Electron-phonon interaction in carbon clathrate hex- $\mathsf{C_{40}}$

We investigate an extended spin ladder with diagonal frustrated exchanges in a wide parameter regime. By representing the model as a sum of semidefinite positive projection operators, we prove that this model has exactly a dimer ground state. Smoothly changing parameters may lead the model cover several exactly known models. Starting from this ladder model, we proposed two two-dimensional net models with exact ground states. The quantum phase transition of the ground state, due to the change of exchange strengths along perpendicular rungs, is also discussed. Received 13 October 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: schen@thphy.uni.duesseldorf.de