Coordinate frame local

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

First of all, one needs to choose the local coordinate frame of a molecule and position it in space. Figure 2a shows the global coordinate frame xyz and the local frame x y z bound with the molecule. The origin of the local frame coincides with the first atom. Its three Cartesian coordinates are included in the whole set and are varied directly by integrators and minimizers, like any other independent variable. The angular orientation of the local frame is determined by a quaternion. The principles of application of quaternions in mechanics are beyond this book they are explained in detail in well-known standard texts... 

The dynamic condition requires that the net force on any portion of the interface has to vanish. In a local coordinate frame attached to an interfacial position, three constraints are derived expressing the force balance for each of the three coordinate directions ... 

Fig. 3. Local coordinate frames and wave functions corresponding to the orbitals doublet eg at neighbour centres (a) ipEe = dzi and (b) if/Ee = dx2 yi.

Coefficients Rlx form an orthogonal matrix iC transforming the (/-orbitals under rotation of the laboratory coordinate frame (LCF) to the local coordinate frame related to the ligand / and constructed such that its Oz axis is going through the metal atom and the ligand atom (DCF - diatomic coordinate frame). The perturbation caused by the ligand has a matrix representation elxx, in the DCF with A = a,7r(x),n(y), 5 xy), S(x2 - y2). These quantities are considered parameters of the AOM. 

The reaction of hybridization tetrahedron on the changes of local geometry can be considered in the linear response approximation eq. (3.133). It is clear that any variation of the valence angle is a sum of equal amounts of hybridization- compatible and hybridization- incompatible deformations. The denominator in the linear response relation eq. (3.133) is the same as for eq. (3.136) while the relevant block of the yxyqE matrix (with q taken as a difference of two opposite valence angles) is proportional to For the methane molecule with the carbon atom put in the origin of the coordinate frame, substitution of matrices of second derivatives for the energy... 

Summation over / in eq. (4.79) is extended to either only occupied or only vacant MOs. Here (/ L) are the coefficients of the Z-th canonical MO in the expansion of the /.-(h local MO. The expansion coefficients of the localized MOs over the canonical MOs are the invariants of the molecular electronic structure as they do not change under the rotation of the coordinate frame. Expansion of the LMOs over AOs has the form ... 

By way of example, we might consider the interaction of a quinoline ligand whose donor nitrogen lone pair is directed exactly toward the metal. The local pseudo-symmetry is C2 and we may label the metal-nitrogen axis as Z, with Cartesian coordinates X and Y lying in, and perpendicular to, the quinoline plane respectively. The central assumption of the AOM is that the d orbital energy matrix is diagonal within this coordinate frame and we write... 

II that the perturbation is diagonal within the local metal-ligand coordinate frame and... 

Velocity time ACFs (v(f)v(O)) for the center-of-mass (COM) of a water molecule have been computed in the local (molecular) coordinate frame, making it possible to look at the COM motion alone each local (Cartesian) coordinate X, Y, and Z independently [65]. All three atoms O, HI, and H2 are placed on the XOZ (molecular) plane and the HI — O —H2 median coincides with the Z axis. 

The radial distribution function is a traditional way of quantifying the local conservation of water molecules during MD simulations. While this measure is suitable for measuring water molecules around spherical elements, things turn out to be tricky for asymmetric, complex stmctures. Another possibility to statistically determine the location of a water molecule and therefore the hydration site is to average its position in relation to the coordinate frame [63]. However, applicability of this approach is limited in case of a flexible protein, as the high level of noise causes information loss. 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

Figure 22 A schematic illustration of the various coordinate frames considered within the two-step model, for the case of a specifically deuterium-labeled methylene segment in the surfactant hydrocarbon chain. The laboratory frame (L) is set by the direction of the external magnetic field, where Zjl is the field direction. In this frame the nuclear quadrupolar moment tensor is diagonal. The director frame ( >) is associated with the micellar aggregate where Z/> specifies the micellar surface normal. It is assumed that the fast local dynamics occur with an essentially cylindrical symmetry around Z/). The molecular frame (Af) corresponds to the principal axis of the electric field gradient tensor. For the case of a methylene segment, Zm specifies the direction maximum component of the field gradient tensor, which is furthermore cylindrically symmetrical around Zm-...

Si i = 1,2,3) are the distances PP R denotes the 3x3 rotation matrix which transforms from the global reference frame / robot platform Fi ,F2 ,F3 to the sensor frame / local camera platform. We outline now the algorithm by which we determine the coordinates x, y, z of the perspective centre as well as the orientation parameters which constitute the 3x3 rotation matrix... 

With kinematic tolerancing, each FE involved in a tolerance chain is associated with a local 4x4 transformation matrix. Each transformation is done with respect to a global coordinate system attached to the geometric entity where the functional requirement (FR) is defined. The effects that small displacements have on functionality are obtained by first-order derivation of the coordinate frames position vectors in the tolerance chain. 

Because SDTIs are each locally expressed with respect to their own coordinate frame, a transformation is necessary to map their effect in FR space. To do so each SDTI is associated with a 6 X 6 Jacobian which structure can be generalized as follows ... 

Translations-Translations Consider the centering pin mechanism in Fig. 2, consisting of a base, a block, and a pin. The upper plane of the base is represented with a magnified bow defect, to which local coordinate frame OpE is associated. The goal is to model the effects of this small geometrical defect on the FR (the pin s axis), to which coordinate frame OpR is also associated. 

Thus, assuming the film in this region to be thin such that the lubrication approximation holds, the spreading dynamics of the drop in the limit L/L quasi-steady limit with respect to a moving coordinate frame translating at a constant dimensionless speed given by Ca are governed by the Bretherton [9] equation ... 

The multipole moment operators in Eq. (7) are still referred to the global coordinate frame. We now transform them to the local or molecule-fixed frame ... 

The second term in the above expression represents a cross-term between the two types of motion, but is zero except when rriL = 0. Unless it is necessary to calculate Jo (a ), or the spin-spin relaxation time, the overall correlation functions will be approximated by linear combinations of the products of the correlation functions for each motion [i.e., retain only the first term in Eq. (8.10)]. To discuss the superimposed rotations model, it is assumed that internal rotations about different C-C bonds are independent and use additional coordinate frames to carry out successive transformations from the local a frame to the molecule-fixed frame. Free rotational diffusion will be used to describe each bond rotation in the following section. 

Associated with the base body and each branch body is a local right-handed coordinate frame with fixed orientation and location (at the centre of mass) in the respective body. 

In robot programming, positions are often expressed relative to local coordinate frame One reason is that it makes it easier to carry out changes. For example, if a robot has to pick several items from a pallet, it is best to express the positions in the coordinate frame of the pallet. If the pallet s position is changed, only the pallet s new position has to be entered, and the pick positions relative to the pallet can remain unchanged. 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

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