Closed-shells total energy

The Separated Molecular Orbitals (SMO) method has been described earlier in details in refs. [16, 17]. At the Hartree-Fock SCF level the closed shell total energy has the following form ... 

The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence electrons necessary to obtain closed-shell electronic structures. Each orbital in the. y-orbital set is assumed to be occupied by a pair of electrons since the 5-orbital energies are low and separate from those of the p-orbital ones, especially for heavy atoms. The total number of valence electrons for the closed-shell structures... 

The i-orbital array of three and four-membered rings is of the Hiickel conjugation. (Scheme 2). The splitting patterns of the orbital energy levels (Scheme 3) show that the total number of valence electrons for the closed-shell structures is 4Af + 2 for the three- N= 0) and four-membered rings (N= 0, 1). 

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... 

These methods can give us useful information on radicals in a manner similar to that for closed-shell systems, provided the exploitation is correct. Of course, in expressions for total energy, bond orders, etc., a singly occupied orbital must be taken into account. One should be aware of areas where the simple methods give qualitatively incorrect pictures. The HMO method, for example, cannot estimate negative spin densities or disproportionation equilibria. On the other hand, esr spectra of thousands of radicals and radical ions have been interpreted successfully with HMO. On the basis of HMO orbital energies and MO symmetry... 

Two isolated reactant molecules in the closed-shell ground state are designated as A and B, whose electronic energies are IFao and Wbo, respectively. Here the term closed-shell implies the structure of a molecule with doubly occupied MO s only. The lowest total energy of the two mutually interacting systems is denoted by W. Then, the interaction energy is defined by... 

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic electron correlation. 

Assuming Eq. (49), this result shows that for canonical NSOs the operator F has an essentially degenerate eigenvalue spectmm that is, all the NO eigenvalues are the same (pi) and are equal to minus the vertical IP [84], Unfortunately, apart from the special case of the HE energy that may be viewed as the simplest 1 -RDM functional, none of the currently known functionals (including the exact functional for the total energy in two-electron closed-shell systems) have effective potentials that satisfy the formal relation (49). 

Table 1 Total energies of the closed-shell systems Au+ and Au (hartree), with the nonrelativistic, DF, and DFB Hamiltonians.

Show that the HF total energy for a closed-shell system may be written in terms of integrals over the orthonormal HF orbitals as ... 

The RHF model leads to a reasonably accurate description of H2 around the equilibrium geometry computed bond distance is 0.735 A (exp 0.746 A) and the bond energy is 84 kcal/mol (exp 109 kcal/mol). It is typical for the RHF model that it is able to describe closed shell systems around their equilibrium geometry rather well. The correlation energy is only a small fraction of the total energy., but it is strongly distance dependent, which explains the error in... 

Here hco = In[j + e, where e refers to photoelectron energy and I ij is the binding energy of the nly-electron in the atom. Performing the summation in (33.9) over j and neglecting the dependence of e on j, we arrive at the following expression for total photoionization cross-section of the closed shell ... 

Hiickel noted that if electron pairs are filled into the energy-level pattern 32 or 33, a closed-shell structure (all electrons paired) will result only when the total number of pairs is odd (total number of electrons = 4re + 2, = 0, 1, 2,...) if the number of pairs is even (total number of electrons = 4re, re = 0, 1, 2,...),... 

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