System, unperturbed

Note that both the numerator and denominator in the final expression are always positive expressions in the case of the denominator, we know this because is the lowest energy eigenvalue of the unperturbed system. (The denominator reduces to a difference in orbital energies.)... 

The superscript (0) here denotes the unperturbed system, molecular orbitals (eq. (3.20)) can be expressed as... 

Where // is the complete Hamiltonian operator for the unperturbed system and the usual quantum-mechanical integrals over all space are indicated. 

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

By essentially the same energy it is meant that the energies of the states of the unperturbed system differ by an amount less than the possible... 

The distance between the two interacting atoms will be denoted by r, and /g will denote the average distance (or the equilibrium distance in a hypothetical vibrationless state) of the unperturbed system. (The distance will be independent of the isotopic mass.) A displacement... 

We have only recently understood the phenomena that control rainwater pH in the natural, unpolluted environment. As pointed out in Section 16.2, these appear to be mainly the cycles of sulfur and nitrogen compounds. A model of the unperturbed system is necessary in order to understand and predict the changes that occur when strong sulfur- and nitrogen-acids are... 

It should be noted, on the other hand, that a symmetry reduction is predicted even in molecules II, VIII and XIII whose peripheral skeletons correspond to 4n-l-2 cyclic polyenes. The transannular bonds in these molecules are different in nature from those mentioned above. For example, the introduction of the transannular bonds between atoms 2 and 8 and between 3 and 7 of cyclododecapentaene to form bowtiene (II) (Fig. 3) brings about the splitting of the top filled degenerate orbitals of the unperturbed system into two levels, one with its energy raised and... 

We start by assuming for a conjugated molecule a fully-symmetrical arrangement of carbon nuclei as an unperturbed system. Electronic wavefunctions and the corresponding energies... 

At the beginning it is necessary to describe the unperturbed system very well, independently of the polarization functions Let us assume that the unperturbed system is reasonably well described by using some finite set of basis functions As... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

Since /(e) is identically zero, the coefficients bk all vanish. Thus, the coefficients of A on the left-hand side of equation (9.21) are equal to the coefficients of on the right-hand side. The coefficients of A give equation (9.18) for the unperturbed system. The coefficients of A yield... 

If the complete set of eigenfunctions for the unperturbed system includes a continuous range of functions, then the expansion of must include these functions. The inclusion of this continuous range is implied in the summation notation. The total eigenfunction tp for the perturbed system to first order in X is, then... 

The perturbation method presented in Section 9.3 applies only to non-degenerate eigenvalues eX of the unperturbed system. When E is degenerate, the denominators vanish for those terms in equations (9.40) and (9.41) in which Ef is equal to E making the perturbations to E and ipn indeterminate. In... 

The unperturbed system is, of course, the hydrogen atom with a point nucleus. (Inside the nuclear sphere, the exponential... 

With the experiment described above in mind, represent the Hamiltonian of the unperturbed system by H° and that of the perturbed system by... 

Assume that the perturbation, XH is small compared with H°, where X is a parameter. As X - 0, the eigenvalues and eigenfunctions are those of the unperturbed system, as given by... 

The dielectric response of a solvated protein to a perturbing charge, such as a redox electron or a titrating proton, is related to the equilibrium fluctuations of the unperturbed system through linear response theory [49, 50]. In the spirit of free energy... 

The statistical perturbation theory arising from the classical work of Zwanzig34 and its detailed implementation in a molecular dynamics program for computation of free energies is described in detail elsewhere.35 36 We give a very brief description of the method for the sake of completeness. The total Hamiltonian of a system may be written as the sum of the Hamiltonian (Ho) of the unperturbed system and the perturbation (Hi) ... 

However, this Hamiltonian is manifestly non-Hermitian (unless S = 0) and therefore cannot correspond to a physical unperturbed system. Neither of the operators H/(0) and H"(°) can serve as a proper unperturbed Hamiltonian for the PMO rationalization (unless S = 0, when both are equivalent to a standard H(0) such as that underlying, e.g., Example 1.1 or Fig. 3.13). 

---