Non-interacting system

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

If H is the one-particle energy operator, the total energy operator for non-interacting systems can be written 2a HafaA or 2a The... 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

Since this Hamilton operator does not contain any electron-electron interactions it indeed describes a non-interacting system. Accordingly, its ground state wave function is represented by a Slater determinant (switching to 0S and (p rather than Osd and % for the determinant and the spin orbitals, respectively, in order to underline that these new quantities are not related to the HF model)... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

An important advantage of MP2 and higher-order perturbation methods is their size-consistency at every order. This is in contrast to many variational Cl methods, for which the calculated energy of two identical non-interacting systems might not be equal to twice that of an individual system. Size-consistent scaling is also characteristic of QCI and CC methods, which are therefore preferable to standard Cl-type variational methods for many applications. 

Kohn and Sham later introduced the idea of an auxiliary non-interacting system with the same electron density as the real system. They were able to express the electron density of the interacting system in terms of the one-electron wavefunctions of the non-interacting system ... 

For a non-interacting system, the space p° is composed of subspaces p , whose associated operators are assumed to satisfy... 

For the case in which M oxidizes the substrate A, the reacting system crosses from the neutral (solid line) reaction profile with its high activation energy to that of the hole-catalyzed reaction (dashed line) as indicated by the arrows. The activation energy for the non-interacting system in the gas phase is given by ... 

So the highest occupied Kohn-Sham orbital has a fractional occupation number Hohenberg-Kohn theorem applied to the non-interacting system. The proof of... 

Eq. (152) describes a non-interacting system. It is clear that is not a subspace of since a linear combination of Slater determinants does not in general yield a single Slater determinant. Applying the same arguments as those of Section 4.1, we conclude that can be decomposed into the orbits (9 ... 

It is of interest to write down the kinetic energy for this wavefunction (i.e., the kinetic energy for a non-interacting system) ... 

In addition, since the non-local part of the 1-matrix for the single Slater determinant (for a non-interacting system) also appears in Eqs. (159) and (160), it is necessaiy that we select a generating function (p n xs) a particular orbit This wavefunction satisfies the condition of minim-... 

Note that, for a non-interacting system of electrons, the kinetic energy is just the sum of the individual electronic kinetic energies. Within an orbital expression for the density, Eq. (8.14) may then be rewritten as... 

As already emphasized above, in principle Fxc not only accounts for the difference between the classical and quantum mechanical electron-electron repulsion, but it also includes the difference in kinetic energy between the fictitious non-interacting system and the real system. In practice, however, most modem functionals do not attempt to compute this portion explicitly. Instead, they either ignore the term, or they attempt to constmct a hole function that is analogous to that of Eq. (8.6) except that it also incorporates the kinetic energy difference between the interacting and non-interacting systems. Furthermore, in many functionals... 

We may perform the same analysis for the allyl radical and the allyl anion, respectively, by adding the energy of 4>2 to the cation with each successive addition of an electron, i.e., H (allyl radical) = 2(a + V2/3) + a and Hn allyl anion) = 2(a + s/2f) + 2a. In the hypothetical fully 7T-localized non-interacting system, each new electron would go into the non-interacting p orbital, also contributing each time a factor of a to the energy (by definition of o ). Thus, the Hiickel resonance energies of the allyl radical and the allyl anion are the same as for the allyl cation, namely, 0.83/1. 

The case of a decomposable matrix (2.6) merely means that one has two non-interacting systems, governed by two M-equations with matrices A and B, respectively. A non-trivial example is a system in which all transitions conserve energy each energy shell E has its own M-equation and its own stationary distribution . The stationary solutions of the total M-equation are linear superpositions of them with arbitrary coefficients nEi... 

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