On the Uniqueness of Molecular Orbitals and limitations of the MO-model.

Transcription

1 On the Uniqueness of Molecular Orbitals and limitations of the MO-model. The purpose of these notes is to make clear that molecular orbitals are a particular way to represent many-electron wave functions. They are not unique and do not have an unambiguous physical meaning. The full wave function does have a physical meaning in the sense that it is the solution of the Schrödinger equation, but one the other hand a wave function is not a directly observable quantity either. Let us first clearly establish the ambiguities in the definition of molecular orbitals by considering the simplest case, the H molecule. We use the symmetry-adapted molecular orbitals σ = g σ = ( s + S + s ) ( s S s ) u in terms of the localized atomic orbitals. The simplest example to illustrate the idea is the M S = component of the triplet excited state of H, which can be written as (, ) σ α() σ ( ) α( )!! ( σ ( ) σ ( ) σ ( ) σ ( )) α ( ) α ( ) Ψ t = By expanding the MO's in terms of AO's we can equally well write (, ) s s s s s s ( S ) ( ( ) ( ) ( ) ( )) ( ) ( ) α α α() α ( ) ( S ) Ψ t = = The expansion is a little work and you should work it out for yourself. Here we have changed the picture in terms of orthonormal molecular orbitals to non-orthogonal localized orbitals that are each centered on a particular atom. The wave function is precisely the same, and there is no magic involved. In this case the wave function remains a single determinant. We will see a case below where, by changing the orbitals you create 4 instead of determinant. In general however, the contents (or normalization) of a Slater-determinant does not change, when we change from one set of orthonormal orbitals to a linear combination of these original orbitals, while maintaining orthonormality. This is precisely the strategy we follow when making hybrid oritals, or when expressing molecular wave functions in terms of localized/hybrid orbitals vs. Molecular orbitals. The key is that while the orbitals change, the wave function

2 (determinant) does not. This statement is completely general, and it explains how we can get such different pictures of molecular wave functions that are totally equivalent. In the Molecular Orbital model the wave function is an antisymmetrized product of oneelectron functions: the molecular orbitals (MO's), often represented as a Slater determinant Φ(,,..., N) = ϕ ( ) ϕ ( )... ϕ ( N) () a b z In this notation: ϕ i indicates MO with spin up, ϕ i indicates the same spatial orbital with spin down, and only the diagonal elements in the determinant... are indicated. The form of the wave function would be exact if Hamiltonian were purely a one-electron operator. Let us indicate it by H $ M, because it is a model Hamiltonian. H $ $ () $ ( )... $ ( ) $ M = f + f + f N = f() i () where i $ f ZA = r r + V$ () (3) R V $ () represents electron repulsion in some average way (e.g screening, or Hartree-Fock theory, or Hückel or Density Functional theory or...). The details can be involved. A A The molecular orbitals are taken to be the solutions of the -electron Schrödinger equation $ () ϕ () = ϕ () ε (4) f a a a It then follows that products of the MO's are exact eigenfunctions of the model Hamiltonian H$ [ ϕ ( ) ϕ ( )... ϕ ( N)] = ( ε + ε +... ε ) [ ϕ ( ) ϕ ( )... ϕ ( N)] (5) M a b z a b z a b z The same is true for Antisymmetric Slater determinants (verify for yourself) and in this way we can obey the Pauli principle, as well as the model Schrödinger equation H$ ϕ ( ) ϕ ( )... ϕ ( N) = ( ε + ε +... ε ) ϕ ( ) ϕ ( )... ϕ ( N) (6) M a b z a b z a b z

3 In LCAO Hartree Fock theory we construct MO's as linear combination of Atomic orbitals r r ϕ ( ) = χ ( ) c. (7) a p pa p The ground state is represented by a single Slater determinant, and the orbital coefficients are optimized to yield the mimimum energy (according to the variational principle). This leads to eqns. -4 where $ f, the Fock operator depends on the molecular orbitals itself. In particular the average electron-electron repulsion potential indicated as V() in Eqn. 3 depends on the electron density, which depends on the molecular orbitals. For this reason one has to iterate the orbitals such that in the end, the orbitals that define the Fock operator, are also precisely the orbitals that satisfy the one-electron Schrödinger equation. The whole scheme is called self-consistent field theory. The details are too involved for now. The most important aspect is that one uses a single determinant as the trial wave function and mimimizes the energy using the variation principle. The rest follows after doing the math appropriately. MO theory (in particular Hartree Fock) works very well for organic molecules around their equilibrium structure. It is quite accurate in its predictions of molecular structure and vibrational frequencies. It is not very satisfactory, however, in its prediction of energetics. For example bindng energies or energy differences between different isomers are not very accurate. There are also cases where MO theory completely breaks down. This means that a single determinant is not good enough to approximate the true wave function. This is true in particular for the description of bond breaking. Let us revisit the prototype chemical bond in H to illustrate the deficiencies of MO theory. The H molecule revisited. The MO description of the ground state of H is given by Φ MO = σ g( ) σ g( ) = σ g( ) σ g( ) ( α( ) β( ) β( ) α( ))/ r σ g( ) = sa + sb ( + S ) ( ( r ) ( r ))

4 Expanding the spatial wave function, and neglecting normalization, we find ( sa( ) + sb( ))( sa( ) + sb( )) = (( sa( ) sa( ) + sb( ) sb( ) + ( sa( ) sb( ) + sb( ) sa( ) Ionic Covalent Φ MO is seen to be an equally weighted sum of ionic and covalent parts! You can only see this clearly if you express things in the AO basis. Combining the spatial function with the spin part, we can write the wave function as a sum of 4 determinants: Φ MO A B = ( s () s ( ) + s () s ( ) + ( s () s ( ) + s () s ( ) Ionic B A Covalent In this case the single determinant picture is not preserved because we had only one orbital to start with: The AO's s A and s B can not be expressed in terms of the σ g orbital alone. You may be surprised that the ionic part is present in the wave function. At the equilibrium distance the MO picture (including the ionic part) is actually a fairly good representation of the exact wave function. However, if the nuclei are pulled apart, the ionic part of the wave function should carry less weight. In the asymptotic region of seperated H atoms the wave function is exactly described by just the covalent part of the wave function. The energetics of the MO wave function is very poor because half of the wave function describes the molecule as H + H. This leads to very poor representation + of the binding energy curve using MO theory. The theory can be much improved by writing HL covalent Φ = C Φ + C Φ cv The superscript denotes Heitler-London who first used this description. Alternatively, one can use an extended version of MO theory, called configuration interaction and write the trial wave function as I ionic Ψ CI = Cgσσ g g + Cuσσ u u At the equilibrium geometry C g, C = 0, while at large separation the two coefficients u have equal weight (but opposite phase). The HL and CI descriptions are completely equivalent. The difference amounts to a different choice of basis set in the variational problem, but the basis functions are just linear combinations of each other, so the same result will be obtained. At each value of the H-H distance one can solve a variational

5 problem for the coefficients, and in this way a faithful representation of the true binding curve is obtained. It is not quantitatively accurate however. This is only achieved if also the higher AO's are considered and long expansions of determinants are considered, that can then be optimized variationally. It is work for computers. From the behavior of the binding energy curve upon including configuration interaction we can deduce the following general consequences for single determinant MO theory: i. Equilibrium bond lengths are generally found to be slightly too short ii. The curvature is too steep too high vibrational frequencies. iii. Poor results far from equilibrium.