Vector orbital

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

NEXAFS data analysis to extract orbital orientation follows simple equations that are trigonometric expansions of the cos 5 relationship described in Section 4.1.3. The typical orientation of interest for organic semiconductors is a surface-relative orientation uniformly distributed about the surface normal. In this case, the azimuth-ally averaged tilt of the orbital can be determined. For a vector orbital, the intensity can be expressed as [31]... 

P is the polarization factor of the beamline 0 is the angle of the beam relative to the substrate plane a is the angle of the vector orbital relative to substrate normal... 

The angle a describes the orientation of a vector orbital, but it is important to understand the meaning of this orientation. It is the azimuthal mean orientation of an ensemble of orbitals within the sampled volume. The underlying distribution of orientations within the ensemble usually cannot be determined. This consideration... 

The azimuthal mean gives rise to an ambiguous quantity — the magic angle of 54.7°. If the vector orbital is tilted at 54.7°, there will be no variation in / with 0. Conversely, if there is no variation in I with 0, Equation 4.2.1 will reveal that a = 54.7°. An infinite number of orientation distributions can result in an azimuthal mean orientation of 54.7°, but the most important of these is complete disorder. If there is no variation in I with 0, one may speculate but not prove that a sample is disordered. 

The simplest case arises when the electronic motion can be considered in temis of just one electron for example, in hydrogen or alkali metal atoms. That electron will have various values of orbital angular momentum described by a quantum number /. It also has a spin angular momentum described by a spin quantum number s of d, and a total angular momentum which is the vector sum of orbital and spin parts with... 

In equation (bl. 15.24), r is the vector coimecting the electron spin with the nuclear spin, r is the length of this vector and g and are the g-factor and the Boln- magneton of the nucleus, respectively. The dipolar coupling is purely anisotropic, arising from the spin density of the impaired electron in an orbital of non-... 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

The electronic energy W in the Bom-Oppenlieimer approxunation can be written as W= fV(q, p), where q is the vector of nuclear coordinates and the vector p contains the parameters of the electronic wavefimction. The latter are usually orbital coefficients, configuration amplitudes and occasionally nonlinear basis fiinction parameters, e.g., atomic orbital positions and exponents. The electronic coordinates have been integrated out and do not appear in W. Optimizing the electronic parameters leaves a function depending on the nuclear coordinates only, E = (q). We will assume that both W q, p) and (q) and their first derivatives are continuous fimctions of the variables q- and py... 

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

As illustrated above, any p2 configuration gives rise to iD , and levels which contain nine, five, and one state respectively. The use of L and S angular momentum algebra tools allows one to identify the wavefunctions corresponding to these states. As shown in detail in Appendix G, in the event that spin-orbit coupling causes the Hamiltonian, H, not to commute with L or with S but only with their vector sum J= L +... 

Their symmetry labels can be obtained by vector coupling (see Appendix G) the spin and orbital angular momenta of the two subsystems. The orbital angular momentum coupling... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

A and B (a simple function of the atomic orbital type). In the case of an sp product, this is a vector of length Dj atomic units pointing along p axis,... 

Figure 1.5 Direction of the angular momentum vector p for an electron in an orbit...

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

We saw in Section 1.3.2 and in Figure 1.5 how the orbital angular momentum of an electron can be represented by a vector, the direction of which is determined by the right-hand screw mle. 

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