What do we talk about when we talk about Empty Orbitals?

My biggest pet peeve regarding the perception of experimental chemists about theoretical chemistry relates to the physical interpretation of orbitals. Every now and then there’s a serious claim about having atomic/molecular orbitals experimentally observed (read: Can Orbitals Be Directly Observed? in this blog for more.) But the discussion becomes even more esoteric when we talk about the existence (potential observation?) of unoccupied orbitals.

So, are orbitals there when no electrons ‘inhabit’ them? The rules of quantum mechanics yield a whole set of orbitals as solutions to the electronic structure of any hydrogen-like atom; therefore, we assign the energy of an empty orbital from the energy it would have if it was occupied. We teach they are there, available for bonding, even when their population is -effectively- zero. In fact, the HOMO-LUMO interaction is the chief interaction searched for and invoked for bonding purposes (intermolecular) as well as for photochemical (intramolecular) ones. The concept of Lewis acidity is deeply rooted in the conception of empty orbitals. The nature and role of empty orbitals is thus a profound epistemological problem.

First, virtual orbitals come, in a nutshell, from the Hartree-Fock equations as a result of the size of the basis set used. For a finite size only the N lowest energy eigenvectors (for an N electrons system) have an occupation number distinct of zero. However, virtual orbitals and unoccupied orbitals are two different concepts in quantum mechanics that can be a bit confusing for those just learning about them. In this post, we will take a closer look at the difference between these two types of orbitals and explain how they relate to the electronic structure of atoms and molecules.

Virtual orbitals refer to the possible energy states that an electron can occupy in a given atom or molecule. These orbitals are not part of the ground state electronic configuration and are said to have higher energy than the occupied orbitals. They are also called “virtual states” because they cannot be directly observed or measured. Virtual orbitals play an important role in chemical reactions and spectroscopy, and are of the utmost importance in the description of excited states of any molecule.

Unoccupied orbitals, on the other hand, are orbitals that are not currently occupied by any electrons in an atom or molecule. These orbitals are also known as “empty” or “vacant” orbitals and they have lower energy than the virtual orbitals. They can be observed, they’re important in the description of chemical bonding and chemical reactivity.

While virtual orbitals and unoccupied orbitals may seem similar, they are actually quite different. Virtual orbitals are used to describe the possible energy states that an electron can occupy and are not directly observable. Unoccupied orbitals, on the other hand, are actual empty energy states that can be observed and play an important role in chemical reactions and bonding. They both play a critical role in understanding the electronic structure of atoms and molecules; virtual orbitals describe the possible energy states that an electron can occupy, while unoccupied orbitals describe the actual empty energy states that can be observed.

The HF method approximates the many-electron wave function by a single Slater determinant built from a set of spin-orbitals. These spin-orbitals are solutions to the Hartree-Fock equations, a set of self-consistent equations that describe the mean-field potential created by the other electrons in the system. The many-electron wave function, or electron density, can be approximated by a single Slater determinant, which is a product of single-particle wave functions or spinors. The Hartree-Fock method then seeks to find the single-particle wave functions that minimize the total energy of the system, by defining a set of spin-orbitals, which are single-particle wave functions that include both spatial and spin information.

Virtual orbitals are defined as the unoccupied spin-orbitals in the Hartree-Fock treatment. They do not correspond to any electron of the system, but they’re used to construct the single Slater determinant and they are also important to provide an estimate of the electronic excited states of the system. They are used in spectroscopy and chemical reactions and have a higher energy than the occupied ones.

_{Featured Image credit: The Orbitron^TM, a gallery of orbitals on the WWW, URL: https://winter.group.shef.ac.uk/orbitron/}