# A Quantum Picture of the Atom, cont.

## Electronic Energy Levels:

We've taken a brief look at the physics underlying atomic structure, focusing on Schrödinger's Equation and the wave picture of electron distribution in atoms.

What we need to explain is the energy distribution of electrons in atoms and how this correlates with atomic properties. First recall the line spectrum of hydrogen and the Bohr model. We are going to keep the concepts of ground state and quantized energy levels from Bohr, after all they worked very well for Hydrogen. But we will need to build a new structure which will give these same predictions but with other factors which explain the details of hydrogen's spectra as well as other atoms. We'll again start by modelling hydrogen.

• We will designate the primary energy level, corresponding to the average radial distance of the electron from the nucleus as a shell, and give it the symbol n. The lowest possible energy level is then the ground state with n = 1.
• The value of n also gives the number of nodes in each of the orbitals in that shell, with each shell having one node at infinity, where:
• A node is a region of zero probability of finding an electron.
• Nodes can have two general geometries:
• radial (or spherical, since they describe a spherical shell at a specific radial distance from the nucleus), with each atom having at least one radial (spherical) node at infinity;
• angular (either planar, e.g. as in the planar p-node and diagonal d-nodes, or cone shaped, e.g. as in the cone-shaped nodes of the dz2 orbitals which results in the donut shaped orbitals).
• Shells with n > 1 may have subshells which are different geometrical patterns of electron distribution. Thus:
• The lowest energy pattern is spherical and given the designation s.
• The next lowest energy distribution is bi-lobed with a planar symmetry. It is given the designation p.
• The third lowest energy distribution has diagonal planes of symmetry and is designated d.
• The fourth lowest energy distribution is designated f. This is the highest subshell type occupied by ground state electrons in any atom, so we will not look any further (an infinite number of subshells exist in theory for excited states, but they are not important to our understanding).
• The average energies of the different subshells are the energy of the shell, thus when subshells are present the energy of the shell is split. For example, in the n=2 shell the 2s orbital becomes lower in energy than the shell, while the 2p orbitals become higher in energy.
• The regions of electron occupancy in subshells are called orbitals.
• For each shell there is one s orbital.
• For atoms electrons will fill lowest energy orbitals first (n = 1, then 2, 3 etc.; s before p, before d etc.).
• For subshells with multiple orbitals electrons will distribute into orbitals singly before doubling up (will have same vale of ms).

# Quantum Numbers

This is an alternate way of designating the electrons in an atom. Each electron will have a unique set of quantum numbers.

 Quantum Number Symbol Characteristic specified Information provided Possible values Principle quantum number n Shell Average distance from nucleus (r) 1, 2, 3, 4, ... Angular momentum (Azimuthal) quantum number l Subshell Shape of orbital 0 (s), 1 (p), 2 (d), 3 (f), ...n - 1 Magnetic quantum number ml Orbital Orientation of orbital - l ... 0 ... +l Spin quantum number ms Electron spin Spin direction ± 1/2

#### Let's go back and look at the shapes of some orbitals:

• For each shell there is one spherical, s orbital.

For each shell with n = 2 or greater there is one s orbital and three p orbitals: px, py, and pz:

For each shell with n = 3 or greater there is one s orbital, three p orbitals and five d orbitals: dxz, dyz, dxy, dx2- y2, and dz2

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