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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.
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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.
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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
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Hamiltonian of hydrogenic atom
The Classical total energy in Cartesian coordinates is
Center of mass motion
Relative motion
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The Schrödinger equation
6-dimensional equation!
Center of mass motion
Relative motion
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Separation of variables
Center of mass motion
Relative motion
Separable into 3 + 3 dimensions
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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
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Further separation of variables
The Schrödinger eq. for atomic structure:
Can we further separate variables? YES
Still 3 dimensional!
( , , ) ( ) ( , )r R r Y
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Further separation of variables
Function of just r Function of just φ and θ
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Particle on a sphere redux We have already encountered the angular
part – this is the particle on a sphere
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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
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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
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The radial part Simplify the equation by scaling the variables
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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
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The Slater-type orbital
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Wave functions
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The radial solutions
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Verification
Let us verify that the (n = 1, l = 0) and (n = 2, l = 1) radial solutions indeed satisfy the radial equation
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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.