Hartree-Fock equation description

You can order the molecular orbitals that are a solution to equation (47) according to their energy. Electrons populate the orbitals, with the lowest energy orbitals first. Anormal, closed-shell, Restricted Hartree Fock (RHF) description has a maximum of two electrons in each molecular orbital, one with electron spin up and one with electron spin down, as shown ... 

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific ... 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

Solutions to the Hartree-Fock equations are exact solutions to an approximate description because ... 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

There are many ways of obtaining an orthonormal set of orbitals. The solution of the Hartree-Fock equations (5.26) for the ground state produces self-consistent occupied orbitals and a set of unoccupied orbitals into which electrons are excited to form the basic configurations rfc),/c 0. The unoccupied orbitals are not optimised in any sense for the description of excited states and in fact provide a bad description since they have boundary conditions that result in their extension too far from the main matter distribution in coordinate space. 

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 correlated methods discussed up to this point provide a delocalized description of the electronic system. The delocalized nature of these methods arises from their use of canonical orbitals (i.e., the eigenvectors of the Hartree-Fock equations) of Eq. (33). To treat large systems, it is better to express the theory in terms of orbitals that are localized in space, extending over only a few atoms. The virtual excitations then occur predominantly locally in the molecule (among localized occupied and virtual orbitals). As a result, the number of excitation amplitudes increases only linearly with system size. 

The energy associated with the removal of an electron from an orbital of a single determinant wavefunction is the diagonal element of the associated e matrix of the Hartree-Fock equation. This energy is only an optimum description of the ionisation energy of the system if the matrix e is diagonal. 

The virtual orbitals generated by the solution of the LCAO approximation to the Hartree-Fock equations are indeed an artifact of the LCAO technique and do not have any physical interpretation except as a residue of those features of the basis functions which are not suitable for the description of the single-determinant model of the electronic structure of the molecule. 

Now that the description of the electronic Hamiltonian, Eq. [4], has been determined and the wavefunction, Eq. [6], has been defined, the effective electronic energy can be found by use of the variational method. In the variational method the best wavefunction is found by minimizing the effective electronic energy with respect to parameters in the wavefunctions. Using this idea, Fock and Slater simultaneously and independently developed what is now known as the Hartree—Fock equations. Note that we now make explicit reference only to the spatial orbitals < >. The only time we make reference to spin is that we will fill the orbitals according to the aufbau principle and place two electrons in each spatial orbital. 

To obtain the photoionization amplitudes, we used our Cl wavefunction for the double-minimum state and a frozen-core Hartree-Fock (FCHF) description of the wavefunction for the ionized state. For the FCHF model the wavefunction is taken to be an antisymmetrized product of HF ion orbitals and a photoelectron orbital that is a solution of a one-electron Schrodinger equation containing the Hartree-Fock potential of the ion core. The HF wavefunction provides a very adequate description of Na over all internu-clear distances of interest. 

While the equations of the Hartree-Fock approach can he rigorously derived, we present them post hoc and give a physical description of the approximations leading to them. The Hartree-Fock method introduces an effective one-electron Hamiltonian. as in equation (47) on page 194 ... 

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

The description above may seem a little unhelpful since we know that in any interesting system the electrons interact with one another. The many different wave-function-based approaches to solving the Schrodinger equation differ in how these interactions are approximated. To understand the types of approximations that can be used, it is worth looking at the simplest approach, the Hartree-Fock method, in some detail. There are also many similarities between Hartree-Fock calculations and the DFT calculations we have described in the previous sections, so understanding this method is a useful way to view these ideas from a slightly different perspective. 

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