Highly-symmetric systems

The reason why complexity and symmetry are linked together is quite straightforward. Indeed, a representation of highly symmetrical systems requires fewer characteristics than that of objects having low symmetry because, if we know the characteristics of one object, we can employ them to represent all those which are symmetrical with the given one. 

For all but the few smallest clusters, the number of possible structures is virtually unlimited. In order to be able to treat the larger systems, quite restrictive assumptions about their geometry has to be made. For those clusters where well-defined equilibrium structures do exist, these are likely to possess a non-trivial point-group symmetry (in many cases the highest possible symmetry). It therefore seemed justified to focus the study on high-symmetric systems. Symmetry can also be used to simplify the calculation of electronic structure, and reduces the number of geometrical degrees of freedom to be optimized. 

A. Highly symmetric systems and the Redfield theory for electron spin relaxation... 

A. Highly Symmetric Systems and the Redfield Theory for Electron Spin Relaxation... 

For instance, GMCSC calculations on the boron anion [2] and on the dilithium molecule [26], both in their ground states, have shown how singleconfiguration wavefunctions, including spin-coupled ones, can be hard-put to provide a robust description of certain highly-symmetric systems. By robust description , we mean one that will not change, at least qualitatively, as more configurations are added to the wavefunction. 

One of the questions implicitly raised by the studies summarized in the previous Section was, of course, whether similar behaviour is exceptional or, in fact, fairly common, at least in highly symmetric systems. 

The exponent m cannot be regarded as a fitting parameter but depends on the symmetry of the system. In most cases, m = 3/2 [16, 140, 158, 166, 167, 174, 175], but m = 2 for highly symmetric systems, such as aligned Stoner-Wohlfarth particles. In particular, the m = 3/2 law is realized for misaligned Stoner-Wohlfarth particles and for most domain-wall pinning mechanisms [5], Experimental values of m tend to vary between 1.5 to 2 [136, 158]. Linear laws, where m = 1, are sometimes used in simplified models, but so far it hasn t been possible to derive them from physically reasonable energy landscapes [5, 16, 176]. The same is true for dependences such as /H- l/H0 [177], where series expansion yields an m = 1 power law. 

The above example referred to an octahedral configuration. Other highly symmetrical systems, for example, tetrahedral arrangements, can also display this effect. For general discussion, see, e.g. References [65-67],... 

The Schrodinger equation is solved for each of the regions, I, II, and III separately for a local potential, and the wavefunctions are matched at the boundaries. This approximation was found to describe the electronic structure quite satisfactorily for highly symmetric systems but gave in some cases very bad results, for example for diatomic molecules and the water molecule [75]. The main reason was the use of the constant potential in region II. Several attempts were made to solve this problem using overlapping spheres [76]. 

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis function may in principle be used expo ... 

The work on the formic acid dimer focused on the double-well potential for a highly symmetric system. An attempt to locate a double-well potential for a less symmetric system was made by Zielinski and Poirier (1984). They studied the formamide dimer and isolated a possible structure for the transition state for a double-proton transfer along the reaction path to the formimidic acid dimer (a dimer of the enol form of formamide) using the 3-21G basis set. The proposed transition state is only slightly less stable than the formimidic acid dimer. In other words, a very asymmetric double-well potential was found with a very shallow well on the formimidic acid dimer side of the reaction. It will be interesting to see the shape of the function for a double-proton transfer between formamide and amidine, which would more closely mimic the double-proton transfer that may be possible for the A-T pair. 

For highly symmetric systems and properly chosen coordinate systems, the matrix containing the polarizabilities become diagonal with only two different values, ay = ctzz and a i = olxx = ayy. Then the average polarizability is... 

Our aim in this chapter will be to establish the basic elements of those quantum mechanical methods that are most widely used in molecular modelling. We shall assume some familiarity with the elementary concepts of quantum mechanics as found in most general physical chemistry textbooks, but little else other than some basic mathematics (see Section 1.10). There are also many excellent introductory texts to quantum mechanics. In Chapter 3 we then build upon this chapter and consider more advanced concepts. Quantum mechanics does, of course, predate the first computers by many years, and it is a tribute to the pioneers in the field that so many of the methods in common use today are based upon their efforts. The early applications were restricted to atomic, diatomic or highly symmetrical systems which could be solved by hand. The development of quantum mechanical techniques that are more generally applicable and that can be implemented on a computer (thereby eliminating the need for much laborious hand calculation) means that quantum mechanics can now be used to perform calculations on molecular systems of real, practical interest. Quantum mechanics explicitly represents the electrons in a calculation, and so it is possible to derive properties that depend upon the electronic distribution and, in particular, to investigate chemical reactions in which bonds are broken and formed. These qualities, which differentiate quantum mechanics from the empirical force field methods described in Qiapter 4, will be emphasised in our discussion of typical applications. 

Perhaps the most familiar (non-trivial) example of the last of the effects listed above is a minimal basis calculation of the MOs of a highly symmetrical system. It is a familiar fact that the Hiickel MOs of the rr-system of benzene also solve the SCF equations when both are expressed in terms of a minimal basis of 2p carbon AOs. The solid-state analogy of this Hiickel/SCF case is the fact that the plane-wave orbitals of a non-interacting electron gas solve the Hartree-Fock equations for the system for exactly the same reason ... 

Mahmoud MA, Chamanzar M, Adihi A, El-Sayed MA. Effect of the dielectric constant of the surrounding medium and the substrate on the surface plasmon resonance spectrum and sensitivity factors of highly symmetric systems silver nanocubes. J Am Chem Soc 2012 134 6434-6442. 

Solution of the PB equation, which is necessary when there are mobile ions surrounding fixed charge distributions, is difficult because of the nonlinearities in the equation associated with the mobile ion Boltzmann factors. The PB equation can only be solved analytically for highly symmetric systems, eg., charged planar plates in electrolytic solutions. For example, consider a single plate with surface charge density... 

In combination with the triangle condition k 6, it is clear that for the crystal-field splitting only the k values 0, 2, 4, 6 have to be considered. It is already mentioned that the parameter shifts the whole configuration. For high symmetric systems, even fewer k values will occur ... 

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