# 5.2. Example chemical calculations

As discussed in *Chapter 4**, Molecular Hamiltonians*, approximating the PES of nuclear motion occurs due to the use of the BO approximation. We can use a semi-empirical method of approximating the PES through experimental data and/or computer simulations.

The PES can be compared to a landscape with mountains and valleys. In practice, as chemists, we want to find the global minimum (ocean floor) not local minima (mountain meadows) of the PES, as seen in *Figure 5.1*. We use the variational method, both classical and quantum, to find the global minimum. This can be compared to a ball rolling around the landscape. If we give the ball a nudge in some direction, generally downward, the ball will wind up in the minimum. We call this gradient descent. The gradient descent can be supplied by numerically changing input values or by an analytic formula of the wave function that describes the PES.

To state that calculation of determining the PES we guess a...