Larger systems

Since Z /c is small with respect to 1, we may expand the square root in the Taylor series, y/ — x = 1 — ----. We obtain 

The Dirac equation represents an approximation- and refers to a single particle. What happens with larger systems Nobody knows, but the first idea is to construct the total Hamiltonian as a sum of the Dirac Hamiltonians for individual particles plus their Coulombic interaction (the Dirac-Coulomb apjmmmation). This is practised routinely nowadays for atoms and molecules. Most often we use the mean-field approximation (see Chapter 8) with the modification that each of the one-electron functions represents a four-component bispinor. Another approach is extreme pragmatic, maybe too pragmatic we perform the non-relativistic calculations with a pseudopotential that mimics what is supposed to happen in a relativistic case. 

---

For larger systems N> 1000 or so, depending on the potential range) anotlier teclmique becomes preferable. The cubic simulation box (extension to noncubic cases is possible) is divided into a regular lattice of n x n x n cells see figure B3.3.7. These cells are chosen so that the side of the cell = L/n is greater than the... 

The flux-flux expression and its extensions have been used to calculate reaction probabilities for several important reactions, including H2+02 H + H2O, by explicit calculation of the action of G in a grid representation with absorbmg potentials. The main power of the flux-flux fomuila over the long mn will be the natural way in which approximations and semi-classical expressions can be inserted into it to treat larger systems. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Unfortunately, the resources required for these numerically exact methods grow exponentially with the number of degrees of freedom in the system of interest. Without the use of clever algorithms to optimize the basis set used [106,107], this limits the range of systems treatable to 4-6 degrees of freedom (3-4 atoms). For larger systems, the MCTDH method [19,20,108] provides a... 

Substantial headway towards longer time scales and larger systems can only be expected from reduction of system complexity. It is here where future... 

Table 1 describes the timing results (in seconds) for a system of 4000 atoms on 4, 8 and 16 nodes. The average CPU seconds for 10 time steps per processor is calculated. In the case of the force-stripped row and force-row interleaving algorithms the CPU time is reduced by half each time the number of processors is doubled. This indicates a perfect speedup and efficiency as described in Table 2. Tables 3, refibm table3 and 5 describe the timing results, speedups and efficiencies for larger systems. In particular. Table 4 shows the scaling in the CPU time with increase in the system size. These results are very close to predicted theoretical results.

It is clear that the power of computers will continue to inaease rapidly, as we have observed since the mid-2()th century. This will allow us to tackle more complicated problems and larger systems, and do so with higher accuracy. 

With the increase in hardware and software, larger systems can be handled with higher accuracy. Much work will continue to be devoted to the study of proteins and polynucleotides (DNA and RNA), and particularly their interactions with more sophisticated methods. Remember proteins and genes are chemical compounds and sophisticated theoretical and chemoinformatics methods should be applied to their study - in addition to the methods developed by bioinfor-maticians. 

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

Among the few systems that can be solved exactly are the particle in a onedimensional box, the hydrogen atom, and the hydrogen molecule ion Hj. Although of limited interest chemically, these systems are part of the foundation of the quantum mechanics we wish to apply to atomic and molecular theory. They also serve as benchmarks for the approximate methods we will use to treat larger systems. 

Using more flexible trial funetions (polynomials in r, Y2, and ri2 perhaps) for /(ri), one ean ealeulate very aeeurate energies for He by the SCF method. This gives us eonfidenee that we ean make a valid generalization to larger systems. 

Size-extensivity is of importance when one wishes to compare several similar systems with different numbers of atoms (i.e., methanol, ethanol, etc.). In all cases, the amount of correlation energy will increase as the number of atoms increases. However, methods that are not size-extensive will give less correlation energy for the larger system when considered in proportion to the number of electrons. Size-extensive methods should be used in order to compare the results of calculations on different-size systems. Methods can be approximately size-extensive. The size-extensivity and size-consistency of various methods are summarized in Table 26.1. 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

The disadvantages are that it is difficult to integrate into a larger system and that the programming can be slow and error-prone. 

Feed systems utilizing gravity are rarely used. Line pressure is usuaHy adequate for smaH systems. AuxHiary pumps are required in larger systems to assure proper flow through aH units and to avoid uneven flow should line pressure decrease as other demands for water or the process stream occur elsewhere in the facHity. 

Within the larger system, research and especially development processes have become business processes, with teams from marketing, manufacturing, and finance participating from the start (8). In some companies, the R D function is so highly integrated with the business that formal R D organizations have vanished many corporate laboratories have been reduced or disbanded, with the researchers moved into separate business divisions (9,10). 

A tmism of computational chemistry is that chemists will always want to model ever larger systems, or smaller systems, at ever more accurate levels of approximation. The total miming time of jobs has, in general, not lowered dramatically. Computational chemists still perform calculations that take several days to complete. However, today the molecules can be much larger and the quaUty of the calculations better. 

The multisubstate approach requires initially identifying all important substates, a difficult and expensive operation. In cases of moderate complexity (e.g., a nine-residue protein loop), systematic searching and clustering have been used [39,66]. For larger systems, methods are still being developed. 

---