Chapter 8 - Orbital Model of Electronic Motion in Atoms and Molecules

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•This chapter is about the Hartree-Fock (HF) procedure. The procedure is based on the variational method, in which the variational function takes the form of a single Slater determinant ψ built of orthonormal molecular spinorbitals: ψ=1N!ϕ1(1)ϕ1(2)…ϕ1(N)ϕ2(1)ϕ2(2)…ϕ2(N)……………………ϕN(1)ϕN(2)…ϕN(N). A molecular spinorbital ϕi(1) is a one-electron function of the electron coordinates x1,y1,z1,σ1. In the Restricted Hartree-Fock (RHF) method, it is the product φi(x1,y1,z1)α(σ1) or φi(x1,y1,z1)β(σ1) of a real molecular orbital φi(x1,y1,z1) and of the spin function α(σ1) or β(σ1), respectively. In the general HF method (GHF), a spinorbital is a complex function, which depends both on α(σ1) and β(σ1). The UHF method uses, instead, real orbitals, which are all different and are multiplied either by α or β(“different orbitals for different spins”).Minimization of the mean value of the Hamiltonian, ε=ψ∣Hˆψψ∣ψ, with respect to the orthonormal spinorbitals ϕi leads to equations for optimum spinorbitals (Fock equations): Fˆ(1)ϕi(1)=εiϕi(1), where the Fock operator Fˆ is Fˆ(1)=hˆ(1)+Jˆ(1)-Kˆ(1), with the Coulombic operator defined by Jˆ(1)u(1)=∑jJˆj(1)u(1) and Jˆj(1)u(1)=∫dτ21r12ϕj∗(2)ϕj(2)u(1), and the exchange operator defined by Kˆ(1)u(1)=∑jKˆj(1)u(1) and Kˆj(1)u(1)=∫dτ21r12ϕj∗(2)u(2)ϕj(1). In the RHF method for closed-shell systems, we assume double occupancy of orbitals i.e., we form two spinorbitals out of each molecular orbital (by multiplying either by α or β). The Fock equations are solved by an iterative approach (Self-Consistent Field, with an arbitrary starting point) and as a result we obtain approximations to:•the total energy,•the wave function (the optimum Slater determinant),•the canonical molecular orbitals (spinorbitals),•the orbital energies.Use of the LCAO expansion for molecular orbitals (MO) leads to the Hartree-Fock-Roothaan equations Fc=Scε. The goal is then to find the LCAO coefficients c. This is achieved by transforming the matrix equation to the form of the eigenvalue problem, and to diagonalize the corresponding Hermitian matrix. The canonical molecular orbitals obtained are linear combinations of the atomic orbitals. The lowest-energy orbitals are occupied by electrons, those of higher energy are called virtual and are left empty. Using the H2+ and H2 examples, we found that a chemical bond results from a quantum effect of an electron density flow toward the bond region. This results from a superposition of atomic orbitals due to the variational principle. In the simplest MO picture:•the excited triplet state has lower energy than the corresponding excited singlet state,•in case of orbital degeneracy, the system prefers parallel electron spins (Hund’s rule),•the ionization energy is equal to the negative of the orbital energy of the removed electron. The electron affinity is equal to the negative of the orbital energy corresponding to the virtual orbital accommodating the added electron (Koopmans theorem).The canonical MOs for closed-shell systems (the RHF method) may be transformed to orbitals localized in the chemical bonds, lone pairs, and inner shells. There are many methods of localization. Different localization methods lead to sets of localized molecular orbitals which are slightly different but their general shape is similar. The localization allows comparison of the molecular fragments of different molecules. It appears that the features of the MO localized on the AB bond relatively weakly depend on the molecule in which this bond is found. This is a strong argument and a true source of experimental tactics in chemistry, which is to tune the properties of particular atoms by changing their neighborhood in a controlled way. Localization may serve to determine hybrids.The molecular orbitals (localized as well as canonical) can be classified as to the number of nodal surfaces going through the nuclei. A σ bond orbital has no nodal surface at all, a π bond orbital has a single nodal surface, a δ bond orbital has two such surfaces.In everyday practice, chemists have to use a minimal model of molecules that enables them to compare the geometry and vibrational frequencies with experiment to the accuracy of about 0.01 Åfor bond lengths and about 10 for bond angles. This model assumes that the speed of light is infinite (non-relativistic effects only), the Born-Oppenheimer approximation is valid (i.e., the molecule has a 3D structure), the nuclei are bound by chemical bonds and vibrate (often harmonic vibrations are assumed), the molecule moves (translation) and rotates as a whole in space. In many cases we can successfully predict the 3D structure of a molecule by using a very simple tool: the Valence Shell Electron Pair Repulsion concept.