Perturbation theory exact

We found a correct account of the broadening magnitude and increase with temperature, out of reach of the renormalized perturbation theory. Exact numerical calculations by Schreiber and Toyozawa53 agree with our data and confirm our values of the exciton-phonon coupling strengths. 

For present purposes it is more useful to concentrate on other approaches, which start from the finite-basis form of the linear variation method. In many forms of variation-perturbation theory, exact unperturbed eigenfunctions are not required and the partitioning of the Hamiltonian into two terms is secondary to a partitioning of the basis. At the same time, as we shall see, it is possible to retrieve the equations of conventional perturbation theory by making an appropriate choice of basis. 

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

This Introductory Section was intended to provide the reader with an overview of the structure of quantum mechanics and to illustrate its application to several exactly solvable model problems. The model problems analyzed play especially important roles in chemistry because they form the basis upon which more sophisticated descriptions of the electronic structure and rotational-vibrational motions of molecules are built. The variational method and perturbation theory constitute the tools needed to make use of solutions of... 

Note, that the eigenfunctions of H are /n with eigenvalues En. Compare this "exact" value with that obtained by perturbation theory in part a. 

Thus, the value of E the first perturbation to the Hartree-Fock energy, will always be negative. Lowering the energy is what the exact correction should do, although the Moller-Plesset perturbation theory correction is capable of overcorrecting it, since it is not variational (and higher order corrections may be positive). 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem which has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution of the already known system. This is described mathematically by defining a Hamilton operator which consists of two part, a reference (Hq) and a perturbation (H )- The premise of perturbation methods is that the H operator in some sense is small compared to Hq. In quantum mechanics, perturbational methods can be used for adding corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

It should be observed that the subscript exact here refers to the lowest eigenvalue of the unrelativistic Hamiltonian the energy is here expressed in the unit Aci 00(l+m/Mz) 1 and Z is the atomic number. If the HE energies are taken from Green et al.,8 we get the correlation energies listed in the first column of Table I expressed in electron volts. The slow variation of this quantity is noticeable and may only partly be understood by means of perturbation theory. 

For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion... 

To illustrate the accuracy of the perturbation theory these results are worth comparing with the well-known values of h and I4 for t = 1 rigorously found from first principles in [8]. It turns out that the second moment in Eq. (2.65a) is exact. The evaluation of I4, however, is inaccurate its first component is half as large as the true one. The cause of this discrepancy is easily revealed. Since M = / and (/) = J/xj, the second component in Ux) is linear in e. Hence, it is as exact in this order as perturbation theory itself. In contrast, the first component in IqXj is quadratic in A and its value in the lowest order of perturbation theory is not guaranteed. Generally speaking... 

In the next section we will show how perturbation theory must be developed to provide an exact value of up. 

This equation is exact, and its kernel is expanded into the series (2.82). The decoupling procedure resulting in (2.24) is equivalent to retention of the first term in this series. Using Eq. (2.86), one may develop a consistent perturbation theory which will take into consideration higher orders of... 

To prove this let us make more precise the short-time behaviour of the orientational relaxation, estimating it in the next order of tfg. The estimate of U given in (2.65b) involves terms of first and second order in Jtfg but the accuracy of the latter was not guaranteed by the simplest perturbation theory. The exact value of I4 presented in Eq. (2.66) involves numerical coefficient which is correct only in the next level of approximation. The latter keeps in Eq. (2.86) the terms quadratic to emerging from the expansion of M(Jf ). Taking into account this correction calculated in Appendix 2, one may readily reproduce the exact... 

The width of this Lorentzian line is half as large as that found in (3.37). This, however, is not a surprise because the perturbation theory equation (3.23) predicted exactly this difference in the width of the line narrowed by strong and weak collisions. This is the maximal difference expected within the framework of impact theory when the Keilson-Storer kernel is used and 0 < y < 1. 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

Although applications of perturbation theory vary widely, the main idea remains the same. One starts with an initial problem, called the unperturbed or reference problem. It is often required that this problem be sufficiently simple to be solved exactly. Then, the problem of interest, called the target problem, is represented in terms of a perturbation to the reference problem. The effect of the perturbation is expressed as an expansion in a series with respect to a small quantity, called the perturbation parameter. It is expected that the series converges quickly, and, therefore, can be truncated after the first few terms. It is further expected that these terms are markedly easier to evaluate than the exact solution. 

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