Atomic Orbitals

Atomic Orbitals S

Atomic Orbitals pz

Atomic Orbitals py

Atomic Orbitals py

Atomic Orbitals for a Carbon Atom

Atomic Orbitals for a Carbon Atom Carbon has 6 electrons but only 4 valence electrons

Ĥ is the Hamiltonian operator that operates on Where do these orbitals come from? The orbitals are actually mathematical functions that satisfy Schrödinger’s equation. Ĥ Y = E Y Ĥ is the Hamiltonian operator that operates on the wave funtions Y to give the Energy E. This is a special form of what is called a wave equation. Y is a wave function, a function that fully describes the electrons of a system in a particular state.

Ĥ Y = E Y Ĥ F2s = E F2s Where do these orbitals come from? The atomic orbitals 1s, 2s, 2p etc are wave functions that describe different states of an electron of a atom with one electron. Ĥ F2s = E F2s

F2s describes where we can find the electron Ĥ F2s = E F2s F2s2 gives the probability of finding the electron. If you integrate over all space the probability is unity. F2s2 = 1 Functions that behave this way are said to be normalized.

E gives the energy of the electron Ĥ F2s = E F2s You can equate the energy E of the electron to its ionization energy. This is the energy needed for the electron to escape from the atom.

Fa Ĥ Fa = Fa E Fa = E Fa2 Fa Ĥ Fa = E How do you actually calculate the Energy of some wave function Fa ? Start with Ĥ Fa = E Fa Multiply by Fa and integrate Fa Ĥ Fa = Fa E Fa = E Fa2 This is a Coulomb integral Gives E of function F Fa Ĥ Fa = E

Fa Ĥ Fb = Eint Another important integral is called an interaction integral. It gives the energy of “interaction” of two different functions. Fa Ĥ Fb = Eint

Y = S ci Fi Now we have the tools we need to describe Molecular Orbitals. We are going to use an approximate method called the LCAO approach. Molecular orbitals, Y, are expressed as Linear Combinations of Atomic Orbitals, F. Y = S ci Fi The coefficients, ci, of each orbital must be determined

Y = S ci Fi Y = c1F1 + c2F2 Let us start with H2 Each hydrogen has a 1s orbital The MOs must be a linear combination of these 1s orbitals. We will call the two 1s orbitals F1 and F2. Y = c1F1 + c2F2

Y = c1F1 + c2F2 Y1 = c1F1 + c1F2 Y2 = c1F1 - c1F2 The H2 molecule has symmetry, this means that the electron density on one end must be the same as the electron density on the other end. This can only happen if the coefficients c1 and c2 are of equal magnitude. There are two ways to do this. Either c2 = c1 or c2 = - c1 . Y1 = c1F1 + c1F2 Y2 = c1F1 - c1F2 or

Y1 = c1F1 + c1F2 Fa Ĥ Fa = E = a = 0 Fa Ĥ Fb = Eint = b How do we determine the Energy? First we need to assume some integral values. Let us assume: Fa Ĥ Fa = E = a = 0 Coulomb integral Fa Ĥ Fb = Eint = b Interaction integral

Y1 = c1F1 + c1F2 Y1 Ĥ Y1 = E1 (c1F1 + c1F2 ) Ĥ (c1F1 + c1F2 ) = E1 ( 2 c12F1 Ĥ F1 + 2 c12F1 Ĥ F2 ) = E1 2 c12 x 0 + 2 c12 x b = 2 c12 b = E1

Y1 = c1F1 + c1F2 Y1Y1 = 1 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1 Fa2 = 1 But what is c1 ? Y1 = c1F1 + c1F2 Y1Y1 = 1 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1 define Fa2 = 1 FaFb = Sab Overlap integral

Y1 Y1 = 1 Y1 = c1F1 + c1F2 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1 But what is c1 ? Y1 Y1 = 1 Y1 = c1F1 + c1F2 (c1F1 + c1F2 ) (c1F1 + c1F2 ) = 1 C12 ( 2 F1 F1 + 2 F1 F2 ) = 1 2 + 2 S12 c1 = 1 c12 ( 2 + 2 S12 ) = 1

E1 = b Y1 = c1F1 + c1F2 E2 = - b Y2 = c1F1 - c1F2 1 c1 = 2 c12 b = E1 2 + 2 S12 c1 = 1 2 c12 b = E1 1 + S12 1 E1 = b Y1 = c1F1 + c1F2 for on your own show: 1 - S12 1 E2 = - b Y2 = c1F1 - c1F2 for

Y2 = c1F1 - c1F2 E2 = - b Y1 = c1F1 + c1F2 E1 = b 1 1 - S12 1 1 + S12 antibonding 1 - S12 1 E2 = - b Y1 = c1F1 + c1F2 bonding 1 + S12 1 E1 = b

This means that He2 would be unstable, because two He atoms, a four electron system would repel each other Antibonding goes up more than: Bonding goes down