Connections between different bases

We conclude this chapter with two final observations. First, the irrep that is the dual of any given D( ] is obtained immediately from (4.3.4) and, since the standard irreps are orthogonal, it is only necessary to multiply the matrices D[a](P) by 1, according to the parity of P, so that 

Secondly, it was stated earlier (p. 95) that the branching diagram spin functions also carried the standard (two-row) irreps of S vri that the Rumer functions also carried irreps, although of non-standard form. The 

5 Standard tableaux for N = 5 and partition [2, 2, 1]. These tableaux are presented as the duals of those in Fig. 4.4, and accordingly appear in inverse last-letter sequence. 

It may in fact be shown that the sum in (4.6.2) contains no terms corresponding to tableaux that stand to the left of T,. in the last-letter sequence. One proof (following from Pauncz, 1979, Section 7.5) depends on adopting the result as an inductive hypothesis, obviously satisfied by nd showing that the permutation that carries 0, into, produces a function with the same property. (Other proofs, based on the algebraic properties of the E (Rutherford, 1948), are more intricate.) The relationship between the Rumer and the standard bases is thus found to be triangular , provided that the basis functions are written in standard order in symbols 

It is then easy to identify the expansion coefficients. Thus f = 0f, 0f-i = a0f- - r b0f, etc., and, on introducing the usual metric 

---

The space filling model developed by Corey, Pauling, and Koltun is also known as the CPK model, or scale model [197], It shows the relative volume (size) of different elements or of different parts of a molecule (Figure 2-123d). The model is based on spheres that represent the "electron cloud . These atomic spheres can be determined from the van der Waals radii (see Section 2.10.1), which indicate the most stable distance between two atoms (non-bonded nuclei). Since the spheres are all drawn to the same scale, the relative size of the overlapping electron clouds of the atoms becomes evident. The connectivities between atoms, the bonds, are not visualized because they are located beneath the atom spheres and are not visible in a non-transparent display (see Section 2.10). In contrast to other models, the CPK model makes it possible to visualize a first impression of the extent of a molecule. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Instruments based on the contact principle can further be divided into two classes mechanical thermometers and electrical thermometers. Mechanical thermometers are based on the thermal expansion of a gas, a liquid, or a solid material. They are simple, robust, and do not normally require power to operate. Electrical resistance thermometers utilize the connection between the electrical resistance and the sensor temperature. Thermocouples are based on the phenomenon, where a temperature-dependent voltage is created in a circuit of two different metals. Semiconductor thermometers have a diode or transistor probe, or a more advanced integrated circuit, where the voltage of the semiconductor junctions is temperature dependent. All electrical meters are easy to incorporate with modern data acquisition systems. A summary of contact thermometer properties is shown in Table 12.3. 

---