Cartesian vector

Orthogonal transfonnation of a Cartesian vector A with components Ai, T2 and. 43 in the system of ol23, under rotation of the coordinate system to nl 2 3 is expressed by the following equation... 

I assume that you are familiar with the elementary ideas of vectors and vector algebra. Thus if a point P has position vector r (I will use bold letters to denote vectors) then we can write r in terms of the unit Cartesian vectors ex, Cy and as ... 

The vector products of the unit cartesian vectors are seen to obey the following relations... 

Any 37/-dimensional Cartesian vector that is associated with a point on the constraint surface may be divided into a soft component, which is locally tangent to the constraint surface and a hard component, which is perpendicular to this surface. The soft subspace is the /-dimensional vector space that contains aU 3N dimensional Cartesian vectors that are locally tangent to the constraint surface. It is spanned by / covariant tangent basis vectors... 

If the set 3N contravariant Cartesian vectors given by the / aj) vectors and K mf vectors are linearly independent, and thus span the full 3N space of Cartesian... 

This may be confirmed by expanding an arbitary contravariant Cartesian vector (with a raised bead index) in a basis of a and m vectors and confirming that one recovers the original vector if such an expansion vector is left-multiplied by the RHS of Eq. (2.149). 

Following the standard RPA formalism, we define externally applied (weakly perturbing) potentials U (U is an n-component vector that can depend on Q but not on monomer orientations) and inter-segment potentials W (u, u ) (nxn matrix) where u and u represent the directions of two test segments. Within the mean field approach, the RPA equations give the mean response of the averaged densities (

is an n-vector) in terms of the response functions for the bare system X0(Q, u,u ) (nxn matrix) and for the interacting system X(Q, u, u ). In this matrix notation approach, bold face characters are used to represent n-vectors, nxn matrices as well as three-dimensional cartesian vectors such as direction u. The RPA equations in the matrix form are ... 

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Table 5.2. The relationship between spherical tensors of ranks 1 and 2 and cartesian vectors V and second-rank tensors T...

This differs from equation (5.110) in both phase and normalisation factors. We have seen that, for k = 1, the spherical tensor corresponds to a cartesian vector the spherical scalar product in this case is the same as the cartesian scalar product of two vectors ... 

The emphasis in other chapters of this book is on initial value solution of classical equations of motion (e.g. the Newton s equations). The Newton s equations are second order differential equations - MX = —dU/dX, where X [X e is the coordinate vector. Throughout this chapter X is assumed to be a Cartesian vector, M is a 3N x 3N (diagonal) mass matrix, and U is the potential energy. A widely used algorithm that employs the coordinates and the velocities (V) at a specific time and integrates the equations of motion in small time steps is the Verlet algorithm [1] ... 

From basic mathematics (e.g., [3]) we recall that a Cartesian vector, as the velocity v, in three dimensions is a quantity with three components, Vx,... 

For a basis set of (re) atomic orbitals which are singly noded in the plane perpendicular to the radial vector (see Fig. 16b), the required Harmonics are the Vector Surface Harmonics146). The two p11 (or d") atomic orbitals at each cluster vertex (i) behave as a pair of orthogonal unit vectors which are tangential to the surface of the sphere, jrf and jif are defined as Jt-symmetry orbitals on the ith cluster atom, pointing in the direction of increasing 0 and cj> respectively, as shown in Fig. 18. At the poles of the cluster sphere these vectors may be related to Cartesian vectors as follows ... 

Here ij, etc., represent diadic products of the elementary Cartesian vectors. The unit vector is... 

Any ordinary vector (e.g. a three-component Cartesian vector) transforms in a similar way... 

Instead of Cartesian tensors one can introduce spherical tensors in the following way. An ordinary Cartesian vector can be replaced by the spherical components... 

Note that in Equation (3.92) we have expressed the wavefunction in terms of a vector, k (which has components in the x and y directions of k and ky) and the Cartesian vector r. 

First, we will review briefly certain aspects of Cartesian vectors. In Cartesian space any point may be uniquely defined by means of a three-component vector from the origin to that point. The position vector of the point (xi, y, zt) is ri. 

Cartesian vectors with three components, so-called 3-vectors, are indicated by boldface type, e.g., the position vector r. Relativistic four-component vectors, so-called 4-vectors, are denoted by normal type. If not otherwise stated, all vectors irrespective of their dimension are assumed to be column vectors. The corresponding row vector is given by the transposed quantity, e.g.,... 

The details of electrical properties require some basic information about electric fields. Fields are usually thought of as Cartesian vectors. They arise from or may be defined from a scalar function, the electric potential, V. The x-component of the electric field is dV/dx, which we will d gnate as Vx. If the potential V is independent of x, then there is no x-component of die field. If V has only a linear dependence on x, the field is uniform in die x-direction. Of course, the elearic potential may depend on the spatial coordinates in some... 

The radial part R i(r) of cancels out in Eqs. 2.6 and 2.7, because and operate only on 9 and 0.) This implies that the orbital angular momentum quantities and are constants of the motion in stationary state with values /(/ -I- l)h and mh, respectively. A common notation for one-electron orbitals combines the principal quantum number n with the letter s, p, d, or/ for orbitals with 1 = 0, 1, 2, and 3, respectively. (This notation is a vestige of the nomenclature sharp, principal, diffuse, and fundamental for the emission series observed in alkali atoms, as shown for K in Fig. 2.2.) An orbital with n = 2,1 = 0 is called a 2s orbital, one with n = 4, / = 3 a 4/ orbital, and so on. Numerical subscripts are occasionally added to indicate the pertinent m value the 2po orbital exhibits n = 2, I = i, and m = 0. Chemists frequently work with real (rather than complex) orbitals which transform as Cartesian vector (or tensor) components. A normalized 2p, orbital is the linear combination... 

These analogies fail to explain the coarse structure observed in diatomic electronic spectra under low resolution (Chapter 4), because the additional nuclear coordinates introduced by the second atom in a diatomic molecule AB create new internal modes that have no counterpart in atoms. The nuclear positions relative to an arbitrary origin fixed in space can be specified using the Cartesian vectors and Rg (Fig. 3.1). They may be equivalently described in terms of the coordinates... 

Normally the F matrix should be 3N — 5) x (3N — 5) = 7 x 7 in the S basis, but we have left out one of the 7r and one of the coordinates.) To derive the G matrix, we begin with the Cartesian basis shown in Fig. 6.8. (We could include four additional Cartesian vectors 9 through 2 pointing out of the paper, but these vectors have projections only along the omitted 7r and Tig modes.) In terms of these, the symmetry coordinates are... 

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