Models single Slater determinant

The remarkable thing is that the HF model is so reliable for the calculation of very many molecular properties, as 1 will discuss in Chapters 16 and 17. But for many simple applications, a more advanced treatment of electron correlation is essential and in any case there are very many examples of spectroscopic states that caimot be represented as a single Slater determinant (and so cannot be treated using the standard HF model). In addition, the HF model can only treat the lowest-energy state of any given symmetry. 

In Chapter 6, I discussed the open-shell HF-LCAO model. 1 considered the simple case where we had ti doubly occupied orbitals and 2 orbitals all singly occupied by parallel spin electrons. The ground-state wavefunction was a single Slater determinant. I explained that it was possible to derive an expression for the electronic energy... 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

According to Lykos and Parr 3>, an unsaturated molecule can be described in this manner by taking for S a (in principle complete) linear combination of Slater determinants built from a orbitals only and for n a similar combination built from n orbitals only. Such a description is more general than the independent-particle model, as it includes the latter as a special case (namely where both and 77 are single Slater determinants). 

The idea behind the Hartree-Fock model is that each electron should be assigned for a single spin-orbital. These spin orbitals are then multiplied together and antisymmetrized to form a single Slater determinant. Most stable molecules have singlet ground states and may be described by a Hartree-Fock wavefunction in which each spacial orbital occurs twice once with a spin and once with /J. Such systems are called closed shell systems. The Hartree-Fock wavefunction for a closed shell system may be written in the form ... 

The Hartree-Fock model is the simplest, most basic model in ab initio electronic structure theory [28], In this model, the wave function is approximated by a single Slater determinant constructed from a set of orthonormal spin orbitals ... 

One way to overcome this difficulty is to use different orbitals for different spins (DODS model). This technique introduced, into the Hartree-Fock scheme, gave rise to the unrestricted Hartree-Fock model (UHF), the wave-function being written as an open shell single Slater determinant[2) ... 

In the Hartree-Fock (HF) model, the n electron wave function. is written as a single Slater determinant of n orthonormal spin orbitals, ( ), ... 

Fig. 10.3. Demonstiation of the power of the Fault exclusion principle, or the Fermi hole formatirai for the molecule in the UHF model (p. 448, a wave function in the form of a single Slater determinant). The two protons (a and b), indicated by occupy positions (0,0,0) and (2,0,01 in a.u.. respective. The space and spin coordinates (the latter shown as arrows) of electrons and 2 [ jq.yi.ri.cri = and JC2, V2,C2,f2 =—3. so they have opposite spins] as well as the spin coordinate...

In the single reference Cl method, the model space (Fig. 10.8) is formed by a single Slater determinant. In the multireference Cl method, the set of determinants constitute the model space. This time, the Cl expansion is obtained by replacement of the spinorbitals participating in the model space by other virtual orbitals. We proceed further as in Cl. 

Fig. 10.8. Illustration of the model space in the multireference Cl method used mainly in the situation when no single Slater determinant dominates the Cl expansion. The orbital levels of the system are presented here. Part of them are occupied in all Slater determinants considered fttKen spinorbitals"). Above them is a region of closely spaced orbital levels called active space. In the optimal case, a significantly large energy gap occurs between the latter and unoccupied levels lying higher. The model space is spanned by all or some of the Slater determinants obtained by various occupancies of the active space levels.

As well known, for the single Slater determinant (SD), that is for independent-particle models, the two-electron RDM is the antisymmetrized product of one-electron RDMs [48, 49] ... 

Slater determinants) in terms of these one-electron functions. We then consider the Hartree-Fock approximation in which the exact wave function of the system is approximated by a single Slater determinant and describe its qualitative features. At this point, we introduce a simple system, the minimal basis (Is orbital on each atom) ab initio model of the hydrogen molecule. We shall use this model throughout the book as a pedagogical tool to illustrate and illuminate the essential features of a variety of formalisms that at first glance appear to be rather formidable. Finally, we discuss the multi-determinantal expansion of the exact wave function of an N-electron system. 

The Hartree-Fock method gives an approximate wave function for the atom of any chemical element fiom the Mendeleev periodic table (orbital picture). The Hartree-Fock method stands behind the orbital model of atoms. The model says essentially that a single Slater determinant can describe the atom to an accuracy that in most cases satisfies chemists. To tell the truth, the orbital model is in principle false, but it is remarkable that nevertheless the conclusions drawn from it agree with experiment, at least qualitatively. It is quite exciting that... 

This expansion shows that there are two errors in any independent electron model. The first error in the wave function, depends on the fact that the spin orbitals calculated are not NSO. The last two terms in the second member of Equation 1.106, together forming Cq, are due to the fact that the occupation numbers are not equal to unity or zero in the first and second part of 8, respectively. This error appears since the true wave function is an expansion in terms of many Slater determinants and cannot be accurately approximated by a single Slater determinant. 

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