Energy variation theory

Equation (ASA. 110) represents the canonical fonn T= constant) of the variational theory. Minimization at constant energy yields the analogous microcanonical version. It is clear that, in general, this is only an approximation to the general theory, although this point has sometimes been overlooked. One may also define a free energy... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

Structural theory may be applied to a consideration of the relative stability of the several complexes on a particular metal, or the variation in energy which follows a change in the metallic atom or its neighbors (ligands). For example, the relative stability of... 

M. Nakata, M. Ehara, and H. Nakatsuji, Density matrix variational theory apphcation to the potential energy surfaces and strongly correlated systems. J. Chem. Phys. 116, 5432 (2002). 

Weiss-Marcus harmonic energy variation theory, 1513, 1519 Wenking, 1118... 

Correspondingly, a typical value for AG°/ES [cf Eq. (9.3)] is 0.5 so that (0r /3 In i) = (2RT/1.5F) = 1.3(RT/F). Although observed values of this coefficient vary from RT/4F to 2RT/F, and sometimes above this, the figure for the majority of electrochemical reactions is very near 2RT/F and thus the formation of the rate— overpotential relation to which this Weiss-Marcus harmonic energy variation theory gives rise is not consistent with experiment (Fig. 9.26). 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Since the exact ground-state electronic wave function and density can only be approximated for most A-electron systems, a variational theory is needed for the practical case exemplified by an orbital functional theory. As shown in Section 5.1, any rule T 4> defines an orbital functional theory that in principle is exact for ground states. The reference state for any A-electron wave function T determines an orbital energy functional E = Eq + Ec,in which E0 = T + Eh + Ex + V is a sum of explicit orbital functionals, and If is aresidual correlation energy functional. In practice, the combination of exchange and correlation energy is approximated by an orbital functional Exc. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

The KR variational principle determines a wave function with correct boundary conditions at a specified energy, the typical conditions of scattering theory. Energy values are deduced from consistency conditions. 

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

The Canonical Variational Theory [39] is an extension of the Transition State Theory (TST) [40,41]. This theory minimizes the errors due to recrossing trajectories [42-44] by moving the dividing surface along the minimum energy path (MEP) so as to minimize the rate. The reaction coordinate (s) is defined as the distance... 

Since the symmetry-adapted perturbation theory provides the basis for the understanding of weak intermolecular interactions, it is useful to discuss the convergence properties of the sapt expansion. High-order calculations performed for model one-electron (Hj) (30), two-electron (H2) (14, 15), and four-electron (He and He-Hz) (31) systems show that the sapt series converges rapidly. In fact, already the second-order calculation reproduces the exact variational interaction energies with errors smaller than 4%. Several recent applications strongly indicate that this optimistic result holds for larger systems as well. 

---