Quantum or Wave Mechanics

Moving Particle Exhibit the Particles of a Wave

• The de Broglie Relationship: if light exhibits properties of particles in motion, then a particle in motion should exhibit the properties of a wave.

• Wave equation: the mathematical equation that describes a wave

  • Neg value = wave trough
  • Pos value = wave crest
  • Can be zero
  • Phase: the sign of the solution
  • Phase changes: changes in sign
  • Occurs at nodes
    • Node: any point where the value of the solution of a wave equation is zero
    • Nodal plane: any plane perpendicular to the direction of propagation (actively spreading) that runs through a node

• Quantum mechanics (wave mechanics): the branch of science that studies particles and their associated waves.

• Solving the Schrödinger equation gives a set of solutions called wave functions. 

• Each wave function is associated with a unique set of quantum numbers and with a particular atomic or molecular orbital. 

• A wave function occupies three-dimensional space and is called an orbital. 

• Each orbital can contain no more than two electrons. 

Wave Function

• Wave function: a set of solutions to the Schrödinger equation that define the energy of an electron in an atom and the region of space it may occupy.
• (wave function)^2
  • Is proportional to the probability of finding an electron at any given point in space
  • Is proportional to the electron density at that point
  • A plot of electron density ((wave function)^2) in a given orbital theoretically reaches to infinity but becomes really small at long distances from the nucleus. 
  • although the value of wave function at any point can be positive or negative, the value of (wave function)^2 will always be positive in an orbital. 
  • the electron density in two regions of an orbital will be equal if those regions have the same absolute value of wave function, regardless of whether that value is negative or positive.
  • (wave function)^2 will be zero at a node
• We should not think about finding an electron as a particle at a particular location in space; rather, we should consider the electron density in various regions of space as the square of the amplitude of the orbital at that position in space 
• The wave nature is reflected by the orbital in which the electron resides. 
• When we describe orbital interactions, we are referring to interactions of waves. 
• Waves interact constructively or destructively (adding or subtracting, respectively). 
• When two waves overlap, positive phasing adds constructively with positive phasing, as does negative phasing with negative phasing. 
• Positive and negative phasing also add destructively, meaning they cancel. 
• It is convenient to add up the electron densities in all of the orbitals in a molecule and then determine which areas of a molecule have larger and smaller amounts of electron density.
  • In general, the greater electron density is on the more electronegative atoms, especially those with lone pairs. 
  • Relative electron density distribution in molecules is important because it allows us to identify sites of chemical reactivity.
  • Many reactions involve an area of relatively high electron density on one molecule reacting with an area of relatively low electron density on another molecule. 

Shapes of Atomic s and p Orbitals

• All s orbitals have the shape of a sphere, with the center of the sphere at the nucleus.
  • These orbitals are completely symmetrical along all axes
• Each 2p orbital consists of two lobes arranged in a straight line with the nucleus in the middle. 
  • The three 2p orbitals are mutually perpendicular and are designated 2px, 2py, and 2pz . 
  • The sign of the wave function of a 2p orbital is positive in one lobe, zero at the nucleus, and negative in the other lobe.
  • The plus or minus is simply the sign of the mathematical function (wave function)2p and has no relationship to energy or electron distribution.
  • The value of (wave function)^2 is always positive, so the probability of finding electron density in the (+) lobe of a 2p orbital is the same as that of finding it in the (-) lobe. 
  • Again, electron density is zero at the node.
• Besides providing a way to determine the shapes of atomic orbitals, the Schrödinger equation also provides a way to approximate the energetics of covalent bond formation.
  • These approximations have taken two forms: 
  •  valence bond (VB) theory 
  • molecular orbital (MO) theory. 
  • Both theories of chemical bonding use the methods of quantum mechanics, but each makes slightly different simplifying assumptions.
  • The VB approach provides the most easily visualized description of single bonds, while the MO method is most convenient for describing multiple bonds