In the quantum mechanical model, an orbital is the volume that encompasses 90% of the probability of where an electron is likely to be found when modeled as a standing wave around the nucleus. These orbitals are based on solutions to the Schrödinger equation. In molecular orbital theory, we want to delocalize the electrons by modeling them as standing waves around multiple atoms in a molecule, thus creating molecular orbitals as opposed to atomic orbitals. Unfortunately, as it turns out, solving the Schrödinger equation for an electron modeled as a standing wave across a molecule can be nearly impossible, even for simple molecules. So instead of solving it directly, we'll use trial functions as a starting point. Imagine you wanted to solve this equation for x. How would you do it? You may not know how to get the exact answer using algebra, but if you had to solve it, you could use the method of successive approximations. In successive approximations, you guess what the answer is; if you're wrong, you guess again. So you might start by guessing the answer is one. If you plug one in for x, you would get 218 equals zero. This is not the right answer. Next, you might guess that x is equal to -1. If you plug this in, you would get 24 equals zero. This is closer. You could continue in this manner until you've found the right answers. In molecular orbital theory, the trial function is the guess that we'll substitute into the Schrödinger equation. To come up with a trial function, we can use a linear combination of atomic orbitals. In valence bond theory, we created hybrid atomic orbitals by combining standard atomic orbitals on a single atom. In linear combination of atomic orbitals, we'll create molecular orbitals by adding together standard atomic orbitals from multiple different atoms in the molecule. For example, if we wanted to create molecular orbitals for two hydrogen atoms, we could add together the 1s orbitals from each hydrogen atom, or we could subtract the 1s orbitals from each hydrogen atom. When adding together the orbitals, anywhere we have electron density overlapping from the two orbitals, we'll consider it to be constructive interference of the two standing waves. When we subtract orbitals, anywhere we have electron density overlapping, we'll consider it to be destructive interference of the two standing waves. As a result, we trade in our two atomic orbitals for two new molecular orbitals. Our bonding molecular orbital is lower in energy than our antibonding molecular orbital. Antibonding molecular orbitals are designated with an asterisk. Our two molecular orbitals are called the sigma 1s bonding molecular orbital and sigma 1s antibonding molecular orbital. Let's use a molecular orbital diagram to represent the hydrogen molecule in molecular orbital theory. We're starting with two hydrogen atoms, which each have one electron in the 1s subshell. We're going to combine these two atomic orbitals on different atoms to produce two molecular orbitals that span the entire molecule. When we combine them, we produce a bonding molecular orbital and an antibonding molecular orbital. The bonding molecular orbital is always lower in energy. The two electrons that had been in the 1s orbitals of the single atoms are now in the bonding molecular orbital. In the process of creating the molecular orbitals, we use up the atomic orbitals just like we did in the hybridization process of valence bond theory. These two atomic orbitals no longer exist. We can calculate the bond order for diatomic molecules in molecular orbital theory by taking the number of electrons in bonding molecular orbitals, subtracting the number of electrons in antibonding molecular orbitals, and dividing by two. A positive bond order indicates that the molecule or polyatomic ion should be stable enough to exist under certain conditions. A bond order of zero indicates it is not energetically favorable to form the molecule, so it should not exist. In general, the higher the bond order, the stronger the bond. Let's apply molecular orbital theory to He2. First, we'll start with the atomic orbitals for our two helium atoms. Each helium atom has two electrons in the 1s subshell. Next, we'll create our molecular orbitals by combining these two atomic orbitals. We create a bonding molecular orbital that's lower in energy and an antibonding molecular orbital that's higher in energy. Creating the molecular orbitals used up the atomic orbitals. To calculate the bond order, we take the number of electrons in bonding molecular orbitals, which is two, subtract the number of electrons in antibonding molecular orbitals, which is also two, and divide by two. Our bond order for He2 is zero. Since we have a bond order of zero, creating the molecule would not lower the overall energy of the system, so it should not be stable enough to exist. Indeed, we have no evidence that He2 does exist. This isn't exactly a victory, though. Given our other theories, we wouldn't expect He2 to exist. To see some of the benefits of molecular orbital theory, let's apply it to He2+. First, we'll start with the atomic orbitals for our two helium atoms. Since there is an overall charge of one plus, we'll remove one of the electrons from our helium atom on the right. Next, we'll create our molecular orbitals by combining the two atomic orbitals. We create a bonding molecular orbital that's lower in energy and an antibonding molecular orbital that's higher in energy. Creating these molecular orbitals used up the atomic orbitals. To calculate the bond order, we take the number of electrons in bonding molecular orbitals, which is two, subtract the number of electrons in antibonding molecular orbitals, which is one, and divide by two. Now our bond order is ½. Since we have a positive bond order, He2+ should be stable enough to exist. And under particular conditions, we do have evidence that He2+ can exist. This is our victory. Valence bond theory and Lewis structures would not predict that He2+ could exist. Although it is more complicated, molecular orbital theory can help us explain some things our simpler theories cannot. If we use linear combination of atomic orbitals to combine 2p orbitals on the two atoms in a diatomic molecule, we can form sigma 2p bonding and antibonding molecular orbitals. The sigma indicates that the electron density is on the line that connects the two nuclei. We create sigma 2p molecular orbitals when we add together the 2p atomic orbitals with the x orientation. We can also create pi 2p bonding and antibonding molecular orbitals when the electron density is not on the line that connects the two nuclei. We create pi 2p molecular orbitals by adding together the 2p atomic orbitals with the z orientation or with a y orientation. When filling molecular orbital diagrams with electrons, we fill molecular orbitals similarly to atomic orbitals in atomic orbital diagrams. We start by filling in the lowest energy available orbital. For degenerate orbitals, we fill one electron per orbital with parallel spins before adding a second electron to an orbital. The relative ordering of energy for molecular orbitals is determined computationally for each molecule. The relative ordering of molecular orbitals for diatomic molecules of elements in the second row is shown here. For Be2, C2, and N2, the pi 2p bonding molecular orbitals are lower in energy than the sigma 2p bonding molecular orbitals. For O2, F2, and Ne2, this order switches. Molecular orbital theory can also help us explain why liquid oxygen is paramagnetic. If we look at the molecular orbital diagram for oxygen, there are two unpaired electrons in the pi 2p antibonding molecular orbitals. The presence of unpaired electrons makes a substance paramagnetic. We would not have predicted the oxygen molecule has unpaired electrons from the Lewis structure or even from valence bond theory. We can also create molecular orbitals for diatomic molecules composed of two different atoms. The relative ordering of energies for the molecular orbitals needs to be determined computationally for each molecule. Molecular orbital theory can also be applied to larger molecules to delocalize electrons across the entire molecule. For example, molecular orbitals can span all three oxygen atoms in ozone or across all six carbon atoms in benzene. Although this model can be more difficult to work with computationally, requiring a higher level of mathematical modeling, it can represent molecules in ways our other models cannot.