Lecture 17
Hydrogenic 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
Proportional to
nuclear charge
2
V 
Attractive
Ze
4 0 r
Inversely
proportional
to distance
Hamiltonian of hydrogenic atom

The Classical total energy in Cartesian
coordinates is
1
1
Ze
2
2
E = mnx n + mex e 2
2
4pe 0 x e - x n
2
2
1
1
Ze
2
2
= MX + mx 2
2
4pe 0 x
Center of mass
Relative
motion
motion
mnx n + mex e
mn me
M = mn + me ; X =
; m=
; x = xe - xn
M
mn + me
The Schrödinger equation
Center of mass
motion
Relative
motion
2
2
2
æ
ö
Ze
2
2
ÑX Ñx ç÷ Y ( X,x ) = EY ( X,x )
2m
4pe 0 x ø
è 2M
6-dimensional equation!
Separation of variables
2
2
2
æ
ö
Ze
2
2
ÑX Ñx ç÷ F ( X ) Q ( x ) = EF ( X ) Q ( x )
2m
4pe 0 x ø
è 2M
-
2
2M
QÑ 2X F -
2
2m
FÑ 2x Q -
Ze2
4pe 0 x
FQ = EFQ
2
1 2
1 2
Ze2
ÑXF ÑxQ = E Separable into
3 + 3 dimensions
2M F
2m Q
4pe 0 x
2
Center of mass
motion
Relative
motion
The Schrödinger equation

-
Two Schrödinger equations
2
2M
2
2m
Ñ F ( X ) = ECOM F ( X )
2
X
Ñ Q(x ) 2
x
Ze2
4pe 0 x
Hydrogen’s gas-phase
dynamics (3D particle in a box)
Q ( x ) = ErelativeQ ( x )
Hydrogen’s
atomic structure
In spherical coordinates centered at the nucleus
2
2
æ
ö
Ze
2
ç - 2m Ñ - 4pe r ÷ Y ( r,q ,j ) = EY ( r,q ,j )
è
0 ø
Further separation of variables

The Schrödinger eq. for atomic structure:
2
2
æ
Ze ö
2
ç - 2m Ñ - 4pe r ÷ Y(r,q ,j ) = EY(r,q ,j )
è
0 ø
Still 3 dimensional!

Can we further separate variables? YES
 ( r ,  ,  )  R ( r )Y ( ,  )
Further separation of variables
2
æ
Ze2 ö
2
ç - 2 m Ñ - 4pe r ÷ R(r)Y (q ,j ) = ER(r)Y (q ,j )
è
0 ø
é
ù
ê
ú
2 ê
¶2 2 ¶ 1 æ 1 ¶2
1 ¶
¶ öú
Ze2
+ 2ç 2
+
sin q ÷ ú RY RY = ERY
ê 2+
2
2 m ê ¶r
r ¶r r è sin q ¶f
sin q ¶q
¶q ø ú
4pe 0r
ê
ú
êë
úû
2
2
æ ¶2 2 ¶ ö
L
Ze2
Yç 2 +
RR 2YRY = ERY
÷
2 m è ¶r
r ¶r ø
2m r
4pe r
=L
0
2
æ ¶2 2 ¶ ö
L
Ze2
+
RY=E
2
2
ç
÷
2 m R è ¶r
r ¶r ø
2 mY r
4pe 0 r
2
2
r æ ¶2 2 ¶ ö
Ze2 r
2
+
R- Er LY = 0
2
ç
÷
2 m R è ¶r
r ¶r ø
4pe 0
2 mY
2 2
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
l(l +1)
LY = EY Û LY = EY Y Û EY =
2mY
2m
2m
2
2
r æ ¶2 2 ¶ ö
Ze2 r
2
+
R
Er
= - EY
2
ç
÷
2 m R è ¶r
r ¶r ø
4pe 0
2 2
r æ ¶2 2 ¶ ö
Ze2 r
l(l + 1)
2
+
R- Er = 2
ç
÷
2 m R è ¶r
r ¶r ø
4pe 0
2m
2 2
æ ¶2 2 ¶ ö
Ze2
l(l + 1)
+
RR+
2
ç
÷
2 m è ¶r
r ¶r ø
4pe 0r
2 mr 2
2
2
2
R = ER
2
Radial and angular components

For the radial degree of freedom, we have a
new equation.
æ ¶2 2 ¶ ö
Ze2
l(l +1)
+
RR+
2
ç
÷
2m è ¶r
r ¶r ø
4pe 0r
2 mr 2
2
This is kinetic energy
in the radial motion
2
R = ER
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
l(l + 1)
+
2
2mr
V 
l
2
2mr
2
 F 
The higher the angular
momentum, the greater the
force in the positive r direction
dV
dr

l
2
mr
2
3

p r
mr
2
3

mv
r
2
The radial part

Simplify the equation by scaling the variables
4pe 0
2Zr
r=
; a0 =
a0
m e2
2
4E
me4
E ¢ = 2 ; e0 =
2 2
Z e0
32p e 0
o
= 1 bohr = 0.529 A
2
= 1 Ry = 13.6 eV
æ ¶2 2 ¶ ö
Ze2
l(l + 1)
+
RR+
2
ç
÷
2 m è ¶r
r ¶r ø
4pe 0r
2 mr 2
2
2
R = ER
¶ 2 R 2 ¶R R l(l + 1)
- 2- +
R = E ¢R
2
r ¶r r
¶r
r
The radial solutions

We need a new set of orthogonal
polynomials:
 R
2



2

2 R
 

R

The solution of this is
l

l ( l  1)

2
Slater-type
orbital
æ rö
- r /2n
Rnl (r) = N nl ç ÷ Lnl ( r )e
è nø
Normalization
R  E R
Associated Laguerre
polynomials
1
E¢ = - 2
4n
The Slater-type orbital
Wave functions
Y ( r,q ,j ) = Rnl ( r ) Ylm (q ,j )
n = 1, 2, 3,…
l = 0, 1, 2,… , n - 1
m = -l, … , l
The radial solutions
æZö
n = 1, l = 0, Rnl = 2 ç ÷
è a0 ø
3/2
e- r /2
1 æZö
n = 2, l = 0, Rnl =
ç ÷
2 2 è a0 ø
3/2
1 æZö
n = 2, l = 1, Rnl =
ç ÷
4 6 è a0 ø
3/2
1 æZö
n = 3, l = 0, Rnl =
ç ÷
9 3 è a0 ø
3/2
æ
r ö - r /4
2
e
çè
÷
2ø
re
- r /4
æ
r 2 ö - r /6
çè 6 - 2 r + 9 ÷ø e
1
E¢ = - 2
4n
E=-
Z 2 me4
32p 2e 02 2 n2
Verification

Let us verify that the (n = 1, l = 0) and (n = 2,
l = 1) radial solutions indeed satisfy the radial
equation
n = 1, l = 0 : R = Ce- r /2
n = 2, l = 1: R = C re
- r /4
¶ 2 R 2 ¶R R l(l + 1)
- 2- +
R = E ¢R
2
r ¶r r
¶r
r
1
E¢ = - 2
4n
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.