Lecture 17Hydrogenic atom

(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the

National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not

necessarily reflect the views of the sponsoring agencies.

Hydrogenic atom

We study the Schrödinger equation of the hydrogenic atom, of which exact, analytical solution exists.

We add to our repertories another special function – associated Laguerre polynomials – solutions of the radial part of the hydrogenic atom’s Schrödinger equation.

Coulomb potential

The potential energy between a nucleus with atomic number Z and an electron is

2

04

ZeV

r

Inversely proportional to distance

Proportional to nuclear charge

Attractive

Hamiltonian of hydrogenic atom

The Classical total energy in Cartesian coordinates is

Center of mass motion

Relative motion

The Schrödinger equation

6-dimensional equation!

Center of mass motion

Relative motion

Separation of variables

Center of mass motion

Relative motion

Separable into 3 + 3 dimensions

The Schrödinger equation

Two Schrödinger equations

Hydrogen’s gas-phase dynamics (3D particle in a box)

Hydrogen’s atomic structure

In spherical coordinates centered at the nucleus

Further separation of variables

The Schrödinger eq. for atomic structure:

Can we further separate variables? YES

Still 3 dimensional!

( , , ) ( ) ( , )r R r Y

Further separation of variables

Function of just r Function of just φ and θ

Particle on a sphere redux We have already encountered the angular

part – this is the particle on a sphere

Radial and angular components

For the radial degree of freedom, we have a new equation.

This is kinetic energy in the radial motion

Original Coulomb potential + a new one

Centrifugal force

This new term partly canceling the attractive Coulomb potential can be viewed as the repulsive potential due to the centrifugal force.

2 2 2 2 2

2 3 32

l dV l p r mvV F

mr dr mr mr r

The higher the angular momentum, the greater the force in the positive r direction

The radial part Simplify the equation by scaling the variables

The radial solutions

We need a new set of orthogonal polynomials:

The solution of this is

2

2 2

2 ( 1)R R R l lR E R

Associated Laguerre polynomials

Slater-type orbital

Normalization

The Slater-type orbital

Wave functions

The radial solutions

Verification

Let us verify that the (n = 1, l = 0) and (n = 2, l = 1) radial solutions indeed satisfy the radial equation

Summary

The 3-dimensional Schrödinger equation for the hydrogenic atomic structures can be solved analytically after separation of variables.

The wave function is a product of the radial part involving associated Laguerre polynomials and the angular part that is the spherical harmonics.

There are 3 quantum numbers n, l, and m. The discrete energy eigenvalues are negative

and inversely proportional to n2.