Linear scaling of the energy

The wave number where viscous forces become very strong is of the same order as 1/tj. The wave number of the energy-containing eddies is of the same order as 1/4 if 4 the linear scale of the energy-containing eddies. 

The essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N, in the system [39,55,63,64], that is,... 

In this chapter, we consider the linear scaling of the energy and other expectation values for many-body systems in atomic and molecular physics and quantum... 

As we have seen in Chapter 3, that linear scaling of the energy with the number of electrons in a given system is the distinguishing feature of many-body theories of electronic stmcture. It is because of their lack of extensivity that Brillouin-Wigner methods have been largely dismissed as the basis for a viable many-body theory... 

The results in Tables 1 and 2 show that the CIM-CCSD and CIM-CR-CC(2,3) calculations for the C H2 +2 systems recover the corresponding canonical CC correlation energies to within 0.14% at the CCSD level and 0.04% at the CR-CC(2,3) level, while offering the nearly linear scaling of the total CPU time with... 

The coordinate scaling of the exchange energy should be linear, i.e. multiplying the electron coordinates with a constant factor should result in a similar linear scaling of the exchange energy. ... 

LINEAR SCALING OF THE OVERLAP AND KINETIC-ENERGY INTEGRALS... 

In the case of, the energy is wrong because the molecular orbital is not a linear combination of atomic orbitals, it is approximated by a linear combination of atomic orbitals. Use of scaled atomic orbitals... 

Thus, values for C°p m T, S°m T, (H°m T - H°m 0) and (G°mT H°m0) can be obtained as a function of temperature and tabulated. Figure 4.16 summarizes values for these four quantities as a function of temperature for glucose, obtained from the low-temperature heat capacity data described earlier. Note that the enthalpy and Gibbs free energy functions are graphed as (// , T - H°m 0)/T and (G T — H q)/T. This allows all four functions to be plotted on the same scale. Figure 4.16 demonstrates the almost linear nature of the (G°m T H°m 0)/T function. This linearity allows one to easily interpolate between tabulated values of this function to obtain the value at the temperature of choice. 

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