Energy first-order effect

Therefore, the principal role of the inclusion of the ionic term in the wave function is the reduction of the kinetic energy from the value in the purely covalent wave function. Thus, this is the delocalization effect alluded to above. We saw in the last section that the bonding in H2 could be attributed principally to the much larger size of the exchange integral compared to the Coulomb integral. Since the electrical effects are contained in the covalent function, they may be considered a first order effect. The smaller added stabilization due to the delocalization when ionic terms are included is of higher order in VB wave functions. 

As the counterion penetrates the plane of the interfacial head groups, the surface pressure will be affected as a first-order effect thus, the expansion of the 7r-A isotherms for the fatty acid monolayers is in the same sequence as the cation sizes noted above. The penetrated counterions must be held with an energy at least comparable to KT since they are not expelled during the kinetic movement of the film molecules, but remain in place and increase the surface pressure. To penetrate the plane of the head groups in the monolayer, the counterions must possess sufficient adsorption energy to overcome the work against the kinetic surface pressure 7tK, such that, according to Davies and Rideal (10) ... 

It will be noticed that two contributions to the vibrational potential energy, of the general form V(R ) and independent of the rotational and spin quantum numbers, have also been included in equation (7.124). These are corrections to Vn(R), the zeroth-order contribution to the electronic energy defined in equation (7.76). The first term, Vr d)(R), is the adiabatic contribution to the electronic energy, which we have discussed in section 2.7. It describes the first-order effect of the nuclear kinetic energy within the... 

For / v, — Vs, J2 can be ignored in comparison with (v, — vs)2 in Eqs. 11.49b and c, so that the changes in energy levels resulting from weak coupling are only the anticipated first-order effects. Initially we use this approximation, hence can use as the eigenfunctions. 

The dipole polarizability also expresses the second-order effect of an electric field on the energy levels of an atom or molecule. We can write, for an atom (which has no dipole moment and therefore, in general, no first-order effect) ... 

While first-order effects of purely inductive substituents on excitation energies of alternant hydrocarbons vanish, higher-order perturbation theory gives nonzero contributions. Thus, Murrell (1963), using second-order perturbation theory, derived the relation... 

Figure 2.29. First-order effects of one-electron heteroatom replacement on frontier orbital energies of a (47V+2)-electron annulene a) for N as an example of an electronegative second-row element (da, < 0, Sfi, = 0) and b) for Si as an example of an electropositive third-row element (da > 0, d/3, > 0).The effects of change in elec-troneg ivity 6a are shown as white vertical arrows (by permission from Michl, 1984).

Next, we consider the even weaker second-order perturbation of the three sublevels of T that is due to Recall that the analogous second-order contributions from are neglected in this approach, as they are generally believed to be small, and the first-order contributions have already been included. It is seen from inspection of Equation 3.3 that the operator can mix each of these three triplet wave functions with singlet, other triplet, and quintet wave functions. This interaction has no first-order effect on the energies -D, -Dy, and -D of the three substates, T,, T, and T, respectively, but in second order. Equation 3.14 with V = IP, they will be affected somewhat and become -D , -Dy, and -Z) . If we continue to define the zero-field splitting parameters by D = 3DJ/2 and E = (DJ - Dy )/2, they can be compared with the values observed. In Section 3.3, we noted the difficulties involved in attempts to evaluate these values accurately in this fashion, due to the very large number of states over which the summation in Equation 3.14 is necessary, and we commented on alternative methods of evaluation such as response theory. 

The deformation parameter rj describes how prolate or oblate the cluster is. This parameter was determined by minimizing the total energy calculated by adding the eigenvalues of the occupied electronic states. For alkali clusters with N less than 100, values up to q = 0.5 are estimated for open-shell clusters. The main first order effects of the ellipsoidal model are energy shifts that are proportional to q. The ellipsoidal model explains well the fine-structure features of the mass spectra [25], that is, those features which are beyond the realm of the spherical jellium model. 

Thus the n energy for double union to benzene is greater than that for single union to hexatriene and the difference is large, being a first-order effect. This... 

Rule 2. (a) Inductive substituents in even AHs, or at inactive positions in odd AHs, have no first-order effect on the n energy. 

It is instructive to compare these results with the behavior of the Is state. In the first place, the effect of a uniform electric field on the Is level is zero, to first order, because (ls i/ ls) vanishes for reasons of symmetry. Only when first-order corrections are made to the 1 s wavefunction are energy effects seen, and these occur in the second-order energy terms. Therefore, the Is state gives a second-order Stark effect, but no first-order effect. The Is state gives no first-order effect because the spherically symmetric zeroth-order wavefunction has no electric dipole to interact with the field. But the proper zeroth-order wave functions for some of the = 2 states, given by Eqs. (12-81) and (12-82), do provide electric dipoles in opposite directions that interact with the field to produce first-order energies of opposite signs. 

Notice that the energy change for the second-order effect goes as F [(Eq. (7-62)], whereas that for the first-order effect goes as F. These dependences are indicative of an induced dipole and a permanent dipole, respectively. The induced dipole for the Is state depends on the field strength F, since, as F increases, the dipole moment increases (due to mixing in higher states). This induced dipole, which depends on F,... 

---