 Welcome to the eleventh video in this series where we'll see how quantum mechanics provides a fundamental explanation for chemistry. A white powder, baking soda, combined with a clear liquid, vinegar, produces a gas and leaves behind a clear liquid. In this kitchen classic we see the essence of chemistry in action, the reaction and transformation of substances. Long before the development of quantum mechanics it was understood that elements are composed of identical atoms, compounds are combinations of elements and definite proportions, and a chemical reaction is a rearrangement of atoms. But the fundamental question remained, what binds atoms together and drives chemical reactions? The generic answer was the somewhat vague idea of chemical affinity. Reacting compounds have a quote affinity for each other, whatever that means. Newton conjectured that particles attract one another by some force, which in immediate contact is exceedingly strong as small distances performs the chemical operations and reaches not far from the particles with any sensible effect. The father of modern chemistry, Antoine Lavocier, was of the opinion that thus the particles of all bodies may be considered as subjected to the action of two opposite powers, the one repulsive, the other attractive, between which they remain in equilibrium. During the 19th century, extensive experimentation with electrochemistry, especially with various types of batteries, strongly indicated an intimate relationship between electricity and chemical affinity. With the discovery of the electron in 1897 the process became clear. Atoms contained positive and negative charges, and electrical forces between those charges are somehow responsible for chemical affinity. For convenience in what follows, we will use so-called atomic units, which are chosen so that H bar, Planck's constant over 2 pi, E, the fundamental charge, and M sub E, the electron mass, are all equal to 1. Also, the force of repulsion between two protons, or two electrons, is simply 1 over the square of the distance between them. To get these conditions, the units of time, length, mass, charge, and energy have to be those shown here. In particular, the unit of length is the so-called Bohr radius, 5.29 times 10 to the minus 11 meters, the nominal radius of the hydrogen atom. In atomic units, the Schrodinger equation for the hydrogen atom has the following simplified form. Let's now look at electrostatic force and potential energy. Suppose two protons are separated by a distance r. They will repel each other with a force 1 over r squared. If they were originally very far apart and we wanted to bring them into this configuration, we would have to do work to push them together. If we define the energy of the system to be zero when they are infinitely far apart at rest, the work we have to do, which is the potential energy of the configuration, is 1 over r. A positive potential energy means energy is required to bring the charges from far away into the configuration. For a proton and electron separated by the same distance, the force has the same magnitude but is attractive instead of repulsive. In this case, we would have to expend work 1 over r to pull the charges away from each other and out to infinity. We say the configuration has a negative potential energy of minus 1 over r. The first molecule to be analyzed using quantum mechanics was the hydrogen molecule ion. This is denoted H2 plus and contains two protons and one electron. With two nuclei and a single electron, it's the simplest of all molecules. We can imagine two hydrogen atoms coming together to form a hydrogen molecule, followed by one of the two electrons being removed through impact, radiation, or some other process. Schematically, we have a system where the mutual repulsion of the two protons is presumably in equilibrium with the proton's attraction to the single electron. If we imagine the three particles at rest with the electron in the middle, a distance z from each proton, the potential energy of the system has a positive contribution of 1 over 2z due to the proton-proton repulsion and a negative contribution of minus 2 over z due to the two proton-electron attractions. This is combined to give a total potential of minus 3 halves over z, which decreases with decreasing z. As a static configuration will always tend towards the state of lowest potential energy, we can see there's no equilibrium arrangement for a non-zero value of z. Suppose instead that the electron's negative charge is effectively spread out over a circle of radius, say 1. This could be due to the electron rapidly moving around in Bohr orbit. The distance between a proton and the negative charge is the square root of z squared plus 1. This changes the negative potential energy term to minus 2 over square root of z squared plus 1. As z decreases, the negative term is never less than minus 2 while the positive term continues to increase. Eventually, the total potential reaches a minimum and then increases. This minimum indicates the value of z in which we have Lavassier's equilibrium between attractive and repulsive forces. We see that the effective spreading out of the electron's charge is a key requirement for the existence of an equilibrium configuration. As shown in this page from the notebook of Niels Bohr, this was at one time considered as a possible mechanism to explain chemical bonds. With the development of the Schrodinger equation, it was realized that the electron in the hydrogen atom is described by a wave function that is spread out in one of the various possible three-dimensional atomic orbitals. A particular importance to us is the ground state, the lowest energy 1s orbital in which the wave function decreases exponentially with distance from the nucleus. The ground state energy of the hydrogen atom is minus 13.6 electron volts. In the case of a molecule, we're looking for molecular orbits, a wave function spread throughout the molecule, such that the distributed charge of the resulting electron cloud satisfies the Schrodinger equation and with the two protons achieves an equilibrium configuration. Recall that the wave function psi, which appears in the Schrodinger equation, is interpreted as a probability amplitude, such that the probability that the electron will be found in some small volume dv is magnitude psi squared dv. The sum of this probability overall space has to be 100% because the electron has to be somewhere. And we write the integral of magnitude psi squared dv equals 1. Here the elongated s, or integral, sign represents the summation operation. The Schrodinger equation has three terms. The first is minus one-half times the Laplacian operator applied to the wave function. As we described in video five, the Laplacian operator is simply a measure of the difference between the wave function at a point and the average value of the wave function on a small sphere surrounding that point. The second term is the potential energy u times the wave function. This is where the specific geometry of the molecule enters the equation. The third term is the total energy e times the wave function. Schrodinger's equation must be solved for both e and the wave function psi. We label the protons a and b, the distance between them r, and the distance from the protons to the electron as r a and r b. The expression for potential energy now has three terms. The repulsion between protons contributes positive one over r. The attraction between the electron and proton a contributes minus one over r a, and between the electron and proton b minus one over r b. So substituting this expression for u, we have the specific form of the Schrodinger equation for the hydrogen molecule ion. Even for the simplest molecule, the full Schrodinger equation is analytically intractable. Central to quantum chemistry calculations is the Born-Appenheimer approximation. This was presented in 1927 in a paper titled On the Quantum Theory of Molecules. The hydrogen molecule ion has three particles, protons a and b, and the electron. Let's label their positions as r1, r2, and r3. Rigorously, the wave function should be a function of all three positions. Its squared magnitude would give the probability that proton a is at position r1, proton b is at position r2, and the electron is at position r3. This is too complicated a problem to solve directly. Born-Appenheimer presented a detailed analysis and methodology for simplifying a molecular wave function that essentially boils down to the following argument. Because they have the same magnitude of electric charge, the protons and electron feel similar magnitude forces. But the protons are more than 1,800 times as massive. So we expect that the electron will move much faster than the protons. Therefore, it's an excellent approximation to treat the protons as fixed in place and have the wave function depend only on the electron position. If the protons move, we assume that the electronic wave function immediately adapts to their new positions. This allows the full molecular wave function to be broken up into an electronic part and a nuclear part. And with this approximation, the Schrodinger equation for the hydrogen molecule ion can be solved exactly for the molecular orbitals. However, this is not true for more complex molecules. So we're going to investigate approximate solutions that can be applied to any molecule. The linear combination of atomic orbitals' molecular orbital method is a physically reasonable way to generate a rough guess for the form of a molecular orbital. Here's the idea. We might imagine the electron and proton A combining to form a hydrogen atom, leaving B as a bare proton. The electron would be in a one-ass atomic orbital centered on proton A. Alternately, the electron could form a hydrogen atom with proton B occupying a one-ass orbital centered on proton B and leaving proton A bare. When the protons are far apart, these are the two physically reasonable scenarios. A linear combination of these two atomic orbitals would represent a superposition of these two cases. If the protons are close enough that the two one-ass atomic orbitals overlap significantly, then we can imagine the electron being able to move from one proton to the other in a single molecular orbital. For reasons of symmetry, a logical choice is to take the CA and CB coefficients here to be equal. We'll set them both equal to one. Here at right, we see the protons separated by eight Bohr radii, each surrounded by a one-ass wave function. At left, we plot the corresponding energy. The total energy is minus 13.6 electron volts, the energy of a normal hydrogen atom and a free proton. As we move the protons closer, the energy decreases until we reach a minimum. After that, the proton-proton repulsion begins to dominate and the potential energy rapidly increases. At the minimum of the energy curve, the separation is about 2.5 Bohr radii and the energy is about minus 15.4 electron volts. The experimental values are 2 Bohr radii and minus 16.4 electron volts. So with this very simple model, we're able to predict the size and energy of the hydrogen molecule ion to within roughly 25% or so. Moreover, in the spirit of the Bohr and Oppenheimer approximation, we can use the energy versus proton-proton distance curve to describe vibrational motion of the two protons. We can estimate the vibrational frequency by examining the energy versus R curve near the minimum. Here we plot three points with the curve minimum in the middle. The R and E values shown are in atomic units relative to this point. The energy stored in a spring stretched or compressed a distance delta R is 1.5k delta R squared, where the spring constant k indicates how stiff the spring is. From our three points, we see that a delta R of 0.1 gives a delta E of roughly 3 times 10 to the minus 4. So k equals 2 times 3 times 10 to the minus 4 over 0.1 squared equals 0.06. The proton mass is 1836 and two objects connected by a spring oscillate at the same frequency as a single object of half the mass, or 918 atomic units. The formula for the vibration frequency of a mass on a spring is 1 over 2 pi times the square root of k over m equals 0.0013. Finally converting from atomic to standard units, the predicted vibration frequency is 5.4 times 10 to the 13th cycles per second, or Hertz. The observed value is 6.6 times 10 to the 13th Hertz. And once again, the prediction of this very simple model is within about 25% of the actual value. Let's look again at the wave function when the two protons are far apart. We've described this as a superposition of the state in which the electron forms a hydrogen atom with proton A, while proton B is bare, and the state in which the electron forms a hydrogen atom with proton B while proton A is bare. A classical model of this case would have the electron orbiting one of the protons. The attractive force of the second proton would at most perturb the electron's orbit, but it would not be enough to pull the electron away from the first proton. So the wave function in which the atomic orbitals have negligible overlap is simply telling us that the electron has 50% probability of forming a hydrogen atom with either of the two protons. The three particles did not form a molecule. When the atomic orbitals have significant overlap, however, their superposition does form a molecular orbital in which the electron is no longer localized near only one of the protons. Classical orbits in this case could have this figure 8 character, or they could encircle both protons in the same direction. But in any case, we would have a true molecule in which the sharing of the electron by the nuclei forms a chemical bond.