19

1. Atomic Concepts Reading for this section: Chapter 1, All sections

1.1 Review of Quantum Mechanical Principles Schrödinger Equation: Q. What is a (classical) Hamiltonian? • H is defined to be the sum of T (kinetic energy) and V (potential energy) • H = T + V • THEREFORE, H is a function of momentum (p) and position (x) coordinates. Example: Harmonic Oscillator (1 degree of freedom)

“Introduction to Quantum Chemistry”, C. E. Dykstra, Prentice-Hall, N.J. 1994. The classical Hamiltonian for this system is:

Q. What is a wavefunction? • A wavefunction is any simple oscillating function (cos, sin, etc…) Example: A(x) is amplitude as a function of position. A0 is the maximum amplitude. x is position, and λ is wavelength.

20

NOTE: Not all mathematical wavefunctions have the appropriate properties required by quantum mechanics!

1. Single-valued 2. First derivative w.r.t. position must be continuous 3. The wavefunction must vanish at any x where the potential is infinite.

Q. What is an eigenvalue? • One feature of a wavefunction is that the wavelength may be extracted from the

function as an eigenvalue.

Quantum Mechanical Postulates 1. For every quantum mechanical system, there exists a wavefunction that contains a

full mathematical description of the system. 2. For every dynamical variable, there is an associated mathematical operation. The

eigenvalue for that operation is the result that would be obtained from measuring that dynamical variable for a particular system.

3. The wavefunction of a system must be an eigenfunction of the Hamiltonian

operator!! The eigenvalue is then the energy!! (i.e., Schrödinger equation!) 1.2 Review of atomic orbitals and their shapes • Atomic orbitals describe the electron distribution as a standing matter wave in the

potential field of a nucleus:

• Derived by solving the Schrödinger equation for the hydrogen atom (see CHEM 386).

21

• Because every Ψ matches an atomic orbital, there is no limit to the number of solutions to the Schrödinger equation!

• Each Ψ describes the wave properties of a given electron in a particular orbital. • Since 2-body problems can be solved, we can fully solve for the H atom in its ground

and all its excited states. (NOTE: 3-body problems cannot be solved explicitly!)

• Mathematical solutions for Ψ are of the general form shown above (in polar spherical

coordinates, NOT Cartesian coordinates xyz) R = radial factors; Y = angular factors

• Each and every orbital is uniquely characterized by its set of quantum numbers:

Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition.

22

• Angular functions describe the shape of the orbital

Source: Purcell + Kotz, Inorganic Chemistry, 1977

• Radial functions describe the radial electron distribution and occurrence of nodal planes

Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition.

23

• The shape of the radial functions and their nodal planes are: Source: Tarr & Miessler, Inorganic Chemistry, 2nd Edition. • Note that according to the Copenhagen Interpretation it is the square of the

wavefunction that has a physical meaning (which one?) and hence is relevant for chemistry.

24

• The resulting shapes of the orbital are:

Note that the orbitals are 3D waves! All atomic orbitals of a given atom must be orthogonal to one another. The surface is typically the 75% probability boundary.

25

• These are 2D “sections” of the 3D orbitals (imagine slicing through an onion):

26

The energies of the orbitals in many electron atoms are a function of nuclear charge:

Source: Shriver & Atkins, Inorganic Chemistry, 3rd Edition. • These energies cannot be calculated exactly as for the H-atom, as no closed form or

analytical solution exists for the Schrödinger equation for three or more particles (same is true in classical mechanics – n-body problem with n > 3)

→ must use approximations ! 1.3 Penetration and shielding

• The subtle dependence of the AO energies can be rationalized through the concepts of penetration and shielding (cf. CHEM 206) leading to an effective nuclear charge Z*.

Z* = Z - S S = shielding parameter

27

• There are two effects that result in an effective nuclear charge: 1) The direct shielding of the outer electrons from the nuclear charge by the “inner” electrons. 2) The penetration of “inner” electron density by “outer” electron density. NOTE: the effective nuclear charge felt by an electron depends on its radial density function!

e.g. 2s vs. 2p electron:

The 2s electron experiences a greater effective nuclear charge because it has a higher probablity density closer to the nucleus.

Similar comparisons can be made for np vs. nd and nd vs. nf orbitals.

The net effect of shielding and penetration is the energetic ordering

ns < np < nd < nf

• Shielding becomes less effective for larger quantum numbers n and l. - Orbital energies increase as (n+l) increases. - For two orbitals with the same (n+l), the one with the smaller n lies lower in energy.

• This results in the filling order:

n+ l = 1 2 3 4 5

Orbital 1s 2s 2p, 3s 3p, 4s 3d, 4p, 5s

• This is the AUFBAU Principle, which together with Pauli’s Exclusion Principle (unique set of quantum numbers n, l, ml, ms for every electron) and Hund’s Rule (maximum multiplicity) determines the electronic ground state configuration.

(see your CHEM 206 lecture notes, and little later in this course as well…)

Related textbook exercises 1.8-1.10, 1.12-1.18.