Scaled density

A system of particles interacting in this way was studied using a microcanonical ensemble at a scaled density of 3.0 which is close to the transitional density for hard oblate ellipsoids with the same ellipticity (see Fig. 3). At a scaled density of 3.0 the system is found to exhibit isotropic. 

Cheng, L, Fentee, P., Nagy, K. L, SCHLEGEL, M. L, StURCHIO, N. C., Molecular-scale density oscillations in water adjacent to a mica surface, Phys. Rev. Lett. 87, 15 (2001) 6103-6104. 

Greeley J, Nprskov JK. 2007. Large-scale density functional theory-based screening of alloys for hydrogen evolution. Surf Sci 601 1590. 

Johnson, B. G., White, C. A., Zhang, Q., Chen, B., Graham, R. L., Gill, P. M. W., Head-Gordon M., 1996, Advances in Methodologies for Linear-Scaling Density Functional Calculations , in Recent Developments and Applications of Modem Density Functional Theory, J. M. Seminario, (ed.), Elsevier, Amsterdam. 

Scuseria, G. E., 1999, Linear Scaling Density Functional Calculations With Gaussian Orbitals , J. Phys. Chem. A, 103, 4782. 

White, C. A., B. G. Johnson, P. W. Gill, and M. Head-Gordon. 1996. Linear scaling density functional calculations via the continuous fast multipole method. Chem. Phys. Lett. 253, 268. 

Figure 49. Solid fraction profiles, glass/plastic viscous limit scaling density ratio mismatched, low velocity case. (From Glicksman et at., 1993a.)...

In this plot, we can see that if we increase the pressure, the energy also will be increased but the rate of this increment will be different for each state. The results discussed for the PIAB model are particular situations of generalizations reported for systems confined with Dirichlet boundary conditions [2]. We must remember these results for further discussion through this chapter. Let us conclude this section with the remark that the state dependence of the effective pressure at the given value of Rc can be analogously understood in terms of the different electron densities and their derivatives at the boundaries. In most general case of atoms and molecules, scaled densities may have to be employed in order to include the excited states. In the next section, we present some basic results on such connections between wave function and electron density. 

Non-dilute solutions also allow for theoretical descriptions based on scaling theory [16, 21]. When the number of polymer chains in the solution is high enough, the different chains overlap. At the overlapping concentration c , the long-scale density of polymer beads becomes uniform over the solution. Consequently c can be evaluated as... 

G. K.-L. Chan, An algorithm for large scale density matrix renormalization group calculations. J. Chem. Phys. 120, 3172 (2004). 

Hachmann, J., Cardoen, W., Chan, G.K.L. Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group. J. Chem. Phys. 2006, 125(14), 144101. 

Name. Approximate Composition. Hardness (Mohs Scale). Density. Usual Form. Arsenic Content, per cent. Approx. 

Fig. 4.3. Linear-scale density plot of the distribution of the ion momentum P = (P,l, P ) in nonsequential double ionization of neon at 8 x 1014 W/cm2 and Hu> = 1.55 eV. From [26]...

Figure 5.9 depicts the scaled density profiles p (z) = p(z)average system density po, all the curves have... 

Kudin, K., Scuseria, G. (2000). Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions efficient evaluation of energy and forces via the fast multipole method. Phys. Rev. B61,16443. 

J. Butter (2004) Large Scale Density Functional Calculations. In Multiscale Modelling and Simulation, eds. S. Attinger and P. Koumoutsakos Springer, Beidelberg. Lecture Notes in Computational Science and Engineering, p. 39... 

White CA, Johnson BG, Gill PMW, Head-Gordon M (1996) Linear scaling density functional calculations via the continuous fast multipole method. Chem Phys Lett 253 268-278 Lee TS, Lewis JP, Yang W (1998) Linear-scaling quantum mechanical calculations of biological molecules the divide-and-conquer approach. Comput Mater Sci 12 259-277 Van Alsenoy C, Yu CH, Peelers A, Martin JML, Schafer L (1998) Ab initio geometry determinations of proteins. 1. Crambin. J Phys Chem A 102 2246-2251... 

Table 1 A comparison of the estimated values of first and second derivatives of scaled density at the nucleus using Equation (6.14) and Equation (6.16), respectively, with the direct numerical calculations. Four sets of results are shown corresponding to different values of confining potential Vq. At the critical radius Rc in each case the energy value is 0. The ij (0), r/ (0) and tf (0) values are given under Density in columns 1 and 4. All values are in au...

Similar to the case of a confined hydrogen atom discussed in Section 6.3, we define the scaled density for the confined isotropic harmonic oscillator by relations (6.13) and (7.1). 

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