Hartree-Fock description

Con versely, an imre.vtrtctcrf Hartree-Fock description implies that there are two different sets of spatial molecular orbitals those molecular orbitals, occupied by electrons of spin up (alpha spin ) and those molecular orbitals, occupied by electrons of spin down (beta spin) as shown next. 

A restricted Hartree-Fock description means that spin-up and spin-down electrons occupy the same spatial orbitals /j—there is no allowance for different spatial orbitals for different electron spins. 

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis,... 

One approach is to construct a more flexible description of electron motions in terms of a combination of Hartree-Fock descriptions for ground and excited states. Configuration interaction (Cl) and Moller-Plesset (MP) models are two of the most commonly used models of this type. The so-called second-order Moller-Plesset model (MP2) is the most practical and widely employed. It generally provides excellent descriptions of equilibrium geometries and conformations, as well as thermochemistry, including the thermochemistry of reactions where bonds are broken and formed. Discussion is provided in Section n. 

We have in this chapter discussed different situations, where the simple single-configurational Hartree-Fock description of the molecular system breaks down. It has been shown that a model can be devised that gives a qualitatively correct wave function in cases where several electronic configurations are close in energy. In the next chapter we shall develop the tools needed to compute such wave functions. 

The dissociation of difluorine is a demanding test case used traditionally to benchmark new computational methods. In this regard, the complete failure of the Hartree-Fock method to account for the F2 bond has already been mentioned. Table 1 displays the calculated energies of F2 at a fixed distance of 1.43 A, relative to the separated atoms. Note that at infinite distance, the ionic structures disappear, so that one is left with a pair of singlet-coupled neutral atoms which just corresponds to the Hartree-Fock description of the separated atoms. 

The complete description of hydrogen bond and van der Waals interactions requires of course the inclusion of electron correlation effects however, almost always, a very useful starting point for subsequent refinements is represented by a Hartree-Fock description, which serves as the basis for both perturbation theory and variational configuration interaction approaches to the treatment of electron correlation. 

There are certain types of system for which this approximate parallelism between the Hartree-Fock and exact potential energy surfaces holds not only in the region of the equilibrium geometry, but over the total surface. In such cases, one can obtain a Hartree-Fock description of the complete chemical change, a description at a level of approximation, which, from the above examples, is clearly adequate for answering many if not most of the questions concerning the static behaviour of the system along the so-called reaction pathway. It is to a discussion of such systems that we now turn. 

Figure 3 The one-electron and two-electron density functions of the Xg ground state of the H2 molecule. The upper plots contain the one-electron and two-electron densities of the uncorrelated Hartree-Fock description in a minimal basis the lower plots contain the corresponding densities of the two-configuration correlated FCI description in the same basis. In all cases, the electron density has been plotted on the molecular axis (one axis for the one-electron densities, two axes for the two-electron densities).

It should be emphasized that Eqs. (20) and (9) define the first-order wave functions not only for different perturbation approaches but, first of all, for different physical situations. The former wave functions correspond to real ground- or excited states of two-electron atoms, whereas all of the latter pair functions correspond to the ground state of an N-electron atom, providing corrections to the Hartree-Fock description of its electron pairs. However, from the mathematical point of view, in both cases we have to deal with equations defining pair functions of identical symmetry characteristic. 

O. Sugino and H. Kamimura. Localized-orbital Hartree-Fock description of alkali-metal clusters. Phys. Rev. Lett. 65, 2696 (1990). 

In a given orbital basis, the Hartree-Fock description divides the orbital space into a set of occupied and virtual spin orbitals. From the Slater determinant any other determinant may be generated by replacing an occupied orbital by a virtual. Formally such operation is performed within the second quantization formalism by using the excitation operators... 

An interesting example of the use of unrestricted wave functions occurs for the methyl radical CH3. This molecule has D3J, symmetry, i.e., it is planar with bond angles of 120. The CH internuclear distance is taken to be 2.039 a.u. The simplest description of the electronic structure of this radical is a restricted Hartree-Fock description, shown in Fig. 3.13. The unpaired... 

Figure 3.13 Restricted Hartree-Fock description of the planar methyl radical.

Most of the developments described above occurred before the advent of the electron in chemistry. Then came the golden years of wave mechanics with one-electron wave functions (orbitals), the Pauli principle, the building-up principle (Ai auprinzip) and, even before the end of the 1920s, the idea that chemistry had now become a question of computation was proposed. Not many years later the best conceivable method of describing atomic and molecular systems in terms of fixed orbitals with one or two electrons in each was invented and named the Hartree-Fock description. The s pearance of electronic computers made the method practical for heavy systems, such as atomic 3d transition-metal ions, as early as about 1960. The results for d spectra were enthusiastically received, mainly b use it was wonderful to see that such calculations could actually be done. On second thought and on further development of computational chemistry, the orbitad or one-electron picture of chemistry became... 

The perturbation operator corresponds to the difference of the instantaneous electron-electron interaction operator and the mean-field electron-electron interaction of the Hartree-Fock description... 

To explicitly obtain the spatial distribution of the photoelectrons for the final electronic state, we assume a frozen-core Hartree Fock description where the final ionized electronic state is represented as an antisymmetrized product of a Hartree Fock ion core wavefunction, c,+, and a photoelectron... 

The second way in which atomic interactions profoundly affect the condensate properties is through their effect on the energy. The effect of atom-atom interactions in the many-body Hamiltonian can be parameterized in the T —> 0 limit in terms of the two-body scattering length. This use of the exact two-body T-matrix in an energy expression is actually a rigorous procedure, and can be fully justified as a valid approximation.One simple theory which has been very successful in characterizing the basic properties of actual condensates is based on a mean-field, or Hartree-Fock, description of the condensate wave function, which is found from the equation ... 

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