Jahn-Teller theorem, orbitally degenerate states

Jahn-TeHer effect The Jahn-Teller theorem states that, when any degenerate electronic slate contains a number of electrons such that the degenerate orbitals are not completely filled, the geometry of the species will change so as to produce non-degenerate orbitals. Particularly applied to transition metal compounds where the state is Cu(II)... 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

Oddershede s calculations were limited by the computational facilities and more recent calculations indicate that while Manne s ordering of the occupied molecular orbitals stays unchallenged, there is little significance to assign to virtual orbitals for an anion [6]. It was pointed out [5] that ionization from a degenerate state would lead to a distortion in the system due to the Jahn-Teller theorem. 

The Jahn-Teller theorem is important in considering the electronic states of polyatomic molecules. Jahn and Teller proved in 1937 that a nonlinear polyatomic molecule cannot have an equilibrium (minimum-energy) nuclear configuration that corresponds to an orbitally degenerate electronic term. Orbital degeneracy arises from molecular symmetry (Section 1.19), and the Jahn-Teller theorem can lead to a lower symmetry than... 

The Hiickel it MOs of a square planar cyclobutadiene are well known. They are the one below two below one set shown in 81. We have a typical Jahn-Teller situation, i.e., two electrons in a degenerate orbital. (Of course, we need worry about the various states that arise from this occupation, and the Jahn-Teller theorem really applies to only one.67) The Jahn-Teller theorem says that such a situation necessitates a large interaction of vibrational and electronic motion. It states that there must be at least one normal mode of vibration that will break the degeneracy and lower the energy of the system (and, of course, lower its symmetry). It even specifies which vibrations would accomplish this. 

In spin-paired d ions, the ground state is Ti and this behaves as a = 1 ion which means that resonance is also not readily seen. Spin-orbit coupling splits this state into Ei, F, F, with F lowest. Since this is singly degenerate, the Jahn-Teller theorem is inapplicable. 

The H3 molecule is an example of a general theorem No nonlinear molecule can have an orbitally degenerate ground state. If calculations indicate that a non-linear molecule would have an orbitally degenerate ground state under a given symmetry, the molecule will in fact distort in such a way that the symmetry is broken. The general statement is referred to as the Jahn-Teller theorem after the scientists who first proved it. 

Interaction between molecular vibrations and the orbital motion of electrons allows excitation of non-totally symmetric molecular vibrations during an electronic transition (breakdown of the Borti-Oppenheimer approximation). As an extreme manifestation of such an interaction, it happens that degenerate electronic states in non-linear molecules become unstable toward distortion (Jahn-Teller theorem). In fact, distortion due to non-totally symmetric vibrations will result in a lower molecular symmetry, with the consequence of lifting the orbital degeneracy [3]. 

---