Direct-diagonalization algorithm calculations

The density-matrix-spectral-moments algorithm (DSMA) °° is an approximate scheme for solving the TDHF equations that allows us to calculate from the source ( W) by solving eq A18 without a direct diagonalization of L. This is accomplished by computing the set of electronic oscillators that dominate the expansion of Without loss of generality, we can take rj itj to be real and express it in terms of our momentum variables as - °°... 

Given the function (4.20), calculate the direction d using the Algorithm 4.4 in xq = 0.5 -with the diagonal matrix of the second derivatives as preconditioner. Next, compare the sequence of y obtained by the CG and PCG procedures. 

Note that the dependence of the shear viscosity on shear rate has been explicitly noted. Alternatively, a non-equilibrium MD simulation can be conducted in which the response of the system to an external perturbation have been considered. The most widely used nonequilibrium approach for viscosity calculations is called "SLLOD" algorithm[146] in which a shear rate is imposed on the system and the resulting stress is computed. The shear viscosity is found at a given shear rate from where Pij is an off-diagonal component of the stress tensor, i is the direction of flow caused by the imposed shear rate, and j is the direction normal to the flow. For the SLLOD algorithm, special "sliding brick" boundary conditions are also typically used, which require modification of the Ewald sum if charged systems are simulated. [147]... 

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