Post on 24-Dec-2015
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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.