulomb matrix. Equal time partition is used to load balance computation of the Coulomb matrix. Equal time partition is a measurement based algorithm for domain decomposition that exploits small variation of the density between self-consistent-field cycles to achieve load balance. Efficiency of the equal time partition is illustrated by several tests involving both finite and periodic systems. It is found that equal time partition is able to deliver 91%-98% efficiency with 128 processors in the most time consuming part of the Coulomb matrix calculation. The current parallel quantum chemical tree code is able to deliver 63%-81% overall efficiency on 128 processors with fine grained parallelism (less than two heavy atoms per processor).     

Efficient boundary basis functions for time-independent quantum scattering calculations

Novel parametrically energy-dependent boundary functions, F, combined with a finite L ² basis set, permit accurate and efficient calculation of scattering wavefunctions for many energies. Both accuracy and efficiency are achieved simultaneously because all the necessary integrals are energy-independent and also certain functionals, (H−E)|F>, in the Schr?dinger equation are allowed to satisfy desirable boundary conditions. In addition, slight modification of the Schr?dinger equation in the boundary region is shown to be useful for improving the numerical accuracy when a cutoff-radius-truncated basis set is used. The advantages of the present approaches are demonstrated for the one-dimensional Eckart barrier problem. Received: 17 June 1998 / Accepted: 13 July 1998 / Published online: 18 September 1998     

Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set

Scaling factors for obtaining fundamental vibrational frequencies from harmonic frequencies calculated at six of the most commonly used levels of theory have been determined from regression analysis for the polarized-valence triple-zeta (pVTZ) Sadlej electric property basis set. The Sadlej harmonic frequency scaling factors for first- and second-row molecules were derived from a comparison of a total of 900 individual vibrations for 111 molecules with available experimental frequencies. Overall, the best performers were the hybrid density functional theory (DFT) methods, Becke's three-parameter exchange functional with the Lee–Yang–Parr fit for the correlation functional (B3-LYP) and Becke's three-parameter exchange functional with Perdew and Wang's gradient-corrected correlation functional (B3-PW91). The uniform scaling factors for use with the Sadlej pVTZ basis set are 0.9066, 0.9946, 1.0047, 0.9726, 0.9674 and 0.9649 for Hartree–Fock, the Slater–Dirac exchange functional with the Vosko–Wilk–Nusair fit for the correlation functional (S-VWN), Becke's gradient-corrected exchange functional with the Lee–Yang–Parr fit for the correlation functional (B-LYP), B3-LYP, B3-PW91 and second-order M?ller–Plesset theory with frozen core (MP2(fc)), respectively. In addition to uniform frequency scaling factors, dual scaling factors were determined to improve the agreement between computed and observed frequencies. The scaling factors for the wavenumber regions below 1800 cm⁻¹ and above 1800 cm⁻¹ are 0.8981 and 0.9097, 1.0216 and 0.9857, 1.0352 and 0.9948, 0.9927 and 0.9659, 0.9873 and 0.9607, 0.9844 and 0.9584 for Hartree–Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2(fc), respectively. Hybrid DFT methods along with the Sadlej pVTZ basis set provides reliable theoretical vibrational spectra in a cost-effective manner. Received: 22 May 2000 / Accepted: 30 August 2000 / Published online: 28 February 2001     

Revisiting superdelocalizability. Mathematical stability of reactivity indices

Superdelocalizability, S _r, defined by Fukui et al. as a variant of reactivity indices in the Hückel molecular orbital scheme for conjugated hydrocarbon molecules is reinvestigated with particular emphasis on its behavior for infinitely large π-electronic systems. Surprisingly, almost all the S _r values of polyacetylene are found to diverge with the size of molecule, while all the S _r values of comb polyene converge to certain values. Similarly, the S _r values of linear polyacene diverge, while those of zigzag polyacene converge. Whether the superdelocalizability for a series of periodic polymers converges or diverges can be predicted, respectively, if the density of states of the infinitely large π-electron network is shown to have non-zero or zero gap at the Fermi level. The behavior of atom-atom polarizability defined by Coulson et al. is also checked and discussed. Received: 27 July 1998 / Accepted: 9 September 1998 / Published online: 23 February 1999     

Generalized extended empirical bond-order dependent force fields including nonbond interactions

We present a general approach for describing chemical processes (bond breaking and bond formation) in materials using force fields (FF) that properly describe multiple bonds at small distances while describing nonbond (Coulomb and van der Waals) interactions at long distances. This approach is referred to as the generalized extended empirical bond-order dependent FF. In this paper we use the Brenner empirical bond-order dependent FF for the short-range interactions and report applications on the energetics and structures of graphite crystal, dynamics of molecular crystals, and distortions of bucky tubes. Received: 7 August 1998 / Accepted: 26 October 1998 / Published online: 16 March 1999