Framework Cartesian

In the framework of the force field calculations described here we work with potential constants and Cartesian coordinates. The analytical form of the expression for the potential energy may be anything that seems physically reasonable and may involve as many constants as are deemed feasible. The force constants are now derived quantities with the following definition expressed in Cartesian coordinates (x ) ... 

This is the operation of inverting all points in a body abont some centre, i.e. if the centre is 0, then any point A is moved to A on the line AO such that OA = OA or put another way, if a set of Cartesian axes have their origin at 0, a point with coordinates (x, y, z) is moved to (— z, —y, —z), If this operation brings the nnclear framework into coincidence with itself, the molecule is said to have a centre of symmetry as a symmetry element and this is symbolized by i (no relation to y —l). In Fig. 2-3.4 we show the inversion operation for an octahedral framework. 

Figure 8 Building a molecular model based on internal geometries (bond lengths /, bond angles 6, and torsional angles ). Each subsequent atom is added to the framework with respect to earlier situated atoms. The convention in many programs is that the x Cartesian axis is the horizontal axis on the computer screen, the y axis is vertical, and the z axis comes out of the computer screen toward the user.

For molecular systems in the vacuum, exact analytical derivatives of the total energy with respect to the nuclear coordinates are available [22] and lead to very efficient local optimization methods [23], The situation is more involved for solvated systems modelled within the implicit solvent framework. The total energy indeed contains reaction field contributions of the form ER(p,p ), which are not calculated analytically, but are replaced by numerical approximations Efp(p,p ), as described in Section 1.2.5. We assume from now on that both the interface Y and the charge distributions p and p depend on n real parameters (A, , A ). In the geometry optimization problem, the A, are the cartesian coordinates of the nuclei. There are several nonequivalent ways to construct approximations of the derivatives of the reaction field energy with respect to the parameters (A1 , A ) ... 

To write down Eq. (4.53) as well as (4.51), we used, after Ref. 54, the distribution function moments presented as Cartesian tensors. However, when solving the orientational problem, it is more natural to use the set of spherical functions. Choosing spherical coordinates for the unit vectors e, it, and h as (Q, cp), (0,0), (v(/,0), respectively, that is, taking as the polar axis of the framework, one gets... 

In order to demonstrate the efficiency of the present theory for systems with many degrees of freedom, we have applied it to a 4-D model of HCN CNH isomerization (i.e., isomerization in a plane). The system is described in terms of two vectors J n=c for the vector from N to C, and Ru for the vector from the center of mass of the system to H. A fixed spatial Cartesian framework is used, with the a -axis set to be parallel to the initial direction of Rn=c, and the y- and 2 -axes perpendicular to it. The center of mass of NC is assumed to be the same as that of the whole system so that the kinetic part of the Hamiltonian is diagonal. The potential energy surface and the dipole moment are taken from [32]. 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

The basic equations of the -method will be presented later within the framework of the more general r -fit problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitch-man s equations (see below) to localize an atomic position requires (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman s equations must be carried out. 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

Despite the interest to obtain AO integral algorithms over cartesian exponential orbitals or STO fimctions [43] in a computational universe dominated by GTO basis sets [2], this research was started as a piece of a latter project related to Quantum Molecular Similarity [44], with the concurrent aim to have the chance to study big sized molecules in a SCF framework, say, without the need to manipulate a huge number of AO functions. 

SO that after a long time all the atoms are at the same average distance from their initial position, in a Cartesian framework with the origin placed at the chain s center of mass. 

One particularly useful feature of the Cartesian index notation is that it provides a very convenient framework for working out vector and tensor identities. Two simple examples follow ... 

In Figure 4.6b we illustrate how the tetrahedral structure of CH4 relates to a cubic framework. This relationship is important because it allows us to describe a tetrahedron in terms of a Cartesian axis set. Within valence bond theory, the bonding in CH4 can conveniently be described in terms of an sp valence state for C, i.e. four degenerate orbitals, each containing one electron. Each hybrid orbital overlaps with the li atomic orbital of one H atom to generate one of four equivalent, localized 2c-2e C—H (T-interactions. 

Finally, we will briefly discuss the properties of polymer blends under shear flow. In small molecule mixtures, shear flow is known to produce an anisotropy of critical fluctuations and anisotropic spinodal decomposition [244, 245], In polymer mixtures, the shear has the additional effect of orienting and stretching the coils, thus making the single-chain structure factor anisotropic. In the framework of the Rouse model these effects have been incorporated into the RPA description of polymer blends [246, 247]. Assuming a velocity field v = yyex, where x, y, z are cartesian coordinates, y the shear rate, and ex is a unit vector in x direction, the single chain structure factor becomes [246, 247]... 

Within this framework the governing equations for boiling two phase flow can be expressed in Cartesian coordinate as follows ... 

The Cartesian coordinates of a molecule are converted to a set of equidistant points arranged orthogonally to each other in three dimensions (x,y,z directions). These points are separated by a distance referred to as the resolution and are all located within the van der Waal s radii of the atoms constituting the molecule (steps A-C, Figure 1). The resulting framework of points will be referred to as the molecular lattice. The number of such points is dependent upon the size of the molecule, the resolution chosen, and the nature of the atoms. 

Although generalized coordinates are often useful for understanding molecular systems, as we shall learn in later chapters, it is desirable to avoid the use of formulations with a configmation-dependent mass matrix since it complicates the design of numerical methods with good conservation properties. This is one reason that molecular simulation methods are typically described in a Cartesian coordinate framework. 

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