Correlation entropy

The correlation entropy is a good measure of electron correlation in molecular systems [5, 7]. It is defined using the eigenvalues of the one-particle density matrix IPDM,... 

Figure 4. Two-site Hubbard model. Upper curve is the entanglement calculated by the von Newmann entropy. The curves 5 1 and 5 2 are the correlation entropies of the exact wavefunction as defined in the text. The dashed line is the 5 2 for the combined wavefunction based on the range of V values. S for the combined wavefunction is zero.

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

Another perspective is offered by considering Type A nondynamical correlation / left-right strong correlation as a manifestation of quantum entanglement. A physically motivated measure of the latter is the von Neumann entropy [13], or the closely related correlation entropy [14, 15]... 

Of the wavefunction-based diagnostics, the only one that has a reasonable correlation with %TAE[T4 + Ts is Tmhlar s Mdiag, which for closed-shell species is effectively the average of HOMO and LUMO natural orbital occupations (as well as the low-n, limit of the HOMO-LUMO correlation entropy). The Tj diagnostic statistically correlates poorly with aU energy-based diagnostics, except... 

Unsurprisingly, the HOMO-LUMO correlation entropy tracks the CASSCF Mdiag very closely, with = 0.953... 

For the values given, correlation coefficients with some diagnostics for nondynamical correlation are as follows Truhlar Mdiag R = 0.923 HOMO-LUMO correlation entropy Scon i = 0.905 CASSCF 1 — C /f = 0.840. Correlation coefficients between (TAEfUHF] — TAE[RHF])/TAE total and various diagnostics are as follows Truhlar Mdiag R = 0.875, = 0.869, CASSCF 1... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The lack of correlation between the flucUiating stress tensor and the flucUiating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for flucUiating hydrodynamics are arrived at by a procedure very similar to that exliibited in the preceding section for difllisioii. A crucial ingredient is the equation for entropy production in a fluid... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Calculate AS° for ionization of each compound. Comment on the contribution of AH° and AS° to the free energy of ionization. Test the data for linear fiee-energy correlations. Are the linear free-energy correlations dominated by entropy or enthalpy terms ... 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

In the PPF, the first factor Pi describes the statistical average of non-correlated spin fiip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on<= of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

For a polytropic process the change of state does not take place at constant entropy, hut for an adiabatic process, it does. Heat may he added to or rejected from a gas in a polytropic process. For a polytropic process, the correlating exponent for the PiVi" component is the exponent n, which becomes an important part of the compressor design, n values are determined from performance testing. 

The decay of the spatial block entropy, which gives the amount of information contained in a block of N contiguous site values ai,...,aN needed to predict the value (Jn+i is considerably slower than [block-length), and is therefore indicative of very long and complex correlations we will come back to this point later in chapter 4, following our discussion of dynamical system theory. 

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