1.4: The quantum mechanical atom

It will take us several weeks to build up the tools and skills to solve the Schrödinger equation for the hydrogen atom. Before we dive in, let's review what we have already learned about these solutions in general chemistry.

Quantum numbers

The various solutions to the hydrogenic (by which we mean atoms having a single electron such as H, He⁺, Li²⁺, and other increasingly unuseful systems) Schrödinger equation are characterized by three quantum numbers: \(n\), \(\ell\), and \(m_\ell\). The results of the Stern-Gerlach experiments and theoretical work by P. A. M. Dirac indicate that electrons need a fourth quantum number: \(m_s\). These numbers obey the following relationships:

• \(n\) is the principal quantum number. For one electron systems it determines the energy of the electron.
  • \(n = 1, 2, 3, ... \infty\)
• \(\ell\) is the angular momentum or azimuthal quantum number and is related to the angular momentum or shape of the orbital.
  • \(\ell = 0 ... n-1\)
• \(m_\ell\) is the magnetic quantum number and is related to the orientation (and sometimes shape) of the orbital.
  • \(m_\ell = -\ell ... \ell \)
• \(m_s\) is the spin quantum number and indicates the orientation of the electron's spin
  • \(m_s = \pm\frac{1}{2} \)

You can explore the shapes and other properties of the hydrogenic wavefunctions.

One of the things we will explore in this course is where these numbers and functions come from.

Electron configurations

While the Bohr model can't handle more than one electron, in principle if we want to know what is going on with an atom (or molecule) with any number of electrons we just have to solve the Schrödinger equation for that situation. If it going to take us a few weeks to learn the tools and skills necessary to do this for a one electron system, you can imagine that doing this for an atom with 2, 3, or 92 electrons might be a bit complicated (let alone doing it for a whole molecule). Indeed, P. A. M. Dirac noted in 1929 that

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

However, it turns out one can get surprisingly far by taking the solutions to the hydrogenic Schrödinger equation and using them as atomic orbitals in multielectron systems without too many difficulties, at least for most of the periodic table. (Things don't get really bad until the actinides, and even there the hydrogenic orbitals yield some reasonable insights for us.) This is the electron configuration system you learned in general chemistry where the neutral atoms have electron configurations like:

H	\(1s^1\)
He	\(1s^2\)
C	\(1s^22s^22p^2\)
Cl	\(\left[\mathrm{Ne}\right]3s^23p^5\)
Ca	\(\left[\mathrm{Ar}\right]4s^2\)
Sc	\(\left[\mathrm{Ar}\right]4s^23d^1\)
Cr	\(\left[\mathrm{Ar}\right]4s^13d^5\)
Zn	\(\left[\mathrm{Ar}\right]4s^23d^{10}\)
Br	\(\left[\mathrm{Ar}\right]4s^23d^{10}4p^5\)

So the \(s\) fills before the \(p\) and so on. The \(4s\) fills before the \(3d\) and so on, in a pattern we can remember due to the shape of the periodic table. Things get weird for chromium (and copper); these more extreme deviations from the "rules" become more and more frequent as the number of electrons in the atoms increases, but even many of the most massive atoms created so far are thought to have "regular" electron configurations.

However, if we step back for a second we can see that the pattern we are relying on get electron configurations, embodied in the shape of the periodic table (such as the fact that the d-block starts with scandium rather than potassium), which in turn can be derived from the periodicity of atomic properties is inverting cause and effect. The atoms have the properties they have because of their electronic structure, not the other way around. So of we really want to fundamentally understand the properties of the elements and the shape of the periodic table we need to learn how to solve the Schrödinger equation.