Relaxation and Convergence

The overall solution procedure and other finer details of under-relaxation and convergence criteria are briefly discussed in Section 6.5. It is useful to briefly discuss the implementation of commonly encountered boundary conditions when solving flow field equations. 

Figure 11.3. Divergent (left) and convergent (right) boundary conditions for the relaxation in the a phase of PVDF (reprinted with permission from Carbeck and Rutledge. Copyright 1996, American Chemical Society).

The practical advantage of these relations is that, in MD simulations, single molecule properties like the self-diffusion coefficient and rotational relaxation times converge much faster than system properties due to additional averaging over the number of molecules in the ensemble. We applied eqs. 10 and 11 to our MD results using data at 800 K as a reference point in order to predict the viscosity over the entire temperature interval. In Fig. 7 we compare the predicted values with those obtained from simulation. It appears that in the temperature interval 600 K to 800 K predictions of Eq. (10) are more consistent with MD results than are the predictions of Eq. (11). This leads us to conclude that the viscosity temperature dependence in liquid HMX is more correlated... 

The sequence of laser excitation followed by collisional relaxation and radiative emission is then iterated to convergence. In the second step, for example, the populations in Eq, (8.26) are taken as the initial populations for two independent computations, one assuming a population of J fD in K0). with Eh) unpopulated, and the second assuming a population of A V in Eh), with IiD) unpopulated. 

Our discussion on the aspiration of linear chain has been basedon the behavior inside the tube. It is also helpful to consider what happens outside (in the entry region). Here, in the most naive picture, we have a convergent flow of velocity v(r)=J/r2, where r is the distance to the center of the entry disc. The velocity gradient is y = Vv J / r3. At a certain radius rc, this is such that yTz = 1, where Tz is the relaxation time of a coil Tz = T)R3 /kT. At distances rdeforms affinely the final lateral dimension of the distorted coil is Tj D/rcRg. When Tj becomes as small as D, the chain can get in. This corresponds to rc Rg and J Jc, where Jc is again given by Eq.(3). Thus this different picture leads to the same conclusion. 

Figure 15 shows the populations of the D and L enantiomers after a single pulse (and before relaxation) and after a convergent sequence of pulses followed... 

Ninomiya makes effective use of the stack and three improvements refining the error estimation, relaxing the convergence criterion and detecting extraordinary points. 

Two types of Cl calculation are presented. The relaxed Cl calculations employ the neutral SCF orbitals in the calculation of the ground-state energy of the neutral, and the ion SCF orbitals are used for the ion ground-state energy. The unrelaxed Cl calculations use a common set of orbitals for both the neutral and ion calculations. The orbitals are taken from a ground-state SCF calculation for whichever of the two is the closed-shell system. The unrelaxed Cl calculations are similar in this respect to the EOM calculations, which utilize only one set of closed-shell SCF orbitals throughout. Comparison of the relaxed and unrelaxed Cl calculations also affords a check of the Cl convergence. A number of different relaxed and unrelaxed Cl calculations are made for each system. The simplest involves... 

In contrast, the coefficients of vectors around/- 12.2 cm decrease as a function of time and converge to certain values depending on the width of the frequency bin. It is instructive that, in the solvated condition (Figure 5.6b), the relaxation of rotation is complete within 200ps while it is slower in vacuum. In particular, the vectors in the narrow bin Af = 2w become almost orthogonal after 200 ps in the solvated... 

When relaxation is taken into account, however, the defect perturbation propagates farther away, and convergence of the defect substitutional energy is slower. 

---