The Hartree Approximation

2 The Hartree and Hartree-Fock approximations 2.2.1 The Hartree approximation 

The simplest approach is to assume a specific form for the many-body wavefunction which would be appropriate if the electrons were non-interacting particles, namely 

Using a variational argument, we obtain from this the single-particle Hartree equations  

The more important problem is to determine how realistic the solution is. We can make the original trial / s orthogonal, and maintain the orthogonality at each cycle of the self-consistency iteration to make sure the final (p s are also orthogonal. Then we would have a set of orbitals that would look like single particles, each Ur) experiencing the ionic potential Vion(r) as well as a potential due to the presence of all other electrons, V/ (r) given by 

This is known as the Hartree potential and includes only the Coulomb repulsion between electrons. The potential is different for each particle. It is a mean-field approximation to the electron-electron interaction, taking into account the electronic charge only, which is a severe simplification. 

If electron-electron interactions must be calculated explicitly, the last entry of Equation (2.5) impedes the separation of the Hamiltonian because each electron is influenced by the positions of all the others. In order to derive a set of one-electron equations, one idea that came to the mind of Hartree [119] is to think of each electron as moving in the field built up by all other electrons. 

For such a scenario, an approximate electron-electron interaction for a given electron only depends on the position of this particular electron alone because the other electrons merely constitute the electronic sea in which this electron propagates. That is equivalent to writing, in atomic units. 

For completeness, we will just mention that the extended Hiickel method also allows one to iterate towards self-consistency, namely if the Coulomb integrals are allowed to better reflect differing atomic potentials (chemists 

The Hartree method is a conceptionally important step for things to come, especially concerning density-functional theory (see Section 2.12), simply because of the idea of self-consistency. The systematic inclusion of the electronic spin and, therefore, electronic exchange, leads to yet another method, which is the most important method of molecular quantum chemistry. 

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The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

The response was calculated in the Hartree approximation, and only a linear response was considered. Suppose an external potential U(r) leads to a change in the electron density 8p. This then... 

This approach is suggestive of the Hartree approximation of atomic and molecular physics. The outer-shell interactions are important, but complicated because of the correlations involved when they are considered directly. The suggested response to this difficulty is to treat these effects as uncorrelated - as a product contribution to the distribution - but with the product factors optimized by (9.53) to be consistent with the basic data. 

Finally, we discuss the effect of nonlinear coupling on domain growth, decoherence, and thermalization. As the wave functionals l/o of Ho are easily found, Eq. (16) leads to the wave functional beyond the Hartree approximation. Putting the perturbation terms (19) into Eq. (16), we first find the wave functional of the form... 

Here, the mean field potential includes the phenomenological isoscalar part Uq x) along with the isovector U (x) and the Coulomb Uc(x) parts calculated consistently in the Hartree approximation Uo(r) and Uso(x) = Uso r)a l are the central and spin-orbit parts of the isoscalar mean field, respectively, and, SPot(r) is the potential part of the symmetry energy. 

Above Tc, Eqs. (22) and (24) yield rather smooth functions of T except the region t 1. The fluctuation region is rather wide since Cy Cy even at T essentially larger then Tc. The appearance of an extra channel of the diquark decay width, beyond the Hartree approximation does not... 

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schrodinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. 

Figure 12. Magnitude of the response function for the stretched I2 target state. The solid line is the result of an exact calculation the dotted line is the result of the use of the Hartree approximation. The parameter c is the coupling constant between the I2 molecule and the bath oscillator, (a) Bath oscillator frequency of 50 cm. (b) Bath oscillator frequency of 100 cm1. (From Ref. 25.)...

The simplest approach to approximating a solution to Eq. 4.3 is to assume that all the electrons move independently of one another. That is, imagine they mumally interact only via an averaged potential energy. This is known as the Hartree approximation. It enables us to write the Hamiltonian for the A -electron system as a sum of N one-electron Hamiltonians, and the many-body wave function as a product of N one-electron wave... 

This HF approximation is the durable and systematic procedures for the search of the possible ground states. (The above approximation to the term is the Hartree approximation, but we use the term HF below for simplicity.) It should be kept in mind, however, that the quantitative aspects of results of this approximation should not be taken literally but that the results will be a basis for the further detailed theoretical studies. 

If the indirect part of the electronic Coulomb repulsion is neglected, we do not get the Hartree approximation as might be expected. Instead, we get a less accurate method, which will be called the neglect of indirect Coulomb repulsion (NICR) method. If only the direct part of the Coulomb repulsion is included, the electron-electron repulsion is... 

Perdew and Zunger (1981), in the Xa-like equivalent of the Hartree approximation, advocate subtracting the total self-interaction of each electron in Xa-like models. This proposal would remove the m dependence of hydrogenic systems. Since the self-interaction of each electron (orthonormal orbital), as well as their sum, is not invariant under a unitary transformation among the orbitals, in contrast to the first-order density matrix and thus Xa-like models, Perdew and Zunger propose picking out a unitary transformation... 

Fig. 44. Proton interaction energy trend along an approach path to the aziridine N. as obtained a) by the electrostatic approximation, b) by the Hartree approximation, c) by SCF computations (GTO wave function)...

The quantum-mechanical SCF method for obtaining the vibrational energy levels is a direct adaptation of the Hartree approximation for electronic struc-true calculations, which dates back to the early stages of quantum theory. The introduction of the method for vibrational modes is, however, rather recent and is due to Bowman and co-workers,6,7 Carney et al.,8 and Cohen et al.9 The semiclassical version of the SCF, the SC-SCF method, proposed by Gerber and Ratner,10 relies on the characteristically short de Broglie wavelengths typical of vibrational motions (as opposed to electronic ones) to gain some further simplification, but is otherwise based on the same physical considerations as the quantum-mechanical approximation. A brief review of the SCF and SC-SCF methods can be found in Ref. (11). 

Degeneracies of the SCF states are an obvious cause for breakdown of the approximation in the form discussed in the previous sections. We discuss now an extension of the method that applies to such cases, that is, to resonances and near-resonances between SCF modes. Just as the vibrational SCF method is an adaptation of the Hartree approximation from electronic structure calculations, so is the generalization discussed here an application of the configuration interaction (Cl) method, which uses for the wavefunctions a linear combination of the strongly interacting SCF states. Quantum Cl for polyatomic vibrations was introduced by Bowman and co-workers,7-21 the semi-classical version is due to Ratner et al.33... 

It seems therefore advisable to try and clarify the analogy between the familiar free Bose gas5) and the n oo G-L system. We shall do it by calculating Tc(aaL) for the latter case. In the limit n - oo the anharmonic term in Equ. (23) can be treated in the Hartree approximation. The transition temperature is then determined by... 

Of course, local fluctuations must stay finite, and thus this result already shows that the fluctuations must lead to a renormalization of the parameters of the effective Hamiltonian. Working this out in the framework of the Hartree approximation yields [58]... 

When solved self-consistently, the electron densities obtained from Eq. [6] can be used in Eq. [5] to calculate the total electronic energy. This is equivalent to the relationship between Eqs. [3] and [4] for the Hartree approach. Unlike the Hartree approximation, however, this expression takes into account exchange and correlation interactions between electrons directly, and requires no other approximations other than the form of the density functional. 

As a simple application of the many-body hamiltonian in form, we consider the problem of N equivalent mutually attracting (gravitating) bosons. That is, we consider the above with all m,- = m, and all Cij = —g for convenience we set m = 1, but retain g through the calculations. This problem is interesting since it is a true many-body problem which can be solved exactly (and easily) for 2 = 1 and for D — oo, though apparently not for D = 3. Here we will solve the problem in the large-2 limit both exactly and in the Hartree approximation. 

In the Hartree approximation, the ground state wavefunction takes the form

effective potential will give all pi the same value p. The interparticle distances will then all be y/2p, since in the D- oo limit... 

What malces this an especially interesting problem to study is the fact that the D = 1 version can also be solved both exactly and in the Hartree approximation [4]. Using the same energy scaling as in Eq. (9), but a distance scaling p, = r,/, the D = 1 problem becomes... 

These strong effects of external screening are well described by the Hartree-Fock model. In the Hartree approximation, the spherical average of the repulsion from the other (K - 1) electrons is subtracted from the potential Z/r (in the atomic units of hartree/bohr) originating in the nucleus considered as a point, and the Schrodinger equation is solved for radial n/-functions in this central field U(r). This needs an iterative process in order to obtain self-consistency between radial func-... 

In fact, the Hartree approximation is the independent-particle model. However, every quantum state in the Hartree wave function satisfies the Pauli exclusion principle such as exchanging the i-th and y-th electrons... 

From now on, we will call the single-particle functions y/a and y/b orbitals . A solution in the form of Equation (4) is called the Hartree approximation. It would be exact if the electrons did not interact. The problem, however, is that not only do electrons interact, they must be indistinguishable from each other. Consequently,... 

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