Force matrix

Fig. 1. Nonbonded force evaluation may be distributed among processors according to atomic coordinates, as in spatial decomposition (left), or according to the indices of the interacting atoms, as in force-matrix decomposition (right). Shades of gray indicate processors to which interactions are assigned.

Methods of decomposing the nonbonded force evaluation fall into two classes, spatial decomposition [15] in which atoms and their interactions are divided among processors based on their coordinates, and force-matrix decomposition [16] in which the calculation of the interaction between a pair of atoms is assigned to a processor without considering the location of either atom (Fig. 1). Spatial decomposition scales better to large numbers of... 

The complexity analysis shows that the load is evenly balanced among processors and therefore we should expect speedup close to P and efficiency close to 100%. There are however few extra terms in the expression of the time complexity (first order terms in TV), that exist because of the need to compute the next available row in the force matrix. These row allocations can be computed ahead of time and this overhead can be minimized. This is done in the next algorithm. Note that, the communication complexity is the worst case for all interconnection topologies, since simple broadcast and gather on distributed memory parallel systems are assumed. 

This algorithm is an improvement over the algorithms described in the previous subsections. The idea behind this algorithm is fairly simple. To ensure load balance, the rows of the force matrix will be allocated in such a way that the load on all processor is equal. 

As before, let Cj, denote tbe cost of force computation on processor i, 0 < i < P — 1). Processor i is assigned U rows of the force matrix and for load balance - , lp will satisfy h algorithm computes a priori the row assignment so that the load sent to processors is balanced. Some typical values are listed in Table 6. 

Lastly, Table 6 describes the assignment of rows to processors for some typical cases, and the load in each case (indicating the number of force interactions computed by each processors in the corresponding case). These are based on equations in Section 3. Several important points can be noted from the results shown in the table. Firstly, it can be observed that in the 4 processor case, processor P3 computes half the maximum number of rows in the force matrix which leads to a load balanced assignment. This would not be the case if processors were assigned equal number of rows. Moreover, when the number of processors is increased from 4 to 16, the load on each processor reduces by a factor of 4, but is still equal on every processor. 

Table 6. Table showing the assignment of rows of the force matrix for 4 processors. 

The eigenvalues and eigenvectors of the mass-weighted force matrix can be obtained by diagonalizing equation (21.5). Then each eigenvalue corresponds to its normal coordinates, Qj,... 

The equation of motion is further generalized by imposition of a 3xn external force matrix o, so that we finally have... 

Values of the force matrix components On and 0)2 are specified by the corresponding Fourier components of the dipole-dipole interaction tensors F P(k)... 

The simplest illustration of molecular vibration is a homonuclear diatomic molecule, which can vibrate in only one direction - parallel to its internuclear axis. If the coordinates of the nuclei are x and X2, the force matrix equation relating restoring forces to displacements is ... 

Strutinsky procedure. Because asymmetric shapes is so shallow, it is worthwhile to deal with octupole correlation effects using a microscopic, two-body interaction treatment of the octupole-octupole residual interaction [CHA80]. The pairing force matrix elements, G. come from a density dependent delta interaction. This set of matrix elements [CHA77] explains many features of the actinides at low and... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

This is the basic equation used to derive normal coordinate analysis [110] as well as to define the vibrational quantities to be calculated using molecular orbital theory [79,94], The coefficients, g , are the forces acting on the nuclei, which are zero at equilibrium geometiy. This leaves the quadratic terms Vs the first term in the change of potential energy with instantaneous vibrational displacement. The quadratic terms Fy, are conveniently ordered as a matrix which is known as the force field or force matrix. These terms correspond to the derivatives of the potential energy V ... 

Rationalization of these discrepancies between calculated and observed molecular properties has been achieved by recognizing that the accuracy problems tend to be systematic for the computational methods [726], For the same levels of theory, the errors in molecular parameters tend to be similar [79] for molecules if a variety of types and size. Building on the ideas of similarities between similar chemical functional groups in different molecules, the SQM procedure develops empirical scaling factors to correct the overestimated harmonic force constants from the computations [77], These scale factors form a diagonal matrix C which modified the calculated force matrix Fcalc according to the relationship [727] ... 

Notes Observed IR frequency assignments for vapor phase formic acid are from R. L. Redington, J. Mol. Spectrosc. 1977,65, 1 71. Calculated frequencies for all four molecules were obtained using a single force matrix scaled with factors listed in Table 1. 

Two categories of force decomposition algorithm have appeared in the literature those that systolically cycle atom data around a ring, and those based on the force-matrix formalism discussed by Plimpton.i o A variety of systolic loop methods have been implemented on both and... 

Force decomposition algorithms based on a block decomposition of the force matrix reduce the memory and communication costs by a factor of /P versus the atom decomposition algorithms. In the LJ benchmark, the force decomposition algorithm continued to speed up, even when hundreds of processors were used. 

The elements of the random-force matrix (F (t), F+(0)) have the following properties under time reversal... 

PEST-ana lysis Five forces matrix Demarcation matrix Core competency assessment... 

FIG. 10 Division of the force matrix among P processors in the atom-decomposition method. 

The block-decomposition, shown in Fig. 11, is actually applied on a permuted force matrix F, which is formed by rearranging the columns of the original F in a particular way. The if) element of F is the force acting on atom i in vector x due to atom j in the permuted vector x. Now the F subblock owned by each processor z is of size (A/P ) x As shown... 

Step (2) is also the same as that of the RD algorithm. The key difference is that now the total force on atom i is the sum of the elements in row i of the force matrix minus the sum of the elements in column i, where i is the permuted position of column i. Thus this step performs a fold of fa iff) within each row (column) of processors to sum-up these contributions. The important point is that now the vector fa iff) being folded is only of length (iV/P ) and only the P elements in one row (eolmnn) are participating in the fold. Thus, this operation scales as NjP instead of iV as in the RD algorithm. Finally, the two eontributions are jointed to yield the total forces (Jf) on the atoms owned by proeessor P. ... 

Moreover, the free motions of the body generate the hnear hydrodynamic added mass and damping effect. In particular, the radiated potential from the body produces a 6 X 6 force matrix. When nondimensionalized by the oscillation frequency (or frequency squared) and structure displacements, the radiated forces give rise to a 6 X 6 added mass coefficient and a 6 x 6 damping coefficient matrix. 

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