Hartree-Fock method Slater determinant

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

In the Hartree-Fock method, the molecular (or atomic) electronic wave function is approximated by an antisymmetrized product (Slater determinant) of spin-orbitals each spin-orbital is the product of a spatial orbital and a spin function (a or ft). Solution of the Hartree-Fock equations (given below) yields the orbitals that minimize the variational integral. Thus the Hartree-Fock wave function is the best possible electronic wave function in which each electron is assigned to a spatial orbital. For a closed-subshell state of an -electron molecule, minimization... 

Y -Hab(Hbb -E)- Hbd Haa -Hab(Hbb -E)"Hh If the Hamiltonian is a one-electron Hamiltonian, for example the Fock operator, the partitioning is done by basis functions, since the latter are usually centered on the atomic nuclei, which belong to donor (d), bridge (b) or acceptor (a). In the Hartree-Fock case, the total wave function is a Slater determinant. There may be problems with symmetry breaking in the symmetric case. Cl that includes the two localized solutions can solve this problem [29-31]. The problem is that the Hartree-Fock method gives energy advantage to a localized state, which holds true also in the unsymmetric case. 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

In 1963. Ldwdin [18] pointed out the occurrence of a sytnmetry dilemma in fte Hartree-Fock method that, even if a symmetiy requirement is self-consistent, it is still a constraint which will Increase the energy . and the associated optimum value of is hence only a local minimum on the other hand, if one looks for the absolute minimum of , the associated Slater determinant may very well be a mixture of various symmetry types. It is evident that some of the optimum values of are not even local mimima, and the study of the nature of the optimum values by means of the second-derlvatles or the Hessians has become one of the most intensely studied problems [19] in the current literature. Most of the papers use very elegant and forceful methods based on the use of second quantization, but it should be observed that the problem may dso be treated in an elementary way [20]. 

Hartree-Fock method (HF). Here the total wave function is assumed to be an antis5nnmetric sum of Hartree functions and can be represented by a Slater determinant... 

As shown in Section 9.1, the A -electron wavefunction is more correctly represented by a Slater determinant of spinorbitals (9.1) rather than a Hartree product of orbitals (9.20), thus accounting automatically for the exclusion principle and the indistinguishability of electrons. The Hartree-Fock method, developed in 1930, is a generalization of the SCF based on Slater determinant wavefimctions. The Hartree-Fock (HF) equations for the spinorbitals (pa have the form... 

The starting point for most high-level molecular computations is the Hartree-Fock method. This is based on a Slater determinant of molecular orbital functions, analogous to Eq (9.1) for atoms ... 

The UHF ansatz is necessary because in case of neutral solitons one has to deal with a doublet state. Thus a DODS (different orbitals for different spins) ansatz, as the UHF one, is necessary to describe the system. However, in the UHF method described so far, one Slater determinant with different spatial orbitals for electrons of different spins is applied, which is not an eigenfunction of S2, i.e. S(S+l)h2. The best way to overcome this difficulty would be to use the PHF (Projected Hartree Fock) method, also called EHF method (Extended Hartree Fock) where before the variation the correct spin eigenfunction is projected out of the DODS ansatz Slater determinant [66,67a]. Unfortunately numerical solution of the rather complicated EHF equations in each time step seems to be too tedious at present. Moreover for large systems the EHF wavefunction approaches the UHF one [68], however, this might be due to the approximations used in [67a]. Another possibility is to apply the projection after the variation using again Lowdin s projection operator [66]. Projection and annihilation techniques were... 

Now we have reduced the problem to solving the Schrodinger equation for just one unit cell (actually even a primitive cell in the cases where it is possible to reduce the unit cell even more), and for separated electronic and nuclear degrees of freedom. Still, as long as we have more than two electrons, the problem is unmanageable. Therefore we must make more approximations in order to get some way to solve the problem. One of the most common approximations is the Hartree-Fock method [8], in which the variational principle is used together with so called Slater determinants of electron orbitals to do calculations. One of the problems with this method is that you have to approximate the orbitals. 

The Hartree-Fock method approximates the true wavefunction by a single Slater determinant (10). A better approximation to the true wave-function is to take a linear combination of many Slater determinants so that the total wavefunction is expressed as ... 

The Hartree-Fock method neglects the electron correlation. The reliability (and applicability) of the HF method itself is rather small. However, the electron correlation can be recovered by post-HF methods based on the wavefunction expansion in terms of Slater determinants constructed from the one-electron Hartree-Fock orbitals. 

The Hartree-Fock method involves the optimization - via energy minimization -of a single Slater determinant wavefunction. This process is usually carried out by solving the single-particle Hartree-Fock equations. From the Hartree-Fock orbitals one can construct the Hartree-Fock density pur-... 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

M-electron wavefunction can be expanded as a linear combination of an infinite set of Slater determinants that span the Hilbert space of electrons. These can be any complete set of M-electron antisymmetric functions. One such choice is obtained from the Hartree-Fock method by substituting all excited states for each MO in the determinant. This, of course, requires an infinite number of determinants, derived from an infinite AO basis set, possibly including continuum functions. As in Hartree-Fock, there are no many-body terms explicitly included in Cl expansions either. This failure results in an extremely slow convergence of Cl expansions [9]. Nevertheless, Cl is widely used, and has sparked numerous related schemes that may be used, in principle, to construct trial wavefunctions. 

The more accurate Hartree-Fock method approximates the wave function as an antisymmetrized product (Slater determinant or determinants) of one-electron spin-orbitals and finds the best possible forms for the spatial orbitals in the spin-orbitals. Hartree-Fock calculations are usually done by expanding each orbital as a linear combination of basis functions and iteratively solving the Hartree-Fock equations (11.12). The Slater-type orbitals (11.14) are often used as the basis functions in atomic calculations. The difference between the exact nonrelativistic energy and the Hartree-Fock energy is the correlation energy of the atom (or molecule). 

In many quantum-mechanical calculations, use is made of the wave functions obtained by the Dirac—Slater and the Hartree—Fock methods for the approximate solution of the Schrddinger equation for free atoms. It woiild be very interesting to determine whether these functions could be refined specifically for crystals and whether the problem could be solved using relatively simple analytic approximations to the calculated functions. In particular, the approximation by Gaussian functions demands attention. 

To solve the corresponding Eq. (6.8), we have at our disposal the variational and the perturbation methods. The latter should have a reasonable starting point (i.e., an unperturbed system). This is not the case in the problem that we want to consider at the moment. Thus, only the variational method remains. If so, a class of the trial functions should be proposed. In this chapter, the trial wave function will have a very specific form, bearing significant importance for the theory. We mean here the so-called Slater determinant, which is composed of molecular orbitals. At a certain level of approximation, each molecular orbital is a parking place for two electrons. We wUl now learn on how to get the optimum molecular orbitals (using the Hartree-Fock method). Despite some quite complex formulas, which will appear below, the main idea behind them is extremely simple. It can be expressed in the following way. 

The Hartree-Fock method is a variational one (p. 232) i.e., it uses the variational wave function in the form of a single Slater determinant and minimizes the mean value of the... 

The reason for any Hartree-Fock method failure can be only one thing the wave function is approximated as a single determinant. All possible catastrophes come from this, and we might even deduce when tiie Hartree-Fock method is not appropriate for description of a particular real system. First, let us ask when a single determinant would be OK Well, out of all Slater determinants tiiat could be constructed from a certain spinorbital basis set, only its energy (i.e., the mean value of tiie Hamiltonian for this determinant) would be close to the true eneigy of... 

Since the Hartree-Fock method looks to be a poor tool for beiyUium, we propose a more reasonable wave function in the form of a Unear combination of the ground-state configuration [Eq. (10.21)] and the configuration given by the following Slater determinant ... 

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... 

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