1.6 Slater Determinant
An appropriate functional form of the wavefunction for a polyelectronic system (not necessarily an atom) with N electrons that satisfies the antisymmetry principle is
However this product of spin orbitals is unacceptable because it does not satisfy the antisymmetry principle; exchanging pairs of electrons does not give the negative of the wavefunction. This formulation of the wavefunction is known as a Hartree product. The energy of a system described by a Hartree product equals the sum of the one-electron spin orbitals. A key conclusion of the Hartree product description is that the probability of finding an electron at a particular point in space is independent of the probability of finding any other electron at that point in space. In fact, it turns out that the motions of the electrons are correlated. In addition, the Hartree product assumes that specific electrons have been assigned to specific orbitals, whereas the antisymmetry principle requires that the electrons are indistinguishable. Recall that for the helium atom, an acceptable functional form for the lowest-energy state, is:
This can be written in the form of 2*2 determinant
The two spin orbitals are:
A determinant is the most convenient way to write down the permitted functional forms of a polyelectronic wavefunction that satisfies the antisymmetry principle. If we have N electrons in spin orbitals 
(where each spin orbital is the product of a spatial function and a spin function) then an acceptable form of the wavefunction is:
As before, X1(1) is used to indicate a function that depends on the space and spin coordinates of the electron labelled '1'. The factor 
ensures that the wavefunction is normalised. This functional form of the wavefunction is called a Slater determinant and is the simplest form of an orbital wave- wavefunction that satisfies the antisymmetry principle. The Slater determinant is a particularly convenient and concise way to represent the wavefunction due to the special properties of determinants. Exchanging any two rows of a determinant, a process which corresponds to exchanging two electrons, changes the sign of the determinant and therefore directly leads to the antisymmetry property. If any two rows of a determinant are identical, which would correspond to two electrons being assigned to the same spin orbital, then the determinant vanishes. This can be considered a manifestation of the Pauli principle, which states that no two electrons can have the same set of quantum numbers. The Pauli principle also leads to the notion that each spatial orbital can accommodate two electrons of opposite spins.
1.7 Basis Sets
Ab-initio theory makes use of the Born-Oppenheimer [Born and Oppenheimer 1927] approximation that the nuclei remain fixed on the time scale of electron movement, that is, that the electronic wave function is unaffected by nuclear motion. It also assumes that basis sets adequately represent molecular orbitals. Each molecular orbital (one electron function) Ψi is expressed as a linear combination of n basis functions Φm.
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(1.17) |
The coefficients cmi are called molecular orbital expansion coefficients or simply MO coefficients. Usually these basis functions are located at the center of atoms and are therefore often called atomic basis functions. The basis functions used in molecular orbital calculations are usually described through an abbreviation or acronym such as "6-31G(d)".
A basis set is essentially a finite number of atomic-like functions over which the molecular orbital is formed via Linear Combination of Atomic Orbitals (LCAO) methods, as summarized in Eq. (1.17).
The first stage of all ab-initio calculations is a single-determinant SCF (Self Consistent Field) calculation. Its quality depends on the basis set i.e LCAO used for the calculation and the computational method employed .The schrodinger equation is solved assuming the electron to move in a field of ‘fixed’ electrons and nuclei. First a set of trial solution (Ψ) is obtained which are used to calculate the Coulomb and Exchange operators. The Hartree Fock Equations are then solved giving a second set of solutionsΨ, which are used in the next iteration This approach i.e (SCF) continues till the energies for all the electrons remains unchanged. The object of all molecular orbital (MO) programs is to build a set of molecular orbitals to be occupied by the electrons assigned to the molecule. In principle this can be achieved by combining any number of different types of electron probability functions, or even by writing one extremely complex function to describe the electron density in each molecular orbital. A far more convenient way is to build up the molecular orbitals from sets of orbitals centered on the constituent atoms. The MO calculation then simply involves finding the combinations of these atomic orbitals that have the proper symmetries and that give the lowest (most negative) electronic energy. This is the linear combination of atomic orbitals (LCAO) formalism.