Hartree-Fock approximation potential

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

Computationally, the simplification introduced by the Hartree-Fock approximation concerns the expression for the component of the potential energy resulting from the repulsion between electrons. These terms in the Hamiltonian depend on the instantaneous positions of all the electrons, because the energy... 

Let us start the discussion with Fig. 15, which presents various levels of the approach to the potential curve of F2. The Hartree-Fock approximation is especially poor in this case because it does not lead to a correct dissociation limit, i.e. to the atoms in the Hartree-Fock ground states. In the case of F2, the proper dissociation is achieved... 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

The motion of each electron in the Hartree-Fock approximation is solved for in the presence of the average potential of all the remaining electrons in the system. Because of this, the Hartree-Fock approximation, as discussed earlier, does not provide an adequate description of the repulsion between pairs of electrons. If the electrons have parallel spin, they are effectively kept apart in the Hartree-Fock method by the antisymmetric nature of the wavefunction, producing what is commonly known as the Fermi hole. Electrons of opposite spin, on the other hand, should also avoid each other, but this is not adequately allowed for in the Hartree-Fock method. The avoidance in this latter case is called the Coulomb hole. 

However, it should be emphasized that use of these less rigorous methods for the calculation of potential-energy surfaces should be viewed with great caution. Even for a system which dissociates properly in the Hartree-Fock approximation, recent research by Kaufman and co-workers [142] has shown that even the INDO method is not capable of giving an accurate or even realistic surface for Li+ + H2 when compared point by point to Lester s accurate Hartree-Fock surface [117]. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

As the opposite to the examples given above, we note now processes that involve a fission of electron-pair bonds. Here the change in correlation energy is extremely large and the Hartree-Fock approximation is inherently incapable of giving a reasonable account of heats of reaction, A very illustrative example is provided by potential curves of diatomic molecules. From Fig, 4,2 it is seen that for larger depar ... 

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