Perturbation theory Perturbative methods

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Claverie P 1971 Theory of intermolecular forces. I. On the inadequacy of the usual Rayleigh-Schrddinger perturbation method for the treatment of intermolecular forces Int. J. Quantum Chem. 5 273... 

Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

Choose the region (single or multiple molecules) of interest for an ab initio calculation from the total molecular system. HyperChem performs the ab initio calculation for the selected region using the perturbation of an electrostatic potential arising from the net charges on the atoms of the un selected part. (For further details of this electrostatic potential perturbation implemented in HyperChem, please see the second part of this book. Theory and Methods). 

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

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). 

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],... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

Perturbative methods (CASPT2 [17], NEVPT2 [18]) add the dynamical correlation in an effective way, using multiconfigurational second-order perturbation theory on the CASSCF input states. These methods have proved to be suitable for studying problems in spectroscopy, photochemistry, and so on [19, 20]. 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

To circumvent this problem avoiding such inefficient exploration of the configurational space, several methods have emerged. Two particularly useful approaches are the Umbrella Sampling and the Statistical Perturbation Theory. Both methods can be used either with Monte Carlo or with Molecular Dynamics simulations. 

The RF interference associated with an infinite number of effective RF fields is a typical problem addressed by the Floqute theory.28 32 To solve the problem, one may, however, face the diagonalization of a matrix with infinite dimensions, which is often unlikely to be done analytically. Certain approximations, such as the perturbation method, may be used.32... 

In general, the results of the perturbation theory based methods, SOPPA, SOPPA(CCSD), MPn and CCSD become smaller with increasing level of theory, whereas the results of the CASSCF/RASSCF calculations go through a maximum for the or °°°CAS wavefunctions depending on the molecule. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

Temperature dependence proton relaxivity 188 relaxation rate 144-5 Temperature-sensitive contrast agents 218-19 Th -" 368 Ti "" 347 Tilt angle 242 Time constant 14 Time correlation functions 76 Time-dependent mechanism 14 Time-dependent perturbation methods 23 Time-dependent perturbation theory 45-8 Titanium(III) 115,134-5,161 TPEN 224 TPPS 219... 

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

Another class of methods uses more than one Slater determinant as the reference wave function. The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above. These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCI), and /V-clcctron valence state perturbation theory (NEVPT) methods.5... 

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