Coordinates Cartesian

The parameters of this matrix are the image / and the vector d written by [dx, dy] in cartesian coordinates or [ r, 0] in polar coordinates. The number of co-occurrence on the image / of pairs of pixels separated by vector d. The latter pairs have i and j intensities respectively, i.e. 

The problem of film flow is formulated on the assumption that the film thickness h is much smaller than the length 1 (in our case h/1 10 ). In Cartesian coordinates with transversal axis y and longitudinal one z we can write the equation for a film flow as follows ... 

Dirac showed in 1928 dial a fourth quantum number associated with intrinsic angidar momentum appears in a relativistic treatment of the free electron, it is customary to treat spin heiiristically. In general, the wavefimction of an electron is written as the product of the usual spatial part (which corresponds to a solution of the non-relativistic Sclnodinger equation and involves oidy the Cartesian coordinates of the particle) and a spin part a, where a is either a or p. A connnon shorthand notation is often used, whereby... 

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

Consider an isotropic medium that consists of independent and identical microscopic cln-omophores (molecules) at number density N. At. sth order, each element of the macroscopic susceptibility tensor, given in laboratory Cartesian coordinates A, B, C, D, must carry s + 1 (laboratory) Cartesian indices (X, Y or Z) and... 

The two coeflScients y and describe the material response and the Cartesian coordinate must be chosen as a principal axis of the material. 

Ciccotti G, Ferrario M and Ryckaert J-P 1982 Molecular dynamics of rigid systems in cartesian coordinates. A general formulation Mol. Phys. 47 1253-64... 

This section deals with the transfonnation of coordinates and forces [U, 47] between different coordinate systems. In particular, we will consider the transfonnation between Cartesian coordinates, in which the geometry is ultimately specified and the forces are calculated, and internal coordmates which allow efficient optimization. 

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... 

Baker J and Hehre W J 1991 Geometry optimization In Cartesian coordinates The end of the Z-matrIx J. Comput. Chem. 12 606... 

Baker J 1992 Geometry optimization in Cartesian coordinates constrained optimization J. Comput. Chem. 13 240... 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

We tieat this case first, since it is simpler than the trigonal case. The molecular displacements are denoted by x and y (with suitable choice of their origins and of scaling). Then, without loss of generality we can denote the positions of the ci pairs in Cartesian coordinates by... 

If the nuclear coordinates are mass-scaled Cartesian coordinates,... 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

The elements of these vectors can be evaluated using an off-diagonal fomt of the Hellmann-Feynmann theorem, which in Cartesian coordinates, Xa, is... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Next, we consider the simple overlap integral of two such basis functions with different powers of Cartesian coordinates and different Gaussian width, centered at different points. Let nuclei 1 locate at the origin, and let nuclei 2 locate at —R, then... 

The three integrals of the Cartesian coordinates have the same form. Take the integral with respect to x as, for example,... 

Figure 6, Equi-nonadiabatic coupling lines for the terms ri2(r,v) and r23(r,y) as calculated for the C2H molecule for a fixed C—C distauce, that is, rcc — 1-35 A. (a) Equi-non-adiabaric coupling term lines for the ri2(r,y). (b) Equi-non-adiabatic coupling term lines for X23(r,y). The Cartesian coordinates (x,y) are related to (q, 9) as follows x — qcosO] y —, sin9, where q and 9 are measured with respect to the midpoint between the two carbons.

In what follows, the 2D space is assumed to be a plane, and therefore we apply either the polar coordinates q, 0) or the Cartesian coordinates (x,y). 

To shift it to some arbirtrary point ( yo,0jo) we first express Eq. (161b) in terms of Cartesian coordinates, and then shift the solution to the point of interest, namely, to (xjo, o)[= ( T/Oi 0yo)]- Once completed, the solution is transformed back to polar coordinates (for details see Appendix F). Following... 

In this appendix, we discuss the case where two components of Xm, namely, x p and XMg (p and q are Cartesian coordinates) are singular in the sense that at least one element in each of them is singular at the point B p = a,q = b) located on the plane formed by p and q. We shall show that in such a case the adiabatic-to-diabatic transformation matrix may become multivalued. 

We start by writing the curl equation in Eq. 057) for a vector t(x,y) in Cartesian coordinates. 

The essential degrees of freedom are found by a principal component analysis of the position correlation matrix Cy of the cartesian coordinate displacements Xi with respect to their averages xi), as gathered during a long MD run ... 

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

Equation (18) is valid when the polarizability of the dielectric is proportional to the electrostatic field strength [4]. The operator V in the Cartesian coordinate system has the form V = dldx,dldy,dldz). 

Consider a system of N particles in d dimensions. Using the standard procedure of integrating over the momenta in Cartesian coordinates, we can write the average of a mechanical property A(r ) as... 

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

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