In terms of this density matrix P, the electronic density p(r) of the molecule can be written as. The n x n dimensional density matrix P can be computed from the set of self-consistent coefficients of atomic orbitals in the occupied molecular orbitals. The SCF LCAO ab initio representation of the molecular electronic density p(r) is a function of the 3D position variable r, and is defined in terms of a set of h atomic orbitals (p,(r), i= 1,2., n. Proof of the variation theorem is given in textbooks on quantum mechanics.18. It tells us that the expectation value for the ground state energy E, (E ), calculated from an approximate wavefunction (P is always larger than the true energy E (Equation 1.14). How can we choose between different approximations And if our trial wavefunction has adjustable parameters (such as the coefficients of atomic orbitals in molecular orbitals see Section 4.1), how can we choose the adjustable parameters best values Here, Rayleigh s variation theorem is of great value. In practice, we will always have to be satisfied with approximate wavefunctions. Inorganic compounds use \(s\), \(p\), and \(d\) orbitals (and more rarely \(f\) orbitals) to make bonding and antibonding combinations.Coefficients of atomic orbitals Observables calculated from approximate wavefunctions as in Equation 1.13 are called expectation values, an expression used in probability theory.The \(y\) and \(z\) axes are define by the other two p orbitals or by being orthogonal to the bonding axis (same thing actually).The origin is define as the central atom.We define a bonding axis because it makes the descriptions of the types of bonds easier - \(\sigma\) bonds occur along the \(z\) axis, \(\pi\) bonds occur in the \(xz\) plane, and \(\delta\) bonds occur through the \(yz\) plane.There is no good reason for any individual s orbital to combine with a particular p orbital.We assign the bonding axis as \(z\) for the purpose of convenience.The three dimensions are usually denoted as \(x,y,z\).This is the same are we get from the valence bond theory for \(HCl\).The nonbonding orbitals are localized on the \(Cl\) atom.Note that there is 1 bond and three pairs of nonbonding electrons.Because these orbitals have the same symmetry (in the point group of the molecule), they can make the bonding and antibonding MO combinations.In contrast, the \(H\) \(1s\) and \(Cl\) \(3p_z\) orbitals both have \(\sigma\) symmetry.The atomic orbitals (also called basis functions) are labeled \(\phi\), for example \(\phi_\) in the MO energy level diagram.Molecular orbitals are also called wavefunctions (\(\psi\)).Sigma Bonding and Antibonding Combinations of an s and p Orbital When they add out of phase, we get destructive interference and a higher energy antibonding orbital.When they add in phase, we get constructive interference and lower energy bonding orbital.LCAO-MO (linear combination of atomic orbitals) - we have seen LCAO before when we built our hybrid orbitals in the previous section.We typically use atomic orbitals (AOs) as a basis for constructing MOs.Shaun Williams, PhD Constructing Molecular Orbitals from Atomic Orbitals
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