Approximation Hartree

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 total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

Fiber R and M Karplus 1990. Enhanced Sampling in Molecular Dynamics Use of the Time-Dependent Hartree Approximation for a Simulation of Carbon Monoxide Diffusion through Myoglobin. Journal of the American Chemical Society 112 9161-9175. 

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... 

Elber, R. Karplus, M., Enhanced sampling in molecular-dynamics - use of the time-dependent Hartree approximation for a simulation of carbon-monoxide diffusion through myoglobin, J. Am. Chem. Soc. 1990,112, 9161-9175... 

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... 

Now, if the many-body (electron) problem can be arranged in such a way that the many-body, nonseparable wave function is expressed in terms of a separable wave function, which depends on N single-particle wave functions (Hartree approximation), i.e.,... 

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. 

The last term is introduced within the self-consistent Hartree approximation (within the functional up to one vertex), //// = 10/9 accounts different coefficients in functional for the self-interaction terms (ri4 - for the given field d and dla)2 d )2 terms), cf [20], We presented da = 2k d t>ke lkfix, ... 

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... 

C. Reduction by Factorization Time-Dependent Hartree Approximation... 

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. 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

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.)...

It must be remembered, however, that the Leibler-type mean field theory [197] is believed to be accurate for the limit of infinite chain length, N—for finite N effects of order parameter fluctuations are important and change the character of the transition from second order to first order even for symmetric composition [185,186,192,210,211]. With a self-consistent Hartree approximation that one believes to be valid for large N, Eq. (41) gets replaced by [234]... 

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. 

A.P. Jansen, A multiconfiguration time-dependent Hartree approximation based on natural single-particle states, J. Chem. Phys., 99 (1993) 4055. 

J.E. Straub, M. Karplus, Energy equipartitioning in the classical time-dependent Hartree approximation, J. Chem. Phys. 94 (1991), 6737... 

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

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