Anisotropy, perturbation theories

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

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). 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

All results obtained below with the thermodynamic perturbation theory are limited to the case of axially symmetric anisotropy potentials (see the Appendix, Section A.2), and all explicit calculations are done assuming uniaxial anisotropy (see the Appendix, Section B). 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

Thermodynamic perturbation theory is used to expand the Boltzmann distribution in the dipolar interaction, keeping it exact in the magnetic anisotropy (see Section II.B.l). A convenient way of performing the expansion in powers of is to introduce the Mayer functions fj defined by 1 +fj = exp( cOy), which permits us to write the exponential in the Boltzmann factor as... 

Let us make the relation between the potential anisotropy and the final state distribution more quantitative. In first-order perturbation theory Equation (5.4f) can be directly integrated yielding the (approximate) expression... 

Using time-dependent perturbation theory and taking full account of the symmetry and commutation relations for the high-order dipolar Hamiltonians, Hohwy et al.61 69 gave a systematic analysis of homonuclear decoupling under sample rotation and proposed a novel approach to the design of multiple-pulse experiments. Based on the theoretical analysis, they proposed a pulse sequence that can average dipolar interaction up to the fifth order. One example of these pulse sequences is shown at the top of Fig. 3. This sequence is sufficiently powerful that it is possible to obtain precise measurement of proton chemical shift anisotropies, as shown in Fig. 3. 

On the theoretical side the H20-He systems has a sufficiently small number of electrons to be tackled by the most sophisticated quantum-chemical techniques, and in the last two decades several calculations by various methods of electronic structure theory have been attempted [77-80]. More recently, new sophisticated calculations appeared in the literature they exploited combined symmetry - adapted perturbation theory SAPT and CCSD(T), purely ab initio SAPT [81,82], and valence bond methods [83]. A thorough comparison of the topology, the properties of the stationary points, and the anisotropy of potential energy surfaces obtained with coupled cluster, Moller-Plesset, and valence bond methods has been recently presented [83]. 

Figure 5.2 i Anisotropy of various contributions to the 3-body forces for the cyclic water trimer . See Fig. 5.19 for definition of a. HL refers to Heitler-London term which prohibits modification of the charge clouds of each molecule in the presence of the others, and SCF-def to the result of such deformation, both at the SCF level. Three-body induction is computed directly via perturbation theory. 

An additional source of chemical shielding anisotropies is that of ab initio theoretical calculations.20 25 There has been considerable progress in this area of molecular quantum mechanics, particularly with the use of gauge-invariant atomic orbitals within the framework of self-consistent-field (SCF) perturbation theory.26 In many cases the theoretical quantities have been extremely accurate and have served not only as a corroboration of experimental quantities but also as a reliable source of new data for molecules of second-row atoms (i.e., Li through F). 

The critical 3p i-2p i spacing is expected to be independent of the nature of the chemical donor, but it differs significantly between Sica and Op. This reflects the fact that the CB minimum for the donors on P site is associated with the Xi camel s back structure. Variational calculations based on k.p perturbation theory have been performed for P-site donor and compared self-consistently with spectroscopic data [40]. The calculations are performed as a function of the ratio of a non-parabolicity parameter9 Q to the separation A between CBs Xi and X3. The authors use an anisotropy parameter p equal... 

---