Computational method

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

In our computation of invariant circles of maps the main cost lies in the 

FIGURE 10 Change of the torus and the angular function with W o. (a) A succession of computed sections of invariant tori for various values of / o (Brusselator, a = 0.0072). The centre point is indicated by (+). (b, c) The occupancy of the converged Jacobian for (u/gjo = 1.186667 and 1.3, respectively. The bumps on some of the circles are artifacts of the mesh and are associated with the almost vertical parts of the nonzero band. They can be eliminated by mesh adaptation. 

An alternative formulation for the torus-computing algorithm is to solve for an invariant circle along with a nonlinear change of coordinates that makes the action of the stroboscopic map conjugate to a rigid rotation on the circle. This is equivalent to the parameterization 

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

Computational methods are increasingly valuable supplements to experiments and theories in the quest to understand complex liquids. Simulations and computations can be aimed at either molecular or microstructural length scales. The most widely used molecular-scale simulation methods are molecular dynamics. Brownian dynamics, and Monte Carlo sampling. Computations can also be performed at the continuum level by numerical solutions of field equations or by Stokesian dynamics methods, described briefly below. 

Computational methods are now used extensively by experimental chemists. Information that can be calculated includes the equilibrium geometry of a molecule, transition state geometries, heats of formation, composition of molecular orbitals, vibrational frequencies, electronic spectra, reaction mechanisms and (from molecular mechanics calculations) strain energies. The last 20 years have witnessed a huge increase in the use of computational methods in chemistry. Two factors have revolutionized the ways in which computational chemistry may be applied. The first is that calculations can now be performed on small computers (including laptops) or small clusters of computers, instead of on a mainframe computer. The second is the development of the computational methods themselves. The importance of 

Development of the theory to allow these intermolecular interactions to be accurately computed is currently (as of 2011) ongoing.  

At a simple level, Huckel MO theory (proposed in the 1930s by Erich Huckel) works well for dealing with the 7r-systems of unsaturated organic molecules. By extending the basis set and including overlap and all interactions (a and tt), Roald Hoffmann showed that extended Huckel theory could be applied to most hydrocarbons. This theory has since been developed further and continues to be a useful method to determine the relative energies of different conformers of an organic molecule. 

The following terms were introduced in this chapter. Do you know what they mean and what the techniques are used for  

Brisdon (1998) Inorganic Spectroscopic Methods, Oxford University Press, Oxford - An OUP Primer covering basic vibrational and electronic spectroscopies and mass spectrometry. 

Computer simulation methods have been extensively used to probe the structural behavior of ionically conducting solids, with particular emphasis on the preferred diffusion mechanisms and the nature of intrinsic (thermally induced) and extrinsic (generated by chemical doping) defects. As discussed elsewhere [6], these techniques can broadly be divided into three categories  

In this section, we introduce computational methods employed in surface adsorption studies. Examples of theoretical studies of the adsorption of molecular oxygen and ORR intermediates on transition metals and alloys will be provided. Insights and findings from theoretical modeling will also be discussed. 

Both cluster and stab methods are employed to investigate interactions between the surface and adsorbates [9, 10, 32]. Usually, the solvent is ignored. These methods work best if a strong chemisorption bond is formed so that the other interactions are secondary. The adsorption energy is calculated using Equation 5.6  

For adsorption on metal, Hammer and Norskov [33] have proposed a model that can predict the adsorption bond strength based on the electronic properties of the metal. According to this model (Equation 5.7), three surface properties contribute to the ability of the surface to make and break adsorbate bonds (1) the energy center Sd of the cf-bands, defined as the centroid of the d-type density of states in an atomic sphere centered at a surface atom, (2) the degree of fitting / of the c/-bands (number of d electrons), and (3) the coupling matrix element V between the adsorbate states and the metal d-states  

An extensive review on structures, configurations and dynamics of bioactive oligosaccharides included a section on computational studies.  

Conformational energy maps for the furanosyl and pyranosyl rings of ribose and 2-deoxyribose in solution, generated with the MM3 force field, indicated the presence of several tautomers in multiple conformations for both compounds. The preferred conformations and energy pseudorotational barriers of 2-deoxy-P-D-ribofuranosylamine in protonated and unprotonated form have been established by use of ab initio molecular orbital and density functional theory calculations. Three-, two- and one-bond spin coupling constants in anhyd- 

Further monosaccharide pyranosyl compounds to have been investigated by use of computational techniques were the anticancer agent etoposide in CD3OD, dry CDCI3 and wet CDCI3, and a series of ellagitannin models with axial chirality in the hexahydroxydiphenoyl moieties, such as compound 3.  

Modifications have been made to the AMBER force field to improve the correlations between calculated and observed molecular properties of a-linked saccharides these led to refinements in solvation studies on maltose, a-, p- and Y-cyclodextrin and two larger cyclodextrins (DP 10 and 21). A molecular dynamics simulation investigation of the solvation patterns of the model disaccharide 4 in aqueous DMSO defined regions in which competition exists 

Carbohydrate Chemistry, Volume 34 The Royal Society of Chemistry, 2003 

One of the fundamental considerations in the application of theoretical techniques in chemistry is the likely consumption of computational resources. Despite the enormous advances in the speed of computing hardware, there are always systems which are too large for a given method/hardware combination. One must therefore stick to the smaller, tractable systems or change the method (or buy the next generation machine with the faster processor ). 

The essence of the problem is well known. Ideally, one would like to apply first principles or ab initio methods since a completely general ab initio scheme can, in principle, handle any chemical system. This cannot be achieved in practice except for the simplest cases - i.e. one and possibly two electron atoms A succession of approximations ensues although the top levels are still often referred to as ab initio. With each successive level of approximation, the calculation becomes easier and hence larger molecules can be treated. The price is a decrease in accuracy or a limitation as to the types of molecule which can be studied or perhaps both. 

At the opposite end of the scale from ab initio theory are the completely empirical approaches such as Molecular Mechanics. MM is very simple and hence very fast allowing large molecules such as proteins to be handled in a reasonable time. The penalty is that MM must be parameterised by recourse to fitting experimental data for a given class of molecule and hence each para-meterisation is more or less specific to that class. New classes, and this includes TM complexes, require new parameterisations. 

Two main aspects of the present contribution can be generalized and formulated as follows. Firstly, we compute both electronic (Sect. 3.2.2) and vibrational (Sect. 3.2.4) contributions to (hyper)polarizability. Thus, we explore the limitations of the currently available computational procedures. Secondly, we associate the linear and nonlinear optical properties of investigated organofullerenes with their electronic structure (Sect. 3.2.3). The purpose of the analysis of the relations between NLO properties and structure of organofullerenes is to make a basis for further rational design of new [60]fullerene derivatives suitable for photonic applications. 

In the presence of static uniform electric field, the total energy of molecular system can be expressed as a Taylor series  

The dipole moment of a molecule in the presence of uniform electric field is given by  

Both Eqs.(3.1) and (3.2) can be used for determination of polarizability (a), first-ip) and second-order hyperpolarizability (7). The averaged (hyper)polarizabilities used throughout the book are defined in the following way [28-31]  

Within the Born-Oppenheimer approximation, the (hyper)polarizabilities can be divided into two parts, namely electronic and vibrational [28-31]  

Many of the sulfur-rich compounds considered in this chapter are unstable reactive species so that important properties such as geometrical structures, vibrational spectra and reaction energies are difficult to obtain experimentally and remain uncertain. In these cases, theory is particularly suited to provide the necessary complementary information to understand and interpret the experimental observations for these systems. 

Before 1980, force field and semiempircal methods (such as CNDO, MNDO, AMI, etc.) [1] were used exclusively to study sulfur-containing compounds due to the lack of computer resources and due to inefficient quantum-chemical programs. Unfortunately, these computational methods are rather hmit-ed in their reliability. The majority of the theoretical studies under this review utilized ab initio MO methods [2]. Not only ab initio MO theory is more reliable, but also it has the desirable feature of not relying on experimental parameters. As a consequence, ab initio MO methods are apphcable to any systems of interest, particularly for novel species and transition states. 

The choice of basis set in ab initio calculations has been the subject of numerous theoretical studies. Early SCF calculations utilized mainly spht-va-lence basis sets such as 3-21G and 4-31G. The importance of inclusion of d polarization functions on sulfur atoms has been stressed by several authors. For instance, Suleimenov and Ha found that the omission of d polarization functions leads to a substantially lower barrier for the internal rotation ( 16 kj mol for the central bond of H2S4) and produces an unreahstically large S-S bond length for the most stable rotamer [4]. In general, the use of 

To obtain a satisfactory evaluation of relative energies, especially for the computation of activation barriers, higher levels of theory than those needed to obtain the underlying geometries are usually required. MP2 is the most economical and popular method of incorporation electron correlation. For a more accurate theoretical estimate, higher-level of correlation treatment such as QCISD(T) or CCSD(T) theory is desirable. 

Extensive comparisons of experimental frequencies with HF, MP2 and DFT results have been reported [7-10]. Calculated harmonic vibrational frequencies generally overestimate the wavenumbers of the fundamental vibrations. Given the systematic nature of the errors, calculated raw frequencies are usually scaled uniformly by a scaling factor for comparison with the experimental data. 

FIGURE 1.21 Trajectories of the intermolecular energies for (a) the strong -complex and (b) the weak / -complex between 0-9-(r rr-butylcarbamoyl)-6 -neopentoxy-cinchonidine and DNB-Leu over the 1 ns simulation period in the polar (water) medium. (Top) Coulomb energy (middle) Van der Waals energy (bottom) total energy. (Reprinted from N.M. Maier et al., J. Am. Chem. Soc., 124 8611 (2002). With permission.) 

The electric-structure-calculation presented here is performed using the CASTEP computer code, which is based on density functional theory, aided by the CERIUS2 graphical front-end. The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. 

The most important ZnS surface is the (110), which is the most common growth surface and is also the perfect cleavage surface. Therefore, the calculation is based on the ZnS (110) surface. The surfaces are cleaved from the bulk ZnS with the optimtun tinit cell volume determined using the GGA with CASTER The Cu and Fe doped surfaces are built by the substitution of Cu or Fe for Zn atom on the cleaved surface. A vacuum spacing of 1.5 nm is inserted in the z-direction to form a slab and mimic a 2D surface. In order to eliminate the interactions between mirror images in the z-direction due to the periodic boimdary conditions, in test calculations, we have done some total energy calculation to find a proper thickness of slab. The result shows that 1.5 run is a desirable thickness. 

In the calculation, we have examined the effect of some parameter on total energy and stmcture of surface and crystal. These parameters include plane wave cutoff energy, k-point and SCF convergence criteria, and the geometry optimization convergence criteria for the geometry optimization task. 

The calculation results indicate that a plane wave cutoff energy of 280 eV and Monkhorst-Paek k-point sampling density of 4 x 4 x 4 are sufficient for the lattice constant and total energy to converge to within 0.0005 A and lO eV respectively. For surface relaxation, a plane wave cutoff of 280 eV and a 4 x 4 x 1 k-point mesh are sufficient to converge the surface geometry to within 0.001 A and relaxed surface energies to within 0.001 Vrc.  

For each system considered, an appropriate computer program was written for solving the equations involved in the modeling presented in Section 4.1. The first-order nonlinear differential equations were solved by numerical integration using the Runge-Kutta procedure [141]. 

In structure 335, one of the acetate and the solvent molecule of 334 are replaced with a hydride and a PPhj ligand. Also, in 335 in place of acetate, a MejCCO Ugand is present. The use of this ligand makes crystal growing and characterization of 335 easier. Structure 335 is another model intermediate. In the actual catalytic cycle where no PPhj is present, a similar hydride species with an alkene in place of PPhj is Ukely to be involved. 

Computational methods are used for structure predictions as well as energy calculations of catalytic intermediates and transition states. They have also been used for predicting optimum Ugand and catalyst structures, as well as conditions under which best performance might be expected. We first discuss two examples to show that there is good agreement between experimental data and computational chemistry-based theoretical predictions. 

As mentioned earlier, in the asymmetric hydrogenation of methyl a-acetamido cinnamate, in situ PHIP-NMR data suggest structure 335 for the dihydride intermediate. Computational energy calculations based on density functional theory (DFT) are also in agreement with such a structure. In this case, to keep the computational requirements to a manageable level, calculations were carried out on a model complex where PHANEPHOS was approximated by two PH3 ligands. 

The reactions shown in (3.5.1) are crucial steps in the catalytic cycles for methanol carbonylation with rhodium or iridium catalysts (see Section 4.2.4). The theoretically calculated free energies of activation (AG ) for 3.36 3.37 3.38 are -19.0 and -28.0 for iridium and 27.0 

The experimentally determined values agree reasonably well with the calculated ones. Also, qualitatively, the theoretical results correctly predict oxidative addition for rhodium but insertion for iridium to be the rate-determining steps. 

The rigorous steady-state and dynamic models used in this book are solved using Matlab programs or Aspen Technology simulation software (Aspen Plus and Aspen Dynamics). 

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Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

Progress in experiment, theory, computational methods and computer power has contributed to the capability to solve increasingly complex structures [28, 29]. Figure Bl.21.5 quantifies this progress with three measures of complexity, plotted logaritlmiically the achievable two-dimensional unit cell size, the achievable number of fit parameters and the achievable number of atoms per unit cell per layer all of these measures have grown from 1 for simple clean metal... 

Nemoshkalenko V V and Antonov V N 1998 Computational Methods in Solid State Physics (Amsterdam Gordon and Breach) An explicit introduction to the all-electron methods. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Ajay and Murcko, 1995] Ajay, ajid Murcko, M. Computational methods to predict binding free energy in ligand-receptor complexes. J. Med. Chem. 38 (1995) 4953-4967... 

J. C. Simo, N. Tarnow, and K. K. Wang. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100 63-116, 1994. 

It was reahzed quite some decades ago that the amount of information accumulated by chemists can, in the long run, be made accessible to the scientific community only in electronic form in other words, it has to be stored in databases. This new field, which deals with the storage, the manipulation, and the processing of chemical information, was emerging without a proper name. In most cases, the scientists active in the field said they were working in "Chemical Information . However, as this term did not make a distinction between librarianship and the development of computer methods, some scientists said they were working in "Computer Chemistry to stress the importance they attributed to the use of the computer for processing chemical information. However, the latter term could easily be confused with Computational Chemistry, which is perceived by others to be more limited to theoretical quantum mechanical calculations. 

The field of chemoinformatics was not founded, nor was it formally installed. It slowly evolved from several, often quite humble, begmnings. Scientists in various fields of chemistry struggled to develop computer methods in order to manage the... 

Chemistry, like any scientific discipline, relies heavily on experimental observations, and therefore on data. Until a few years ago, the usual way to publish information on recent scientific developments was to release it in books or journals. In chemistry, the enormous increase in the number of compounds and the data concerning them resulted in increasingly ineffective data-handling, on the side of the producers as well as the users. One way out of this disaster is the electronic processing, by computer methods, of this huge amount of data available in chemistry. Compared with other scientific disciplines that only use text and numbers for data transfer, chemistry has an additional, special challenge molecules. The molecular species consist of atoms and bonds that hold them together. Moreover, compounds... 

The 3D structure of a raolectile can be derived either from experiment or by computational methods. Regardless of the origin of the 3D model of the molecule under consideration, the user should alway.s be aware of how the data were obtain-... 

An extensive series of studies for the prediction of aqueous solubility has been reported in the literature, as summarized by Lipinski et al. [15] and jorgensen and Duffy [16]. These methods can be categorized into three types 1 correlation of solubility with experimentally determined physicochemical properties such as melting point and molecular volume 2) estimation of solubility by group contribution methods and 3) correlation of solubility with descriptors derived from the molecular structure by computational methods. The third approach has been proven to be particularly successful for the prediction of solubility because it does not need experimental descriptors and can therefore be applied to collections of virtual compounds also. 

The investigation of molecular structures and of their properties is one of the most fascinating topics in chemistry. Chemistry has a language of its own for molecular structures which has been developed from the first alchemy experiments to modem times. With the improvement of computational methods for chemical information processing, several descriptors for the handling of molecular information have been developed and used in a wide range of applications. 

IL iiicliiilcs general references to aid you in more ileiailed study. Although these references arenot comprehensive, some are key references. Other references provide examples of research problem s n sing these computational methods. 

The classical introduction to molecular mechanics calculations. The authors describe common components of force fields, parameterization methods, and molecular mechanics computational methods. Discusses th e application of molecular mechanics to molecules comm on in organic,and biochemistry. Several chapters deal w ith thermodynamic and chemical reaction calculations. 

AMI is generally the most accurate computational method included in IlyperChem and is often the best method for collecting qiian tiiative in formation. PM3 is function ally similar to AM 1. but uses an alternative parameter set (see PM3" on page 150). 

An N-atom molecular system may he described by dX Cartesian coordinates. Six independent coordinates (five for linear molecules, three fora single atom) describe translation and rotation of the system as a whole. The remaining coordinates describe the nioleciiUir configuration and the internal structure. Whether you use molecular mechanics, quantum mechanics, or a specific computational method (AMBER, CXDO. etc.), yon can ask for the energy of the system at a specified configuration. This is called a single poin t calculation. ... 

Editor) 1997. Computational Methods for the Analysis of Molecular Diversity. Perspectives in Drug "Muery and Design Volumes 7/8. Dordrecht, Kluwer. 

C and T Lengauer 2000. Computational Methods for the Structural Alignment of Molecules. nal of Computer-Aided Molecular Design 14 215-232. 

Donea, J., 1992. Arbitrary Lagrangian-Eulerian finite element methods. In Belytschko, T. and Hughes, T. J. R. (eds), Computational Methods for Transient Analysis, Elsevier Science, Amsterdam. 

Zienkiewicz, O. C. et al, 1985. Iterative method for constrained and mixed approximation, an inexpensive improvement to f.e.m. performance. Comput. Methods Appl. Meek Eng. 51, 3-29. 

Donea, J. and Quartapelle, L., 1992. An introduction to finite element methods for transient advection problems. Comput. Methods Appl Meek Eng. 95, 169-203. 

Fox, L. and Mayers, D.F., 1977. Computing Methods for Scientists and Engineers, Clarendon Press, Oxford. 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

Carley, A. F. Morgan, P. H., 1989. Computational Methods in the Chemical Sciences. Halsted Press (Wiley) New York. 

Greenwood, H. H., 1972. Computing Methods in Quantum Organic Chemistry. Wiley Interscience, New York. 

Recently, molecular dynamics and Monte Carlo calculations with quantum mechanical energy computation methods have begun to appear in the literature. These are probably some of the most computationally intensive simulations being done in the world at this time. 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

If you cannot specifically answer these questions, then you have not formulated a proper research project. The choice of computational methods must be based on a clear understanding of both the chemical system and the information to be computed. Thus, all projects start by answering these fundamental questions in full. The statement To see what computational techniques can do. is not a research project. However, it is a good reason to purchase this book. 

There are quite a number of ways to effectively change the equation in an SCF calculation. These include switching computation methods, using level shifting, and using forced convergence methods. 

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