Subsystems uniform scaling

As in the uniform scaling case the subsystem scaled wavef unctions in Eq. (48) are the grounds states of the respective scaled effective hamiltoniams ... 

Examples of a non-uniform scaling of the subsystem electronic charges ame represented by the paths 2-6 in Fig. 2 we have summarized in the figure Eqs. (44) and (53)-(55) identifying the subsystem scaled external potentials corresponding to the four points defining these scaling trajectories. 

The distinction between intensive (/ ,) and extensive (X,) (vs. other ) properties is quite important. Qualitatively, one can say that intensive properties Rt are uniform everywhere in the system, whereas extensive properties Xt are additive in subsystems ( scale with the size of the system ). 

Figure 2. Uniform (1) and nonuniform (2-6) scaling paths of electronic charges in the complementary subsystems of M = (i4 B). The subsystem external potentials of the scaled hamiltonians for the starting and end points of each path are also listed, with the upper (lower) entries corresponding to ti (B).

The Hellmann-Feynman theorem for the uniform, A-sca-ling scheme of electronic interactions/charges in the scaled ground state 4 Ip, P ] = of the two subsystems in M(, A) reads ... 

If a system is not uniform, it is not in thermodynamic equilibrium then can obey law of Maxwell s speeds distribution. However, if "equilibrium absence" is not big, can considered like good approximation, all little volume (microscopic scale) is in equilibrium (considered like subsystem). This is for two reasons. First little portions of gas contain a big number of molecules. Second the necessary time for established the equilibrium in a little volume is brief in comparison with necessary time for that transport processes get equilibrium in little volume with rest of system (it is true when concentration, temperature, etc. gradients are not too much big). In consequence, can supjpose that is local thermodynamic equilibrium so speed distribution in any volume element (macroscopic) of medium is Maxwellian, although density, temperature and macroscopic velocity change the position (Duderstadt Martin, 1979 Schwabl, 2002 Bhatnagar et al., 1954 Pai, 1981 Maxwell, 1997 Sued, 2001 Succi, 2002 Cercignani, 1975 Lebowitz Montroll, 1983). 

As stated previously, the dimensionality of each subsystem Fock matrix F may be reduced from N to W , where N is the number of basis functions contained in subsystem H . The overall cost of diagonalization, then, is simply the sum of the (W ) subsystem expenditures. For uniformly sized subsystems, this means that the o N ) cost of diagonalization may be reduced to o N). Thus the advantage of D C over the standard MO approach becomes increasingly dramatic as N grows. There are of course a number of other aspects of a D C calculation besides matrix diagonalization, and these tasks may or may not exhibit linear scaling. However, as we will demonstrate, the vast majority of the computational expense is tied up in procedures that are linearized by the D C approach. 

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