Energy first-order

However, in practical terms, this analyticity means condensation of the atomic properties in the molecule in the associated single-density matrix y(x,x ) and its development in Taylor series around the diagonal point, X = x development performed only to second order. In the correspondent plan of the density functionals of the atomic energies, first order will correspond to electronegativity and second order will correspond to chemical strength. ... 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

In all methods, the first-order interaetion energy is just the differenee between the expeetation value of the system Hamiltonian for the antisyimnetrized produet fiinetion and the zeroth-order energy... 

Figure Al.5.3 shows that, as in interactions between other species, the first-order energy for Fte-Fle decays exponentially with interatomic distance. It can be fitted [70] within 0.6% by a fimction of the fonn... 

Figure Al.5.2 First-order Coulomb (O) and exchange-repulsion ( ) energies for Fle-FIe. Based on data from Komasa and Thakkar [70].

The small additions to all themiodynamic potentials are the same when expressed in temis of appropriate variables. Thus the first-order correction temi when expressed in temis of Vand P is the correction temi for the Helmlioltz free energy A ... 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

A3.13.3.2 THE MASTER EQUATION FOR COLLISIONAL AND RADIATIVE ENERGY REDISTRIBUTION UNDER CONDITIONS OF GENERALIZED FIRST-ORDER KINETICS... 

There is one special class of reaction systems in which a simplification occurs. If collisional energy redistribution of some reactant occurs by collisions with an excess of heat bath atoms or molecules that are considered kinetically structureless, and if fiirthennore the reaction is either unimolecular or occurs again with a reaction partner M having an excess concentration, dien one will have generalized first-order kinetics for populations Pj of the energy levels of the reactant, i.e. with... 

Figure Bl.12.1. (a) Energy level diagram for an/= nueleus showing the effeets of the Zeeman interaetion and first- and seeond-order quadnipolar effeet. The resulting speetra show statie powder speetra for (b) first-order perturbation for all transitions and (e) seeond-order broadening of the eentral transition, (d) The MAS speetnim for the eentral transition. ...

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Frenkel D 1986 Free-energy computation and first-order phase transitions Moiecuiar Dynamics Simuiation of Statisticai Mechanicai Systems ed G Ciccotti and W G Hoover (Amsterdam North-Holland) pp 151-88... 

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