Space fixed axe

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

General Note In this section the director n is treated mathematically as a unit vector, with components n, ni along space-fixed axes Xi, X2, X3. 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

A position vector p is a quantity which defines the location of some point P in three-dimensional physical space (see Fig. 5-2.1). If O is the origin of some set of space-fixed axes, the length p of OP and the direction of OP with respect to these axes constitute the position vector. If the set of space-fixed axes are mutually perpendicular, the position... 

In Section 1.19 we classified the electronic wave functions of homonuclear diatomic molecules as g or u, according to whether they were even or odd with respect to inversion g and u refer to inversion of the electronic coordinates with respect to the molecule-fixed axes. This is to be distinguished from the inversion of electronic and nuclear coordinates with respect to space-fixed axes, which was discussed in this section. The electronic Hamiltonian for a diatomic molecule is... 

Next we consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia. To achieve inversion of all particles with respect to space-fixed axes, we first rotate all the electrons and nuclei by 180° about the c axis (which is perpendicular to the molecular plane) we then reflect all the electrons in the molecular ab plane. The net effect of these two transformations is the desired space-fixed inversion of all particles. (Compare the corresponding discussion for diatomic molecules in Section 4.7.) The first step rotates the electrons and nuclei together and therefore has no effect on the molecule-fixed coordinates of either the electrons or the nuclei. (The abc axes rotate with the nuclei.) Thus the first step has no effect on tpel. The second step is a reflection of electronic spatial coordinates in the molecular plane this is a symmetry plane and the corresponding operator Oa has the possible eigenvalues +1 and — 1 (since its square is the unit operator). The electronic wave functions of a planar molecule can thus be classified as having... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

Figure 1.3. Transformation from space-fixed axes X, Y, Z to molecule-fixed axes using the spherical polar coordinates R,Q,

We wish to divide XT into a part describing the nuclear motion and a part describing the electronic motion in a fixed nuclear configuration, as far as possible. Equations (2.36) and (2.37) do not themselves represent such a separation because 3 is still a function of R,

partial differential operators with respect to these coordinates. The obvious way to remove the effects of nuclear motion from. >iel is by transforming from space-fixed axes to molecule-fixed axes gyrating with the nuclei. 

However, it is more convenient to use curvilinear cartesian coordinates to describe the rotational motion of the nuclei. To this end we relate a set of rotating, molecule-fixed axes to the space-fixed axes by the three Euler rotations. In our experience the Euler angles and the rotations based upon them are not easily visualised Zare [7] has given as good a description as any. Figure 2.2 defines the Euler angles [Pg.45]

Figure 4.1. Electron and proton coordinates. X, Y, Z are space-fixed axes with origin at the nucleus.

It can be shown that a sequence of rotations about axes fixed in space is equivalent to the same sequence of rotations but performed in the reverse order about axes which rotate with the body (provided that the body-fixed and space-fixed axes coincide initially). 

Figure 5.3. The Euler angles , LX OX" = 0 and LY Oy = x-...

In molecular quantum mechanics, we often find ourselves manipulating expressions so that one of a pair of interacting operators is expressed in laboratory-fixed coordinates while the other is expressed in molecule-fixed. A typical example is the Stark effect, where the molecular electric dipole moment is naturally described in the molecular framework, but the direction of an applied electric field is specified in space-fixed coordinates. We have seen already that if the molecule-fixed axes are obtained by rotation of the space-fixed axes through the Euler angles (, 6, /) = >, the spherical tensor operator in the laboratory-fixed system Tkp(A) can be expressed in terms of the molecule-fixed components by the standard transformation... 

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are... 

Since we shall be interested in the electric field gradient with respect to a molecule-fixed coordinate system, we need to transform (8.492) from space-fixed to molecule-fixed axes the relationships between the two are illustrated in figure 8.53. Denoting molecule-fixed axes with primes, and space-fixed axes without primes, the spherical harmonic addition theorem gives the result ... 

Throughout this book we have used, at different times, space-fixed or molecule-fixed axis systems, with arbitrary origin, origin at the molecular centre of mass, origin at the nuclear centre of mass, or origin at the geometrical centre of the nuclei. We use CAPITAL letters for SPACE-FIXED axes, and lower case letters for molecule-fixed axes. The various origins are denoted by primes as follows. 

Ra = position vector of nucleus a P, = momentum conjugate to Ra / , = position vector of ith electron Pi = momentum conjugate to Rt Si = spin of ith electron X, Y, Z = space-fixed axes... 

When dealing with components of vector quantities we usually use subscripts X, Y, Z or x, y, z for space-fixed or molecule-fixed components, the origin of coordinates usually being denoted in the primary subscripted symbol. For the electron spin we use capital. S, for space-fixed axes and small s for molecule-fixed it is not necessary to distinguish the origin of coordinates. A difficulty with this notation is that, in conformity with common practice, we also use the symbol S to denote the total spin (E,.y or X,.S, ).Wc hope... 

The two first equations follow from the commutation relations of the angular momentum components, Jf F = X, Y, Z, along space fixed axes. The latter two are not as obvious as they may appear at first glance. Since Jx and Jy do not commute with h we cannot factorize the vector space to treat // and Jz in a separated basis of vectors, (In, />, such as it is usually assumed when discussing the two-dimensional harmonic oscillator. Jy + iJx cannot be used as ladder operators of Jz, and similarly it may be shown that the usual ladder operators85 for H are inapplicable as well, since they do not commute with/2. 

Fig. 3. The polarization ellipse referred to space-fixed axes x and y. The propagation direction z is out of the plane of the paper, n is the ellipticity and 0 is the azimuth...

The central construct of the expert system DOCENT which we are developing is its capacity to represent a macromolecule by "generalized cylinders", and to permit manipulation of the cylinders directly, instead of adjusting the molecular internal coordinates or space-fixed axes. The inverse problem, to recover reasonable values of the underdetermined atomic coordinates from the disposition of the generalized cylinders, is posed. 

X, Y, and Z are the space fixed axes with running index F and a, b, and c are the rotating molecular axes with running index y. The angular momentum is given in units of h. 

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