Total electronic energy, definition

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

The numerical determination of E grr by the use of many-body theory is a formidable task, and estimates of it based on E j and E p serve as important benchmarks for the development of methods for calculating electron correlation effects. The purpose of this work is to obtain improved estimates of Epp by combining the leading-order relativistic and many-body effects which have been omitted in Eq. (1) with experimentally determined values of the total electronic energy, and precise values of Epjp. We then obtain empirical estimates of E grr for the diatomic species N2, CO, BF, and NO using Epip and E p and the definition of E g in Eq. (1). 

Two key pieces of theoretical data are required to obtain an empirical estimate of the correlation energy from the experimental data collected in the preceding section the total molecular Hartree-Fock energy and the relativistic corrections . It is implicit in the definition of the correlation energy presented in Eq. (1) that the total electronic energy ofa given molecule, Ef, may be divided into three constituent parts,... 

From its definition in eqn (6.71) in terms of an integral of an energy density, it is clear that, like other atomic properties (eqn (6.41)), the sum of the energies of the atoms in a system equals the total electronic energy... 

In previous section, by considering the electrostatic energy of the quantum dot charging we have determined the tunnel curves using the phenomenological approach. A strict definition of the tunnel curves as total electronic energy at a fixed dot location between leads is implied by the Born-Oppenheimer adiabatic strategy. For the quantum-mechanical computation of the tunnel curves, the information about (1) the spatial profile of electrostatic potential and (2) the electron and ion distributions of the SET is required as an input. 

The second approach to defining the absolute hardness rj has a companion parameter taken from density functional theory, called the electronic chemical potential ji. The value of pis essentially the same as the negative of %, as defined in Equation 3.2, and the value of rj is essentially the same as in the more approximate definition in Equation 3.3 but both are defined differently. If the total electronic energy of an atom or molecule is plotted as a function of the total number of electrons N, the graph takes the form of Fig. 3.2 in which the only experimental points are at integral values of but between them it is convenient to... 

It is important to keep in mind the difference in the definition of the general (space-spin) quantities for the general case and the special (space-only) quantities for the closed shell case. In particular the expression for the total electronic energy in the closed-shell case is... 

Because an independent electron model, such as our EBO treatment, neglects the explicit repulsion between the electrons, meaning that all Jjj, and Kj in definition 14 for the orbital energies Sj are set equal to zero, it follows that the total electron energy of the hydrocarbon is given by... 

To interpret the electronic structure of a material, it is often useful to understand what states are important in the vicinity of specific atoms. One standard way to do this is to use the local density of states (LDOS), defined as the number of electronic states at a specified energy weighted by the fraction of the total electron density for those states that appears in a specified volume around a nuclei. Typically, this volume is simply taken to be spherical so to calculate the LDOS we must specify the effective radii of each atom of interest. This definition cannot be made unambiguously. If a radius that is too small is used, information on electronic states that are genuinely associated with the nuclei will be missed. If the radius is too large, on the other hand, the LDOS will include contributions from other atoms. 

Since it can be shown that "( ), like the original Hamiltonian H, commutes with the transformation operators Om for all operations R of the point group to which the molecule belongs, the MOs associated with a given orbital energy will form a function space whose basis generates a definite irreducible representation of the point group. This is exactly parallel to the situation for the exact total electronic wavefunctions. 

We have already examined a few graph invariants which, according to the above definition, could be included among the topological indices of benzenoid molecules. These are the number of Kekule structures, the eigenvalues, spectral moments, (coefficients of) the characteristic and sextet polynomials, to mention just some. The total 7t-electron energy is surveyed elsewhere in this volume. 

An inspection of Fig. 3 clearly shows that the total Jt-electron energies of isomeric benzenoid hydrocarbons possessing equal numbers of Kekule structures still vary to some limited extent. The structural invariants causing this variations are not satisfactorily understood, but suspect no. 1 is the number of bay regions. (For the definition of bay regions see [1] or [12].)... 

---