Mechanics molecular

Molecular mechanics (MM) calculations have been employed for determining dihedral angles and to establish a comparison with values calculated from coupling constants, during conformational studies of tricyclic and tetracyclic quinolizidine alkaloids. The MM results had to be treated with care, as they sometimes predicted ring conformations different to those supported by experimental data 1999JST215 . 

Molecular mechanics as a minimization of strain energy makes a rigid distinction between steric and electronic effects. Electronic effects are introduced in the form of empirical constants such as characteristic bond lengths and angles, the corresponding force constants, torsional rigidity of even-order bonds, planarity of aromatic systems and the coordination symmetry at transition-metal centres. These constants are accepted, without proof, to summarize the ensual of electronic interactions and used without further optimization. 

Any molecular parameter which, in a trial structure, has a value at variance with the characteristic electronic standard, adds to the strain energy. It is considered a steric effect and subject to optimization. At convergence the actual molecular structure is recovered, providing that all empirical constants had been specified correctly. 

Because of the empirical specification of its electronic interaction parameters molecular mechanics is routinely discounted as devoid of theoretical underpinning. This conclusion is certainly too harsh. Apart from connectivity, the electronic basis of all other force-field concepts has already been demonstrated  

Bond length The Heitler-London method allows the calculation of all first-order diatomic interactions using valence-state wave functions as defined in terms of characteristic ionization radii. 

Bond order Any chemical bond is stabilized by a pair of shared electrons. When there is an excess of valence electrons over bonding pairs, bond dissociation energy is increased by the screening of nuclear repulsion, and the bond length is contracted accordingly, in discrete steps. 

The molecular mechanics (MM) or force field method is an empirical method based on classical mechanics and adjustable parameters. It has the disadvantage of being limited in its application to certain kinds of compounds for which the required parameters have been determined (experimentally or by theoretical calculations). Its advantage is a considerably shorter computation time in comparison with other procedures having the same purpose. This method has been shown to be very reliable and efficient in determing molecular geometries, energies, and other properties for a wide variety of compounds. 

In the early molecular mechanics studies in both inorganic and organic chemistry the strain energy E/totai is defined as arising from four principle energy terms (Eq. 2.1), 

More recently a number of additional components have been added to the calculation of the strain energy. Out-of-plane deformation terms Eg have been included in models of aromatic or sp2 hybridized systems (Eq. 2.6), 

The set of functions together with the collection of terms that parameterize them ( b, r0, etc.) is referred to as the force field. In some cases force field parameters can be related to experimentally determinable values. For example, the bond stretching force constant kb is approximately equivalent to the vibrational force constant derived from an infrared spectrum. However, in general the force field terms are derived empirically with the target of reproducing experimental structures and energy distributions. 

Once a model and a force field have been chosen for a particular problem, the goal of molecular mechanics is to find the geometry with the minimum strain energy. This can be achieved by a variety of mathematical techniques described else- 

The classical equations of motion used in molecular mechanics (MM) are only slightly more difficult to solve than simple additive bond energy equations hence MM calculations are fast and not very demanding of computer resources. In molecular mechanics, one determines the structure of a molecule from a knowledge of the force field, a collection of empirical force constants governing, in principle, all classical mechanical interatomic interactions within the molecule. In practice, it is not feasible for a parameter set to include all possible interactions within a complicated molecule. One hopes that all significant interactions have been included in the force field. 

The principle of classical equation solving, which is at the heart of MM, can be appreciated by imagining two objects, connected by a spring, executing simple harmonic motion collinear with the spring. If the force constant k of the spring is known, it is possible to calculate the potential energy V of the system at any separation x of the objects as 

the objects in the last paragraph are atoms and the springs are chemical interactions, including the chemical bonds that hold the molecule together along with other repulsive and attractive interactions within the molecule. At the potential energy minimum, the position of each atom will be as near to the bottom of its parabolic potential well as it can be without undue distortion of the forces acting on all the other atoms. 

AjSOm 298 = steric energy(tram) - steric energy(c/s) 

The enthalpy of isomerization as written is negative because the tram form is more stable than the cis. 

The quantum mechanical methods described so lar all properly treat the electrons as quantum particles. A vastly simpler approach toward obtaining molecular structures and energies is to treat atoms as classical particles. The potential energy is then just a function of the nuclear coordinates. MM, also referred to as force field methods, involves specifying the various functions used to relate nuclear positions to energy and fitting these functions to experimental data. It is a highly empirical approach, dependent on the choice of reference data, the functional form, and selection of parameters. 

The typical MM approach is to divide the total energy into components related 

Each of these terms will be briefly discussed in turn. 

The first term stretch is the energy associated with stretching a bond. Typically, one assumes a harmonic potential for this stretching, making an analogy between a chemical bond and a classical spring. This term can be written as 

The second term is the energy associated with bending about the angle formed of atoms ABC. This too is generally regarded as behaving like a spring  

Why force field . In many situations it is necessary to know about the forces between atoms. This is the case for molecular dynamics, but also for many molecular mechanics applications. According to Eq. (17), the forces F are calculated as the negative derivative of the potential energy E with respect to the coordinates r  

All the structural models in this book are computer-drawn. To make sure they accurately portray bond angles, bond lengths, torsional interactions, and steric interactions, the most stable geometry of each molecule has been calculated on a desktop computer using a commercially available molec-ular mechanics pYogmm based on work by N. L. Allinger of the University of Georgia. 

Computer programs make it possible to portray accurate representations of molecular geometry. 

FIGURE 4.18 The structure ofTamiflu (oseltamivir phosphate), an antiviral agent active against type A influenza, and a molecular model of its minimum-energy conformation as calculated by molecular mechanics. 

22 Name the following compound, identify each substituent as axial or equatorial, and tell whether the conformation shown is the more stable or less stable chair form (yellow-green = Cl)  

23 A trisubstituted cyclohexane with three substituents— red, green, and blue—undergoes a ring-flip to its alternative chair conformation. Identify each substituent as axial or equatorial, and show the positions occupied by the three substituents in the ring-flipped form. 

You were probably taught very early in your professional career that skills in quantum ehemistry are a prerequisite for the study of atomie and moleeular phenomena. I must tell you that this isn t completely true. Some moleeular phenomena can be modelled very accurately indeed using classieal meehanies, and to get us started in our study of moleeular modelling, we are going to study moleeular mechanics. This aims to treat the vibrations of eomplex moleeules by the methods of classical mechanics, and as we shall see, it does so very sueeessfully. 

In a nutshell (MM2 force field) No electrons, purely mechanical model Structural accuracy bond lengths, 0.01 A bond angles, 1° torsion angles, a few degrees Conformational energies accurate to 1 kcal mol-1 Vibrational frequencies accurate to 20-30 cm-1. 

We don t give a damn where the electrons are. Words to the author, from the president of a well-known chemical company, emphasizing 

The value of S-EWetching in the force field equation (see equation (5.1)) for a structure is given by the sum of appropriate expressions for E for every pair of bonded atoms in the structure. For example, using the Hooke law model, for a molecule consisting of three atoms bonded a-b-c the expression would be  

The other energy terms in the force field equation for a structure are treated in a similar manner using expressions appropriate to the mechanical or electrical 

Ehend = e(0 — 6o)2 where 0O is the ideal bond angle, that is, the minimum energy positions of the three atoms. 

Evaw = s[% 12 - 2( )6] where rm n is the distance between two atoms i and j when the energy is at a minimum e and r is the actual distance between the atoms. This equation is known as the Lennard-Jones 6-12 potential. The ( )6 term in this equation represents attractive forces, whilst the ( )12 term represents short range repulsive forces between the atoms. 

Electrostatic coulombic interactions Attractive or repulsive interactions 

Several attempts have been undertaken to derive such MM force constants for modeling zeolite frameworks [39]. Typical examples are the rigid ion and the shell model which assume that the character of the bonds in the lattice is largely ionic. Within the rigid ion model developed by Jackson and Catlow [40], the potential energy is given by 

An extension of the rigid ion model is the shell model taking additional ionic polarizabihties into account [46]. Whereas for silicon and aluminum atoms usually a low polarizability is assumed and they are, therefore, treated as rigid cations in the shell model, for oxygen anions and extra-framework cations an additional term of the form 

In summary, a high diversity of potential models for molecular mechanics calculations of zeolites hitherto exists. From the theoretical point of view, an appropriate force field should be able to predict structures and vibrations with similar accuracy. On the other hand, the structure of a system under study is determined by the energy minimum, whereas normal modes are dependent on the curvature (second derivative) of the potential energy surface. Consequently, force fields obviously successful in predicting structural features might not automatically be appropriate for simulating vibrational spectra. The only way to overcome this difficulty is to include experimental spectroscopic data into the parametrization process [60]. Alternatively, besides structures and energies a matrix of force constants obtained in quantum mechanical calculations can be included into the quantum mechanical data base used to tune the parameters of the potential function [51]. 

See Westheimer, F. H. in Newman, M. S., Ed. Steric Effects in Organic Chemistry John Wiley Sons New York, 1956 p. 523 f Westheimer, F. H. Mayer, J. E.. Chem. Phys. 1946, 14, 733 Westheimer, F. H. /. Chem. Phys. 1947,15, 252. Similar ideas were developed independently by Hill, T. L. /. Chem. Phys. 1946, 14, 465. 

This definition is revised slightly from that given by Hagler, A. T. Stem, P. S. Lifson, S. Ariel, S. 

In order to carry out a molecular mechanics calculation, all the force constants for each type of atom under consideration must be known. In the case of MM2, only some atom types were included at the time of development of the procedure. Others were added later, in some cases by other researchers. Lipkowitz has provided a summary of parameter sources Lipkowitz, K. B. QCPE Bulletin 1992,12, 6. 

Unlike conformational analysis, in which relative energies are compared by assigning fixed amounts of strain to specific interactions (such as butane gauche interactions), molecular mechanics determines the energy of a conformation by computing the value of a mathematical function. An equation such as equation 3.3 can be used to calculate the total steric energy of the molecule as the sum of a number of different kinds of interactions. Most molecular mechanics methods include as a 

For example, the MM3 calculation of the heat of formation of Qo was much closer to the experimental value than were the values calculated by the molecular orbital methods to be discussed in Chapter 4. Beckhaus, H.-D. Riichardt, C. Kao, M. Diederich, F. Foote, C. S. Angeiv. Chem. Int. Ed. Engl. 1992, 31, 63. 

a molecule is modeled as a collection of masses (atoms) and springs (bonds), with additional forces added to describe other interactions such as hydrogen bonding, electrostatics, and dispersion forces. Although such simulations have been done using carefully constructed mechanical models [22], MM has been most successfully implemented computationally. The present discussion will focus on the MM3 method [13], since it is popular and is implemented in a number of software packages. Beware that not all implementations of MM3 provide thermochemical information. 

Sample results and approximate computational times for various computations of gas-phase thermochemistry 

Siegel, J., Gutierrez, A., Schweizer, W. B., Ermer, O., and Mislow, K. Static and Dynamic Stereochemistry of Hexaisopropy Ibenzene A Gear-Meshed Hydrocarbon of Exceptional Rigidity. /. Am. Chem. Soc., 108, 

Can we put this analysis on a quantitative basis Could we develop some sort of equation that relates the extent of distortion to the energy of the molecule We can, and the method is called molecular mechanics. Here we will lay out the basic tenants of molecular mechanics and provide a description of its strengths and weaknesses. The method is now quite common and easily implemented for sizable molecules on a standard personal computer. It is a powerful aid to experimentalists in all fields of organic chemistry, as well as in molecular-scale studies of biology and materials science. It should be appreciated from the start, however, that the method has significant limitations and is susceptible to misuse. 

An excellent source of much thermochemical data is S. J. Ashcroft and C. T. Mortimer, Thermochemistry of transition metal complexes. Academic Press, London and New York 1979, although, today, computer databases are better if access can be gained to one. 

The classic reference for the material in all but the last two sections of this chapter is P. George and D. S. McClure in Prog. Inorg. Chem. (1959) 1, 381. 

The approach to crystal field stabilization energies adopted in this chapter is one that bypasses the problem of pairing 

Molecular mechanics is covered in Molecular Mechanics Calculations as a Tool in Coordination Chemistry R. D. Hancock, Prog. Inorg. Chem. (1989) 37, 187 Methods for molecular mechanics modelling of coordination compounds B. P. Hay, Coord. Chem. Rev. (1993) 126,177 and a review with a wide variety of examples The relation between ligand structures, coordination stereochemistry, and electronic and thermodynamic properties by P. Comba, Coord. Chem. Rev. (1993) 123, 1. 

1 Explain carefully what is meant by crystal field stabilization energy and compare this term with the alternative, ligand field stabilization energy used by some textbooks. 

The force constants are conveniently represented in a matrix F, each of whose rows and columns corresponds to one of the coordinates of the chosen set. The F matrix is symmetrical for water (see Fig. 2.1) the numbers are [3]  

Given the definition of the force constants, the total vibrational potential energy is 

In the early 1950s, spectroscopic and diffraction data on molecular structure were beginning to accumulate, but there was no safe and sound method for predicting 

There are several conceptual arguments against this molecular mechanics construction, and there were many difficulties in its actual implementation. First, molecular vibrations are quantized, and are not near an equipartition regime at ordinary temperatures, so vibrational energies cannot be considered as a continuum. Second, organic molecules are multiform, while experimental force constants are available only for a handful of simple systems. Third, it must be assumed that equilibrium structural parameters and force constants are transferable among chemical systems, because if new parameters are to be found for each new molecule to be treated, the method loses much if not all of its practical appeal. Then, of course, the whole machinery rests upon a judicious choice of these transferable parameters. 

The strain energy in terms of parameters related to vibrational spectroscopy contains terms for bond stretching, bond angle bending, and dihedral vibration (torsion). 

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A considerable number of experimental extensions have been developed in recent years. Luckliam et al [5] and Dan [ ] review examples of dynamic measurements in the SFA. Studying the visco-elastic response of surfactant films [ ] or adsorbed polymers [7, 9] promises to yield new insights into molecular mechanisms of frictional energy loss in boundary-lubricated systems [28, 70]. 

Lowack K and Helm C A 1998 Molecular mechanisms controlling the self-assembly process of polyelectrolyte multilayers Macromolecules 31 823-33... 

At any geometry g.], the gradient vector having components d EjJd Q. provides the forces (F. = -d Ej l d 2.) along each of the coordinates Q-. These forces are used in molecular dynamics simulations which solve the Newton F = ma equations and in molecular mechanics studies which are aimed at locating those geometries where the F vector vanishes (i.e. tire stable isomers and transition states discussed above). 

A very recent overview, including efforts to interface semi-empirical electronic structure with molecular mechanics treatments of some degrees of freedom is given by ... 

The situation is more complicated in molecular mechanics optimizations, which use Cartesian coordinates. Internal constraints are now relatively complicated, nonlinear functions of the coordinates, e.g., a distance constraint between atoms andJ in the system is — AjI" + (Vj — + ( , - and this... 

The complexity of polymeric systems make tire development of an analytical model to predict tlieir stmctural and dynamical properties difficult. Therefore, numerical computer simulations of polymers are widely used to bridge tire gap between tire tlieoretical concepts and the experimental results. Computer simulations can also help tire prediction of material properties and provide detailed insights into tire behaviour of polymer systems. A simulation is based on two elements a more or less detailed model of tire polymer and a related force field which allows tire calculation of tire energy and tire motion of tire system using molecular mechanisms, molecular dynamics, or Monte Carlo teclmiques 1631. 

Flelm C A, Israelachvili J N and McGuiggan P M 1989 Molecular mechanisms and forces involved in the adhesion and fusion of amphiphilic bilayers Science 246 919-22... 

The molecular mechanism of ion enliancement proved to be both subtle and complex. The complexity of the... 

For this reason, there has been much work on empirical potentials suitable for use on a wide range of systems. These take a sensible functional form with parameters fitted to reproduce available data. Many different potentials, known as molecular mechanics (MM) potentials, have been developed for ground-state organic and biochemical systems [58-60], They have the advantages of simplicity, and are transferable between systems, but do suffer firom inaccuracies and rigidity—no reactions are possible. Schemes have been developed to correct for these deficiencies. The empirical valence bond (EVB) method of Warshel [61,62], and the molecular mechanics-valence bond (MMVB) of Bemardi et al. [63,64] try to extend MM to include excited-state effects and reactions. The MMVB Hamiltonian is parameterized against CASSCF calculations, and is thus particularly suited to photochemistry. 

Singh, U.C., Kollman, P.A. A combined ab initio quantum mechanical and molecular mechanical method for carrying out simulations on complex molecular systems Applications to the CH3CI 4- Cl exchange reaction and gas phase protonation of polyethers. J. Comput. Chem. 7 (1986) 718-730. 

Field, M.J., Bash, P.A., Karplus, M. A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations. J. Comput. Chem. 11 (1990) 700-733. 

Bakowies, D., Thiel, W. Hybrid models for combined quantum mechanical and molecular mechanical approaches. J. Phys. Chem. 100 (1996) 10580-10594. 

Stanton, R.V., Heutsough, D.S., Merz, K.M. Jr An Examination of a Density Functional-Molecular Mechanical Coupled Potential. J. Comput. Chem. 16 (1995) 113-128. 

Grubmiiller et al., 1996] Grubmiiller, H., Heymann, B., and Tavan, P. Ligand binding and molecular mechanics calculation of the streptavidin-biotin rupture force. Science. 271 (1996) 997-999... 

A number of issues need to be addressed before this method will become a routine tool applicable to problems as the conformational equilibrium of protein kinase. E.g. the accuracy of the force field, especially the combination of Poisson-Boltzmann forces and molecular mechanics force field, remains to be assessed. The energy surface for the opening of the two kinase domains in Pig. 2 indicates that intramolecular noncovalent energies are overestimated compared to the interaction with solvent. 

Additionally, as in all Tl-based approaches, the free energy differences are linear functions of the potential. Thus non-rigorous decompositions may be made into contributions from the different potential energy terms, parts of system and individual coordinates, providing valuable insight into the molecular mechanisms of studied processes [8, 9, 10). 

In molecular mechanics and molecular dynamics studies of proteins, assig-ment of standard, non-dynamical ionization states of protein titratable groups is a common practice. This assumption seems to be well justified because proton exchange times between protein and solution usually far exceed the time range of the MD simulations. We investigated to what extent the assumed protonation state of a protein influences its molecular dynamics trajectory, and how often our titration algorithm predicted ionization states identical to those imposed on the groups, when applied to a set of structures derived from a molecular dynamics trajectory [34]. As a model we took the bovine... 

Very recently, we have developed and incorporated into the CHARMM molecular mechanics program a version of LN that uses direct-force evaluation, rather than linearization, for the fast-force components [91]. The scheme can be used in combination with SHAKE (e.g., for freezing bond lengths) and with periodic boundary conditions. Results for solvated protein and nucleic-... 

The two ways of learning - deductive and inductive - have already been mentioned. Quite a few properties of chemical compounds can be calculated explicitly. Foremost of these are quantum mechanical methods. However, molecular mechanics methods and even simple empirical methods can often achieve quite high accuracy in the calculation of properties. These deductive methods are discussed in Chapter 7. 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

To see the contributions to the molecular mechanics potential energy function and their mathematical representation... 

To get an overview of the currently available software for molecular mechanics calculations with their strengths, weaknesses, and application areas... 

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