Computation scheme for variable density

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 - °°... 

The ideal variable to measure is one that can be monitored easily, inexpensively, quickly, and accurately. The variables that usually meet these qualifications are pressure, temperature, level, voltage, speed, and weight. When possible the values of other variables are obtained from measurements of these variables. For example, the flow rate of a stream is often determined by measuring the pressure difference across a constriction in a pipeline. However, the correlation between pressure drop and flow is also affected by changes in fluid density, pressure, and composition. If a more accurate measurement is desired the temperature, pressure, and composition may also be measured and a correction applied to the value obtained solely from the pressure difference. To do this would require the addition of an analog or digital computer to control scheme, as well as additional sensing devices. This would mean a considerable increase in cost and complexity, which is unwarranted unless the increase in accuracy is demanded. 

In order to overcome the limitations of currently available empirical force field param-eterizations, we performed Car-Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car-Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above details on computational parameters can be found in specific papers [13]. 

Another approach to the C matrix construction is a CSF-driven approach proposed by Knowles et al.. With this approach, the density matrix elements dlgrs ars constructed for all combinations of orbital indices p, q, r and s, but for a fixed CSF labeled by n. Each column of the matrix C is constructed in the same way that the Fock matrix F is computed except that the arrays D" and d" are used instead of D and d. As with the F matrix construction described earlier, there are two choices for the ordering of the innermost DO loops. One choice results in an inner product assembly method while the other choice results in an outer product assembly method. The inner product choice, which does not allow the density matrix sparseness to be exploited, results in SDOT operations of length m or about m, depending on the integral storage scheme. The outer product choice, which does allow the density matrix sparseness to be exploited, has an effective vector length of n, the orbital basis dimension. However, like the second index-driven method described above, this may involve some extraneous effort associated with redundant orbital rotation variables in the active-active block of the C matrix. 

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