# Is it correct to talk about an empty orbital?

Professor A. J. Kirby mentions:

The properties of an orbital are those of an electron contained in it. It is normal practice, illogical though it may sound, to talk of 'vacant orbitals'.The properties of vacant orbitals are those calculated for electrons occupying them.

Since an orbital isn't defined until it is occupied by an electron, would it still be correct to say that an empty orbital (such as a LUMO) can interact with other filled orbitals?

• Orbitals are only approximate functions, efforts to describe things not known precisely. Nov 27 '17 at 15:45
• @Mithoron The orbital functions of a hydrogen or Rydberg atom are very precise descriptions, and so are, with some effort, the more complex approaches for multielectron systems, aren't they?
– Karl
Nov 27 '17 at 19:53
• @Karl There are so many levels of theory that I'd sooner flip out then remembered all of them :D , and these are always approximations, it's only a matter of specific case if they're precise enough. Nov 27 '17 at 20:02
• Simply take "vacant" or "empty" as "potentially occupied" (or overlapping with an occupied orbital, to form a new pair of un/occupied orbitals).
– Karl
Nov 27 '17 at 20:11
• @Mithoron What I meant to say is that for many practical purposes, they are. Although not many people have practical uses for hydrogen or Rydberg atoms. ;-)
– Karl
Nov 27 '17 at 20:17

"The properties of an orbital are those of an electron contained in it. It is normal practice, illogical though it may sound, to talk of 'vacant orbitals'. The properties of vacant orbitals are those calculated for electrons occupying them."

I consider this a bit of a truism, especially the last sentence. IUPAC defines as an orbital:

Wavefunction depending explicitly on the spatial coordinates of only one electron.

which as noted in this answer falls short of considering spin. So let's use that definition for spatial orbitals. (For a spin-orbital, the analogous definition would be: depending explicitly on the spatial coordinates and spin coordinate ...) At this point, let us point out that a spin-orbital is a one-electron wavefunction. It is not an observable. One can observe the density or even the spin-density by means of X-ray diffraction, indirectly by NMR, ESR etc. (Another truism: orbitals of a single-electron systems, such as the hydrogen atom, are an important, but also somewhat trivial exception.) The density can be calculated from a many-electron wavefunction (WF) and a popular way of obtaining such a WF is combining several orbitals. This involves a lot more theory and mathematics than introductory chemistry courses can show.

So one can decide that if no electron is present, there is no orbital and stand on sound mathematical ground. Then again, I can just put an electron there by means of mathematics. Depending on the "computation-chemistry-method", for instance when employing a basis set (which is by far the most common approach), one can calculate properties even for empty orbitals such as the orbital energy (not an observable), which can be used to (approximately) describe electronic excitations (an observable). Computational chemists have resolved to call the empty orbitals "virtual" to bridge the gap between the two opposing views and use them as mathematical tools to describe excited states or to improve the quality of the description of the ground state.

Since an orbital isn't defined until it is occupied by an electron, would it still be correct to say that an empty orbital (such as a LUMO) can interact with other filled orbitals?

Let's consider the formation of a bond between two hydrogen atoms in a special way: reversing a heterolytic splitting, which we will briefly compare with one reversing a homolytic splitting. In order to decide this question, one needs to split the non-existent hair dividing the following two views: a) Since there are no empty orbitals, in the heterolytic case, the proton will distort the orbitals on the anion until the bond is formed. Does the orbital on the former proton now magically appear? b) There is an empty orbital on the proton. However, at large distance, its effect on the anion can be simulated by an electric field, which should not carry orbitals (one can assume that the metal plates creating the field are far away and crank up the charge). If a situation involving an empty orbital cannot be distinguished from one where there isn't one, is it really there?

Of course, the end result of either way of looking at things is the same hydrogen molecule that we also get from combining two hydrogen atoms the standard way. I thus suggest to abandon the view of orbitals carried around by atoms (except in the computational chemist way, as will be outlined below). Rather, I suggest to think of the effective potential felt by a newly added electron - where would it go? Regardless of how the nuclei got to where they are now, where do the electrons go? "Unoccupied orbital" is then a useful shorthand for the relevant regions.

(At this point, I consider the original question answered. I will elaborate a bit on how I got here.) The last question of the previous paragraph is one way of looking at the algorithm of the Hartree-Fock procedure (the first step of wavefunction-based quantum chemistry, which is comparable to density functional theory, DFT, in this regard). We have some "basis set" (and we do not care what is looks like right now) containing candidates for orbitals, nuclei (i.e. charges and positions) and a number of electrons. The first step is to evaluate the potential/the forces acting on the candidate orbitals, which is nuclei only at this point$^1$. One then linearly combines the candidate orbitals (that's LCAO right there) to form the best choice one can make. Obeying the aufbau principle, one fills the candidate orbitals until all electrons have got an orbital. One then updates the potential (which now also considers the electrons), combines again, fills the new candidates, updates the potential again and so forth until the changes between iterations are small. The result is then evaluated in terms of energy, and possibly electron density or other observables. By doing it this way, one arrives at the familiar MO picture without assuming an electron distribution or specific orbitals filled in a certain way on certain atoms.

Let's get back to the basis set. In the previous paragraph, no assumption was made on its nature. The initial candidate orbitals could be any shape (as long as they are reasonable from a mathematical point of view, for instance, they must be twice differentiable, square-integrable etc.) and many different choices exist. Experience shows that using functions resembling the exact orbitals of hydrogen are very useful for molecular chemistry$^2$. This is because they combine results of decent accuracy with computational efficiency and chemist-friendly starting points of interpretation - such as HOMOs and LUMOs that can be related to individual atoms, flawed though as that line of reasoning may be (as I have hoped to show with this wall of text).

Footnotes:
$^1$ A bit of a white lie: this is only the special situation of the so-called core guess. But done here for illustration.

$^2$ Solid state chemistry is a different matter, where standing waves are the norm.

It depends what you mean by correct.

If we are being technical, it isn't valid to speak of orbitals at all (besides the trivial case of a one electron system, like a hydrogen atom), as they only emerge as an approximation to the real description of the atom/molecule, which is the wavefunction that satisfies the Schrodinger equation. Specifically, orbitals have meaning if we make the assumption that a many electron wavefunction can be formed as the antisymmetrized product of many one electron wavefunctions. This is useful because we can solve the one electron Schrodinger equation exactly, but have yet to develop a way to analytically solve for even a two electron system.

Now the use of orbitals is convenient, not only because it allows us to obtain approximate solutions to the Schrodinger equation, but also because they often facilitate qualitative understanding of certain molecular properties. For example, if we use atoms and their atomic orbitals as building blocks, we can use valence bond, MO theory, or even wavefunction based methods to construct qualitatively correct geometries, absorption/emission spectra, etc for molecules composed from those atoms.

In terms of your original question, it can be useful to consider the LUMO to describe the behavior of a molecule. The only caveat is that the features of the LUMO are meant to describe an electron in it, not an empty orbital. So, hypothetically you could obtain the electron affinity for a molecule by determining the energy of the LUMO (this is a version of Koopmans' Theorem). However, these results would only be valid under the assumption that orbitals that describe the molecule don't change when another electron is added, which is typically a poor assumption.

Similarly, it might be qualitatively convenient to say that the excited state of some molecule is obtained by moving an electron from the the HOMO to the LUMO, but this again ignores the likelihood that the orbitals would relax to better suit the new electronic configuration.

In general, unoccupied orbitals don't have as much meaning as their occupied counterparts. When I, and I assume others, initially learned about orbitals, I pictured them as bubbles that contained electrons, but this is not the case. A slightly better, but still incorrect conception I had was that orbitals were like preexisting orbits or paths in which an electron would travel if it encountered them. But these orbitals are descriptors of the electron itself; they don't exist on their own. The reason an unoccupied orbital is something of a misnomer is that an orbital is meant to describe the probability distribution of an electron, but since an unoccupied orbital isn't describing any of the electrons in the molecule, its unclear what meaning can be drawn from it.

• Just to be even more technically correct, is it not valid to speak of orbitals in the one electron case? Nov 27 '17 at 19:45
• @orthocresol you are correct, I should make that distinction more clear. Using orbitals is making the assumption that the many electron wavefunction can be made as an antisymmetrized product of many one-electron wavefunctions (i.e. orbitals). This assumption is exact when we only have one electron, as in this case the exact wavefunction is already a one electron wavefunction. The orbital picture would also be exact if the Hamiltonian only depended on single particle interactions. Nov 27 '17 at 19:56