Hartree-Fock determinantal energy

Hartree-Fock Potential Energy Surface Calculations Asymptotic Behaviour of Single Determinantal Wavefunctions.—In general, a single determinantal wavefunction, whether or not it is at the Hartree-Fock limit, does not provide an adequate description of a molecular system over the complete range of intemuclear separations, because of the failure of such a function to describe... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

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. 

Answer. Orbitals are one-electron wave functions, ). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), 1(1) 2(2) 3(3) NfNe). However, the fact that electrons are fermions also imposes the requirement that the many-electron wave function be antisymmetric toward the exchange of any two electrons. All of the physical requirements, including the indistinguishability of electrons, are met by a determinantal wave function, that is, an antisymmetrized sum of Hartree products, ( 1,2,3,..., Ne) = 1(1) 2(2) 3(3) ( ). If (1,2,3,...,Ne) is taken as an approximation of (1,2,..., Ne), i.e., the Hartree-Fock approximation, and the orbitals varied so as to minimize the energy expectation value,... 

Hartree-Fock theory makes the fundamental approximation that each electron moves in the static electric field created by all of the other electrons, and then proceeds to optimize orbitals for all of the electrons in a self-consistent fashion subject to a variational constraint. The resulting wave function, when operated upon by the Hamiltonian, delivers as its expectation value the lowest possible energy for a single-determinantal wave function formed from the chosen basis set. 

Table 1 Correlation Corrections (eV) for Various Energy Components with Respect to Kohn-Sham and Hartree-Fock One-Determinantal Wavefunctions, for N2 at Three Internuclear Distances3...

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

In this section, we propose to illustrate how the availability of the CRAY has assisted progress in the area of molecular electronic structure. We shall concentrate on two recent advances, namely, the evaluation of the components of the correlation energy which may be associated with higher order excitations, in particular triple-excitations with respect to a single-determinantal, Hartree-Fock reference function, and the construction of the large basis sets which are ultimately going to be necessary to perform calculations of chemical accuracy, that is one millihartree. 

In equation (17) 5 is defined as the overlap of two electronic determinantal wave functions S = z,R, P z,R,P ) and the energy is E = Pl[/2Mi + (z,7 ,Pl/7eiecl, ./ ,E)/(z,7 ,PIz,7 ,PX This level of theory can be characterized as fully non-linear time-dependent Hartree-Fock for quantum electrons and classical nuclei. It has been applied to a great variety of problems involving ion-atom [12,14,15,23-25], and ion-molecule reactive collisions... 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Hartree-Fock theory for the ground state whose zero-order description is Ohf-The Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are determinantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The pertui-bation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

As to the correlation energy, all integrals in Eq. (11) are over non-correlated orbitals. It is true that by taking the derivative of the average of a multiplet system we are implicitly by-passing the limitation of the spin-correlated Hartree-Fock single determinantal formalism, but with so many approximations involved in the SCF—Ajj, method it is not easy to judge how important is this particular feature. 

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