Unperturbed state

This simply means that one must be willing to identify one of the unperturbed states as the best approximation to the state being sought. This, of eourse, implies that one must therefore strive to find an unperturbed model problem, eharaeterized by H that represents the true system as aeeurately as possible, so that one of the Ok will be as elose as possible to /k. 

Moreover, it is assumed that, as the strength of the perturbation is redueed to zero, /k reduees to one of the unperturbed states k and that the full eontent of k in /k is eontained in the first term /k ). 

In the spirit of the time-independent perturbation treatment, I write as a linear combination of the unperturbed states... 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

In general the transitions appearing between the unperturbed states in such perturbation theories are of no physical significance they are simply a result of our attempt to express the true eigenstates of the true perturbed hamiltonian in terms of convenient but erroneous eigenstates of the unperturbed erroneous hamiltonian. If we were able to find the true eigenstates—mid this is, of course, possible in principle— no such transitions would be discovered and the apparent time-dependence would disappear. 

The system to be treated consists of two nuclei A and B and two electrons 1 and 2. In the unperturbed state two hydrogen atoms are assumed, so that the zeroth-order energy is 2WH-If the first electron is attached to nucleus A and the second to nucleus B, the zeroth-order eigenfunction is ip (1)

zeroth-order energy, so that the system... 

The Helium Molecule and Molecule-ion.—The simplest example of a molecule containing a three-electron bond is the helium molecule-ion, in which a Is eigenfunction for each of two identical atoms is involved. The two unperturbed states of equal energy are He He+ and He-+ He. The formation of this molecule might be represented by the equation He Is2 >5 + He+ Is 5 —>- He (Is + ls) 2 Three dots in a horizontal line placed between the two atomic symbols may be used to designate a three-electron bond He He+. 

In the theory of radiationless transitions as covered in this paper (6,4g), the two non BO terms are treated as perturbations (not externally plied, but arising as imperfections within this model of molecular structure) that can induce transitions between unperturbed states each of which is taken to be a specific Bom-Oppenheimer product state ... 

The same parameter for the unperturbed state. Corresponding parameters for the distribution of chain segments about the center of gravity of the molecule, in the Gaussian approximation. 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

Here, theta state [21]. It is now well established that chains in the bulk undiluted state are in the unperturbed state. Equation (4a) represents the probability distribution of the vectorial quantity r. A less detailed form of representation is the distribution zt (r) showing the probability that the magnitude r of r has a certain value irrespective of direction. Thus, the probability that the chain end-to-end length is in the range r to r+dr irrespective of its direction is... 

Although the disorientation process might be looked upon as a melting process, eq. (9.54) does not give a good estimate of the entropy of fusion of polymers. A more reasonable approach is to introduce the mean square displacement length of the actual chain in its unperturbed state [7], The number of such units would then replace r in eq. (9.54), but one still needs to determine the coordination number z in an independent manner. 

Fig. 4. Two-angle fibers can be easily deformed via the bending and twisting of their linkers. This can be most easily seen for the special case of a planar zig-zag fiber under an external tension F, which extends the fiber via the bending of its linkers from its unperturbed state with contour length Lo (a) to a stretched state of length L>Lo (b).

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

In the rod limit, Le = L, and de = d. On the other hand, in the coil limit, Le - V6de, if we assume the chain to be in the unperturbed state. It is to be noted that the axial ratio of the fuzzy cylinder is greater than unity even in the coil limit. At intermediate N, L and the axial ratio Le/de may be calculated, respectively, from the Kratky-Porod equation [102,103] for the (unperturbed)... 

From Eq. (72) we see that the contribution to the MCD intensity from the perturbation to the transition density can be identified with the MCD due to the mixing of the excited state J with other excited states. The remainder of the MCD intensity from terms and spin-orbit-induced C terms is due to the perturbation of the integrals over the electric dipole moment operator (Eq. 52). The perturbed integrals thus include the contribution to the MCD from the mixing of excited states with the ground state. The perturbed integrals are written in terms of unperturbed orbitals (Eqs. 53 and 54) rather than unperturbed states or transition densities as this form is much easier to compute. With some further effort the contribution to the MCD from the perturbed integrals can also be analyzed in terms of transitions. 

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

---