Hartree-Fock theory quantum mechanics

The value of 3 and its dispersion can be theoretically calculated from equation 6, provided a complete set of electron states of the system is known. Such quantum mechanical calculations have been developed based on molecular Hartree-Fock theory including configuration interactions( 1 3). A detailed theoretical analysis of 3 and contributing 1T -electron states has been presented for several important molecular structures. 

Density functional theory (DFT),32 also a semi-empirical method, is capable of handling medium-sized systems of biological interest, and it is not limited to the second row of the periodic table. DFT has been used in the study of some small protein and peptide surfaces. Nevertheless, it is still limited by computer speed and memory. DFT offers a quantum mechanically based approach from a fundamentally different perspective, using electron density with an accuracy equivalent to post Hartree-Fock theory. The ideas have been around for many years,33 34 but only in the last ten years have numerous studies been published. DFT, compared to ab initio... 

A good first approach to a quantum mechanical system is often to consider one-electron functions only, associating one such function, a spin-orbital , with one electron. Most popular are the one-electron functions which minimize the energy in the sense of Hartree-Fock theory. Alternatively one can start from a post-HF wave function and consider the strongly occupied natural spin orbitals (i.e. the eigenfunctions of the one-particle density matrix with occupation numbers close to 1) as the best one-electron functions. Another possibility is to use the Kohn-Sham orbitals, although their physical meaning is not so clear. 

Koopmans Theorem applies to Hartree-Fock theory by virtue of the particular method for evaluating the quantum mechanical exchange interaction. In Density Functional Theory, a different method is employed. Hence, HF orbitals are not the same as DFT orbitals and Koopmans Theorem does not apply. This can be illustrated with reference to Slater s Xu (i.e. DFT exchange only) model [15]. 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

At present, a newcomer to solid-state chemistry might therefore believe that this science must have been a key proponent in challenging quantum mechanics (and quantum chemistry, too) for the solution of solid-state chemical problems. Strangely, this is not at all the case. Let us remind ourselves that the puzzle of chemical bonding was ingeniously clarified in 1927, not for a crystalline solid but for the hydrogen molecule. The rapidly emerging scientific discipline, quantum chemistry, also focused on the molecular parts of chemistry both because of technical and "political" reasons first, the most important quantum-chemical workhorse (Hartree-Fock theory) has been particularly resistant to adaptation to the solid state (see Section 2.11.3) and, second, we surely must be aware of the fact that the solid-state chemical commimity is limited in size such that the number of "customers" for quantum chemists is relatively small. As a sad consequence, the solid-state chemists have been left alone for some... 

As indicated before, the ab initio electronic-structure theory of solid-state materials has largely profited from density-functional theory (DFT), and the performance of DFT has turned out well even when the one of its molecular quantum-chemical competitors - Hartree-Fock theory - has been weakest, namely for metallic materials. For these, and also for covalent materials, DFT is a very reasonable choice. On the other hand, ionic compounds (with both metals and nonmetals present) are often discussed using only the ionic model, on which most of Section 1.2 was based, and the quantum-mechanical approach is not considered at all, at least in introductory textbooks. Nonetheless, let us see, as a first instructive example, how a t5q)ical ionic material can be described and understood by the ionic and the quantum-chemical (DFT and HF) approaches, and let us also analyze the strengths and weaknesses. 

Early work based on a statistical treatment of the core electrons was published by Hellmaim (1935) and Gombas (1935) for molecular and solid-state physics, respectively. Quantum-mechanical justifications in the framework of the Hartree and Hartree-Fock theories were given by Fenyes (1943) and Szepfalusy (1955, 1956), respectively. The first derivation of the pseudopotential approach within the Hartree-Fock formalism which came to general attention is due to Phillips and Kleinman (1959) and was later generalized by Hazi and Rice (1968). The accuracy and limitations of the pseudopotential approach... 

---