Hessian Approach Research Articles

Full optimization of Jastrow–Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules

We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.

Calculating Reaction Rates with Partial Hessians: Validation of the Mobile Block Hessian Approach.

In an earlier paper, the authors have developed a new method, the mobile block Hessian (MBH), to accurately calculate vibrational modes for partially optimized molecular structures [J. Chem. Phys. 2007, 126 (22), 224102]. The proposed procedure remedies the artifact of imaginary frequencies, occurring in standard frequency calculations, when parts of the molecular system are optimized at different levels of theory. Frequencies are an essential ingredient in predicting reaction rate coefficients due to their input in the vibrational partition functions. The question arises whether the MBH method is able to describe the chemical reaction kinetics in an accurate way in large molecular systems where a full quantum chemical treatment at a reasonably high level of theory is unfeasible due to computational constraints. In this work, such a validation is tested in depth. The MBH method opens a lot of perspectives in predicting accurate kinetic parameters in chemical reactions where the standard full Hessian procedure fails.

Partial Hessian Vibrational Analysis of Organic Molecules Adsorbed on Si(100)

A partial Hessian approach is used to study the vibrational spectroscopy of acetylene, ethylene, benzene, 1,3-butadiene, and naphthalene adsorbed on the Si(100) surface. The partial Hessian approximation introduces a very small error in the computed frequencies and intensities of the adsorbed molecules and results in a significant reduction in computational cost. Exploiting this reduced computational cost, molecular adsorbates on large cluster models of the Si(100) surface can be studied with density functional theory. Using clusters with two and four silicon−silicon dimers, the effects of surface coverage and the spectroscopy of large adsorbates that bind to more than one dimer are studied. For benzene and butadiene, molecules adsorbed on neighboring silicon dimers in the same dimer row can have a significant effect on the infrared spectrum in the C−H stretching region, changing both the frequency and intensity of the computed vibrational modes. A weaker effect is observed for molecules adsorbed in adjacent dimer rows. Comparison of the computed spectra with experiment shows that several binding configurations are present for benzene. For 1,3-butadiene, the adsorption product is dominated by the product of a Diels−Alder [4+2] addition. However, the product may have the hydrogens of the sp2 carbons cis or trans with respect to each other. Larger cluster models of the surface also allow large adsorbates that can bind to multiple silicon dimers to be studied. The infrared spectroscopy of naphthalene bound to one, two, and four silicon dimers is computed. Comparison with experiment shows that naphthalene preferentially binds within a dimer row.

Cartesian formulation of the mobile block Hessian approach to vibrational analysis in partially optimized systems

Partial optimization is a useful technique to reduce the computational load in simulations of extended systems. In such nonequilibrium structures, the accurate calculation of localized vibrational modes can be troublesome, since the standard normal mode analysis becomes inappropriate. In a previous paper [A. Ghysels et al., J. Chem. Phys. 126, 224102 (2007)], the mobile block Hessian (MBH) approach was presented to deal with the vibrational analysis in partially optimized systems. In the MBH model, the nonoptimized regions of the system are represented by one or several blocks, which can move as rigid bodies with respect to the atoms of the optimized region. In this way unphysical imaginary frequencies are avoided and the translational/rotational invariance of the potential energy surface is fully respected. In this paper we focus on issues concerning the practical numerical implementation of the MBH model. The MBH normal mode equations are worked out for several coordinate choices. The introduction of a consistent group-theoretical notation facilitates the treatment of both the case of a single block and the case of multiple blocks. Special attention is paid to the formulation in terms of Cartesian variables, in order to provide a link with the standard output of common molecular modeling programs.

Update of parton distributions at NNLO

We present a new set of parton distributions obtained at NNLO. These differ from the previous sets available at NNLO due to improvements in the theoretical treatment. In particular we include a full treatment of heavy flavours in the region near the quark mass. In this way, an essentially complete set of NNLO partons is presented for the first time. The improved treatment leads to a significant change in the gluon and heavy quark distributions, and a larger value of the QCD coupling at NNLO, αS(MZ2)=0.1191±0.002(expt.)±0.003(theory). Indirectly this also leads to a change in the light partons at small x and modifications of our predictions for W and Z production at the LHC. As well as the best-fit set of partons, we also provide 30 additional sets representing the uncertainties of the partons obtained using the Hessian approach.

Vibrational modes in partially optimized molecular systems

In this paper the authors develop a method to accurately calculate localized vibrational modes for partially optimized molecular structures or for structures containing link atoms. The method avoids artificially introduced imaginary frequencies and keeps track of the invariance under global translations and rotations. Only a subblock of the Hessian matrix has to be constructed and diagonalized, leading to a serious reduction of the computational time for the frequency analysis. The mobile block Hessian approach (MBH) proposed in this work can be regarded as an extension of the partial Hessian vibrational analysis approach proposed by Head [Int. J. Quantum Chem. 65, 827 (1997)]. Instead of giving the nonoptimized region of the system an infinite mass, it is allowed to move as a rigid body with respect to the optimized region of the system. The MBH approach is then extended to the case where several parts of the molecule can move as independent multiple rigid blocks in combination with single atoms. The merits of both models are extensively tested on ethanol and di-n-octyl-ether.

Enhancing structure relaxations for first-principles codes: An approximate Hessian approach

We present a method for improving the speed of geometry relaxation by using a harmonic approximation for the interaction potential between nearest neighbor atoms to construct an initial Hessian estimate. The model is quite robust, and yields approximately a 30% or better reduction in the number of calculations compared to an optimized diagonal initialization. Convergence with this initializer approaches the speed of a converged BFGS Hessian, therefore it is close to the best that can be achieved. Hessian preconditioning is discussed, and it is found that a compromise between an average condition number and a narrow distribution in eigenvalues produces the best optimization.

Computation of the amide I band of polypeptides and proteins using a partial Hessian approach

A partial Hessian approximation for the computation of the amide I band of polypeptides and proteins is introduced. This approximation exploits the nature of the amide I band, which is largely localized on the carbonyl groups of the backbone amide residues. For a set of model peptides, harmonic frequencies computed from the Hessian comprising only derivatives of the energy with respect to the displacement of the carbon, oxygen, and nitrogen atoms of the backbone amide groups introduce mean absolute errors of 15 and 10 cm(-1) from the full Hessian values at the Hartree-Fock/STO-3G and density functional theory EDF16-31G(*) levels of theory, respectively. Limiting the partial Hessian to include only derivatives with respect to the displacement of the backbone carbon and oxygen atoms yields corresponding errors of 24 and 22 cm(-1). Both approximations reproduce the full Hessian band profiles well with only a small shift to lower wave number. Computationally, the partial Hessian approximation is used in the solution of the coupled perturbed Hartree-Fock/Kohn-Sham equations and the evaluation of the second derivatives of the electron repulsion integrals. The resulting computational savings are substantial and grow with the size of the polypeptide. At the HF/STO-3G level, the partial Hessian calculation for a polypeptide comprising five tryptophan residues takes approximately 10%-15% of the time for the full Hessian calculation. Using the partial Hessian method, the amide I bands of the constituent secondary structure elements of the protein agitoxin 2 (PDB code 1AGT) are calculated, and the amide I band of the full protein estimated.

Nonlinear fitness–space–structure adaptation and principal component analysis in genetic algorithms: an application to x-ray reflectivity analysis

Two novel genetic algorithms implementing principal component analysis and an adaptive nonlinear fitness–space–structure technique are presented and compared with conventional algorithms in x-ray reflectivity analysis. Principal component analysis based on Hessian or interparameter covariance matrices is used to rotate a coordinate frame. The nonlinear adaptation applies nonlinear estimates to reshape the probability distribution of the trial parameters. The simulated x-ray reflectivity of a realistic model of a periodic nanolaminate structure was used as a test case for the fitting algorithms. The novel methods had significantly faster convergence and less stagnation than conventional non-adaptive genetic algorithms. The covariance approach needs no additional curve calculations compared with conventional methods, and it had better convergence properties than the computationally expensive Hessian approach. These new algorithms can also be applied to other fitting problems where tight interparameter dependence is present.

On the accuracy of covariance matrix: Hessian versus Gauss–Newton methods in atmospheric remote sensing with infrared spectroscopy

Most retrieval schemes use a linear approximation of the radiative transfer function within each iteration as well as for error analysis. Like most standard methods, the improved Hessian method relies on a quadratic form of the cost function and linear approximation in the error analysis. Often, there is no robust criterion in determining step size that can be used to calculate covariance matrix by discrete perturbation of the cost function in the Hessian approach. The Hessian method improved recently, however, overcomes this problem by employing adaptive algorithm which uses small step sizes in steep directions and large step sizes in flat directions of the cost function. The results of retrievals of atmospheric trace gases from simulated limb emission spectra show that Gauss–Newton algorithm and the improved Hessian generally give nearly identical volume mixing ratios and error covariance matrices in the original state vector space. Due to interlevel correlations, however, the agreement in the uncertainities in the original state vector coordinate system is partly lost in a space in which the elements of state vector are independent after orthogonal coordinate transformation. The significant discrepancies between the estimated uncertainities by the two methods are found to be related with elements of state vector that are dominantly controlled by flattest eigenvector directions of the inverse covariance matrix. The improved Hessian method determines the uncertainities in those shallowest directions with better accuracy than Gauss–Newton approach. The performance of the Hessian method is also found to be better in resolving structures related to the shallowest eigenvector directions as revealed by better vertical resolutions in the retrieved profiles of the trace species.

Calculation of magnetizabilities using GIAO current density distributions

Calculations of SCF molecular magnetizabilities in terms of GIAO magnetic current density distributions are presented. for incomplete basis sets, this “GIAOCD” method is not, in general, equivalent to the conventional GIAO magnetic Hessian approach to calculating molecular magnetizabilities. However, even for basis sets of practical size the two methods consistently yield very similar results. The value of the GIAOCD method relative to the GIAO magnetic Hessian method is that it is easier to implement, requires less computation and relates the molecular magnetizabilities directly to the physical magnetic current density distributions. The GIAOCD method is compared to the CSGT and IGAIM methods previously developed by the author and is found to be superior for smaller basis sets.

Electronic structure of strained polymers: 3. Scaled Hessians for polyethylene

Abstract Earlier estimates for the Young's modulus of polyethylene based on the AM1 semi-empirical molecular orbital method were 15% larger than the largest experimental value. Neglect of electron correlation in the single determinant wavefunction for polyethylene leads to a Hessian matrix which is too stiff. Here we apply the biased Hessian approach of Dasgupta and Goddard1 to the polyethylene chain at equilibrium and under tensile and compressive strain. The chain modulus is 340 GPa with the biased Hessian, in agreement with extant experimental results. Although the variation of the vibrational spectrum with strain is reduced with biasing, the stress-dependence of vibrational frequencies is unchanged.

Reduced successive quadratic programming strategy for errors-in-variables estimation

Parameter estimation problems in process engineering represent a special class of nonlinear optimization problems, because the maximum likelihood structure of the objective function can be exploited. Within this class, the errors in variables method (EVM) is particularly interesting. Here we seek a weighted least-squares fit to the measurements with an underdetermined process model. Thus, both the number of variables and degrees of freedom available for optimization increase linearly with the number of data sets. Large optimization problems of this type can be particularly challenging and expensive to solve because, for general-purpose nonlinear programming (NLP) algorithms, the computational effort increases at least quadratically with problem size. In this study we develop a tailored NLP strategy for EVM problems. The method is based on a reduced Hessian approach to successive quadratic programming (SQP), but with the decomposition performed separately for each data set. This leads to the elimination of all variables but the model parameters, which are determined by a QP coordination step. In this way the computational effort remains linear in the number of data sets. Moreover, unlike previous approaches to the EVM problem, global and superlinear properties of the SQP algorithm apply naturally. Also, the method directly incorporates inequality constraints on the model parameters (although not on the fitted variables). This approach is demonstrated on five example problems with up to 102 degrees of freedom. Compared to general-purpose NLP algorithms, large improvements in computational performance are observed.