What is the energy of an atomic orbital?

When we refer to energy of an atomic orbital what is this energy ? is potential energy ? but if is potential energy then shouldn't be considered the whole potential energy of the atom ( potential energy "belong" to systems ) rather than potential energy of the orbital ? can someone makes clear what this energy is (potential or kinetic or both) and explain it in terms of a first semester chemistry student ?i should also add that many textbooks refer the energy of an atomic orbital as the energy of the electron...and this is when i am going crazy.. Thanks in advandance

• Dec 31, 2018 at 18:10

Let’s first discuss hydrogen. The “energy of an orbital” is the total energy of the hydrogen atom when the electron is in that orbital. Ignorning the rest energy of the electron and the proton (which are the same regardless the state of the atom), this has three main contributions; in order of decreasing magnitude, they are: the electrostatic potential energy (about -27.2 eV for the ground state), the kinetic energy of the electron (about +13.6 eV for the ground state), and the kinetic energy of the proton (about 2000 times smaller). There are smaller contributions like spin-spin coupling that can often be ignored.

The kinetic energy of the proton can either be neglected, or taken into account by using a “reduced mass” for the electron. The proton is so much heavier than the electron that it is almost stationary. So it is common to just think of the energy as being the energy of the electron, but it is really the energy of the whole atom.

Any atom with more than one electron is complicated, because there are forces between electrons. When we use hydrogen-like orbitals to describe such atoms, we either ignore inter-electron electrostatic energy or approximate the effect of the inner electrons as simply reducing the effective charge of the nucleus, as felt by outer electrons.

• thanks for the answer ...i always had in mind in what reference frame we calculate the stationary proton ? we use proton point of view as the reference frame or we treat the atom as the electron revolving around the nucleus in same way we do with sun-earth system ? do protons also revolve around the "barycentre" of electron-proton system ? Dec 30, 2018 at 20:51
• The proton reference frame is not an inertial frame because the proton is moving around. In the semi-classical Bohr model, the proton orbits around the center of mass (barycenter). But the way it works in the Schrodinger equation is that the wavefunction is a wavefunction depending on the separation vector between the proton and the electron. You start by including a kinetic term for the proton, but it turns out to be equivalent to using a reduced mass for the electron, just as with the Kepler problem. Dec 30, 2018 at 21:15

Okay, I get your issue. Indeed is not very simple, because it is all about quantum mechanics, which work quite differently than classical physics. But I'll try to make it "simple".

1. They say "energy" because it is "total energy". Not just kinetic or potential. It's total energy.

1. You'll hear both "energy of an orbital" and "energy of an electron". Both are right somehow. To be precise, it's the electron which has the energy, but the electron has that energy because it is in that orbital.

So the orbitals can be empty, and nothing happens. Orbitals are like "possible places" (possible states). Those "places" can be empty or not.

When an electron occupies those states, the electrno acquires that energy. If an electron iss found in the first orbital, it will have a certain amount of energy.

So we can associate that value to orbital 1. That's why we say "energy of orbitals". It actually means "energy that electrons have when they're in that state".

Then, electrons can be in other states, even a mixture of them. Weird things of QM haha. But that's pretty much it, made veery simple.

Edit:

Okay, in a simple sentence: no, the places do not have energy. Energy is a property of particles and waves. An electron has a certain energy when it "is in the state of the second orbital". Electrons (or the wavefunctions) have energy.

When you go deeper in QM, you'll see that not all states have a defined energy. That means that there are two types of states:

• Some of them have defined energy. If the electron is in that state, it will certainly have that energy. If you have an energy-meter, you'll always get that value.
• There are (so many) states with no defined energy. That means that you can measure energy and get different values everytime. These are random, but with certain probabilities. (And once you measure it, you change the system, but that's another story haha).

What I mean is that

There are some states such that, if the electron is there, the electron will always have that energy.

So, it is natural to "associate" that energy as "energy of the orbital". But it actually means that "when the electron gets there, it will acquire that energy". So the energy is in the electron, but it is ruled by where it is.

It's like wearing the uniform. You wear it, but you only wear it when you are at the correct place. (Hmm I like this analogy haha).

So orbitals have no energy. Orbitals are the places. But... they have an energy associated because the electrno will have that energy when it gets there.

Secondly. Well, kind of. We still work with the usuall classic formulas: $$\frac{1}{2}mv^2 + K\frac{Qq}{r}$$, but now they mean a different thing. First, we write KE in terms of momentum:

$$E_k=\frac{p^2}{2m}$$

And we do this ebcause in QM, these are no longer numbers, but operators. And the equation is no longer $$E_k+E_p=E$$, but the Schrödinger's equation. For an electron in Coulomb's potential, it is

$$\frac{-\hbar^2}{2m} \nabla^2\psi + K\frac{Qq}{r}\psi = E\cdot\psi$$.

As you can see, it's all about wavefunctions $$\psi$$. So making a clear image for beginners is quite hard haha.

• thanks for the answer..but i have little more questions.. these ''places" have that energy without the electron ? or we treat them as what would be the energy if the electron where there ? also for calculating total energy we still use the equation of coulomb electric potential energy and the classic formula for kinetic energy 1/2mv^2 ? i am getting confused because in chemical reactions and in hydrogen molecule formation we usually draw a potential energy diagram...QM is weird but damn it works Dec 30, 2018 at 20:46
• Sure it works haha. Well, since your questions seen to be long, I'll rather edit the question instead of expanding this comment. Dec 30, 2018 at 21:11