Scaled quantum mechanical methods

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

A full-scale treatment of crystal growth, however, requires methods adapted for larger scales on top of these quantum-mechanical methods, such as effective potential methods like the embedded atom method (EAM) [11] or Stillinger-Weber potentials [10] with three-body forces necessary. The potentials are obtained from quantum mechanical calculations and then used in Monte Carlo or molecular dynamics methods, to be discussed below. 

All the macroscopic properties of polymers depend on a number of different factors prominent among them are the chemical structures as well as the arrangement of the macromolecules in a dense packing [1-6]. The relationships between the microscopic details and the macroscopic properties are the topics of interest here. In principle, computer simulation is a universal tool for deriving the macroscopic properties of materials from the microscopic input [7-14]. Starting from the chemical structure, quantum mechanical methods and spectroscopic information yield effective potentials that are used in Monte Carlo (MC) and molecular dynamics (MD) simulations in order to study the structure and dynamics of these materials on the relevant length scales and time scales, and to characterize the resulting thermal and mechanical proper-... 

One approach to a better understanding of the properties of the solid surface is to model the electron structure with quantum mechanical methods, which is a useful complement to experimental techniques. It allows direct observation of atomic-scale phenomena in complete isolation, which cannot be achieved in current experimental studies. 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

However, the most reliable and widely used is the method developed by Puley [16], which is now referred to as Scaled Quantum Mechanical Force Field (SQMF). In SQMF a scale constant Xs is ascribed to each internal coordinate qH such that the corrected (scaled) force-constants are calculated according to the equation ... 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

Normally the scaling factors are extracted by minimizing the squared deviation (4) considered as a functional R A) of the variable set A, - The frequency parameters z alc now correspond to the harmonic normal frequencies calculated with the scaled quantum-mechanical force-field (6). The first and second derivatives of R( A) with respect to the scaling factors can be calculated analytically [17,18], which permits to implement rapidly converging minimization procedures of the Newton-Gauss type. Alternative iterative minimization methods were also proposed [19]. 

The transferability of the scale factors is based on the fact that a given quantum-mechanical method produces almost equivalent relative errors of the quantum-mechanical force constants for a series of related molecules, and hence equivalent scale factors [5]. 

V. Scaled Quantum Mechanical Force-Field Method.240... 

Scaled quantum-mechanical force fields for furan (and thiophene) and its isotopomers have been calculated with the B3LYP/6-31G method. Corresponding MP2 and FIE calculations gave less satisfactory results. Excellent agreement... 

Optimized geometries, scaled quantum-mechanics (SQM) force fields, and the corresponding vibrational frequencies, IR absorption intensities, and scale factors were calculated for thiazole and the [2(2)-H], [4-H-2], and [2,5-H-2(2)] isotopomers of thiazole using the DFT and B3LYB/6-31G methods <1995JCM354, 1995JCM174>. 

Car-Parrinello methods contrasted wilhslalic (0 Ktemperature) computational quantum mechanical methods They can treat entropy accurately without the need to use models such as the harmonic approximation for degrees of freedom of atomic motions. They can be used to sample potential energy surfaces on picosecond time scales, which is essential for treating liquids and aqueous systems. Tliey can be used to sample reaction pathways or other chemical processes with a minimum of a priori assumptions. In addition, they can be used to find global minima [in conjunction with methods of optimization such as simulated annealing (Kirkpatrick et at, 1983)] and to step out of local minima. 

White CA, Johnson BG, Gill PMW, Head-Gordon M (1996) Linear scaling density functional calculations via the continuous fast multipole method. Chem Phys Lett 253 268-278 Lee TS, Lewis JP, Yang W (1998) Linear-scaling quantum mechanical calculations of biological molecules the divide-and-conquer approach. Comput Mater Sci 12 259-277 Van Alsenoy C, Yu CH, Peelers A, Martin JML, Schafer L (1998) Ab initio geometry determinations of proteins. 1. Crambin. J Phys Chem A 102 2246-2251... 

Formal chemisorption theory has also been used to described a number of other important chemisorption phenomena, such as the stabilizing effects of neighboring electropositive adsorbates (K, Na), the destabilizing effects of electronegative adsorbates (Cl, F), surface relaxation and surface reconstruction [18], More recently. Hammer and Nbrskov [17] applied formal theory to elegantly explain the results from a series of large-scale periodic density functional quantum chemical calculations for adsorption on transition metal and bimetallic surfaces. Chemisorption theory will undoubtedly continue to play an important role in describing relevant concepts in chemisorption and surface reactivity well into the future. More quantitative results from theory, however, will require more sophisticated quantum mechanical methods. 

The application of ab initio methods in the calculation of harmonic force fields of transition metal complexes has been hampered by the size of these systems and the need to employ costly post-Hartree-Fock methods, in which electron correlation is taken into account. Thus, the fruitful symbiosis between ab initio theory and experiment, to determine empirically scaled quantum mechanical force fields, has been virtually absent in studies of transition metal complexes. 

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