 So far, we've seen that by forming a molecular orbital from a linear combination of 1s atomic orbitals and calculating the molecular energy as a function of the nuclear separation, we're able to predict the hydrogen molecule ion size, energy, and vibrational frequency to within about 25%. We haven't yet made clear how the energy is calculated, so let's look at that now. Schrodinger's equation for the hydrogen molecule ion contains three terms corresponding to kinetic energy, potential energy, and total energy. There will be solutions corresponding to various energy levels. Let's call the lowest energy E0. This is the ground state, the unexcited normal state of the molecule. The corresponding wave function is psi0. There are also excited states with energy E1, E2, etc., and corresponding wave functions. In video 9, we introduced Dirac notation. The quantum state of the system is abstractly represented by a so-called cat with a symbol between a vertical line and an angled bracket. We represent the kinetic and potential energy operations in the Schrodinger equation as an operator, H hat, the so-called Hamiltonian operator. The equation says that the molecule's stationary states, the states of definite energy, are those for which the Hamiltonian operator applied to the wave function produces the wave function multiplied by the state's energy. We can take the projection of each of these wave functions onto itself equal to one. From the mathematical theory of so-called Hermitian operators, of which our H hat is a specific example, we're guaranteed that the stationary states are quote orthogonal, meaning that the projection of one state onto another is zero. To calculate this projection, we sum a product of the two wave functions over all space. Let's take the Schrodinger equation and project both sides onto the wave function. We can solve this for the energy En. If we apply this formula to an arbitrary wave function U, we get a formula for the expected value, or average, of the energy of that wave function. This is the energy that appears in the energy versus nuclear separation curve. Our formula for average energy is at the heart of one of the most powerful techniques in quantum chemistry, the so-called variational method. Here's how it works. We want to approximate a molecule's ground state, psi zero. But the Schrodinger equation is too difficult to solve directly. Let U be any trial wave function. Just our best guess, it doesn't have to be a solution of the Schrodinger equation. The theory of Hermitian operators tells us that no matter what U is, it can be represented as a linear superposition of the stationary states, psi zero, psi one, and so on, where C zero, C one, etc., are some constants. The projection of U onto itself is the sum of the squared magnitudes of these constants. The projection of H hat U onto U is the sum with each term multiplied by the corresponding energy. If we replace E one by E zero, we reduce this value, unless C one is zero, in which case there will be no change. If we replace E two by E zero, we reduce this value, unless C two is zero, in which case there will be no change, and so on. Now, the last expression is simply E zero times the projection of U onto itself. And this is less than or equal to the first expression. Equality holds only if all the constants except C zero are zero. Because of this, we are guaranteed that our average energy expression is greater than or equal to the ground state energy E zero, with equality if and only if our trial wave function U is proportional to the ground state wave function psi zero. In the variational method, we choose a trial wave function that contains one or more adjustable parameters. We vary those free parameters to minimize the expected value of energy. The lower that energy is, the closer our wave function should be to the true ground state. In fact, we were already using the variational method when we varied the nuclear separation to find an energy minimum. The point of the previous discussion is that we're justified in putting arbitrary parameters into the wave function and adjusting them to get minimum energy. In 1928, Finkelstein and Horowitz used this idea to improve the linear combination of one-acetomic orbitals model we've looked at so far in this video. That molecular orbital is E to the minus rA plus E to the minus rB, where rA and rB are the distances from the electron to protons A and B. Finkelstein and Horowitz multiplied these distances by a parameter z, which controls the size of the one-acetomic orbitals. For z equals one, they're the same functions we've been using. Decreasing z expands the orbital, increasing z compresses the orbital. If we examine the variation of energy with nuclear separation, r, for different values of the parameter z, we obtain different curves with different energy minimums at different values of r. The blue diamonds correspond to z equals 1.13, with minimum at r equals 2.2. The red circles correspond to z equals 1.23, with minimum at r equals 2. The green squares correspond to z equals 1.33, with minimum at r equals 1.85. As we might expect, larger z values correspond to a smaller molecule. The best z and r values are those which give the absolute minimum energy. We find as did Finkelstein and Horowitz, optimum values z equals 1.23 and r equals 2. This r is equal to the experimentally determined value. However, the energy of about minus 16 electron volts is not equal to the observed value, minus 16.4 electron volts, implying that this model can be improved. The pace of breakthroughs in the application of quantum mechanics to chemistry was rapid. In 1926, Schringer presented his equation and the exact solution of the hydrogen atom. In 1927, Born and Oppenheimer published their method for separating the electronic and nuclear components of the wave function. In that same year, Brau presented the exact solution of the Schringer equation for the hydrogen molecule ion. In agreement with the experiment, he found the protons to be separated by two Bohr radii with a total energy of minus 16.4 electron volts. The details involved some fairly sophisticated mathematics and some of the steps had to be done numerically. In the following year, Finkelstein and Horowitz published their variational model. Now, it might seem like a step backward to publish an approximate solution to a problem for which the exact solution is known. In fact, it was very important. The hydrogen molecule ion is the only molecule for which the Schringer equation can be solved exactly. On the other hand, the variational method is applicable to any molecule. Seeing how closely this approximate method can reproduce the exact result forms a very important test case. In 1929, Guillemot and Zener presented a two-parameter variational model that matched the exact solution's nuclear separation and energy values to three digits. The Guillemot and Zener trial wave function extends that of Finkelstein and Horowitz by adding terms with a parameter z' that effectively stretches each of the 1s atomic orbitals toward the location of the other proton. This model is what's referred to as polarization. If z' equals 0, we have the Finkelstein and Horowitz model in which each of the 1s atomic orbitals is spherically symmetric. A non-zero z' value causes the atomic orbitals to be stretched toward the other proton. This is physically reasonable. If a spherically symmetric hydrogen atom is approached by a bare proton, we would expect the electron to be drawn or polarized towards the other proton. Here are the energy versus r-curves for the three approximate models we've discussed. The green triangles are the basic combination of 1s orbitals we explored first. The blue squares are the Finkelstein and Horowitz model and the red circles are the Guillemot and Zener model. The last two both predict the correct value of r equals 2. The Guillemot and Zener model has an energy that's extremely close to the observed value implying that it gives the most accurate representation of the ground state wave function. The energy of a hydrogen atom and a separate proton is minus 13.6 electron volts. The difference between this and the minus 16.4 electron volt energy of the hydrogen molecule ion is the binding or dissociation energy of the chemical bond. To pull one of the protons out of a hydrogen molecule ion requires this energy, namely 2.8 electron volts. Let's compare the wave functions of the three models of the hydrogen ion molecule that we've developed in this video. The simple sum of atomic orbitals model gives us rough agreement with the observed nuclear separation and energy. This same wave function is shrunk to the observed size of the molecule by the Z parameter in the Finkelstein and Horowitz model. However, the energy remains about 0.4 electron volts greater than the observed value. Finally, the addition of a second parameter in the Guillemot and Zener model, accounting for polarization, results in close agreement of both molecular size and energy. This gives us an accurate picture of the chemical bond formed by the sharing of an electron among two nuclei. The accurate description of the hydrogen molecule ion was a great triumph for quantum mechanics. The inner secrets of the atom and the chemical bond had finally been discovered. While realizing a much difficult work lay ahead, it seemed that an accurate model of reality at the small scales was in hand. As expressed by Paul Dirac in 1929, the fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.