Ch 9pages 469-476

Lecture 23 – The Hydrogen Atom

Energy levels for a quantum particle in a box

Summary of lecture 22

En h

mLn 2 2

28

Energy levels for a quantum linear oscillator

)2

1( nhvEn

Energy levels for Bohr’s atom

2 4

2 2 20

1

8n

Z e mE

h n

The Quantum Hydrogen Atom

We shall now revisit the hydrogen atom, an atom containing a nucleus of charge Z and a single electron. We have considered the hydrogen atom before in our discussion of the Bohr model and on energy quantization. That model derived an expression for the quantized energies associated with particular electron orbits. The energy expressions is:

R is called Rydberg constant and expresses the energy is in terms what is required to remove an electron from an atom:

R=2.18x10-18 J=13.6 eV/molecule (electron volt)

2 4

2 2 2 20

1 1

8n

Z e mE R

h n n

The Quantum Hydrogen Atom

Of course the Bohr model uses a quantization scheme that only yields energies, but not electronic wave functions; it was derived years before wave mechanics was introduced. We shall now reexamine the hydrogenic atom using the Schrodinger equation. The time independent Schroedinger equation for the hydrogen atom is:

Here the potential energy is the Coulombic attraction between the positively charged nucleus and the negatively charged electron

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

The Quantum Hydrogen Atom

This time-independent Schroedinger equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

222

cos

sinsin

cossin

zyxr

rz

ry

rx

The Quantum Hydrogen Atom

This time-independent Schroediner equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

p+

e-

The Quantum Hydrogen Atom

This time-independent Schroediner equation is never solved in Cartesian coordinates. Like other central force problems, this equation is solved by converting to spherical coordinates:

Once the Schroedinger equation is converted to spherical coordinates, it can be solved by separation of variables by assuming that the wave function is a product of two functions, one of which described the radial component and the other the angular component of the wave function

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

222

cos

sinsin

cossin

zyxr

rz

ry

rx

,,, ,, mlln YrRr

The Quantum Hydrogen Atom

Like other three dimensional problems, the hydrogen atom wave function is parametrized by three integers n, l, and m that arise out of the solutions of the differential equations that are separated out of the Schroedinger equation

They are called principal, angular and magnetic quantum number. A wave function with a given set of quantum numbers is called an orbital. The first term is called radial wave function, while the second is the angular wave function and is expressed in terms of well-known functions called spherical harmonics

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

,,, ,, mlln YrRr

Radial Wave Function

The wave function is the solution of the radial wave equation. The radial function is the dependence of the wave function on the electron-nuclear distance r. The integer n=1, 2, 3 is called the principal quantum number. It is used to calculate the energy:

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

,,, ,, mlln YrRr

rR ln,

In order to be correct, one would have to use the reduced mass, but for the hydrogen atom, the reduced mass is the electron mass to within less than 0.1%. The energy is quantized, but only in terms of one integer n derived from the radial equation. This occurs because the potential energy is only dependent on r.

2 4

2 2 20

1

8n

Z e mE

h n

Radial Wave Function

The second integer l is the angular momentum quantum number, which varies from 0 to n-1. As we shall see in a moment, the quantum number l quantizes the total angular momentum L according to the equation:

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

,,, ,, mlln YrRr

It is conventional to designate wave functions corresponding to l=0 as s, l=1 as p, and l=2 as d, l=3 as f. s orbitals are easiest to discuss because the electron density is only dependent on r (spherical symmetric).

12,1,04

12

22 nl

hllL

Radial Wave Function

The radial probability distribution function is the probability that an electron is located in a volume 4r2 dr. This function is proportional to:

zyxEzyxr

Zezyx

zyx

h,,,,

4

1,,

8

2

02

2

2

2

2

2

2

2

,,, ,, mlln YrRr

)()( 2.

2 rRrrP ln

One of several ways to quantify the size of an orbital is to determine the radius that encloses 90% of the total electron density. This is given by the equation:

R

sn drRr0

2,

2 90.04

Radial Wave Function

The best quantitative measure of the size of an orbit valid for all values of n and l is the average distance of an electron from the nucleus:

,,, ,, mlln YrRr )()( 2.

2 rRrrP ln

Note the leading term is just the orbital radius obtained from Bohr theory:

n

ll

Z

nar ln

)1(1

2

11

20

,

ah

e mm0

02

2100 53 10

.

Radial Wave Function

Hydrogen-like orbitals are used to describe the properties of many molecules, for which the Schrodinger equation cannot be solved analytically. It is useful to plot some radial wave functions and probabilities in terms of a0=0.53 A (0.53x10-8

cm) (Bohr radius) and =r/a0. The radial wave functions have

n-1 nodes. For n=1, l=0(s), there are 0 nodes. For n=2, l=0, the node is at r=2a0. For n=3 there are nodes at r=1.9a0 and 7.1a0.

,,, ,, mlln YrRr

Radial Wave Function

,,, ,, mlln YrRr

3/2

2/3

0,3

3/2

2/3

0,3

2/

2/3

0,2

3/2

2/3

0,3

2/

2/3

0,2

2/3

0,1

3081

1

6681

1

62

1

21827381

1

222

1

2

ea

ZR

ea

ZR

ea

ZR

ea

ZR

ea

ZR

ea

ZR

d

p

p

s

s

s

Radial Wave Function

,,, ,, mlln YrRr

1s Radial Wave Function & Probability

0

1

2

3

4

5

r (Angstroms)

R(1

s) a

nd

P(1

s)

R1s

P1s

2s Radial Wave Function and Probability

- 0.5

0

0.5

1

1.5

2

r (Angstroms)

R(2s)

P (2s)

3s Radial Wave Function & Probability

-0.5

0

0.5

1

r (Angstroms)

R(3

s) &

P(3

s)

R(3s)

P(3s)

Orbital Shapes

s (l = 0) orbitals

r dependence only

as n increases, orbitals contain n-1 nodes

• Note that the “1s” wavefunction has no angular dependence (i.e., and do not appear).

0

3 32 2

1

1 1Zr

as

o o

Z Ze e

a a

* Probability =

Probability is spherical

Orbital Shapes

Angular Wave Function

,,, ,, mlln YrRr

Y() is the part of the wave function dependent on the equatorial and azimuthal angles and and is called the angular wave function; these are quite famous functions called spherical harmonics. The angular wave function has two more quantum numbers: l and m. These quantum numbers arise because two more physical quantities are quantized. Recall that the principal quantum number n exists because the energy is quantized.

Angular Wave Function

,,, ,, mlln YrRr

The quantum numbers l and m exist as a result of the quantization of the angular momentum of the electron. The angular momentum is defined as the cross product of the radial vector that defines the classical orbital radius of the electron, the momentum vector

L r p r mv

and is thus itself a vector with both direction and magnitude. Both properties are quantized. The l quantum number exists as a result of the quantization of the magnitude of the angular momentum, which is quantized according to the rule:

1...2,1,0;4

)1(2

22

nl

llhL

Angular Wave Function

,,, ,, mlln YrRr

The final quantum number m exists because the orientation or direction of the angular momentum is also quantized. This can be expressed by saying that the z component of the angular momentum Lz can only assume certain values

according to the equation:

The m quantum number varies from –l to +l.

Lmh

m l l l lZ 2

1 1

; , ,... , .

Angular Wave Function

,,, ,, mlln YrRr

The wave functions Yl,m() are not easily visualized in any

simple coordinate system. Instead, linear combinations of the Yl,m() wave functions are constructed that are easily

visualized in spherical coordinates. For the n=2 p orbitals, these functions are designated Yl,x(),Yl,y(),and Yl,z,(),

also called the px, py, and pz orbital wave functions

The p orbital wave functions have a simple sine/cosine dependence on the azimuthal and equatorial angles. For example, the wave function for a 2px orbital has the form:

),()( ,,2,1,2 xppxmln rR

cossin

4

3

62

102/

0

2/3

0

aZrea

Zr

a

Z

Angular Wave Function

The form of the angular portion of this function indicates that electron density will be greatest where sin and cos are large, i.e. at =/2 and =0. In other words, around the x axis. Similarly, for the py orbital density is greatest around the y

axis, and for the pz orbital, density is greatest where =0,

which is near the z axis. Tables of angular wave functions can be found in various texts on quantum mechanics.

),()( ,,2,1,2 xppxmln rR

cossin

4

3

62

102/

0

2/3

0

aZrea

Zr

a

Z

Quantum Numbers and Orbitals

n l Orbital ml # of Orb.

1 0 1s 0 12 0 2s 0 1

1 2p -1, 0, 1 33 0 3s 0 1 1 3p -1, 0, 1 3

2 3d -2, -1, 0, 1, 2 5

Multi-electron Systems

• Can set up the Schrodinger Equation problem for multi-electronic systems

• Can not solve it analytically (basic math principles) as there are too many variables and not enough equations

• Can solve using more complicated methods “Self-consistent field method”

• Result: A series of wave function not unlike that of the one electron system

• Orbitals of multi-electronic systems do not differ significantly from “hydrogenic” wave functions

• Add in a fourth quantum number (magnetic spin)

Angular Wave Function

The form of the angular portion of this function indicates that electron density will be greatest where sin and cos are large, i.e. at =/2 and =0. In other words, around the x axis. Similarly, for the py orbital density is greatest around the y axis, and for the pz

orbital, density is greatest where =0, which is near the z axis.

),()( ,,2,1,2 xppxmln rR

cossin

4

3

62

102/

0

2/3

0

aZrea

Zr

a

Z

2p (l = 1) orbitals

not spherical, but lobed

labeled with respect to orientation along x, y, and z

32

22

1cos

4 2zpo

Ze

a

Orbital Shapes

3p orbitals

more nodes as compared to 2p (expected)

still can be represented by a “dumbbell” contour

3

22 3

3

26 cos

81zpo

Ze

a

Orbital Shapes

3d (l = 2) orbitals

labeled as dxz, dyz, dxy, dx2-y2 and dz2.

Orbital Shapes

4f (l = 3) orbitals

• exceedingly complex probability distributions

Orbital Shapes