Ab-initio method

Ab initio methods are characterized by the introduction of an arbitrary basis set for expanding the molecular orbitals and then the explicit calculation of all required integrals involving this basis set. 

Ab initio calculations can be performed at the Hartree-Fock level of approximation, equivalent to a self-consistent-field (SCF) calculation, or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrodinger equation (within the clamped-nuclei Born-Oppenhe-imer approximation). 

HyperChem s ab initio calculations solve the Roothaan equations (59) on page 225 without any further approximation apart from the use of a specific finite basis set. Therefore, ab initio calculations are generally more accurate than semi-empirical calculations. They certainly involve a more fundamental approach to solving the Schrodinger equation than do semi-empirical methods. 

In an ab initio method, all the integrals over atomic orbital basis functions are computed and the Fock matrix of the SCF computation is formed (equation (61) on page 225) from the integrals. The Fock matrix divides into two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix elements 

An ab initio calculation involves the calculation of the following types of integrals  

The constant value, E, is termed the eigenvalue and this value is, in fact, the energy of the system in quantum mechanics. T is usually termed the wavejunction. The operator H Hamiltonian) in Equation (1), like the energy in classical mechanics, is the sum of kinetic and potential parts. Equation (1) is usually so complicated that no analytical solutions are possible for any but the simplest systems. However, numerical techniques, to be briefly discussed is this section, enable Equation (1) to be converted to an algebraic matrix eigenvalue equation for the energy, and such equations can be effectively handled by powerful computers today. 

The potential energy surface, i.e., the variation of the energy of a system as a function of the positions of all its constituent atoms, is fundamental to the quantitative description of chemical structures and reaction processes. The quantum mechanical evaluation of potential surfaces is based on the use of the Born-Oppenheimer approximation (e g., see Hehre et al. 1986 Lasaga and Gibbs 1990). In the Bom-Oppenheimer approximation, the positions of the nuclei in the system, R, are fixed and the wave equation is solved for the wavefunction of the electrons. The energy, E, will then be a function of the atomic positions E R), i.e., the solutions will produce a potential surface. If we know E R) accurately, we can predict the detailed atomic forces and the chemical behavior of the entire system. 

One of the common schemes used to solve the Schrodinger equation assumes that the electrons can be approximated as independent particles that interact mainly with the nuclear charges and with an average potential from the other electrons. With this approximation, the Schrodinger equation becomes a set of independent one-electron equations, and the usual separation of variable method can be used. In this case, can be written as a product of functions of only one electron coordinates, Wt x,y,z), an approximation that is called the Hartree-Fock approximation or HF. 

The one-electron functions, Wj, are called molecular orbitals. These molecular orbitals form the basis for the conceptual treatment of bonding in molecules (see Hehre et al. 1986 Lasaga and Gibbs 1990). In practice, the molecular orbitals are expanded as a sum over some set of prescribed atomic orbitals, (pp. (i.e., the usual Is, 2s, 2p, 3s, 3p, 3d,. .. functions), centered on each atom in the system  

The set of coefficients, Cpi, are obtained from optimizing the solution to the Schrodinger equation. This process leads to a matrix equation for the Cpp One problem with this scheme is that the unknown coefficients, Cpi, appear also in the definition of the average 

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The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Calculated transition structures may be very sensitive Lo the level of theory employed. Semi-empirical methods, since they are parametrized for energy miriimnm structures, may be less appropriate for transition state searching than ab initio methods are. Transition structures are norm ally characterized by weak partial" bonds, that is, being broken or formed. In these cases UHF calculations arc necessary, and sometimes even the inclusion of electron correlation effects. 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

In the most elementary models of orbital strueture, the quantities that explieitly define the potential V are not eomputed from first prineiples as they are in so-ealled ab initio methods (see Section 6). Rather, either experimental data or results of ab initio ealeulations are used to determine the parameters in terms of whieh V is expressed. The resulting empirieal or semi-empirieal methods diseussed below differ in the sophistieation used to inelude eleetron-eleetron interaetions as well as in the manner experimental data or ab initio eomputational results are used to speeify V. 

Each of these tools has advantages and limitations. Ab initio methods involve intensive computation and therefore tend to be limited, for practical reasons of computer time, to smaller atoms, molecules, radicals, and ions. Their CPU time needs usually vary with basis set size (M) as at least M correlated methods require time proportional to at least M because they involve transformation of the atomic-orbital-based two-electron integrals to the molecular orbital basis. As computers continue to advance in power and memory size, and as theoretical methods and algorithms continue to improve, ab initio techniques will be applied to larger and more complex species. When dealing with systems in which qualitatively new electronic environments and/or new bonding types arise, or excited electronic states that are unusual, ab initio methods are essential. Semi-empirical or empirical methods would be of little use on systems whose electronic properties have not been included in the data base used to construct the parameters of such models. 

In all semi-empirieal models, the quantities that explieitly define the potential V are not eomputed from first prineiples as they are in so-ealled ab initio methods. Rather, either... 

More elaborated treatments have also been applied ab initio methods by Bouscasse (130) and Bernardi et al. (131) then the all-valence-electrons methods, derived from PPP. by Gelus et ai. (132) and by Phan-Tan-Luu et al. (133) and CNDO methods by Bojesen et al. (113) and by Salmona et al. (134). 

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