Free calculation

In this section we aim to introduce some of the main theoretical ideas which underlie the strategies for modelling liquid crystal molecules. It is clear that there are a very wide range of methods available and we will not attempt to be comprehensive. Instead, we will begin with a brief overview of traditional semi-empirical approaches and then progress to concentrate on treating fully predictive parameter-free calculations of molecular electronic structure and properties in some depth. 

Scheffler M, Dabrowski J. 1988. Parameter-free calculations of total energies, interatomic forces and vibrational entropies of defects in semiconductors. Phil Mag A 58 107-121. 

A test of the computational strategy outlined in the previous paragraph has been performed on a set of synthetic noisy structure factor amplitudes. The diffraction data were computed from the same model density for L-alanine at 23 K as the one used for the noise-free calculations described in Section 3.1. 

Within the computational scheme described in the course of this work, the available information about the atomic substructure (core+valence) can be taken into account explicitly. In the simplest possible calculation, a fragment of atomic cores is used, and a MaxEnt distribution for valence electrons is computed by modulation of a uniform prior prejudice. As we have shown in the noise-free calculations on l-alanine described in Section 3.1.1, the method will yield a better representation of bonding and non-bonding valence charge concentration regions, but bias will still be present because of Fourier truncation ripples and aliasing errors ... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Although Mie s theory was first published in 1908, computations of scattering coefficients were not tabulated to any extent until the 1940s (Lowan, 1948), and then the available tables were quite limited. Nevertheless, considering that each data point represented many hours of error-free calculation with a desk calculator, the accuracy of these early tables is indeed remarkable. 

Here 5 4/y j denotes so-called a reduced matrix element (RME) which depends only on the quantum numbers S,S. Because the Cl wavefunction employed in our spin-free calculations is a linear combination of CSFs which are eigenfunctions of S-, so that Ms = S, it is advantageous to evaluate the RMEs only for the states Ms = S. 

The first striking feature in figure 3 is the enormous PCE in AuCl reflecting the gold maximum of relativistic effects also with respect to property calculations. The absolute amount of the EFG PCE is larger than the relativistic contribution ( ) to the property and nearly as large as the correlation contribution underlining the importance of PCE-free calculations. In copper and silver the PCE is also comparable to the... 

Starting with our field-free calculations we have implemented the calculations of these finite-z-interval integrals. It turned out that these were best performed numerically, also in the interstitial region where the basis functions are represented analjrtically. The reason is that the planar boundary of our integration region is only with great difficulties combined with our representation for the basis functions in spherical coordinates. 

Finally, we stress that our study does not answer all questions. Thus, the fact that we use an approximate density functional in our parameter-free calculations may be one source of errors in the calculated quantities, although this problem is only marginally related to that of a proper treatment of the external field in an electronic-structure method. Second, our method is still in its infancy and many tests are required before it can be established whether it is a useful approach. Third, we have presented a method for directly including a DC field in the calculations, whereas other approaches, based on perturbation theory, also allow for the treatment of AC fields. Our approach allows for an alternative control of the results of the latter in the limit of vanishing frequencies, but it still is an open question how the results can be used in improving the perturbation-theoretic approaches. 

Reprinted from Ref [175] with kind permission of The American Institute of Physics DK3 spin-free calculation DK SOC treatment spin-free calculation - - BP SOC treatment DK3 spin-free calculation - - BP SOC treatment with the X6p orbital... 

Niehaus et al. implemented the method of Jamorski et al diseussed above in the parameterized density-funetional program that we diseussed in Seetion 4 in order to ealeulate exeitation energies. They explored the performanee of their method on a number of smaller moleeular systems and compared the results with those of parameter-fiee density-functional calculations and with experimental values. Some representative results are shown in Table 15, where it is seen that their method performs well also in comparison with the parameter-free calculations. The Table also shows, however, that a perfect agreement with experimental values is by no means automatically obtained, independent of the theoretical approach that is being used. 

As in field-free calculations, there are very many basis sets that may be used in different circumstances. A totally uncoupled representation is often convenient because the matrix elements needed are usually quite simple. For example, for collisions of a molecule in a state with an unstructured atom we need basis functions to handle the molecular rotation n, electron spin s, and end-over-end angular momentum... 

The error resulting from quantization can be modeled as a random variable uniformly distributed over the appropriate error range. Therefore, calculations with roundoff error can be considered error-free calculations that have been corrupted by additive white noise. The mean of this noise for rounding is... 

Another potential application field are intermolecular complexes and clusters [140]. Here, the local correlation method has some conceptual advantages, insofar that it eliminates the notorious basis-set superposition error (BSSE) [141]. This allows for the BSSE-free calculation of interaction energies structures of large molecular clusters i.e. involving many monomers) without the expensive counterpoise correction that is required in conjuction with canonical methods. There are many more application areas one could think of. 

Free Calculation of Response Functions in Time-Dependent Density-Functional Theory. 

At the beginning of this chapter it was asserted that a considerable amount of work could be saved by performing a spin separation in the Dirac equation, using the spin-free part variationally, then treating the spin-dependent terms as a perturbation. We are now in a position to assess the reduction of work that would actually be obtained by performing a spin-free calculation. 

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