Hessian Calculations Research Articles

Experimental and ab initio structural study of estertin compounds, X 3SnCH 2CH 2CO 2Me: Crystal structures of Cl 3SnCH 2CH 2CO 2Me at 120 K and Br 3SnCH 2CH 2CO 2Me at 120 and 291 K

The MeO 2CCH 2CH 2 ligand in X 3SnCH 2CH 2CO 2Me, (X = Cl, Br or I), acts as a C, O-chelating group, via the carbonyl group, both in the solid state and in solutions in non-coordinating solvents. The crystal structures of 1 (X = Cl), a redetermination at 120 K and of I (X = Br), at 298 and 120 K, are reported. Comparison of the intramolecular Sn O bond lengths in solid 1 (X = Cl, Br or I) indicates that the strength of the Sn O interaction increases in the order X = I < Br < Cl. Furthermore, the strength of the Sn O bond is greater in Cl 3SnCH 2CH 2CO 2Me than in the corresponding ketotin compound, Cl 3SnCMe 2CH 2COMe, another chelated complex. Mixtures of 1 (X = Cl) and 1 (X = Br or I) undergo exchange reactions in solution, as shown by NMR spectra, to give all possible halide derivatives, (Cl n X 3−nSnCH 2CH 2CO 2Me: n = 0–3: X = Br or I). A series of electronic structure calculations on ( 1: X = F, Cl, Br, I, SCN; R = Me) have been carried out at various levels of theory, including RHF and MP2. Hessian calculations have also been performed on these optimized geometries in order to obtain internal coordinate force constants. Comparisons of the theoretical and experimental structures of 1 (X = Cl, Br and I) are reported.

Enabling ab initio Hessian and frequency calculations of large molecules

A linear scaling method, termed as cardinality guided molecular tailoring approach, is applied for the estimation of the Hessian matrix and frequency calculations of spatially extended molecules. The method is put to test on a number of molecular systems largely employing the Hartree-Fock and density functional theory for a variety of basis sets. To demonstrate its ability for correlated methods, we have also performed a few test calculations at the Moller-Plesset second order perturbation theory. A comparison of central processing unit and memory requirements for medium-sized systems with those for the corresponding full ab initio computation reveals substantial gains with negligible loss of accuracy. The technique is further employed for a set of larger molecules, Hessian and frequency calculations of which are not possible on commonly available personal-computer-type hardware.

Growth pattern and electronic properties of acetonitrile clusters: A density functional study

We report a systematic theoretical study on the growth pattern and electronic properties of acetonitrile clusters [(CH(3)CN)(n) (n = 1, 9, 12)] using density functional approach at the B3LYP6-31++G(d,p) level. Although we have considered a large number of configurations for each cluster, the stability of the lowest energy isomer was verified from the Hessian calculation. It is found that the lowest energy isomer of the dimer adopts an antiparallel configuration. For trimer and tetramer, cyclic ring structures were found to be favored over the dipole stabilized structure. In general, it is found that the intermolecular CH...N interactions play a significant role in the stabilization of the cyclic layered geometry of acetonitrile clusters. A critical comparison between trimer and tetramer clusters suggests that the three member cyclic ring is more stable than four member rings. The growth motif for larger clusters (n = 5-9, 12) follows a layered pattern consisting of three or four membered rings, which, in fact, is used as the building block. Based on the stability analysis, it is found that clusters with an even number of molecular entities are more stable than the odd clusters, except trimer and nonamer. The exceptional stability of these two clusters is attributed to the formation of trimembered cyclic rings, which have been found to form the building blocks for larger clusters.

Parallel implementation of electronic structure energy, gradient, and Hessian calculations

ACES III is a newly written program in which the computationally demanding components of the computational chemistry code ACES II [J. F. Stanton et al., Int. J. Quantum Chem. 526, 879 (1992); [ACES II program system, University of Florida, 1994] have been redesigned and implemented in parallel. The high-level algorithms include Hartree-Fock (HF) self-consistent field (SCF), second-order many-body perturbation theory [MBPT(2)] energy, gradient, and Hessian, and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] energy and gradient. For SCF, MBPT(2), and CCSD(T), both restricted HF and unrestricted HF reference wave functions are available. For MBPT(2) gradients and Hessians, a restricted open-shell HF reference is also supported. The methods are programed in a special language designed for the parallelization project. The language is called super instruction assembly language (SIAL). The design uses an extreme form of object-oriented programing. All compute intensive operations, such as tensor contractions and diagonalizations, all communication operations, and all input-output operations are handled by a parallel program written in C and FORTRAN 77. This parallel program, called the super instruction processor (SIP), interprets and executes the SIAL program. By separating the algorithmic complexity (in SIAL) from the complexities of execution on computer hardware (in SIP), a software system is created that allows for very effective optimization and tuning on different hardware architectures with quite manageable effort.

Γ -point lattice free energy estimates from O(1) force calculations

We present a new method for estimating the vibrational free energy of crystal (and molecular) structures employing only a single force calculation, for a particularly displaced configuration, in addition to the calculation of the ground state configuration. This displacement vector is the sum of the phonon eigenvectors obtained from a fast-relative to, e.g., density-functional theory (DFT)-Hessian calculation using interatomic potentials. These potentials are based here on effective charges obtained from a DFT calculation of the ground state electronic charge density but could also be based on other, e.g., empiric approaches.

A Nonlinear Iterative Reconstruction and Analysis Approach to Shape-Based Approximate Electromagnetic Tomography

A nonlinear Helmholtz-equation-modeled electromagnetic tomographic reconstruction problem is solved for the object boundary and inhomogeneity parameters in a damped Tikhonov-regularized Gauss-Newton (DTRGN) solution framework. In this paper, the object is represented in a suitable global basis, whereas the boundary is expressed as the zero level set of a signed-distance function. For an explicit parameterized boundary-representation-based reconstruction scheme, analytical Jacobian and Hessian calculations are made to express the changes in scattered field values w.r.t. changes in the inhomogeneity parameters and the control points in a spline representation of the object boundary, via the use of a level-set representation of the object. Even though, in this paper, a homogeneous dielectric is considered and a spline representation has been used to represent the boundary, the formulation can be used for a general global basis representation of the inhomogeneity as well as arbitrary parameterizations of the boundary, and is generalizable to three dimensions. Reconstruction results are presented for test cases of landminelike dielectric objects embedded in the ground under noisy data conditions. To confirm convergence and, at times, to know which of the obtained iterates are closer to the actual unknown solution, using a perturbation theory framework, a local (Hessian-based) convergence analysis is applied to the DTRGN scheme for the reconstruction, yielding estimates of convergence rates in the residual and parameter spaces.

Parameter Set Uniqueness and Confidence Limits in Model Identification of Insulin Transport Models from Simulation Data

The use of patient models describing the dynamics of glucose, insulin, and possibly other metabolic species associated with glucose regulation allows diabetes researchers to gain insights regarding novel therapies via simulation. However, such models are only useful when model parameters are effectively estimated with patient data. The use of least squares to effectively estimate model parameters from simulation data was investigated by observing factors that influence the accuracy of estimates for the model parameters from a data set generated using a model with known parameters. An intravenous insulin pharmacokinetic model was used to generate the insulin response of a patient with type 1 diabetes mellitus to a series of step changes in the insulin infusion rate from an external insulin pump. The effects of using user-defined gradient and Hessian calculations on both parameter estimations and the 95% confidence limits of the estimated parameter sets were investigated. Estimations performed by either solver without user-supplied quantities were highly dependent on the initial guess of the parameter set, with relative confidence limits greater than +/-100%. The use of user-defined quantities allowed the one-compartment model parameters to be effectively estimated. While the two-compartment model parameter estimation still depended on the initial parameter set specification, confidence limits were decreased, and all fits to simulation data were very good. The use of user-defined gradients and Hessian matrices results in more accurate parameter estimations for insulin transport models. Improved estimation could result in more accurate simulations for use in glucose control system design.

Convergence Acceleration of Direct Trajectory Optimization Using Novel Hessian Calculation Methods

Sparse sequential quadratic programming (SQP) has offered fast and robust convergence of trajectory optimization based on direct collocation. However, the conventional approach of calculating the Hessian of the Lagrangian is sometimes inefficient in view of the computational time. Therefore, this paper proposes two novel Hessian calculation methods that exploit the doubly-bordered block diagonal structure of the Hessian. Through applications to the constrained brachistochrone problem and the space shuttle reentry problem, the proposed methods were demonstrated to show faster convergence speeds as compared with the conventional methods.

A theoretical study on SCN − + [rad]XH reactions (X = O, S): Hemi bonded vs. H-bonded intermediate

Ab initio molecular electronic structure calculations are employed to elucidate the structure, bonding and spectra (IR and UV–VIS) of various intermediates formed on the reaction of OH, SH radicals and atomic O with thiocyanate anion (SCN −) and thiocyanate radical (SCN ) in gas phase. Geometry optimizations are carried out following second order Moller–Plesset perturbation theory (MP2) adopting 6-311++G(d,p) basis set. Hessian calculations are performed at the same level of theory to check the nature of the equilibrium geometry as well as to generate IR spectrum. Visualization of the highest doubly occupied molecular orbital (HDOMO) in a few SCNX − adducts (X = OH, SH) is able to depict clearly the presence of a two-center three-electron ( 2c-3e) σ bond between O…S, S…S and S…N pairs. Excited state calculations are performed at configuration interaction with single-electron excitation (CIS) level for all the species. Strong optical absorption predicted for 2c-3e bonded species is due to σ → σ ∗ electronic transition. Structures having 2c-3e bonding in SCNX − systems (X = OH and SH) show strong C N stretching IR band. In case of neutral SCNX and NCSSH adducts (X = OH and SH), a strong C N stretching band and weak O H or S H stretching bands are obtained. Strong O N or O S stretching bands are observed in case of anionic ONCS − or OSCN − adducts. But strong C N stretching band is predicted for radical adducts, ONCS or OSCN .

A parallel distributed data CPHF algorithm for analytic Hessians

One of the most commonly used means to characterize potential energy surfaces of reactions and chemical systems is the Hessian calculation, whose analytic evaluation is computationally and memory demanding. A new scalable distributed data analytic Hessian algorithm is presented. Features of the distributed data parallel coupled perturbed Hartree-Fock (CPHF) are (a) columns of density-like and Fock-like matrices are distributed among processors, (b) an efficient static load balancing scheme achieves good work load distribution among the processors, (c) network communication time is minimized, and (d) numerous performance improvements in analytic Hessian steps are made. As a result, the new code has good performance which is demonstrated on large biological systems.

Structural Investigation of Asymmetrical Dimer Radical Cation System (H2O−H2S)+: Proton-Transferred or Hemi-Bonded?

Ab initio molecular orbital and hybrid density functional methods have been employed to characterize the structure and bonding of (H2O-H2S)+, an asymmetrical dimer radical cation system. A comparison has been made between the two-center three-electron (2c-3e) hemi-bonded system and the proton-transferred hydrogen-bonded systems of (H2O-H2S)+. Geometry optimization of these systems was carried out using unrestricted Hartree Fock (HF), density functional theory with different functionals, and second-order Møller-Plesset perturbation (MP2) methods with 6-311++G(d,p) basis set. Hessian calculations have been done at the same level to check the nature of the equilibrium geometry. Energy data were further improved by calculating basis set superposition error for the structures optimized through MP2/6-311++G(d,p) calculations. The calculated results show that the dimer radical cation structure with H2O as proton acceptor is more stable than those structures in which H2O acts as a proton donor or the 2c-3e hemi-bonded (H2O thereforeSH2)+ system. This stability trend has been further confirmed by more accurate G3, G3B3, and CCSD(T) methods. On the basis of the present calculated results, the structure of H4OS+ can best be described as a hydrogen-bonded complex of H3O+ and SH with H2O as a proton acceptor. It is in contrast to the structure of neutral (H2O...H2S) dimer where H2O acts as a proton donor. The present work has been able to resolve the ambiguity in the nature of bonding between H2O and H2S in (H2O-H2S)+ asymmetrical dimer radical cation.

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.

SG‐0: A small standard grid for DFT quadrature on large systems

We report the development of a new standard quadrature grid for DFT calculations. Standard Grid 0 (SG-0) is designed to be approximately half as large as, and to provide approximately half the accuracy of, the established SG-1 grid. It is based on MultiExp and Lebedev quadrature for radial and angular coordinates, respectively. We find that SG-0 is typically 50% faster than SG-1 for energy, gradient, and hessian calculations for the exchange-correlation energy. This leads to a 35-38% speedup in the total gradient and hessian computations, and we particularly recommend its use for preliminary calculations on moderately large biochemical systems. It has been implemented as the default grid for DFT calculations in the Q-Chem 3.0 package.

Automated sensitivity analysis of stiff biochemical systems using a fourth-order adaptive step size Rosenbrock integration method

Sensitivity analysis is one of the most effective approaches for studying mathematical models of biochemical systems. A stiff Rosenbrock integrator has been developed for sensitivity analysis using a direct sensitivity approach. Automated sparse Jacobian and Hessian calculations of the coupled system (the original model equations and the sensitivity equations) have been implemented in the freely available software package CellSim. The accuracy and efficiency of the integrator are tested extensively on the complex mitogen-activated protein kinase (MAPK) pathway model of Bhalla and Iyengar. Both time-dependent concentration and parameter-based sensitivity coefficients are measured using several integration schemes. The method is shown to perform sensitivity analysis in a manner that is cost effective with moderate accuracy. The error control strategy between the decoupled direct method and the Rosenbrock with direct method is discussed and their computational accuracies are compared. The method is used to analyse the positive feedback loop within the MAPK signal transduction pathway.

A new real-time perspective on non-linear model predictive control

This work presents a new formulation of continuous-time non-linear model predictive control (NMPC) in which the parameters defining the input trajectory are adapted continuously in real time. Continuous implementation of the control as the input parameterization is being optimized reduces the impact of computational delay, in particular in response to process disturbances. By eliminating the typical correspondence between the time partitions used for input parameterization and implementation, and instead parameterizing the input over arbitrary intervals of variable length, a means is provided to reduce the overall number of optimization parameters (and hence the dimension of the required gradient and Hessian calculations) without adversely affecting stability or optimality.

Vibrational spectra of tris(dmit) complexes of main group metals: infrared, Raman and ab initio calculations

Infrared Fourier Transform investigation of several metal tris-complexes of 1,3-dithiole-2-thione-4,5-dithiolate (dmit) ligand have been recorded within a theoretical–experimental investigation of the vibrational molecular spectra of crystalline [NEt 4] 2[Sn(dmit) 3] and [NEt 4][Sb(dmit) 3] compounds. For the [Sn(dmit) 3] −2 anion we recorded as well the Raman Fourier Transform spectra. Ab initio calculations have been carried with several ECPs, basis sets and methodologies (RHF and DFT) in order to assess family and methodological errors precisely. Geometry optimization and subsequent hessian calculation lead to the vibration frequencies reported. These calculated frequencies and intensities assisted the fundamental, overtones and combinations bands assignments. Remarkable agreement has been found between the experimental geometries and frequencies to those here calculated. Besides the bands traditionally studied for the dmit compounds, as C C and C S stretchings, also the region below 500 cm −1 was evaluated, allowing to characterize several modes involving angular deformation of the dmit as the M S stretching of these octahedral distorted systems.

PREFERENTIAL POLYMERIZATION OF 5-(AMINOMETHYL)-2-FURANCARBOXYLIC ACID (AMFC) INTO A CYCLIC TRIPEPTIDE

Theoretical justification for the preferential formation of an 18-membered cyclic oligopeptide 1 based on 5-(aminomethyl)-2-furancarboxylic acid (AMFC, 2), is attempted using ab initio quantum chemical calculations. The free energy values obtained from Hessian calculations show that while the formation of the higher order cyclized oligopeptides [ HN – H 2 C –( furan )– CO ]n with n=3,4, is energetically favorable, the formation of the cyclic dipeptide 3 is thermodynamically unfavorable. Inclusion of the solvent in calculations enhances this preference further. A plausible explanation for this is the proximity of repulsive charges on the furan O atoms in the dimer 3. Details of geometric parameters and Mulliken charges support this possibility. The repulsion is enhanced, as low frequency modes of nuclear motion that bring these O atoms closer are active at room temperature. [Formula: see text]

5-Hydroxycyclooctanone–hemiacetal rearrangement: ab initio mechanistic study

Ab initio theoretical studies are carried out to investigate the relationship between reactivity and conformation in 5-hydroxy cyclooctanone. Calculations at MP2/6-31G* and B3LYP/6-31G* levels are reported along with AIM calculations for critical points of vanishing density gradients. AIM results establish interaction between the carbonyl carbon and the nucleophilic oxygen from the hydroxyl group. The transition state geometry is investigated in detail. The analysis indicates that the geometric locations of the atoms involved in one of the local minima of 5-hydroxy cyclooctanone are conveniently arranged for the cyclization. Intrinsic reaction coordinate profile is obtained to monitor the intramolecular nucleophilic attack of the hydroxyl group at the C atom of the carbonyl moiety with a concerted shift of H atom from the hydroxyl group to the carbonyl oxygen. Thermodynamic data from Hessian calculations indicate much higher stability of the cyclized hemiacetal formed. The same degree of thermodynamic stability of the cyclized product may not be possible if such a geometric arrangement is not available.

B3LYP/6-311++G ** study of α- and β- d-glucopyranose and 1,5-anhydro- d-glucitol: 4C1 and 1C4 chairs, 3,OB and B3,O boats, and skew-boat conformations

Geometry optimization, at the B3LYP/6-311++G ** level of theory, was carried out on 4 C 1 and 1 C 4 chairs, 3,O B and B 3,O boats, and skew-boat conformations of α- and β- d-glucopyranose. Similar calculations on 1,5-anhydro- d-glucitol allowed examination of the effect of removal of the 1-hydroxy group on the energy preference of the hydroxymethyl rotamers. Stable minimum energy boat conformers of glucose were found, as were stable skew boats, all having energies ranging from ∼4–15 kcal/mol above the global energy 4 C 1 chair conformation. The 1 C 4 chair electronic energies were ∼5–10 kcal/mol higher than the 4 C 1 chair, with the 1 C 4 α-anomers being lower in energy than the β-anomers. Zero-point energy, enthalpy, entropy, and relative Gibbs free energies are reported at the harmonic level of theory. The α-anomer 4 C 1 chair conformations were found to be ∼1 kcal/mol lower in electronic energy than the β-anomers. The hydroxymethyl gt conformation was of lowest electronic energy for both the α- and β-anomers. The glucose α/β anomer ratio calculated from the relative free energies is 63/37%. From a numerical Hessian calculation, the tg conformations were found to be ∼0.4–0.7 kcal/mol higher in relative free energy than the gg or gt conformers. Transition-state barriers to rotation about the C-5–C-6 bond were calculated for each glucose anomer with resulting barriers to rotation of ∼3.7–5.8 kcal/mol. No energy barrier was found for the path between the α- gt and α- gg B 3,O boat forms and the equivalent 4 C 1 chair conformations. The α- tg conformation has an energy minimum in the 1 S 3 twist form. Other boat and skew-boat forms are described. The β -anomer boats retained their starting conformations, with the exception of the β- tg- 3,O B boat that moved to a skew form upon optimization.

Efficient characterization of stationary points on potential energy surfaces

Traditional methods for characterizing an optimized molecular structure as a minimum or as a saddle point on the nuclear potential energy surface require the full Hessian. However, if f denotes the number of nuclear degrees of freedom, a full Hessian calculation is more expensive than a single point geometry optimization step by the order of magnitude of f. Here we present a method which allows to determine the lowest vibrational frequencies of a molecule at significantly lower cost. Our approach takes advantage of the fact that only a few perturbed first-order wave functions need to be computed in an iterative diagonalization scheme instead of f ones in a full Hessian calculation. We outline an implementation for Hartree–Fock and density functional methods. Applications indicate a scaling similar to that of a single point energy or gradient calculation, but with a larger prefactor. Depending on the number of soft vibrational modes, the iterative method becomes effective for systems with more than 30–50 atoms.