Orthogonal relations Cartesian coordinates

Now we demonstrate the system of coordinates, where the ellipsoids of rotation and hyperboloids of one sheet form two mutually orthogonal coordinate families of surfaces. First, we introduce the Cartesian system at the center of the mass and suppose that semi-axes of the ellipsoid of rotation obey the condition brelation between coordinates of the Cartesian and cylindrical... 

In a Cartesian coordinate system, by applying the definition of a cross product in this orthogonal system, the unit vectors ex, ey, ez are related as follows ... 

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

Let qi,q2,q3) be curvilinear orthogonal coordinates connected with the Cartesian coordinates x,y,z) by the vector relation r = r qi, q2, qa), where r is the radius vector of the point P considered. The Cartesian coordinates are then related to the generalized curvilinear coordinates by ... 

A central advantage of the q representation is that these coordinates are related to the mass-weighted Cartesian coordinates via an orthogonal transformation. 

We will consider the general class of orthogonal curvilinear coordinates, designated q, q2, and <73, whose coordinate surfaces always intersect at right angles. The Cartesian coordinates of a point in three-dimensional space can be expressed in terms of a set of curvilinear coordinates by relations of the form... 

There are several coordinate systems that have to be dealt with. Ultimately, in order to carry out the minimization process, the total energy is best expressed in terms of Cartesian coordinates. However, a general unit cell or lattice is characterized by non-orthogonal basis vectors. A cylindrical coordinate system is used to represent the molecular helix. The intramolecular energy is expressed in terms of valence coordinates. Thus transformations must be set up that relate the Cartesian coordinates to the helix parameters, the unit cell parameters and the valence coordinates. The helix operations and the unit cell parameters are considered first. 

Coordinate system In three-dimensional Euclidean space, which we use in geodesy for solving most of the problems, we need either the Cartesian or a curvilinear coordinate system, or both, to be able to work with positions. The Cartesian system is defined by an orthogonal triad of coordinate axes a curvilinear system is related to its associated generic Cartesian system through some mathematical prescription. 

Atomic polar tensors may also be represented in mass-weighted Cartesian coordinates q as defined by relations (2.9). The mass-weighted Cartesian displacemoits q referring to a space-fixed Cartesian fiame are related to the normal coordinates Q ( by an orthogonal transformation [4]... 

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