Spinorbital, molecular

Then the expectation value, when the ground-state determinantal function W0) is constructed from the set of occupied molecular spinorbitals [Pg.303]

The electronic wave function of a molecule containing N electrons depends on 3N Cartesian coordinates of the electrons and on their N spin coordinates (for each electron, its o = or — ). Thus, it is a function of position in 4A-dimensional space. This function will be created out of simple bricks , i.e., molecular spinorbitals. Each of those will be a function of the coordinates of one electron only three Cartesian coordinates and one spin coordinate (cf.. Chapter 1). A spinorbital is therefore a function of the coordinates in the 4-D space, and in the most general case, a normalized spinorbital reads as (see Fig. 8.1) ... 

These 0,- are called canonical spinorbitals and are the solution of the Foek equation, and e,-is the orbital energy eorresponding to the spinorbital 0,-. It is indicated in brackets that both the Fock operator and the molecular spinorbital depend on the coordinates of one electron only (represented by electron 1). 

It a shell IS not closed, it is called open. We assume that there is a umque assignment for which molecular spinorbitals within a closed shell are occupied in the ground state. The concept of the closed shell is approximate because it is not clear what it means when we say that the HOMO-LUMO energy distance is large or small. ... 

Mendeleev periodic table (p. 446) minimal model of a molecule (p. 489) molecular spinorbital (p. 394) molecular orbital (p. 420) occupied orbital (p. 409) open shell (p. 411) orbital centering (p. 422) orbital localization (p. 467) orbital size (p. 424) penetration energy (p. 454)... 

Slater determinants are usually constructed from molecular spinorbitals. If, instead, we use atomic spinorbitals and the Ritz variational method (Slater determinants as the expansion functions), we would get the most general formulation of the valence bond (VB) method. The beginning of VB theory goes back to papers by Heisenbeig, the first application was made by Heitler and London, and later theory was generalized by Hurley, Lennard-Jones, and Pople. The essence of the VB method can be explained by an example. Let us take the hydrogen molecule with atomic spinorbitals of type liaO and Vst (abbreviated as aa and b ) centered at two nuclei. Let us construct from them several (non-normalized) Slater determinants, for instance ... 

Generally, we construct the Slater determinants < >/ by placing electrons on the molecular spinorbitals obtained with the Hartree-Fock method, in most cases, the set of determinants is also limited by imposing an upper bound for the orbital energy. In that case, the expansion in Eq. (10.30) is finite. The Slater determinants < >/ are obtained by the replacement of oecupied spinorbitals with virtual ones in the single Slater determinant, which is the Hartree-Fock function... 

The problem of many-body correlation of motion of anything is extremely difficult and so far unresolved (e.g., weather forecasting). The problem of electron correlation also seemed to be hopelessly difficult. It still remains that way however, it turns out that we can exploit a certain observation made by Sinanoglu. This author noticed that the major portion of the correlation is included through the introduction of correlation within electron pairs, next through pair-pair interactions, then pair-pair-pair interactions, etc. The canonical molecular spinorbitals, which we can use, are in principle delocalized over the whole molecule, but practically the delocalization is not so large. Even in the case of canonical spinorbitals. and certainly when using localized molecular spinorbitals, we can think about an electron excitation as a transfer of an electron... 

In the Cl formalism, the Slater determinants are built from the molecular spinorbitals. 

We start from the Slater determinant built of N molecular spinorbitals. Any of these is a linear combination of the spinorbitals of the donor and acceptor. We insert these combinations into the Slater determinant and expand the determinant according to the first row (Laplace expansion, see Appendix A available at booksite. elsevier.com/978-0-444-59436-5 on p. el). As a result, we obtain a linear combination of the Slater determinants, all having the donor or acceptor spinorbitals in the first row. For each of the resulting Slater determinants, we repeat the procedure, but focusing on the second row, then the third row, etc. We end up with a linear combination of the Slater determinants that contain only the donor or acceptor spinorbitals. We concentrate on one of them, which contains some particular donor and acceptor orbitals. We are interested in the coefficient C d) that multiplies this Slater determinant number i. 

The Hartree-Fock procedure is a variational method. The variational function takes the form of a single Slater determinant [Pg.422]

Each of the determinants is constructed from molecular spinorbitals which are not fixed (as in the Cl method) but are modified in such a way as to have the total energy as low as possible. 

The Cl (Configuration Interaction) approach is a Ritz method (Chapter 5) which uses the expansion in terms of known Slater determinants. These determinants are constructed from the molecular spinorbitals, usually occupied and virtual ones, produced by the... 

Let us recall what the Hartree-Fock exchange energy looks like [Chapter 8, eq. (8.35)]. The Kohn-Sham exchange energy looks, of course, the same, except that the spinorbitals are now Kohn-Sham, not Hartree-Fock. Therefore, we have the exchange energy Ex as (the sum is over the molecular spinorbitals ) y SMO SMO... 

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