QM Description of Orbitals

Quantum Mechanical Description of electrons in orbitals

Electrons distribute themselves in atoms and molecules according to Quantum Mechanics.   The Schrodinger equation is the starting point for determining the distribution of electrons;

H(op)ψ = Eψ  (Schrodinger equation)

Here H(op) is the Hamiltonian operator, E the energy of the system and Ψ the wavefunction.

Hamiltonian Operator

The solution to the Schrodinger equation yields the wavefunctions (orbitals) and a set of 4 quantum numbers for each electron in the atom.  Quantum numbers describe the state of the electrons and each electron in an atom or a molecule has a unique set of these four quantum numbers.  The quantum numbers are n, l, ml and ms.

  • The principal quantum number (n) indicates the energy and size of the orbital and corresponds to rows on the periodic table.  For example, a hydrogen atom has n=1  since it is in the first row.
  • The angular momentum quantum (l) (or azimuthal) specifies the shape of the orbital.  An s orbital corresponds to l = 0, while a p orbital has l = 1 and so on.  Thus an electron in a p orbital of boron atom would have n = 2 (2nd row) and l=1.
  • Magnetic quantum number ml is related to the orientation of the orbitals in space.  Possible values for this quantum number depends on l.  There are 2l+1 possible values of ml.  For example an s orbital has l = 0 there are 2(0)+1 = 1 possible orientations in space.  Not very surprising considering an s orbital is spherically symmetric.  On the other hand, a p orbital has l=1 and therefore 2(1)+1 = 3 possible values of ml.  These correspond to the degenerate px, py, and pz orbitals and have values of -1, 0, +1.  Thus an electron in the 2px orbital of a boron atoms has n = 2, l = 1 and ml =-1
  • The spin quantum number ms can have two possible values +1/2 and -1/2.  this corresponds to the two possible spin states (spin up or down) an electron can have.  Thus an electron in the 2py orbital of carbon would have n = 2, l = 1, ml = 0, ms = +1/2. The following table summarizes the possible values of the quantum numbers for the first three rows.
n l ml Number of
orbitals		 Orbital
Name		 Number of
electrons
1 0 0 1 1s 2
2 0 0 1 2s 2
  1 -1, 0, +1 3 2p 6
3 0 0 1 3s 2
  1 -1, 0, +1 3 3p 6
  2 -2, -1, 0, +1, +2 5 3d 10
4 0 0 1 4s 2
  1 -1, 0, +1 3 4p 6
  2 -2, -1, 0, +1, +2 5 4d 10
  3 -3, -2, -1, 0, +1, +2, +3 7 4f 14

 

Action
  • Click on the 1s (or 2s or 3s) from the "Choose an orbital" and examine the shape of the s orbitals.
  • Do the same for a p orbital and then the d orbitals and again examine the shapes.

 
Take Note
  • Orbitals presented here represent a volume of space within which an electron would have a certain probability of being based on particular energy states and atoms
  • s orbitals are centered around the nucleus and are therefore the electrons in them are closer to the nucleus.
  • p orbitals have two lobes that project away from the nucleus, thus electrons in p orbitals spend more time further away from the nucleus than s electrons.