The application of quantum mechanics to the calculation of rate constants was developed in a formalism that is called time-dependent perturbation theory. It is actually very different from static or time-independent perturbation theory. It seems appropriate to introduce both of them together in order to point out these differences. Our first presentation in this section concerns normal perturbation theory and its application. All perturbation theory is based on the idea that one applies a relatively small change to an existing solution in order to describe some aspect of reality. In the case of perturbation theory we will illustrate it using the concrete example of anharmonic corrections to the harmonic oscillator. We can also use a standard application that includes relatistic and spin-orbit corrections to the wave function. In normal perturbation theory one can speak of first order, second order, and even higher order corrections that are usually each orders of magnitude smaller than the previous one. The exception to this generalization occurs when the first order perturbation theory term is zero by symmetry or mathematical necessity. For example, the cubic correction to the harmonic potential is zero to first order since the integral used to calculate it is zero. One must second order perturbation theory in this case. Finally, we note that one difference in the application of time-dependent perturbation theory concerns the order. Although we will develop "first order" time-dependent perturbation theory, there is no common second order application. In fact, the entire notion of order is different in the time-dependent case. However, we will begin with the time-independent perturbation theory in the following presentation.
Time-dependent quantum mechanics
In order to describe phenomena such as the interaction of light with molecules we will need to develop a time-dependent version of the Schrodinger equation. In this form the solution consists of a static wave function and an evolution operator.
The Fermi Golden Rule
The application of time-dependent perturnbation theory leads to an expression for the rate constant for absorption of radiation known as the Fermi Golden Rule. In this expression the rate constant is proportional to the square of the transition dipole moment. There is also an energy matching function known as a delta function. This form of the rate constant needs to be modified for application to real molecules since the delta function is an infinitely narrow energy-matching function. The dephasing rate in the excited state requires that there be a level width that is determined by the Uncertainty Principle. We show that a Lorentzian is the appropriate form for the function to describe the linewidth. Line broadening is treated in greater detail in the next lecture. In this presentation we also make a connection with experiment. It is shown that the square of the transition dipole moment is proportional to the Einstein absorption coefficient, B12. `The integrated area under the absorption curve determined experimentally can be realted to the square of the transition dipole moment making a crucial connection between theory and experiment.
The emission rate constant
The emission rate constant requires a definition of a photon density of states. For absorption we can accept that a photon is present. We do not worry about where it came from. Hoowever, when a molecule emits radiation it starts out from an excited state and a photon must be created. To account for this we define a photon density of states and show how it is consistent with the Einstein theory of absorption and emission discussed in the first lecture.
