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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.

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 with a single electron has n=1 since it is in the first row.
  • The angular momentum quantum (l) (or azimuthal) specifies the orbital shape.  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 depend 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.  That is 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 an electron's two possible spin states (spin up or down).  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.
  • What do you notice as you go from 1s to the 2s to the 3s?
  • Checkout the p and d orbitals examining their 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 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.

 

Questions

Describe the shapes of the s and p atomic orbitals. How do these shapes influence the geometry of molecules in organic chemistry?

  • Solution: The s orbital is spherical, allowing for symmetrical electron distribution. The p orbitals are dumbbell-shaped, oriented along different axes (x, y, z), which is crucial for understanding the directional nature of covalent bonds in organic molecules. The shapes of these orbitals determine the geometry and bond angles in organic molecules.

How do atomic orbitals contribute to the formation of sigma (σ) and pi (π) bonds in organic molecules? Provide examples.

  • Solution: Sigma bonds are formed by the head-on overlap of orbitals (s-s, s-p, or p-p), allowing for free rotation around the bond axis. Pi bonds result from the side-to-side overlap of p orbitals, restricting rotation and leading to double or triple bonds. For example, the double bond in ethene consists of one sigma and one pi bond. 

Discuss the role of atomic orbitals in determining the reactivity of organic compounds. How do the electron configurations of these orbitals influence chemical reactions?

  • Solution: The reactivity of organic compounds is influenced by the availability of electrons in atomic orbitals. Electrons in higher energy orbitals (e.g., sp2) make a molecule more reactive. Additionally, electrons in atomic orbitals can participate in resonance structures, stabilizing the molecule and affecting its reactivity.