Phases of Atomic or Molecular Orbitals

What does phase mean in orbitals?

I know that phases are separated by nodes. They are in some way related to wavefunctions, I can't understand, how? How can wavefunctions be negative, since they are complex numbers (as I know)?

• Phase doesn't mean much; the difference in it does. BTW, I vaguely remember hearing that the eigenfunctions for a bounded state can always be chosen so as not to be complex. Nov 10 '17 at 21:03
• This post of mine may shed some light on your question: sciencemadness.org/talk/…
– Gert
Nov 10 '17 at 21:50

Complex Numbers

As you point out, general a wavefunction if complex-valued. A complex number and be described by two real numbers and hence are of depicted on the 2D complex plane (sometimes called the argand plane).

Numbers in the complex plane can be expressed in multiple forms. The one usually taught first is as the sum of a real part and imaginary part:

\begin{align} z = x + iy\;{} & ; x,y \in \mathbb{R}\\ \Re(z) = x\; {} & ; \Im(z) = y \\ |z| = {} & \sqrt{x^2+y^2} \end{align}

Alternatively it can be described by a magnitude ($r$) and a phase ($\phi$):

\begin{align} z = r e^{i\phi}\; {} & ;r\in\mathbb{R}^{+0}, \phi\in[0:2\pi)\\ \Re(z) = r \cos(\phi) \; {} &; \Im(z) = r \sin(\phi) \\ |z|{} & = r \end{align} Molecular Orbitals

As we just showed they both have the same probability density, but when we begin to combine them, the phases dictate how much the wavefunctions constructively/destructively interfere. We can combine linear combinations of atomic orbitals to make molecular orbitals.

Consider adding together two adjacent wavefunctions with the same phase on different atoms: The adjacent regions of different phase cancel one another out. When the probability density is examined: we can see than the chance of the particle being between the two atoms is very low and it is localised around each nuclei. This would be an anti-bonding molecular orbital.

On the other hand, adding together two adjacent wavefunctions with the opposite phase on different atoms: The adjacent regions of similar phase add together. When the probability density is examined: we can see than the chance of the particle being between the two atoms is very much higher and it is localised in a bond. This would be a bonding molecular orbital.

This is why we talk about in phase and out of phase orbital when talking about bonding - the regions of in phase overlap of wavefunctions increases the probability that a particle will be found in there, and out of phase overlap decreases the probability.

The physical result is the wide variety of bonding phenomena we see in nature

• covalent bonds: local in phase overlap
• anti-bonding orbitals: local out of phase overlap
• conjugation: extended in phase overlap etc...