The S function

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

The linear independence of this sort of set is discussed in Section 5.4.5. 

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The other limiting case corresponds to the limit of large values of Ra. In order to remain concise, this has not been illustrated. For F = 0 one finds three values of zg for which S = S = 0. The first one is zq = 0 and the remaining two are close to zg = R/2 and zg = -R/2 (the two positions of the nuclei). They correspond to the broken symmetry behaviour analyzed in section 2. For very large values of F>0, no value of zg satisfies the equation, and the S function does not attain negative values. This means that the energy E(zg) has no extremum. 

Figure 2. Illustration of the S-functional (23) for (a) the external driving field and the eigenvalues (b) jj, = —1, (c) (i = l, and (d) (j, = i. (The latter case is discussed in detail in Section IIIC.)...

The combinational contribution to AG,n for PMMA particles stabilized by PIB in 2-methylbutane is shown plotted as a function of temperature in Figure 3(a). The values of the parameters used in Equations 2 and 3 were u = 8 x 10- g cm-, a = 300 nm, >2 = 1.09 cm g- and V = 116.4 cnr mole- . The thickness of the steric barrier,L, was taken to be 25 nm and the particle separation, do, was fixed at 30 nm. It can be seen from Figure 3(a) that AGj (comb) is a positive quantity that becomes more positive as the temperature increases, indicating that in the absence of other contributions to AG, the particle would become more stable with increasing temperature. In the above calculation, we have assumed that the S function, Equation 3, remains invariant with temperature, which is incorrect. 

However, the change is thought to be small and we feel justified in ignoring it. Similarly, the S function is pressure dependent, but since the change is also expected to be small, we consider AG (comb) to be essentially independent of pressure. 

This is verified by substitution of the S functions which shows that... 

Notice that this indicates that orbital 1 is a combination of the s functions on He only (dissociating properly to He + H+). 

Such an operator is indeed the first derivative of the familiar impulse or Dirac S function. It can, like the S function, be represented as the limiting... 

If spreadsheets are to be used it is prudent to ensure that any macros and procedures are correct and that the in-built statistical functionality is appropriate It is very easy to select the s function instead of s i. Remember that... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

If we now proceed further to the fluorine molecule, both the 7r-antibonding orbitals will be doubly occupied. As with the s functions, the configuration (YW-bondmg)2(YV-antitonding)2 can be transformed into two jr-lone-pair orbitals, one on each atom. Similarly with the ny orbitals. The localised description of Fa, therefore, has four localised jr-lone-pairs, there being only one single bonding orbital. 

Fig. 5.16 One basis function can be used to shift another in a given direction (to polarize it). In minimizing the energy, the program adjusts the relative contributions of the two functions to shift the electron density where it is needed to get the minimum energy, p Functions are also commonly used to polarize the s functions on hydrogen atoms, but the main use of polarization functions is the utilization of d functions on heavy atoms (atoms other than H and He)...

Next, the formal expansion of the S-function into a Taylor s series about the centre of mass qa of the ath macromolecule can be used, retaining only the first two terms of the expansion... 

It is instructive to compare the T-dependences of s and A in this elementary case. The s function is represented schematically together with the A function in Fig. 1. One has simply, for T < Tx, s = A = A0, and for T > T2, s = A = A At T = T0, [dPNdT1] = 0, [dsldT = 0, and then s is maximum at the transition [15]. In consequence, when a material presents a well-defined semiconducting state with a constant energy gap 2A at low temperature, any subsequent increase in the slope function s upon heating is in fact the indication of a decrease in this gap. These opposite behaviors of s and A have sometimes been the source of misleading interpretations in the past. 

Champagne has expressed her data in terms of a function p/F (in our terminology), which differs from the /S-function by a constant only. 

For the hybridization of one s and two p electrons, the s function is divided equally amongst the three resulting hybrid functions, so that into each hybrid electron orbital there enters one third of the s cloud i.e. fl2 1/3). The first orbital may therefore be represented by... 

In many applications, it is more computationally convenient to express the S functions in terms of the scalar products between the unit local axis vectors (xj, yi, and X2, ya, i) and the unit intermolecular vector R. This set of variables is highly redundant, but easily calculated from the local axis vector information in most simulations. As an illustration. Table 1 gives the S functions that are important in describing the anisotropy of the atom-atom repulsion between an N atom in pyridine and a hydrogen-bonding proton of methanol. 

Table 1 Examples of a Few of the S Functions That Can Be Used to Express the Orientational Dependence of the Intermolecular Potential ...

These are expressed in terms of scalar products between the unit axis system vectors on sites 1 and 2 (on different molecules) and the unit vector 6. from site 1 to 2. The S functions that can have nonzero coefficients in the intermolecular potential depend on the symmetry of the site. This table includes the first few terms that would appear in the expansion of the atom-atom potential for linear molecules. The second set illustrate the types of additional functions that can occur for sites with other than symmetry. These additional terms happen to be those required to describe the anisotropy of the repulsion between the N atom in pyridine (with Zj in the direction of the conventional lone pair on the nitrogen and yj perpendicular to the ring) and the H atom in methanol (with Z2 along the O—H bond and X2 in the COH plane, with C in the direction of positive X2). The important S functions reflect the different symmetries of the two molecules.Note that coefficients of S functions with values of k of opposite sign are always related so that purely real combinations of S functions appear in the intermolecular potential. 

The sum runs over all atoms fi. Each term includes only contributions from those pairs of atomic orbitals in which at least one of the partners is located at the atom in question. The x, y, and z components of the vector operator which differs from the angular momentum operator if only by the factor hli, are d/d, d/dr], and d/di, respectively, where t], and t, are the angles of rotation around the x, y, and z axes passing through the -th nucleus. The action of this operator on atomic orbitals located on atom p. is as follows The s functions are annihilated, and for p functions. 

Huzinaga s primitive optimized set for second row elements.A DZ type basis is derived by contracting (12s8p) —> [5s3p], and a TZ type is derived by contracting (13s9p) —> [6s4p]. The latter contraction is 6,3,1,1,1,1 for the s-functions and 4,2,1,1,1 for the p-functions, and is often used in connection with the Pople 6-3IG when second... 

The derivative of the S function is also a useful concept. It satisfies (from integration by parts)... 

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