Electronic excitation energy transfer A Förster energy transfer demonstration experiment 4

Electronc excitation energy transfer A Förster energy transfer demonstration experiment K. Lutkouskaya, G. Calzaferri J. Phys. Chem. B 2006, 110, 5633

D D* 1 R6 A* A D* D Energy transfer Light quantum DE = hn 1. Radiationless transfer of electronic excitation energy Förster energy transfer, principle electronic excitation Light quantum DE = hn D D* D D* 1 R6 Energy transfer A* D D* A

2 Comment on the dipole-dipole interaction and on the orientation factor k VDi-Di = interaction energy between two dipoles m1=l1q and m2=l2q - + m1 m2 R l2 l1 n = unit vector in direction of R. VDi-Di = the first term in a Taylor's series expansion of the electrostatic interaction between two neutral molecules. We express VDi-Di between two dipoles in polar coordinates. We introduce the factor k to describe the angle dependence VDi-Di = VDi-Di (R,q1,q2,f12). To do this, we assume two fixed positive charges ea and eb at distance R, each of them compensated by a negative charge –ea and –eb.

Potential energy: The first 4 terms represent the mutual interaction of two dipoles. Approximate expression for this interaction by assuming: (a) R is constant (R changes only slowly with respect to the movements of the electrons) (b) the distances between ea and –ea, and also between eb and –eb, are very short with respect to the distance R between the objects 1 and 2 (R >> r1a and R >> r2b). This means, that the term indicated in blue color is of constant value (does not depend on R and also not on r12, r2a , r1b). This is the condition for a dipole-dipole interaction. Hence, the interaction energy Vdd between two dipoles can be expressed as: Expansion in a series along the Cartesian coordinates:

Neglecting higher terms, we get: This equation is equivalent to: We now express the dependence of Vdd on the coordinates (R,q1,q2,f12): z1 = l1 cos(q1) z2 = l2 cos(q2) x1 = l1 sin(q1) cos(f1) x2 = l2 sin(q2) cos(f2) y1 = l1 sin(q1) sin(f1) y2 = l2 sin(q2) sin(f2) x1 x2 + y1y 2 -2 z1 z2 = l1l2 {sin(q1) sin(q2)[cos(f1) cos(f2)+ sin(f1) sin(f2)]-2 cos(q1) cos(q2)} cos(f1-f2) cosf12=cos(f1-f2)

k12 = orientation factor. It describes the dependence of the dipole-dipole inter-action energy Vdd = Vdd (R,q1,q2,f12) on the relative orientation of the 2 dipoles. Some values for k (q1,q2,f12 = 0): It is often more convenient to set the origin of the coordinate system in the middle of the dipoles and to use the following equivalent picture. q1 q2 F12=0 k12

D*(0’) + A(0) ® D(0) + A*(2’) etc. The energy transfer rate constant kEnT for electronic excitation energy of the type: D*(0’) + A(0) ® D(2) + A*(0’) D*(0’) + A(0) ® D(1) + A*(1’) D*(0’) + A(0) ® D(0) + A*(2’) etc. kEnT can be expressed by means of Fermi’s golden rule: = measure of the density of the iner-acting initial D*…A and final D…A* states. r is related to the overlap between the emission spectrum of the donor and the absorption spectrum of the acceptor.

The spectral overlap integral is usually The dimension of S(n) is equal to that of n-1. Hence, expressing the spectral overlap integral in wave numbers, and using , we get: The spectral overlap integral is usually abbreviated with the symbol J: The above formula is correct if the dimension of [J] is chosen to be cm6mole-1 For chemists the more natural way to choose the dimension of the spectral overlap integral is: [J] = [cm3M-1], [M] =[mol L-1]. kEnT for energy transfer must then be expressed as follows: