The Hydrogen Atom

Model The “orbitals” we know from general chemistry are wave functions of “hydrogen-like” atoms Hydrogen-like: any atom, but it has only one electron Atomic charge Z can be anything in other words If we can understand atomic orbitals, we can use them to: Build up molecules Understand electronic spectroscopy

Hydrogen-Like Atom an electron the nucleus is at the origin electron is “confined” to the atom by a “spherically symmetric” potential r

Spherical Polar Coordinates It’s easier to study this “planetary like” models in terms of spherically polar coords. (r,  ) than Cartesian coords. (x, y, z) x y z   r  can vary form 0 to   can vary form 0 to  r can vary from 0 to ∞ x = r sin(  cos(  y = r sin(  sin(  z = r cos(  Cartesians to polar polar to Cartesians

The Schrodinger equation for the Hydrogen atom: Hydrogen Atom

The Schrodinger equation for the Hydrogen atom: Hydrogen Atom Luckily this splits up by separation of variables: For the Radial part of the Schrodinger equation: Gives R n,l (r)

Hydrogen Atom For the Angular parts of the Schrodinger equation: Gives  l,m (  ) Gives  m (  ) Quantum number rules for hydrogen atom: n = 1, 2, 3, 4, … l = 0, 1, 2, 3, …, n-1 m = -l, …, 0, …, l

Hydrogen Energies Summary Energies: Energies only depend on principle quantum number n Rydberg constant R H in J Orbital energy degeneracies: For every n, there are n-1 values of l For every value of l the are 2l+1 values of m l

Orbital Energies Orbital Energy in units of R H (J) 1s1s 0 2s2s 0 2p2p 01 3s3s 0 3p3p d3d 01 2 RHRH …

Hydrogen Orbitals Summary Wave functions:  n,l,m (r,  ) = R n,l (r) Y l,m (  are called orbitals Orthogonalized modified associated Laguerre functions Spherical harmonic functions We used a Numerov solution to get these Tells what happens “inside” the orbital There are n-l-1 radial nodes We “Monte Carlo sampled” their probability density to look at them Text book pictures of orbitals (i.e. the outer shells) There are l angular nodes

Rn,l(r)Rn,l(r) n = 1 l = 0 R 1,0 (r)r 2 |R 1,0 | 2

Rn,l(r)Rn,l(r) n = 2 l = 0 R 2,0 (r)r 2 |R 2,0 | 2

Rn,l(r)Rn,l(r) n = 2 l = 1 R 2,1 (r)r 2 |R 2,1 | 2

Rn,l(r)Rn,l(r) n = 3 l = 0 R 3,0 (r)r 2 |R 3,0 | 2

Rn,l(r)Rn,l(r) n = 3 l = 1 R 3,1 (r)r 2 |R 3,1 | 2

Rn,l(r)Rn,l(r) n = 3 l = 2 R 3,2 (r)r 2 |R 3,2 | 2

Sampled Orbital Probability Densities