Perturbation theories simplicity

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

The next lowest unperturbed energy level however, is four-fold degenerate and, consequently, degenerate perturbation theory must be used to determine its perturbation corrections. For simplicity of notation, in the quantities and we drop the index n, which has the value... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

In the canonical partition function of (5.1), we have for simplicity ignored combinatorial prefactors. Free energy perturbation theory [12] relies on evaluating effectively the ratio of the partition functions to obtain the free energy difference between the initial and final states corresponding to coupling parameters A = 1 and 0 (see also Chap. 2),... 

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

Feg). Subsequently, thermodynamic properties of spins weakly coupled by the dipolar interaction are calculated. Dipolar interaction is, due to its long range and reduced symmetry, difficult to treat analytically most previous work on dipolar interaction is therefore numerical [10-13]. Here thermodynamic perturbation theory will be used to treat weak dipolar interaction analytically. Finally, the dynamical properties of magnetic nanoparticles are reviewed with focus on how relaxation time and superparamegnetic blocking are affected by weak dipolar interaction. For notational simplicity, it will be assumed throughout this section that the parameters characterizing different nanoparticles are identical (e.g., volume and anisotropy). 

In first-order perturbation theory (and restricted for simplicity, to two-electron configurations) the Eq. (9) is simplified if it is possible to neglect one of the two terms Hi or H2 as a perturbing term (this is of course possible if the two differ greatly in magnitude). Three cases may therefore be distinguished (couplings) ... 

With the ansatz (11.2) we reorder bare perturbation theory in powers of u. For simplicity taking A = 1, from Eq. (4.16 ii) we find... 

Quasi-degeneracy Effects.—The convergence properties of the non-degenerate formulation of the many-body perturbation theory deteriorate when quasidegeneracy is present in the reference spectrum. In view of its simplicity, however, there is considerable interest in exploring the range of applicability of the nondegenerate formalism. 

Ab initio QED calculations for heavy few-electron atoms are generally performed by perturbation theory. In recent research (Yerokhin et al. 2000, 2001), in the zeroth approximation the electrons interact only with the Coulomb field of the nucleus. To zeroth order the binding energy is given by the sum of one-electron binding energies. The interelectronic interaction and the radiative corrections are accounted for by perturbation theory in the parameters 1/Z and a, respectively. Since 1/Z a for very-high-Z ions, for simplicity we can classify all corrections by the parameter a. 

In the above discussion we have not considered dissipative processes. To include these processes we can, in the framework of the above applied perturbation theory, aside from the perturbation operator U, include a time-independent operator which induces transitions between states tE o- In this case the tensor ej, ij is again defined by the expression (7.52), where in the resonant denominators the energy Huj must be replaced by a complex quantity tuv + ih ylu , k), 7 = 7 + iy", with I7I -C uj. Knowledge of the function 7(0 , k) is important, for example, in the analysis of a lineshape. Below we take into account the exciton-phonon interaction and, for simplicity, consider only the first order of perturbation theory. 

Connection with conventional perturbation theory and the Bloch method. - For the sake of simplicity we wiU now choose the reference set dp so that it consists of p eigenfunctions to the unperturbed... 

In high magnetic fields, M, Wq and the quadrupole shift can be calculated using perturbation theory. Taking = 0 for simplicity, the first-order shift is... 

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