Coordinates orthonormalized coordinate system

Imagine a particle with position vector ra in a three-dimensional (3D) orthonormal coordinate system with axes x, y, z and a wave function P(ia) = I ixa, ya, za)- If the particle is rotated about one of the axes x, y, or z through an infinitesimal angle 8cp, the new position coordinates r a are given in first order by... 

Procedure Briefly, the procedure developed consists of the transformation of the positional parameters of the M4Y4 atoms, together with their standard deviations, to an orthonormal coordinate system. One then performs a least squares analysis in order to adjust rotations about the orthonormal axes to obtain the best fit to an assumed symmetry. One can then test the variance of an observation of unit weight by an appropriate chi square test. Further details are given by Averill et al.2 ... 

The static depolarized light scattering provides information on the fluctuations in the Sector field, which in turn are resisted by the molecular field. The fluctuations in the n(r) at position r may be separated into two components Sni(r) in the plane formed by q and the undistorted director n, and Sn2(r) in the direction perpendicular to that plane. These are conveniently described in terms of an orthonormal coordinate system spanned by the unit basis vectors Co, and Or, where eo = ii, = eTxn, and ex = (q x n)/ q x n. Then, with i, = e -i and s = e S, where i and s are unit vectors along the polarization of the incident and scattered beams, respectively, the fluctuations in the dielectric tensor may be expressed in the form [134, 135, 137]... 

In PCA, for instance, each pair j, k of loading vectors is orthogonal (all scalar products bj h/, are zero) in this case, matrix B is called to be orthonormal and the projection corresponds to a rotation of the original coordinate system. 

Let the coordinate system have 3 parallel to h, X normal to one plane, and 2 normal to the other (a local orthonormal coordinate system). Then 3 can be... 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

The coordinates of a vector x in an m dimensional space, e.g., an m x 1 mixture spectrum measured at m = 700 wavelengths, can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lowerdimensional space. Clearly, we cannot imagine a 700-dimensional space. It is... 

Since the coordinate system is simply rotated, not shrunk or expanded in any way, the matrix A must be orthonormal ... 

Thereby the matrix C contains in its third column C3 the normalized eigenvector mi of the eigenvalue Ai (that means the direction r of the rotation axis). Both the columns ci and C2 will be selected in such a way that together with C3 they describe an orthonormal system (now the direction of the z —axis of this new coordinate-system (x ,y ,z ) is the Erection r of the rotation axis, see fig. 3). The angle of rotation will be simply obtained by... 

The vectors (functions) of a coordinate system may in some cases be given naturally by the problem, and these are not always normalized or orthogonal. For computational purposes, however, it is often advantageous to work in an orthonormal coordinate system. We first note that normalization is trivially obtained by simply scaUng each vector by the inverse of its length. 

We will need the unitary transformations exp (lA) and exp (iS). They are very convenient, since when starting from some set of the orthonormal functions (spinorbitals or Slater determinants) and applying this transformation, we always retain the orthonormality of new spinorbitals (due to A) and of the linear combination of determinants (due to S). This is an analogy to the rotation of the Cartesian coordinate system. It follows from the above equations that exp (/A) modifies spinorbitals (i.e., operates in the one-electron space), and exp (i S) rotates the determinants in the space of many-electron functions. 

The pair (x, y) define the branching plane or g-h plane. The remainder of the intersection adapted coordinate system, w , i = l-(Ar " — 2), spans the seam space. These — 2 mutually orthonormal vectors need only be orthogonal to the branching space. It is also convenient to define... 

It always proves possible to rotate the XYZ axes into a new body-fixed coordinate system with orthonormal axes a, b, and c (called the principal axes) which diagonalize the inertia tensor,... 

An example of an orthonormal set is our coordinate system, consisting of three vectors at right angles of each other. In principal component analysis it will be shown that there ate other coordinate systems that are more convenient to use. 

To express a vector or matrix in coordinates of a coordinate system K Oi Xi yi z they will be denoted with upper left index, e.g. v = ( f The transformation matrix Kj for sixdimensional velocities and forces between coordinate system K Oj, Xj, yj Zj) and K Oi Xi yi Zi) consists therefore of the known 3x3 orthonormal transformation matrix T... 

For orthogonal coordinate systems, a set of unit vectors ia = a a I can be defined. Hence, at each point F on A/ one can define a set of orthonormal basis vectors (ii, i2, n/), where ii and i2 are lying in the tangent plane and n/ is the unit normal to A/. In this frame the vector c associated with points in the surface can be expressed by... 

Notation. We will use boldface italic letters to denote vectors and tensors. We adopt the summation convention for repeated indices, imless stated otherwise. Most often, vectors are denoted by lowercase boldface italic letters, and second-order tensors, or 3x3 matrices, by lowercase boldface Greek letters. Fourth-order tensors are usually denoted by uppercase boldface italic letters. We will make use of a Cartesian coordinate system with an orthonormal basis ei, ej, e. Where it is necessary to show components of a vector or a tensor, these will always be relative to the orthonormal basis e, 2, 3. Throughout this work we will identify a second-order tensors r with a 3x3 matrix. We will always use 1 < / <3, to denote the components of the vector a, and the components of the... 

To introduce the concepts of tensor scattering amplitude and amplitude matrix it is necessary to choose an orthonormal unit system for polarization description. In Sect. 1.2 we chose a global coordinate system and used the vertical and horizontal polarization unit vectors Ba and e,g, to describe the polarization state of the incident wave (Fig. 1.9a). For the scattered wave we can proceed analogously by considering the vertical and horizontal polarization unit vectors and eg. Essentially, (e/t, 6/3,60.) are the spherical unit vectors of fee, while er,eg,e ) are the spherical unit vectors of fcs in... 

Kuppermann A 1996 Reactive scattering with row-orthonormal hyperspherical coordinates. I. Transformation properties and Hamiltonian for triatomic systems J. Phys. Chem. 100 2621... 

In general, Eq. (4.2) has many acceptable eigenfunctions 4 for a given molecule, each characterized by a different associated eigenvalue E. That is, there is a complete set (perhaps infinite) of 4, with eigenvalues ,. For ease of future manipulation, we may assume without loss of generality that these wave functions are orthonormal, i.e., for a one particle system where the wave function depends on only three coordinates. 

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