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Orbitals - 2. Shape
Node
  • Where an electron cannot be located
  • The more nodes an orbital has, the higher its energy
S orbital
  • Sphere
  • One phase
  • No nodes
P orbital
  • Dumbbell
  • Two phases
  • one node
Molecular orbitals (MOs)
  • Formed from linear combination of atomic orbitals (LCAO)
  • Bonding orbitals result from combining in-phase orbitals
  • Antibonding orbitals result from combining out-of-phase orbitals
  • Number of MOs in molecule = number of AOs from atoms

Orbitals - 2. Shape

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Shape
An orbital's shape can be represented as a three-dimensional container that is likely to be holding an electron.
  • Make a set of "x, y, z" Cartesian coordinates.
  • Show the S orbital as a sphere centered at the origin.
  • Along the x-axis of the first coordinate system, show the "px orbital" as a dumbbell with its center at the origin.
The center point separating the two halves of the dumbbell is known as a node, or a point where an electron cannot be located.
  • The more nodes an orbital has, the higher its energy. The p orbitals, which have one node, are higher energy than the s orbitals, which do not have any nodes. The two halves of the dumbbell are opposite phases, where the sign of the phase is either positive or negative.
Dumbbell
  • Shade the lobe of the dumbbell that is in the positive region of the x-axis to indicate that its sign is positive.
  • Draw an analogous dumbbell on the y-axis to represent the py orbital on the second coordinate system, and the pz orbital on the z-axis in the third coordinate system.
Molecular orbital
The building blocks of molecular orbitals are the atomic orbitals we have just examined. Once we combine atomic orbitals into molecular orbitals, we can determine the location and energies of the electrons in the same way as with atomic orbitals.
  • Write that molecular orbitals come from a linear combination of atomic orbital (LCAO).
The combinations can be formed one of two ways: in-phase mixing, or out-of-phase mixing. Bonding orbitals result from combining in-phase orbitals. The energies of bonding orbitals are lower than of the atomic orbitals of the separated atoms.
  • Draw two S orbitals of the same phase overlapping. This in-phase overlap represents constructive interference, so electrons are likely to be found here.
  • Illustrate the molecular orbital formed as an ellipsoid. Antibonding orbitals result from combining out-of-phase orbitals. Antibonding orbitals have energies that are higher than those of the associated atomic orbitals.
  • Draw two S orbitals of different phase overlapping. This time, the region of overlap represents destructive interference. No electrons will be found here.
  • Illustrate the molecular orbital formed as two half-spheres of different phase separated by a node.
Orbital interaction diagram
Let's use these molecular orbitals to show how the atomic orbitals from elemental hydrogen combine to make dihydrogen, H2, in an orbital interaction diagram.
  • Indicate with an upwards arrow the direction of increasing energy.
  • Draw the 1s orbital of the first hydrogen on the left. Show the orbital as both a shape—a shaded sphere—and an energy level, a horizontal line with a single electron of "up" spin.
  • Draw the 1s orbital of the second hydrogen on the right, keeping it level with the first. Since we have one atomic orbitals from each of the elemental hydrogens, they will combine to form two molecular orbitals in H2.
  • Write that:
    of molecular orbitals in a molecule = # of atomic orbitals from the atoms
  • Draw the bonding molecular orbital with its energy level below the 1s atomic orbitals.
    • Add the shape, which is a shaded ellipsoid.
    • Draw the antibonding molecular orbital at an energy level above the atomic orbitals.
    Add its shape, which is a shaded half-sphere next to an unshaded half-sphere.
    • Add a dotted line from each of the 1s orbitals down to the bonding orbital and up to the antibonding orbital.
    These lines indicate the contributions of the atomic orbitals to each molecular orbital.
    Now that the diagram has been constructed, let's put the electrons from the hydrogen atoms in the available molecular orbital.
    • Draw the first electron in the bonding molecular orbital with an up spin.
    • Draw the second electron in the same molecular orbital as the first, but with a down spin.
    We can see that this configuration corresponds to the lowest-energy state because it matches with the Aufbau principle and the Pauli exclusion principle from earlier.

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