Hartree-Fock equation total energy

If one uses a Slater detemiinant to evaluate the total electronic energy and maintains the orbital nomialization, then the orbitals can be obtained from the following Hartree-Fock equations ... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

In dealing with the MO-LCAO wave function no additional assumptions concerning the vibronic matrix elements are necessary. The evaluation of the total molecular energy exactly copies the lower sheet of the adiabatic potential. This is a consequence of the well-known fact that the Hartree-Fock equations are equivalent to the statement of the Brillouin theorem the matrix elements of the electronic Hamiltonian between the ground-state and... 

As Dewar points out in ref. [30a], this derivation is not really satisfactory. A rigorous approach is a simplified version of the derivation of the Hartree-Fock equations (Chapter 5, Section 5.2.3). It starts with the total molecular wavefunction expressed as a determinant, writes the energy in terms of this wavefunction and the Hamiltonian and finds the condition for minimum energy subject to the molecular orbitals being orthonormal (cf. orthogonal matrices, Section 4.3.3). The procedure is explained in some detail in Chapter 5, Section 5.2.3)... 

The next step in deriving the Hartree-Fock equations is to express the energy of the molecule or atom in terms of the total wavefunction T the energy will then be minimized with respect to each of the component molecular (or atomic an atom is a... 

Using this wavefunction the total energy of the electron system is minimized with respect to the choice of spin-orbitals under the constraint that the spin orbitals are orthogonal. The variational procedure is applied to this minimization problem and the result is the so called Hartree-Fock equations ... 

They computed the total energy for a variety of box sizes and different values of Uo by numerically solving the Hartree-Fock equation. In order to test the accuracy accomplished by this method, they computed the energy for the case of an impenetrable box (Uo = oo), finding slightly lower energies than those reported by Ludena [102]. They also obtained the pressure numerically. 

The best possible variationally determined wavefunction of this form is that in which both the spacial orbitals total wavefunction be normalized. For such a wave-function, the variational method leads to orbital equations similar to the Hartree-Fock equations (10). These are.-26... 

The Density Functional theorem states that the total ground state energy is a unique functional of the electron density, p [40]. This simple but enormously powerful result means that it is possible, in principle, to provide an exact description of all electron correlation effects within a one-electron (i.e. orbital-based) scheme. Khon and Sham (KS) [41] have derived a set of equations which embody this result. They have an identical form to the one-electron Hartree Fock equations. The difference is that the exchange-correlation term, Vxc, is not the same. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

How do we know that iterations improve psi and epsilon This is not always the case, see e.g. [Ij], p. 35, but in practice initial gness solutions to the Hartree-Fock equations usually converge fairly smoothly to give the best wavefunction and orbital energies (and thus total energy) that can be obtained by the HF method from the particular kind of gness wavefunction (e.g. basis set section 5.2.3.6e). 

Once the closed-shell Hartree-Fock equations including electrostatic solute-solvent interaction have been solved, the total energy of the solvated molecule at the SCF level is obtained by the expression... 

This explicit removal of the self-interaction amongst the electrons is a great strength of the (algebraic approximation to the) Hartree-Fock equations. We shall see later that separate approximations to parts of the total interaction energy of a system of electrons do not have this convenient property and may often include spurious energies of interaction between different parts of a given electron . The so-called self-interaction correction (SIC) must be invoked in such... 

The variational method of generating the Schrodinger equation and the Hartree-Fock equation contains a possible ambiguity in the sense that the total kinetic energy of a single electron is given by both of... 

Now that we have derived the unrestricted Hartree-Fock equations, we can write down expressions for the unrestricted orbital energies, total unrestricted energy, etc. First, we need to define a few terms. The kinetic energy and nuclear attraction of an electron in one of the unrestricted orbitals jf or is the expectation value... 

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