Hartree model

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

I will refer to the Hartree model from time to time in the text. Hartree s energies were in poor agreement with experiment. With the benefit of hindsight he should have allowed for indistinguishability and the Pauli principle. This was Fock s contribution to the field he wrote the wavefunction as what we would now recognize as a Slater determinant. Such a wavefunction automatically satisfies the Pauli principle. 

In the 1930s Douglas Hartree and his father William Hartree used a mechanical analog computer to explore the idea that an electron in an atom moves partly in the attractive potential of the nucleus and partly in an averaged repulsive potential due to all the other electrons. Later, V. Fock added to the Hartrees model an exchange term due to the effects of the antisymmetry of the many-electron wave function. The Hartree-Fock approximation is still a basic tool of quantum chemistry. 

Douketis C, Socles G, Marchetti S, Zen M and Thakkar A J 1982 Intermolecular forces via hybrid Hartree-Fock SCF plus damped dispersion (HFD) energy calculations. An improved spherical model J. Chem. Phys. 76 3057... 

Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

Extended Hiickel theory Generalised valence bond model Hartree-Fock... 

In this model, the Hartree-Foek orbitals of the parent moleeule are used to deseribe both the parent and the speeies generated by eleetron addition or removal. It is said that sueh a model negleets orbital relaxation whieh would aeeompany the eleetron addition or removal (i.e., the reoptimization of the spin-orbitals to allow them to beeome appropriate to the daughter speeies). 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

DFT methods are attractive because they include the effects of electron correlation—the fact that electrons in a molecular system react to one another s motion and attempt to keep out of one another s way—in their model. Hartree-Fock calculations consider this effect only in an average sense—each electron sees and... 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

In order to do so, you will need to perform Hartree-Fock NMR calculations using the 6-311+G(2d,p) basis set. Compute the NMR properties at geometries optimized with the B3LYP method and the 6-31G(d) basis set. This is a recommended model for reliable NMR predictions by Cheeseman and coworkers. Note that NMR calculations typically benefit from an accurate geometry and a large basis set. 

The predicted energy of this structure is approximately -215.76438 Hartrees, yielding a reaction barrier of about 86.6 kcal/mol. This value is more in line with expectations, although it is on the high side. Rotation of a double bond is a problem which often requires a higher level of theory than Hartree-Fock (for example, CASSCF) for accurate modeling. 

Each cell in the chart defines a model chemistry. The columns correspond to differcni theoretical methods and the rows to different basis sets. The level of correlation increases as you move to the right across any row, with the Hartree-Fock method jI the extreme left (including no correlation), and the Full Configuration Interaction method at the right (which fuUy accounts for electron correlation). In general, computational cost and accuracy increase as you move to the right as well. The relative costs of different model chemistries for various job types is discussed in... 

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

A variety of theoretical methods have been developed which include some effects of electron correlation. Traditionally, such methods are referred to as post-SCF methods because they add correlation corrections to the basic Hartree-Fock model. As of this writing, there are many correlation methods available in Gaussian, including the following ... 

Hartree-Fock theory is very useful for providing initial, first-level predictions for many systems. It is also reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states. As such, it is a good base-level theory. However, its neglect of electron correlation makes it unsuitable for some purposes. For example, it is insufficient for accurate modeling of the energetics of reactions and bond dissociation. 

Prior to the widespread usage of methods based on Density Functional Theory, the MP2 method was one of the least expensive ways to improve on Hartree-Fock and it was thus often the first correlation method to be applied to new problems. It can successfully model a wide variety of systems, and MP2 geometries are usually quite accurate. Thus, MP2 remains a very useful tool in a computational chemist s toolbox. We ll see several examples of its utility in the exercises. 

CBS models typically include a Hartree-Fock calculation with a very large basis set, an MP2 calculation with a medium-sized basis set (and this is also the level where the CBS extrapolation is performed), and one or more higher-level calculations with a medium-to-modest basis set. The following table outlines the components of the CBS-4 and CBS-Q model chemistries ... 

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