Large systems

Periodic boundary conditions force k to be a discrete variable with allowed values occurring at intervals of lull. For very large systems, one can describe the system as continuous in the limit of i qo. Electron states can be defined by a density of states defmed as follows ... 

Makri N 1999 Time dependent quantum methods for large systems Ann. Rev. Phys. Chem. 50 167... 

The accuracy of most TB schemes is rather low, although some implementations may reach the accuracy of more advanced self-consistent LCAO methods (for examples of the latter see [18,19 and 20]). However, the advantages of TB are that it is fast, provides at least approximate electronic properties and can be used for quite large systems (e.g., thousands of atoms), unlike some of the more accurate condensed matter methods. TB results can also be used as input to detennine other properties (e.g., photoemission spectra) for which high accuracy is not essential. 

The first part of the method involves sorting all the atoms into their appropriate cells. This sorting is rapid, and may be perfonned at every step. Then, within the force routine, pointers are used to scan tlirough the contents of cells, and calculate pair forces. This approach is very efficient for large systems with short-range forces. A certain amount of unnecessary work is done because the search region is cubic, not (as for the Verlet list) spherical. 

Darden T, York D and Pedersen L 1993 Particle mesh Ewald—an N.log(N) method for Ewald sums in large systems J. Chem. Phys. 98 10089-92... 

The fomialism outlined in the previous sections is very usefiil for small systems, but is, as explained, impractical for more than six to ten strongly interacting degrees of freedom. Thus, alternate approaches are required to represent dynamics for large systems. Currently, there are many new approaches developed and tested for this purpose, and these approaches are broadly classified as follows ... 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

While simulations reach into larger time spans, the inaccuracies of force fields become more apparent on the one hand properties based on free energies, which were never used for parametrization, are computed more accurately and discrepancies show up on the other hand longer simulations, particularly of proteins, show more subtle discrepancies that only appear after nanoseconds. Thus force fields are under constant revision as far as their parameters are concerned, and this process will continue. Unfortunately the form of the potentials is hardly considered and the refinement leads to an increasing number of distinct atom types with a proliferating number of parameters and a severe detoriation of transferability. The increased use of quantum mechanics to derive potentials will not really improve this situation ab initio quantum mechanics is not reliable enough on the level of kT, and on-the-fly use of quantum methods to derive forces, as in the Car-Parrinello method, is not likely to be applicable to very large systems in the foreseeable future. 

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

For large systems comprising 36,000 atoms FAMUSAMM performs four times faster than SAMM and as fast as a cut-off scheme with a 10 A cut-off distance while completely avoiding truncation artifacts. Here, the speed-up with respect to SAMM is essentially achieved by the multiple-time-step extrapolation of local Taylor expansions in the outer distance classes. For this system FAMUSAMM executes by a factor of 60 faster than explicit evaluation of the Coulomb sum. The subsequent Section describes, as a sample application of FAMUSAMM, the study of a ligand-receptor unbinding process. 

The Monte Carlo approach, although much slower than the Hybrid method, makes it possible to address very large systems quite efficiently. It should be noted that the Monte Carlo approach gives a correct estimation of thermodynamic properties even though the number of production steps is a tiny fraction of the total number of possible ionization states. 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

Much work remains to be done in the development of this approach to explore the advantages and limitations of the method. The method will be extended to force fields that include torsional terms large systems such as biological macromolecules will also be treated. 

Brooks, B. R., Janezic, D., Karplus, M. Harmonic Analysis of Large Systems I. Methodology. J. Comput. Chem. 16 (1995) 1522-1542 Janezic, D., Brooks, B. R. Harmonic Analysis of Large Systems II. Comparison of Different Protein Models. J. Comput. Chem. 16 (1995) 1543-1553 Janezic, D., Venable, R. M., Brooks, B. R. Harmonic Analysis of Large Systems. HI. Comparison with Molecular Dynamics. J. Comput. Chem. 16 (1995) 1554-1566... 

The computational efficiency is a major advantage of CSP and CI-CSP, and we expect that in the forthcoming few years CSP-based methods will be extensively used as practical tools for the study of an increased range of dynamical processes in large systems. 

M. Hochbruck, Ch. Lubich, and H. Selhofer Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. (1998) (to appear)... 

The quantum mechanical techniques discussed so far are typically appUed to moderate-sized molecules (up to about 100 atoms for ab-initio or DFT and up to 500 for semi-empirical MO techniques). However, what about very large systems, such as enzymes or DNA, for which we need to treat tens of thousand of atoms. There are two possible solutions to this problem, depending on the application. 

Isolated gas ph ase molecules are th e sim plest to treat com pii tation -ally. Much, if not most, ch emistry lakes place in the liq iiid or solid state, however. To treat these condensed phases, you must simulate continnons, constant density, macroscopic conditions. The usual approach is to invoke periodic boundary conditions. These simulate a large system (order of 10" inoleeti les) as a contiruiotis replication in all direction s of a sm nII box, On ly th e m olceti Ics in the single small box are simulated and the other boxes arc just copies of the single box. 

Hacgri Jr., and K. Korscll,, /. Comp. Ckern. 3, 385 (1982), However, in practical applications that th reshold must he tightened as large systems arc considered, in order to reduce the accumulation of... 

The traditional way to provide the nuclear coordinates to a quantum mechanical program is via a Z-matrix, in which the positions of the nuclei are defined in terms of a set of intei ii.il coordinates (see Section 1.2). Some programs also accept coordinates in Cartesian formal, which can be more convenient for large systems. It can sometimes be important to choow an appropriate set of internal coordinates, especially when locating rninima or transitinn points or when following reaction pathways. This is discussed in more detail in Section 5.7. 

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