Quantum mechanics Hartree-Fock method

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

A measure of the extent to which any particular ab initio calculation does not deal perfectly with electron correlation is the correlation energy. In a canonical exposition [79] Lowdin defined correlation energy thus The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. This is usually taken to be the energy from a nonrelativistic but otherwise perfect quantum mechanical procedure, minus the energy calculated by the Hartree-Fock method with the same nonrelativistic Hamiltonian and a huge ( infinite ) basis set ... 

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. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

C. Density functional theory Density functional theory (DFT) is the third alternative quantum mechanics method for obtaining chemical structures and their associated energies.Unlike the other two approaches, however, DFT avoids working with the many-electron wavefunction. DFT focuses on the direct use of electron densities P(r), which are included in the fundamental mathematical formulations, the Kohn-Sham equations, which define the basis for this method. Unlike Hartree-Fock methods of ab initio theory, DFT explicitly takes electron correlation into account. This means that DFT should give results comparable to the standard ab initio correlation models, such as second order M(j)ller-Plesset (MP2) theory. 

Computational quantum mechanical methods, such as the Hartree-Fock method (Hehre et al., 1986 Szabo and Ostlund, 1989 Levine, 2000), were developed to convert the many-body Schrodinger equation into a singleelectron equation, which can then be solved tractably with modern computational power. The single-electron equation is an approach by which the state (or wavefunction) of each electron is computed within the field... 

Density functional methods is an attractive alternative to Hartree Fock method due to its computational efficiency. There are density functional methods with different correlation functions [42-48]. Recently several calculations [49-53] were done on zeolites of varying cluster models with density functional methods. Periodic density functional method similar to periodic Hartree Fock is also being used recently [54]. Thus the quantum chemical technique has been at the center stage among the other computational methods in improving our understanding of stmeture and reactivity of zeolites. We apply this powerful technique to study the mechanism of zeolite synthesis. 

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. 

The application of ab initio methods in the calculation of harmonic force fields of transition metal complexes has been hampered by the size of these systems and the need to employ costly post-Hartree-Fock methods, in which electron correlation is taken into account. Thus, the fruitful symbiosis between ab initio theory and experiment, to determine empirically scaled quantum mechanical force fields, has been virtually absent in studies of transition metal complexes. 

In many quantum-mechanical calculations, use is made of the wave functions obtained by the Dirac—Slater and the Hartree—Fock methods for the approximate solution of the Schrddinger equation for free atoms. It woiild be very interesting to determine whether these functions could be refined specifically for crystals and whether the problem could be solved using relatively simple analytic approximations to the calculated functions. In particular, the approximation by Gaussian functions demands attention. 

The van der Waals parameters have usually been obtained by fitting to crystal structure data and to heats of sublimation of molecular crystals,77,93-97,101-103 but it is also possible to obtain parameters from quantum mechanical calculations. Because van der Waals interactions include dispersion, it would be necessary to use a correlated quantum mechanical method. Unfortunately, density functional methods do not describe dispersive interactions, and post-Hartree-Fock methods, capable of capturing this physical effect, are computationally expensive. The dispersion interactions are therefore often ignored. The repulsive part of the van der Waals interaction, on the other hand, can be determined from test particle calculations - using the Hartree-Fock method. 

Finding and describing approximate solutions to the electronic Schrodinger equation has been a major preoccupation of quantum chemists since the birth of quantum mechanics. Except for the very simplest cases like H2, quantum chemists are faced with many-electron problems. Central to attempts at solving such problems, and central to this book, is the Hartree-Fock approximation. It has played an important role in elucidating modern chemistry. In addition, it usually constitutes the first step towards more accurate approximations. We are now in a position to consider some of the basic ideas which underlie this approximation. A detailed description of the Hartree-Fock method is given in Chapter 3. 

Hartree-Fock method-Quantum mechanics approach that computes the energy of a molecular system with a single determinant wavefunction a trial wavefunction is iteratively improved until self... 

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