Exact Born-Oppenheimer molecular orbitals

The factor is the same factor that occurs in hydrogen-like atomic orbitals. As in that case, we can choose either the complex functions or the real functions mx and < >my. The other factors are more complicated and we do not display the formulas representing them. These molecular orbitals are called exact Born-Oppenheimer molecular orbitals. They are not exact solutions to the complete Schrodinger equation, but they contain no approximations other than the Born-Oppenheimer approximation. 

The exact Born-Oppenheimer molecular orbitals and 2 both correspond to A. = 0. A nonzero value of A corresponds to two states because m can be positive or negative. 

We can obtain two independent linear combinations from two independent basis functions, and we seek two that are approximations to the exact Born-Oppenheimer molecular orbitals i/iog and V 2cr ... 

The exact Born-Oppenheimer orbitals for hJ are expressed in an unfamiliar coordinate system and we did not explicitly display them. It will be convenient to have some easily expressed approximate molecular orbitals that resemble the correct molecular orbitals. We define molecular orbitals that are linear combinations of atomic orbitals, abbreviated LCAOMOs. If /i, f2, /s,... are a set of basis functions, then a linear combination of these functions is written as in Eq. (16.3-34) ... 

In order to obtain a molecular orbital with the same symmetry properties as the exact Born-Oppenheimer orbital ir2a, we must choose... 

CSFs into the wavefunction expansion. Although unattainable in molecular calculations, the second limiting case, corresponding to full Cl for a complete orbital set, is called the complete Cl expansion s. The eigenvalues of the complete Cl expansion are the exact energies within the clamped-atomic-nucleus Born-Oppenheimer approximation. A correspondence may then be established with the bracketing theorem between the lowest eigenvalues of a limited CSF expansion and those of the exact complete Cl expansion. This is illustrated schematically in Fig. 2. 

The Slater determinant is the central entity in molecular orbital theory. The exact -electron wave function of a stationary molecule in the Born-Oppenheimer approximation is a 4-dimensional object that depends on the three spatial coordinates and a spin coordinate of the N electrons in the system. This object is of course too complicated for any practical application and is, in first approximation, replaced by a product of N orthonormal 4-dimensional functions that each depend on the coordinates of only one of the electrons in the system. 

The term "ab initio means "from first principles" it does not mean "exact" or "true". In ab initio molecular orbital theory, we develop a series of well-defined approximations that allow an approximate solution to the Schrodinger equation. We calculate a total wavefunc-tion and individual molecular orbitals and their respective energies, without any empirical parameters. Below, we outline the necessary approximations and some of the elements and principles of quantum mechanics that we must use in our calculations, and then provide a summary of the entire process. Along with defining an important computational protocol, this approach will allow us to develop certain concepts that will be useful in later chapters, such as spin and the Born-Oppenheimer approximation. 

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