The Born-Oppenheimer Approximation

The full hamiltonian for a diatomic or polyatomic molecule is:

The T stands for total. Nuclear coordinates are indicated by Q and electronic coordinates are indicated by q. The hamiltonian is written in atomic units and rotation and translation have been eliminated. The hamiltonian contains the nuclear kinetic energy operator T_n(Q) and the electronic kinetic energy operator T_n(Q). These are:

The potential energy terms involve nuclear-nuclear repulsion V_nn (Q), electron-electron repulsion V_ee(q), and the nuclear-electron attraction V_en(q,Q). Taken together we can write all potential terms as V(q,Q):

In a more compact notation, we express the full hamiltonian as:

where H_el(q,Q) is the electronic hamiltonian. Note that H_el includes V_nn(Q).

The Schrödinger equation for the total hamiltonian is:

In order to solve this equation various approximations must be made. The most central is the Born-Oppenheimer approximation. The B-O approximation is based on the difference in mass of the electron and mass of the nucleus. Protons and neutrons are 2000 times more massive than electrons. Therefore, electrons move rapidly compared to the nuclei. Thus, we picture electronic states changing as a function of the nuclear coordinate Q. To determine the energy as a function of Q we may perform a series of calculations at fixed values of Q to build up a potential energy surface. In the B-O approximation we can write the total wavefunction as:

where the index k refers to the electronic state and v refers to the vibrational state.

Definition of the vibrational wavefunction

The function c_kv(Q) is a vibrational wavefunction. It is a function of the nuclear coordinates only. In a diatomic there is only one vibrational mode. That mode consists of the stretch along the internuclear axis. In polyatomic molecules there are 3N-6 modes in non-linear molecules. Each of these vibrational modes has a wavefunction. Adding the index n to indicate the vibrational mode we have c_knv(Q) for the vibrational wavefunction.

Definition of the electronic wavefunction

The electronic wavefunction f_k(q,Q) depends parametrically on the nuclear coordinate Q. This means that the wavefunction changes as the nuclear coordinate changes. Thus, the energy and the bonding change as a function of the nuclear coordinate. We can define an electronic Schrödinger equation as:

where U(Q) is the potential energy surface for the electronic state. The equation must be solved at each value of Q in order to obtain a complete description of the electronic wavefunction.

The significance of the potential energy surface

The potential energy surface U(Q) is the energy of the electronic wavefunction at a particular nuclear coordinate. For a molecule in a bound state the potential energy surface has the general form shown below.

The potential shown is a bonding potential. The depth of the potential at the equilibrium bond distance is equal to the bond dissociation energy. The Born-Oppenheimer approximation requires that the electronic wavefunction be a slowly varying function of the nuclear coordinates. The potential energy surface U(Q) should also be a slowly varying function (true everywhere except near Q = 0). The steep increase in energy as Q approaches zero is due to internuclear repulsion.

The nuclear equation

We assume that the electronic equation commutes (approximately) with nuclear kinetic energy operator.

We use this expression in the Schrödinger equation

to obtain the nuclear Schrödinger equation.

The nuclear equation can be solved exactly in the harmonic approximation. For a nondegenerate ground state g we can approximate:

where k_g is the harmonic force constant. Note that this expression is also a term in a series expansion of U(Q). The energies E_gv and wavefunctions c_gv(Q) are those of the harmonic oscillator.

The transition moment

Consider an electronic transition from ground state g to excited state e. The Born-Oppenheimer wavefunctions are:

The transition moment is separated into electronic and nuclear parts:

The Condon approximation

We assume that electronic motion is rapid compared to nuclear motion. Thus, electronic excitation occurs rapidly compared to nuclear position shifts. Q is a constant during an electronic transition and we set Q = Q₀ to symbolize the fixed nuclear position.

The transition moment can also be written as:

The electronic and nuclear contributions to the transition moment have the names:

The transition probability is proportional to the square of the transition moment so

where |V₀|² is the electronic factor and FC is the Franck-Condon factor. The electronic factor determines the magnitude of the transition probability. When summed over all relevant vibrational states the Franck-Condon factor is a shape factor.

The Franck-Condon factor

The FC factor is:

It is determined by the overlap of ground state vibrational wavefunctions with excited state vibrational wavefunctions. Note that it is the sum over all vibrational levels (not just the individual levels v and v' that we have considered up to now). This can be seen most easily in the harmonic approximation as shown below.

A Franck-Condon transition is a vertical transition. In the figure the nuclear position is fixed at Q₀ = 0. In the case shown the only vibrational wavefunction in the ground state is á0|. The temperature is T = 0K. The transition probability to each of the excited state wavefunctions depends on the nuclear overlap. These can be calculated individually as:

á0|0ñ , á0|1ñ , á0|2ñ , á0|3ñ , etc.

The number on the left refers to the ground state quantum number and the number on the right refers to the excited state quantum number. We call D the displacement in nuclear coordinates along a vibrational coordinate. The nuclear overlap terms depend on the magnitude of the displacement D. This can be illustrated by projecting the ground state and excited state vibrational wavefunctions onto the same axis.

The á0|0ñ nuclear overlap factor is:

The á0|1ñ nuclear overlap factor is:

We will show that the FC factor is normalized. Thus, it contributes to the shape of an absorption band but not to its overall intensity. The lineshape consists of lines weighted by |á c_gv| c_ev' ñ|². Line broadening can also occur due to the finite lifetime and dephasing processes to give these individual vibronic lines a finite width.

To see that the FC factor is normalized to a value of one we can take the extreme limit where D = 0. In this case there is no displacement and the excited state vibrational wavefunctions are the same as the ground state vibrational wavefunctions. Thus, the only non-zero term in the FC factor will be |á0| 0ñ|². For D = 0 this is simply: