Orbitals

Sections

Overview
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Overview

Orbitals

• Regions of space where electrons can be found

Features of orbitals

• Occupancy
• Energy
• Shape

Pauli Exclusion Principle

• A single orbital can contain a maximum of two electrons if they have different spins (down or up)

Aufbau Principle

• Electrons occupy the lowest energy orbital first

Hund's Rule

• Degenerate orbitals (orbitals with the same energy level) are occupied first by single electrons with up spin before they are occupied by paired electrons

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

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• Here, we will learn about atomic and molecular orbitals.
  • First, we will see how individual atoms hold their electrons in orbitals surrounding the nucleus.
  • Then, we will see how these atomic orbitals overlap to share their electrons in covalent bonds.

• Start a table to list the key properties of orbitals.

• Denote that orbitals are regions of space where electrons can be found. There are three essential features of every orbital:
  • Occupancy, which refers to the number of electrons in an orbital.
  • Energy – energy increases with increasing distance from the nucleus. 1s is the lowest, followed by 2s, followed by 2p, and so on.
  • Shape, which we represent as a three-dimensional container that demarcates the limits in which we are likely to find an electron.

Let's depict each of these properties in orbital diagrams to see how they relate to atomic orbitals (the orbitals of single atoms).

First, we will address occupancy.

• Write that according to the Pauli exclusion principle, a single orbital can contain a maximum of two electrons if they have different spins.

• Show that an electron's spin can be pictured as:
  • "up," or "down"

Let's show an orbital diagram for helium, which has a ground state electron configuration [1s^2] with both electrons in the 1s orbital.

• Depict the 1s orbital as a horizontal line. This shows it has a discrete energy level.
  • Signify the first electron's "up" spin.
  • And the second's "down" spin.

So next, let's factor in the property of energy. To do so, we'll consider how to position electrons in multiple orbitals with different energies on a vertical scale.

• Write that according to the Aufbau [Off-bau] principle, electrons occupy the lowest energy orbital first
  • They will fill orbitals in a way that achieves the most stable, lowest-energy configuration.
  • Consider that they will also fill orbitals with the same energy level (degenerate orbitals) in the lowest energy manner, as well.

• Write that according to Hund's rule, degenerate orbitals (orbitals with the same energy level) are occupied first by single electrons with "up" spin before they are occupied by paired electrons.
  • Having the same spin ensures that the electrons will not pair until all the orbitals have been singly occupied.

• Now, let's apply the Aufbau principle and Hund's rule in the orbital diagram for oxygen, which has the configuration [1s^2 2s^2 2p^4].

• Indicate the direction of increasing energy.

• Draw the 1s orbital.

• Draw the 2s orbital above the 1s orbital.

• Draw the three 2p orbitals as three lines side-by-side, above the 2s orbital.

• Now, add the electrons for each orbital.
  • Show the two 1s electrons as a pair in the 1s orbital, first.
  • Show the two 2s electrons as a pair occupying the 2s orbital, second.
  • In accordance with Hund's rule, add one electron with "up" spin to each of the three p orbitals.
  • Then add the last electron with "down" spin to pair with one of the other three.

Although orbital diagrams typically only show occupancy and energy, orbital shape can also be included as a working model to visualize the electron organization.

• First, let's show the s orbital.

• Make a set of x, y, z-Cartesian coordinates.

• Draw the s orbital as a sphere centered at the origin.

Next, let's show the p orbitals. Before we draw them in their planes, let's show their shape.

• Draw a dumbbell: one half positive, the other negative.
  • Indicate that the node is the center point, which separates the two halves of the dumbbell.

• Write that the node is the point where an electron cannot be located.

• Write that 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.

• Make three more sets of coordinates.

• Along the x-axis of the first coordinate system, show the px orbital as a dumbbell with its center at the origin.

• Now, 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.

Let's turn our focus to molecular orbitals. 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 orbitals (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.

• Then, draw the 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 phases

• Then, redraw them 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.

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 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.

• To indicate the contributions of the atomic orbitals to each molecular orbital:
  • Add a dotted line from each of the 1s orbitals down to the bonding orbital.
  • Add a dotted line from each of the 1s orbitals up to the antibonding orbital.

Now that the diagram has been constructed, let's put the electrons from the hydrogen atoms in the available molecular orbital.

• Draw both electrons in the bonding molecular orbital with paired spins.
  • 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.

As another example,

• As an illustration of degenerate orbitals, show the direction of increasing energy, and let's compare:
  • Iron in its elemental form elemental iron, [4s^2 3d^6] with
  • Iron in its +3 (ferric) oxidation state, [4s^0 3d^5].

• Draw the orbital diagram for the valence electrons of elemental iron:
  • Draw the 4s orbital.
  • Draw the five 3d orbitals above the 4s orbital.

• Now that we have constructed the diagram, we can bring in the electrons.
  • Show the two 4s electrons as a pair in the 4s orbital.
  • Now, in accordance with Hund's rule, add the six 3d electrons one-by-one to the 3d orbitals:
  • First, add one electron with "up" spin to each of the five orbitals.
  • Lastly, add the sixth electron with "down" spin to pair with one of the other five.

• Next, let's draw the orbital diagram for ferric iron.
  • Indicate that the 4s orbital is empty because electrons are removed from the orbital with the highest quantum number first.
  • Show that the five 3d electrons each singly occupy a 3d orbital with "up" spin.
  • The second configuration, in which each electron is in a different orbital, is energetically stable because it allows the electrons to be as far apart as possible.