The Basis Set Approximation

The last two sums are identical and the energy difference becomes eq. (3.47). 

For small highly symmetric systems, such as atoms and diatomic molecules, the Flartree-Fock equations may be solved by mapping the orbitals on a set of grid points, and these are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis functions may in principle be used  

ELECTRONIC STRUCTURE METHODS INDEPENDENT-PARTICLE MODELS 

The first criterion suggest the use of exponential functions located on the nuclei, since such functions are known to be exact solutions for the hydrogen atom. Unfortunately, exponential functions turn out to be computationally difficult. Gaussian functions are computationally much easier to handle, and although they are poorer at describing the electronic structure on a one-to-one basis, the computational advantages more than make up for this. For periodic systems, the infinite nature of the problem suggests the use of plane waves as basis functions, since these are the exact solutions for a free electron. We will return to the precise description of basis sets in Chapter 5, but for now simply assume that a set of Mbasis basis functions located on the nuclei has been chosen. 

Each MO (j) is expanded in terms of the basis functions conventionally called atomic orbitals (MO = LCAO, Linear Combination of Atomic Orbitab), although they are generally not solutions to the atomic HF problem. 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

The S matrix contains the overlap elements between basis functions, and the F matrix contains the Fock matrix elements. Each element contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over 

For use in Section 3.8, it can also be written in a more compact notation 

The total energy (3.32) in term of integrals over basis functions is given as 

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The smallest basis sets are called minimal basis sets. The most popular minimal basis set is the STO—3G set. This notation indicates that the basis set approximates the shape of a STO orbital by using a single contraction of three GTO orbitals. One such contraction would then be used for each orbital, which is the dehnition of a minimal basis. Minimal basis sets are used for very large molecules, qualitative results, and in certain cases quantitative results. There are STO—nG basis sets for n — 2—6. Another popular minimal basis set is the MINI set described below. 

The central tenet of CC theory is that the full-CI wave function (i.e., the exact one within the basis set approximation) can be described as... 

To this point, this is an exact formalism (apart from the basis set approximation in 0 >) and gives the same results (and costs as much) as FCI. In practice, of course, approximations must be made in order to have computationally tractable methods, just as is necessary for the ground state CC treatment. Conceptually, the most straightforward set of approximations run parallel to the approximations for the ground state ... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

The optimization procedure is canied out to find the set of coefficients of the eigenvector that minimizes the energy. These are the best coefficients for the chosen linear combination of basis functions, best in the sense that the linear combination of arbitrarily chosen basis functions with optimized coefficients best approximates the molecular orbital (eigenvector) sought. Usually, some members of the basis set of funetions bear a eloser resemblanee to the true moleeular orbital than others. If basis function a +i. 

Energy, geometry, dipole moment, and the electrostatic potential all have a clear relation to experimental values. Calculated atomic charges are a different matter. There are various ways to define atomic charges. HyperChem uses Mulliken atomic charges, which are commonly used in Molecular Orbital theory. These quantities have only an approximate relation to experiment their values are sensitive to the basis set and to the method of calculation. 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals approximation (LCAO). The combination of atomic orbitals chosen is called the basis set. 

A basis set is a mathematical representation of the molecular orbitals within a molecule. The basis set can be interpreted as restricting each electron to a particular region of space. Larger basis sets impose fewer constraints on electrons and more accurately approximate exact molecular orbitals. They require correspondingly more computational resources. Available basis sets and their characteristics are discussed in Chapter 5. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

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