Wave-function based methods Hartree-Fock

This is not the place for a full overview of the wave function based post Hartree-Fock methods currently applied for the calculation of intermolecular interactions and in particular molecule/surface interactions. Table 3 contains a brief characterization of the most widely applied schemes. The two most popular methods are MP2 (second order Moller-Plesset perturbation theory), because it covers large part of electronic correlation at comparably low ex-... 

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

Over the decade 1995-2005, ab initio quantum chemistry has become an important tool in studying imidazole derivatives. Two highly productive approaches are often utilized for the calculations the wave function-based methods (e.g., Hartree-Fock theory and second-order Moller-Plesset perturbation theory (MP2)) and the density functional theory (DFT) based methods (e.g., gradient-corrected (BLYP) and hybrid (B3LYP) methods). 

We end this short review on ab initio wave function-based methods by noting all of them can be applied to a cluster model, whereas only the Hartree-Fock can be extended to account for periodic symmetry [68]. 

Approximating the Schrodinger Equation. - With the approaches we have discussed so far one attempts to solve the exact Schrodinger equation as accurately as possible. All quantities are formulated in terms of the many-particle wavefunctions and, consequently, these approaches are called wave-function-based methods. Their main disadvantages are that the equations are highly complicated and, therefore, that the solutions become more and more approximate the more complex the system of interest is. Typically, the computational efforts scale as N4 for the Hartree-Fock approximation and up to N1 for the methods that include correlation effects. 

It is apparent that the Hartree-Fock level is characterized by an enormous average deviation from experiment, but standard post-HF methods for including correlation effects such as MP2 and QCISD also err to an extent that renders their results completely useless for this kind of thermochemistry. We should not, however, be overly disturbed by these errors since the use of small basis sets such as 6-31G(d) is a definite no-no for correlated wave function based quantum chemical methods if problems like atomization energies are to be addressed. It suffices to point out the general trend that these methods systematically underestimate the atomization energies due to an incomplete recovery of correlation effects, a... 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The description above may seem a little unhelpful since we know that in any interesting system the electrons interact with one another. The many different wave-function-based approaches to solving the Schrodinger equation differ in how these interactions are approximated. To understand the types of approximations that can be used, it is worth looking at the simplest approach, the Hartree-Fock method, in some detail. There are also many similarities between Hartree-Fock calculations and the DFT calculations we have described in the previous sections, so understanding this method is a useful way to view these ideas from a slightly different perspective. 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

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