Ordering parameter perturbation theory

The theory involves four parameters related to the use of low order perturbation theory ... 

Truncation of the Many-body Perturbation Expansion General Remarks.—Perturbation theory forms the basis of a unique approach to the calculation of accurate expectation values in that it provides a clearly defined order parameter indicating the relative importance of various terms. This order... 

Perturbation theory provides a clearly defined order parameter in the expansion for expectation values giving a least-biased indication of the importance of various terms. To quote Brandow 107... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

The first step in quantitative description of pure polyamorphic fluid is a selection of the model that can qualitatively describe a possible multiplicity of critical points in wide range of temperatures and pressures. A great many of explanations of multicriticality in monocomponent fluids (perturbation theory models semiempirical models lattice models, two-state models, field theoretic models, two-order-parameter models, and parametric crossover model has been disseminated after the pioneering work by Hemmer and Stell Here we test more extensively the modified van der Waals equation of state (MVDW) proposed in work and refine this model by introducing instead of the classical van der Waals repulsive term a very accurate hard sphere equation of state over the entire stable and metastable regions... 

We return here to mean field theory with a scalar order parameter thick film geometry, assuming hard walls or surface against vacuum, respectively) at z = 0 and z = L. Starting again from eq. (14), we may disregard the x and y-coordinates [as in our treatment of the interfacial profile, eqs. (177)—(181)], but now we have to add a perturbation 2Fv(bare) to... 

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. 

Becke s three parameters Lee, Yang, Parr functional Complete Active Space Perturbation Theory 2nd Order Complete Active Space SCF Coupled Cluster Single Double CCSD (Triple)... 

In order to derive expressions which can be used for calculating magnetic properties at the quasi-relativistic level of theory, we analogously replace p in equations (13)-(18) by the corresponding generalized momentum %. The expressions are inserted into equation (20) and one proceeds as described in Section 3. The total quasi-relativistic energy as a function of the perturbation strength parameter then becomes... 

At the ZORA level, the sum of the A, B and X operators vanishes, thus only the first term and half of the second and the third terms in equation (62) are nonzero, whereas at the lORA level also the two last and energy dependent renormalization terms must be considered. At the general quasi-relativistic level of theory all terms contribute to the magnetic properties. Expressions for second-order magnetic properties can be obtained by differentiating the energy expression (59) twice with respect to the perturbation-strength parameters of the vector potentials. 

The ordering parameter perturbation theory can be extended to the case of more perturbations simultaneously present. The new assumption is the following form of the Hamiltonian... 

The equilibrium order determined within the mean-field theory is perturbed due to thermal fluctuations which give rise to collective excitations. Except in the close vicinity of the phase/structural transitions, the thermal fluctuations of the order parameter can be assumed small, and the free energy of the fluctuations can be considered a correction to the mean-field free energy. In such a case, the fluctuations of the liquid-crystaUine order are described consistently by a harmonic Hamiltonian of the form... 

Perturbation theory is no longer applicable when the field-induced deformations reach giant values 10 (see fig. 10). In strong fields Ho > 15 kOe) one must take into account the effect of deformation in the energy spectrum of Tm + ions, i.e., one should use solutions of self-consistent field equations for the electronic order parameters Op) in this case the dependence of magnetostriction on H becomes nonlinear, which was observed experimentally by Al tshuler et al. (1980) and Bondar et al. (1988). 

The critical behaviours of higher order satellites near a continuous phase transition are of particular interest, as they provide the opportunity to study the crossover exponents of the respective symmetry breaking perturbations in the spin Hamiltonian. Each order of satellite has an associated order parameter critical exponent given by / = 2 - a -where a is the specific heat exponent, and crossover exponent As an example, if the transition is described by the 3DXY model, then the exponent 2 measures the crossover caused by a perturbation of uniaxial symmetry. For this model, theory predicts that = o P, with... 

Relationships between the Six Leslie Coefficients and Three Molecular Parameters in the Doi Theory By limiting their analysis to the first-order perturbation from the equilibrium state, Kuzuu and Doi (1984) derived the Ericksen-Leslie equation from the Doi theory under weak velocity gradient and obtained the following relationships between the six Leslie coefficients and the three molecular parameters (concentration, molecular weight, and the order parameter) appearing in the Doi theory ... 

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