Computational bottleneck

H. B. Schlegel and M. J. Frisch, Computational Bottlenecks in Molecular Orbital Calculations, in Theoretical and Computational Models for Organic Chemistry, ed. S. J. Formosinho et. al. (Kluwer Academic Pubs., NATO-ASI Series C 339, The Netherlands, 1991), 5-33. 

The first summation requires electron repulsion integrals with four virtuaJ indices. Efficient algorithms that avoid the storage of these integrals have been discussed in detail [20]. For every orbital index, p, this OV contraction must be repeated for each energy considered in the pole search it is usually the computational bottleneck. 

Presently, only the molecular dynamics approach suffers from a computational bottleneck [58-60]. This stems from the inclusion of thousands of solvent molecules in simulation. By using implicit solvation potentials, in which solvent degrees of freedom are averaged out, the computational problem is eliminated. It is presently an open question whether a potential without explicit solvent can approximate the true potential sufficiently well to qualify as a sound protein folding theory [61]. A toy model study claims that it cannot [62], but like many other negative results, it is of relatively little use as it is based on numerous assumptions, none of which are true in all-atom representations. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

The quantum system. The quantum system in the laboratory knows its own Hamiltonian and solves its own Schrodinger equation with full precision and as rapidly as possible when exposed to a laser field e(t). The quantum system acts as an excellent analog computer, circumventing the field design difficulties of Hamiltonian uncertainty and computational bottlenecks. 

Before considering several new and exciting applications of COS-MO-RS in drug design, we will discuss some important computational aspects for the application of COSMO-RS in ADME prediction and drug design, and some special methods and software developed to overcome the computational bottleneck. 

There are some benefits in using STO instead of GTO fiinctions, the problem consists on how to solve the many center integral computation bottleneck. A possible way will be discussed now, using CETO s. 

An important feature when biochemical systems are involved is the performance of the method with very large solutes the PCM computational bottlenecks have been analysed for this problem, and specific algorithms have been elaborated in order to extend this treatment also to solutes with hundreds or thousands of atoms, as briefly resumed in the following. 

In the past, even simple models of biological processes have presented a computational bottleneck since they are largely made up of sets of nonlinear differential or partial differential equations for which analytical solutions are not usually available. However this bottleneck has... 

The presence of /,/ and components requires an iterative solution of this equation—an approach that necessitates storage of the T3 amplitudes in each iteration This scheme is unreasonable because the number of such amplitudes would rapidly become the computational bottleneck as the size of the molecular system increased. This problem may be circumvented, however, by utilizing the so-called semicanonical molecular orbital basis in which the occupied-occupied and virtual-virtual blocks of the Fock matrix are diagonal. In this basis, the two final terms in the T3 equation above vanish, and the conventional noniterative computational procedure described earlier in the chapter may be employed. 

There are two major computational bottlenecks in KS-DFT and PIF calculations [15] evaluation of the KS (or Fock) matrix elements and solution of the self-consistent field (SCF) equations. The latter requires diagonalization of the Fock... 

Furthermore, in the multivariable problem, while three to five variables can be handled relatively easily, one reaches a computational bottleneck for larger problems. This can be possibly resolved by considering some of the new developments in HMM training algorithms [254, 71],... 

With an O(N) XC algorithm such as the one presented here, the Coulomb potential is the computational bottleneck for large molecules in the closing section of this chapter we shall mention a promising means of addressing this problem which we are currently pursuing. 

Now, these equations have been written with formal integration, but of course in the numerical implementation only the evaluation of the integrand at the grid points is required. Therefore, it is evident that the derivatives lyl, (Vpo)M and (Vpp)tyl can be evaluated on the grid independently of the indices pv, and so the four-index problem is decomposed into two independent two-index procedures, avoiding the potential computational bottleneck. By comparison, the resolution of the identity technique proposed by Komornicki and Fitzgerald [66] gives an approximate result in terms of a product of one two-index and two three-index quantities. 

The small number of coefficients needed can be pre-tabulated and held in memory, and we retain the computational simplicity of the Cartesian formulation along with the vital transformation properties of the spherical Gaus-sians. The coefficients [A, B p, P , i, j,k , i, j, kf ,s,t, ] are simple to construct, and the accumulation of sums like (208) can mostly be done in integer aritb-metic. The extensive cancellation which occurs for higher angular momentum spinors can therefore be done exactly without rounding error. The computational bottlenecks encountered in our preliminary work with the complex recurrence relations for direct constraction of Eg[A,B ,p,P ,n,l,m ,r, 1, m s,t,u] given by [107] are completely eliminated. The calculation of these coefficients and the spinor coefficients of the next section now constitutes a trivial part of the computational load. 

H. B. Schlegel and M. J. Frisch, Computational Bottlenecks in Molecular Orbital Calculations, in Theoretical and Computational Models for Organic Chemistry, ed. 

The inversion of the D matrix used in PCM (see eq.23) can represent a computational bottleneck. This inversion may be avoided making use of an expansion interpolation procedure which maintains the quality of the results at a lower computational cost. The matrix (DA ) is rewritten in terms of other matrices in the following way ... 

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