Potential Energy Surfaces and Local Optimization
Introduction to Potential Energy Surface: a diatomic
The concept of a potential energy surface (PES) is critical for computational
material science research. The PES describes how the potential energy of the atoms in a
molecular system changes depending on the position of the atoms. We will explore
multiple PESs throughout the course for various chemical systems.
Let’s start by considering the simplest example: a diatomic molecule (i.e. two
atoms that want to form a bond). A potential energy surface of a diatomic is shown
below in Figure 1a. The x-axis is the bond length of the molecule, or the distance
between the two atomic nuclei. As the atoms move closer together the potential energy
goes to infinity. This makes sense since 2 atoms cannot be at the same place! As you
pull the atoms apart the potential energy asymptotically reaches the energy of two
individual atoms that are not interacting. The energy to break this bond is the difference
between the potential energy of the two individual non-interacting atoms and the
potential energy at the optimal bond length (the blue point in Fig. 1a).
a) Diatomic molecule
b) Noble gas
Figure 1. Two examples of 1D PES are shown above one of which is a diatomic molecule (a) and the other
is two noble atoms interacting(b).
Hydrogen, Oxygen, and Nitrogen are common examples of chemical elements
that naturally form diatomic molecules. Other 1D examples are two noble gas atoms
(e.g. Argon) that do not form bonds. A PES of two noble gas atoms interacting is shown
in Fig. 1b. Similar to the diatomic molecule, the potential energy goes to infinity as you
bring the atoms closer together. Unlike the diatomic, the most stable points on the PES
are when two Argon atoms are far enough apart that they are no longer interacting (e.g.
red point in Fig. 1b).
To see other 1D examples of PESs, we can examine concepts in physics. The ideas
behind a PES where we create a landscape based on the location of the nuclei, can be
applied to other problems you have likely seen in physics. For example, a springs PES
!
can be written as, 𝑉 𝑥 = ! 𝑘(𝑥 − 𝑥!"# )! where x the springs position and xopt represents
the zero force positions. Shown in figure 2 is a picture illustrating this potential where V
is potential energy, K is kinetic energy, and E represents the total energy assuming the
spring is moving from position a to –a.
Figure 2. Illustration of a spring’s PES in terms of the positions of the mass as well as its
corresponding kinetic energy (K) and total energy (E).
How to determine the PES
The general shape of the potential energy surface for a diatomic is intuitive.
However, the properties of this surface will change depending on the chemical species
involved. For example, the optimal bond lengths of oxygen (O2) and nitrogen (N2) are
different. Also, the potential energy surface for more complex chemical species may not
be so trivial. We need accurate and efficient methods for determining the PES. In
general, there are two ways of doing this:
1) The first way to find a PES is to approximately solve Schrödinger’s equation,
ĤΨ=EΨ, where Ĥ is the Hamiltonian operator used to describe the total energy
and E is the total energy. We will go over the details on Schrödinger’s equation
and the Hamiltonian operator in a few weeks. These calculations are referred to as
electronic structure calculations since they take into account all protons and
electrons in a chemical system. In general, the Schrödinger equation cannot be
solved for chemical systems with more than one electron! It is very expensive
(i.e. takes a lot of computational time) to approximately solve this equation, but
this will give you the most accurate PES. We will use electronic structure
calculations when we investigate new materials for application or when we try to
model experimental work.
2) The second way to find the PES is to use an empirical potential, which is a
mathematical function describing how the potential energy changes with the
positions of atoms. This function could be approximated from experimental
results or from more rigorous theoretical calculations like approximately solving
Schrödinger’s equation. Empirical potentials are cheap (i.e. takes less time on a
computer), but not as accurate. We will use empirical potentials when we are
investigating current methodologies or developing new methods. A simple and
widely used empirical potential is the Lennard-Jones (LJ) potential. For a
diatomic molecule, the potential energy surface the LJ potential is as follows,
(! δ $12 ! δ $6 +
VLJ (r) = 4ε *# & − # & *)" r % " r % -,
where r is the bondlength, and ε and δ are parameters that can be tuned to a
particular chemical system. We will talk about high dimensional empirical
potentials in the future.
Local Optimization on a PES
Now that we have been introduced to the PES and method we can use to generate this
landscape for chemical systems, we next will learn about methods for solving for
properties of a chemical system. First we will learn how to do geometry optimization,
i.e. finding local minima on a PES. This means we will be given a starting point on the
PES and just roll down hill to the nearest local minimum. In the picture below, I show
the optimized geometry for the L-J potential
This point is a critical point, a point on a function where the derivative equals zero. It is
a local minimum, meaning that all points close by are higher in energy. In chemistry
terms, this is the ideal bond length for a diatomic molecule.
Why do local optimization in chemistry
The reason for doing geometry optimization in chemistry is simple; it is because low
energy structures of a molecular system are the most probable structures. Specifically the
probability, p, that you will find a particular configuration of an atomic system, r, is:
𝑝(𝑟) ∝ 𝑒 !!(!)/!! !
where V(r) is the potential energy at position r, T is temperature, and kb is a constant
called the Boltzmann constant. This probability distribution is known as the Boltzmann
distribution. We will talk about the Boltzmann distribution again in a few weeks. For
now, what this distribution is telling us is the structure of a chemical system you are most
likely to find in nature is the lowest energy structure.
The ideas behind local optimization
You have probably found local minima or critical points in calculus by taking the
derivative of a function, setting it equal to zero and finding its solutions. By going
through this procedure you are solving for the critical points analytically. Finding
analytical solutions is ideal, but there is rarely a simple analytical expression to describe a
PES in which the critical points can be determined. Thus, we will use numerical methods
in this course. Numerical methods use mathematical approximations to calculate
properties you have studied in mathematics. In this case we will take an initial point on
the PES and roll downhill until a local minimum is reached.
Introduction to Steepest Descent
Suppose you were standing on top of a mountain, and you decided to throw a ball down
the mountain. Which path would the ball take? The ball would take the path which is
steepest. Why is that? Because, this is the direction of the net force due to both gravity
and the normal force (Recall: inclined plane problems in physics).
We can think of chemical systems in the same way as a ball moving down hill. In a
chemical system, we can determine the force from the potential energy surface. The
relationship for a one dimensional example is as follows,
∂V (r)
F =−
∂r
Now, how can we use this information to simulate following the path of steepest descent
on a computer? First lets go over a general procedure for optimization algorithms.
General procedure for optimization algorithms
The general procedure for numerical geometry optimization is as follows:
1. Calculate the force (or negative of the gradient) on all atoms for some
configuration of an atomic system.
2. If the force is less than threshold, you have found a critical point! STOP.
3. If not, move the atoms such that they go towards a critical points
4. Repeat.
Procedure for gradient descent (aka steepest descent)
Now the question is this: how can we use information we know about the system to move
towards a critical point? One of the simplest optimizers is gradient descent or steepest
descent. In this method, we move the atoms in the direction of the force:
rn+1 = rn + α Fn
Thus, we would move along F (the force) in step 3 of the procedure for local
optimization. If we were to roll a ball down a hill, it would take this path, the path of
steepest descent. The assumption we are making here is that the force at a point on the
PES is similar to the force nearby. How nearby will be determined by the parameter α.
The steepest descent method is known as a first-order optimization indicating that the
method only uses the first derivative. In your lab 1-part A, you will implement the
steepest descent method and test it’s performance on a 1D potential. In the future we will
learn how to implement this method for more complex chemical systems and benchmark
it with other local optimization methods. If you choose to do the extra credit assignment,
you can learn about Newton’s Method, a second order optimization method.