Cartesian frame

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

The geometries of HF, H2O, NH3 and H2S were taken from Ref. 33, of H2CO from Ref. 34, and of HCl and PH3 from Ref. 35. The molecular geometry of H2CS (the nitroanilines) have been optimized at the RHF/6-3 IG (RHF/6-31G) level by using the GAUSSIAN94 program (see Scheme 1 for their orientation in the Cartesian frame). 

If the right-handed Cartesian frame is chosen (Fig. 2) and is assumed as... 

Fig. 2. Instantaneous snapshots (t = 0) of the right-circularly polarized light travelling in the right-handed Cartesian frame...

Classical dipole radiation. Usually, we will be dealing with radiation emitted (or absorbed) by dipoles. In the simplest case, a dipole may be thought of as a point charge q positioned a distance r away from the origin of a Cartesian frame the dipole moment p is then given by... 

We note that the scattering angle x an<3 the polar angle at t —> oo are related according to % + 23(oo) = n. The dipole moment fi(R) is parallel to the intermolecular axis. Introducing a Cartesian frame with its x-axis along the apse line and the y-axis in the collision plane as illustrated in Fig. 5.7, we get from the definition of for i = 1,... 

The new relative and center of mass velocities were transformed back to the cartesian frame to give a new set of starting velocities. 

Two terms in Eqs. (17) and (18) are worthy of special note. In Eq. (17) the term pvj/r is the centrifugal force. That is, it is the effective force in the r direction arising from fluid motion in the 0 direction. Similarly, in Eq. (18) pvrvg/r is the Coriolis force, or effective force in the 0 direction due to motion in both the r and 0 directions. Both of these forces arise naturally in the transformation of coordinates from the Cartesian frame to the cylindrical polar frame. They are properly part of the acceleration vector and do not need to be added on physical grounds. 

The aim is to tip the net magnetization M0 toward the xy horizontal plane of the stationary Cartesian frame of reference and measure the resulting component of magnetization in that plane. Rf electromagnetic... 

To simplify visualization of the problem, the static Cartesian frame is replaced by a frame rotating around the z axis at the carrier frequency (< ,). The situation is similar to that of an observer who is rotating with the frame as though riding a carousel. 

Solution Equation (13.13) can be used to calculate the flux on a circular small electrode by summing along z, the effect of elementary rectangular strips. In this case, x contained in the definition of Si must be replaced by x — Xi), which actually corresponds in the local Cartesian frame of reference to the distance from the leading edge of any elementary strip. The position of the leading edge x (2) is a function of R and z as... 

Assuming that the liquid travels between the surfaces without any change in velocity, as viewed from a nonrotating Cartesian frame of reference, the time required to travel between the cone surfaces can be written as... 

Let us consider surfaces in a Cartesian frame, whence these results can be generalized to any set of three coordinates x in an arbitrary coordinate system fixed in space. A surface in 3D space can generally be defined in several different ways. Explicitly, z = F x,y), implicitly, f x,y,z) = 0 or parametrically by defining a set of parametric equations of the form x = x C, rf), y = y C, v), z = z (, p) which contain two independent parameters Q, p called surface coordinates or curvilinear coordinates of a point on the surface. In this coordinate system a curve on the surface is defined by a relation f Q, p) = Q between the curvilinear coordinates. By eliminating the parameters Q, p one can derive... 

Fig. 5.4.1 Relationship between the Cartesian laboratory coordinate frame with axes x and y and the rotated Cartesian frame with axes r and s. The field gradient is applied parallel to r at an angle

The coherently pseudorotating molecule then interacts with a phase-locked pair of electronically resonant light pulses. The laser electric field, polarized in the I direction of a space-fixed cartesian frame, gives an interaction of the form... 

The unit vectors in (3.6) and (3.7) belong to a molecule-fixed cartesian frame (i,j,k) in which k is normal to the molecular plane. The ring... 

In order to find the equation of motion of the atoms in a molecule we need to express the kinetic and potential energies as a function of the atomic coordinates. The coordinates that we shall use describe the displacements of the atoms from their equilibrium positions, . Here ui), (ui)y, (ui) 2 are the magnitudes of displacements of the atom I in a molecule from its equilibrium position, referred to the Cartesian frame. Using the time derivatives of these coordinates, we can write the kinetic energy of a molecule, E]f° containing Adatom atoms with masses mi. 

In this specific case the Cartesian frame of the tensor is referred to hydrogen in a C-H bond with its direction along the j axis, the x axis lies perpendicular to y in the benzene plane and z is normal to the plane. The anisotropy is very clear, with the greatest value being six times that of the smallest. 

The y and z parameters in (42) measure the computational effort to rotate from the first cartesian frame to the second, and from the second to the third, respectively, and to accumulate them therein. These are relatively substantial tasks and this explains the large y and z parameters in Table I for PH. 

If this equation is satisfied, a unidirectional rotation occurs when the kinetic energy of the rotor is greater than maxg V(0). A given atom of the molecule shown in Fig. 1 will follow an x(t) = dcos(0(f) andy(f) = dsin(0(f)) motion in a Cartesian frame. 

This S.l. result corresponds to the expression originally derived in Gaussian units by Andrews and Harlow (1983). As it stands, all molecular and radiative parameters in the above expression are referred to a single Cartesian frame of reference, making the result directly applicable only to rigidly oriented molecules or solids. For application to fluid samples, further processing of the result to account for molecular orientation is necessary before this result and others below can be applied the procedi re is discussed in detail in Section VII. 

The second problem is pictorial. How can these phenomena be presented in the conventional, static, Cartesian frame of reference We avoid the complexities by using a rotating frame of reference. 

---