Molecular-orbitals Pauli principle

Don t confuse the state wavefunction with a molecular orbital we might well want to build the state wavefunction, which describes all the 16 electrons, from molecular orbitals each of which describe a single electron. But the two are not the same. We would have to find some suitable one-electron wavefunctions and then combine them into a slater determinant in order to take account of the Pauli principle. 

According to the Pauli exclusion principle, each molecular orbital can accommodate up to two electrons. If two electrons are present in one orbital, they must be paired. 

To compensate for the drastic assumptions an effective Hamiltonian for the system is defined in such a way that it takes into account some of the factors ignored by the model and also factors only known experimentally. The HMO method is therefore referred to as semi-empirical. As an example, the Pauli principle is recognized by assigning no more than two electrons to a molecular orbital. 

Molecular-orbital theory treats molecule formation from the separated atoms as arising from the interaction of the separate atomic orbitals to form new orbitals (molecular orbitals) which embrace the complete framework of the molecule. The ground state of the molecule is then one in which the electrons are assigned to the orbitals of lowest energy and are subject to the Pauli exclusion principle. Excited states are obtained by promoting an electron from a filled molecular orbital to an orbital which is normally empty in the ground state. The form of the molecular orbitals depends upon our model of molecule formation, but we shall describe (and use in detail in Sec. IV) only the most common, viz., the linear combination of atomic orbitals approximation. 

The molecular orbital describes the "motion" of one electron in the electric field generated by the nuclei and some average distribution of the other electron. It is in the simplest model occupied by zero, one, or two electrons. In the case of two electrons occupying the same orbital, the Pauli principle demands that they have opposite spin. 

Exact solutions such as those given above have not yet been obtained for the usual many-electron molecules encountered by chemists. The approximate method which retains tile idea of orbitals for individual electrons is called molecular-orbital theory (M. O. theory). Its approach to the problem is similar to that used to describe atomic orbitals in the many-electron atom. Electrons are assumed to occupy the lowest energy orbitals with a maximum population of two electrons per orbital (to satisfy the Pauli exclusion principle). Furthermore, just as in the case of atoms, electron-electron repulsion is considered to cause degenerate (of equal energy) orbitals to be singly occupied before pairing occurs. 

The problems that are connected with the solution of the electronic structures of molecules are in principle the same as those which occur in the treatment of atomic structures. The single-electron orbitals for molecules are called molecular orbitals, and systems with more than one electron are built up by filling the molecular orbitals with electrons, paying proper attention to the Pauli principle. Thus, we always require that the total wave function be antisymmetric. 

We can now explain something that puzzled Lewis-—the fact that a bond normally consists of a pair of electrons. The Pauli exclusion principle allows only two electrons (with paired spins) to occupy any one molecular orbital. Hence, a single bond between two atoms consists of two paired electrons in a bonding orbital. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

In Equation 6.16, K represent certain electronic configurations, and usually have the form of so-called Slater determinants that are in accord with the Pauli principle and the antisymmetry condition (the sign of K changes if the coordinates of two electrons are interchanged see Atkins [1983], Jensen [1998], Klessinger [1982], Levine [1991], Springborg [2000]). These //-electron Slater determinants are built from orbitals as one-electron functions t, the latter of which are multi-center molecular orbitals in the case of molecular orbital schemes (as opposed to, for example, valence-bond schemes where the one-electron functions are much more localized). 

The distribution of -electrons over the tt-MOs is regulated by the Aufbau principle and the Pauli rule. The highest energy occupied MO thereof is called the HOMO (highest occupied molecular orbital). The lowest unoccupied MO thereof is called the LUMO (lowest unoccu-... 

The Hiickel molecular orbital (HMO) model of pi electrons goes back to the early days of quantum mechanics [7], and is a standard tool of the organic chemist for predicting orbital symmetries and degeneracies, chemical reactivity, and rough energetics. It represents the ultimate uncorrelated picture of electrons in that electron-electron repulsion is not explicitly included at all, not even in an average way as in the Hartree Fock self consistent field method. As a result, each electron moves independently in a fully delocalized molecular orbital, subject only to the Pauli Exclusion Principle limitation to one electron of each spin in each molecular orbital. 

In the H2 molecule the lowest energy electron configuration is obtained by placing both electrons in the aR s molecular orbital with their spins paired so as to satisfy the Pauli exclusion principle. The Slater determinant for this arrangement is... 

In a Lewis structure, the double bond of an alkene is represented by two pairs of electrons between the carbon atoms. The Pauli exclusion principle tells us that two pairs of electrons can go into the region of space between the carbon nuclei only if each pair has its own molecular orbital. Using ethylene as an example, let s consider how the electrons are distributed in the double bond. 

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