You have synthesized silver dimer (Ag₂2) clusters and want to study their catalytic activity using scan tunneling microscopy (STM). Knowing that the Lennard-Jones (L-J) potential is a good first-order approximation for modeling atomic interactions: (b) Use the L-J potential to determine the natural resonance frequency of a silver dimer. The L-J potential can be written in the form: 4(r) = 48 [(9)-(9)] or more simply B p(r)= + where (r) is the inter-atomic potential, & is the depth of the potential well, is the separation between the Ag atoms when the inter-atomic potential is zero, and r is the inter-atomic separation. In the simpler form A and B can be treated as the strength of the attractive and repulsive interaction, respectively, where A = 40 and B=4c02. The values for and σ for silver are given below. EAg=-55.276 x 10-21 J Ag = 2.644 x 10-10 m

Now consider the London interaction between the phenyl groups of two Phe residues (see Problem P14B.5). (a) Estimate the potential energy of interaction between two such rings (treated as benzene molecules) separated by 0.4 nm. For the ionization energy, use I = 5.0 eV. (b) Given that force is the negative slope of the potential, calculate the distance-dependence of the force acting between two non-bonded groups of atoms, such as the phenyl groups of Phe, in a polypeptide chain that can have a London dispersion interaction with each other. What is the separation at which the force between the phenyl groups (treated as benzene molecules) of two Phe residues is zero? Hint: Calculate the slope by considering the potential energy at r and r + δr, with δr << r, and evaluating {V(r + δr) − V(r)}/δr. At the end of the calculation, let δr become vanishingly small.

Estimate the magnitude of the transition dipole moment of a charge-transfer transition modelled as the migration of an electron from a H1s orbital on one atom to another H1s orbital on an atom a distance R away. Approximate the transition moment by −eRS where S is the overlap integral of the two orbitals. Sketch the transition moment as a function of R using the expression for S given in Table 9C.1. Why does the intensity of a charge-transfer transition fall to zero as R approaches 0 and infinity?

15. Modeling ligand binding. A ligand is bound to a protein with a spring-like square-law energy e(x), where x is the distance between the ligand and protein as shown in the figure. e(2) = }ca² Protein Ligand (a) For constant (T,V,N), write an expression for the probability distribution p(x) of the ligand separation from the protein. (b) Sketch a plot of p(x) vs x. Write an expression for the average location of the ligand, (x). (d) Write an expression for the second moment of the location of the ligand, (x2). (e) Calculate the average energy, (e) of the system.