Scaling, exponential

It is obvious, and verified by experiment [73], that above a critical trap concentration the mobility increases with concentration. This is due to the onset of intertrap transfer that alleviates thermal detrapping of a carrier as a necessary step for charge transport. The simulation results presented in Figure 12-22 are in accord with this notion. The data for p(c) at ,=0.195 eV, i.e. EJa—T), pass through a minimum at a trap concentration c—10. Location of the minimum on a concentration scale depends, of course, on , since the competition between thermal detrapping and inter-trap transport scales exponentially with ,. The field dependence of the mobility in a trap containing system characterized by an effective width aeff is similar to that of a trap-free system with the same width of the DOS. 

Systematic searches exhaustively sample conformational space by sequentially incrementing the torsional angles of aU of the rotatable bonds in a given molecule. This conceptually simple approach is straightforward to implement, but scales exponentially with respect to the number of rotatable bonds. To control the exponential increase in the number of potential conformers obtained, systematic searches are usually combined with tree-based search techniques taken from computer science. Even the best implementations of systematic searches become impractical beyond several rotatable bonds (typically greater than 10). Stochastic searches are based on probabiHstic theories and are better suited to calculations... 

Elucidation of glycan composition is a combinatorial problem where the number of compositions that must be tested scales exponentially with the number of different monomers that can form the solution. Therefore, the composition for molecules of high mass can take a very long time to be derived. The number of solutions will also grow at a similar ratio, thus offering an enormously large amount of alternatives to the user. Furthermore, when adducts and other mass spectrometric losses are taken into account, a number of compositions may be indistinguishable within a relatively low mass delta threshold (less than 0.05 Da). Therefore, it is often... 

The maximum number of walkers that can be used scales exponentially with the dimensionality of the free energy that has to be reconstructed. In practical applications, the number of walkers can be used for reconstructing a free energy in two, three and for dimensions is of the order of 10, 100 and 1000, respectively. This feature of the algorithm allows to reconstruct free energies as a function of several variables in a clock time that depends only on the maximum number of available processors. 

Here M is the number of uncorrelated Monte Carlo samples. Similarly the error for the numerator increases exponentially and the time needed to achieve a given relative error scales exponentially in N and / . 

If our priors - as they are called in the domain of Bayesian inference, for which see Appendix 18.1- admit a very large number of imaginable genetically regulated metabolisms (say, a number that scales exponentially with genome size, such as the number of possible combinations of enzymes), a strict neo-Darwinian view of metabolic universals becomes almost an anthropic null model, because any particular combination has near-zero probability of discovery and is preserved either because no competitive scheme was ever found by the genome or as a result of its... 

Direct, easy methods are those ones which locate the resonances and only the resonances. Of course the scattering methods give the most detailed information possible about the scattering at a resonance and indeed the two methods can be used in a complementary fashion. Both types of calculations scale exponentially with the number of coupled degrees of freedom and therefore they can quickly become computationally intractable. Thus, for both the scattering and direct methods we have been interested in reduced-dimensionality strategies to reduce the number of coupled degrees of freedom. 

A special case of full Cl is the complete active space self-consistent field (CASSCF) or fully optimized reaction space (FORS) approach in which one defines an active space of orbitals and corresponding electrons that are appropriate for a chemical process of interest [20]. The FORS wavefunction is then obtained as a linear combination of all possible electronic excitations (configurations) from the occupied to the unoccupied (virtual) orbitals in the active space, so a FORS wavefunction is a full Cl within the specified active space. Since a full Cl provides the exact wavefunction for a given atomic basis, there is no need to re-optimize the component molecular orbitals. On the other hand, a FORS wavefunction generally corresponds to an incomplete Cl, in the sense that only a subset of configuration (or determinant) space is included. Therefore, one also optimizes the molecular orbital coefficients to self-consistency. The calculation of a full Cl wavefunction is extremely computationally demanding, scaling exponentially with... 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

The use of quantum chemistry to obtain the individual rate coefficients of a free-radical polymerization process frees them from errors due to kinetic model-based assumptions. However, this approach introduces a new source of error in the model predictions the quantum chemical calculations themselves. As is well known, as there are no simple analytical solutions to a many-electron Schrodinger equation, numerical approximations are required. While accurate methods exist, they are generally very computationally intensive and their computational cost typically scales exponentially with the size of the system under study. The apphcation of quantum chemical methods to radical polymerization processes necessarily involves a compromise in which small model systems are used to mimic the reactions of their polymeric counterparts so that high levels of theory may be used. This is then balanced by the need to make these models as reahstic as possible hence, lower cost theoretical procedures are frequently adopted, often to the detriment of the accuracy of the calculations. Nonetheless, aided by rapid and continuing increases to computer power, chemically accurate predictions are now possible, even for solvent-sensitive systems [8]. In this section we examine the best-practice methodology required to generate accurate gas- and solution-phase predictions of rate coefficients in free-radical polymerization. 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

Of course, quantum chemistry is not without limitations. Since the multielectron Schrodinger equation has no analytical solution, numerical approximations must instead be made. In principle, these approximations can be extremely accurate, but in practice the most accurate methods require inordinate amounts of computing power. Furthermore, the amount of computer power required scales exponentially with the size of the system. The challenge for quantum chemists is thus to design small model reactions that are able to capture the main chemical features of the polymerization systems. It is also necessary to perform careful assessment studies, in order to identify suitable procedures that offer a reasonable compromise between accuracy and computational expense. Nonetheless, with recent advances in computational power, and the development of improved algorithms, accurate studies using reasonable chemical models of free-radical polymerization are now feasible. 

Alternatively, the whole reactive surface can be determined through a sufficiently large number of single point calculations and (quasi) classical trajectory studies can be performed on an analytic representation of this predetermined PES. However, the fitting of high dimensional surfaces is a very time consuming procedure. Every degree of freedom has to be discretized with a sufficient number of points M and the number of required calculations scales exponentially with the number of atoms N. A... 

One of the most remarkable applications of quantum computing is the ability to simulate others quantum systems. In fact, classical computers cannot be used to simulate a quantum system efficiently [24,25]. The basic problem is the dimension of the Hilbert s space, that is 2 for a simple system containing n particles with only two degrees of freedom. It is obvious that as the number of particles increases, this problem becomes intractable, since it scales exponentially. 

The zero-shear viscosity t]o of linear polymers scales exponentially with molecular weight [102] above the critical chain length Me, but LCB polymers repeatedly deviate from this dependency. In comparison to linear polymers of similar M, polymers with low levels of LCB exhibit enhanced zero-shear viscosity values and, in a qualitative sense, C-NMR-based LCB content often [85, 92, 93], but not always [100], correlates well with the viscosity increase. For long-chain branched LDPE, the t]q in comparison to linear polyethylene of similar is lower [103, 104]. A zero-shear viscosity t]o value higher than that of the corresponding linear polymers of similar M , is reported to occur at an LCB content of 0.2 LCB/10,000 C but the increase becomes more pronounced as the LCB content grows [85, 91, 92, 105,106]. This feature of low amounts of LCB has also been utilized to explore the extent of metallocene LCB [13, 85, 106, 107]. 

The representation Eq. (5.1) is sometimes also called the standard form of the wavefunction. Given that on average N basis functions per DOF are sufficient for an accurate description of the amount of information that needs to be stored and processed scales exponentially with N, where N is usually of the order of 10. The standard form therefore de facto limits the size of the molecules that can be treated to about 4 atoms, i.e., 6 internal DOF. 

Over the last 20 years powerful methods to solve the TDSE have been developed [46, 47]. These are based on using a grid-based representation of the wavefunction and Hamiltonian and have provided detailed descriptions of non-adiabatic events. Unfortunately, such numerically exact solutions of the TDSE require huge computer resources as they scale exponentially with the number of degrees of freedom and approximations must be introduced to treat systems with more than 20 atoms, which include the majority of photochemistry. 

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