Bohr model only works for one electron atom, or ions. i.e. H, He+, Li2+, ….

It can’t account for electron-electron interactions properly

The wave “like” properties of electron need to be explored to do job properly.

Electrons as Waves

Louis de Broglie : All matter has a corresponding wave character, with a wavelength determined by its momentum p (= mv)

l = h/p =h/mv

Example: Electron moving at 0.1000 C:

l = (6.626*10-34 Js)/(9.109*10-28 kg)*(0.1000*2.998*108 m/s)

= 2.423*10-14 m = 242.3 pm

Example: A 0.100 kg baseball moving at 150. km/hr:

l = (6.626*10-34 Js)/(0.1 kg)*(41.7m/s)

=1.59*10-34 m

v = (150000 m)/(3600 s) = 41.7 m/s

Correspondence Principle

Macroscopic bodies don’t feel the effect of quantum mechanics due to their large masses and slow motion

Ex) The wavelength of the base ball is insignificant on the scale of the base ball

Microscopic bodies do feel the effect of quantum mechanics strongly due to their small masses and fast motion

Ex) The wavelength of the electron ball is very large on the scale of the size of the electron i.e. 10-30 m

WavefunctionsStanding waves

Ex) String on a guitar

Only a few wave forms are suitable, which is determined by length

l = 2L, L, 2L/3 ….

Wavefunction: Y(x)

depends on number of lobes n =1, 2, 3, …

Y(x) = sinq

Y(x) = sin(2q)

Y(x) = sin(3q)

Therefore Y(n,x) is s series of solutions, each with a different energy E(n)

q =px/2L

x

A Wave in Orbit

A circular path imposes a length on the wave form, allowing for only an whole number of nodes.

A Little MathF(y) - is function that depends on y

F(G(x)) - is the function, F, that acts on G(x), where G(x) s a function of x.

F(y) = cy - the function F acting on y to give back y multiplied by some constant c

Ex) F(y) = 3 y requires that c = 3

Ex) F(y) = 3 y, and y = G(x) = 4x2+2

F(G(x)) = 3 (4x2+2) = 12x2+2 = 3 G(x) where c = 3

F(G(x)) = cG(x) - The function F acting on G(x) gives back G(x) multiplied by some number c

H(Y(n))- means that H is a function that acts on Y(n)

- H is a function that calculates the total energy using, Y(n)

- The result is the Energy, E(n), which depends on n, and the original wavefunction Y(n)

H( Y(n) ) = E(n) Y(n) r = 3-D coordinates of electron

Y(n) - is the wavefunction corresponding to the electron

The Schrödinger Equation

- H contains electron-electron, electron-nuclear iteration, and kinetic energy terms.

H( Y(n, l,m, s) ) = E(n) Y(n, l, m, s)

The wavefunction depends on four quantum number, each associated, with a different property on the electron.

n – Principle Quantum Number

l – Angular momentum Quantum Number

Determines which shell the electron is in and the energy of the electron, E(n)

n = 1, 2, 3, 4, …

E(n) = -Ry Z/n2

Subshells exist for each shell differing in the angular momentum value.

l = 0, 1 …n-1

m- Magnetic Quantum NumberRelated to the orientation in space that of the orbital.

m = -l …+l

s - Spin Quantum NumberRelated to symmetry of wavefunction

s = 1/2, -1/2

L = hl/2p

Wavefunctions of H

n = 1 m = 0

Lets for the moment ignore spin

l = 0

States of m are labeled as:

l = 0 S

l = 1 P

l = 2 D

l = 3 F

Therefore this state is: 1s0 = 1s

n = 2 l = 0 m = 0

n = 2 l = 1 m= 1, 0, -1

2s

2p1, 2p0, 2p-1

2px, 2py, 2pz

Wavefunction of H

n = 3 l = 0 m = 0

n = 3 l = 1 m = 1,0, -1

n = 3 l = 2 m= 2,1,0, -1,-2

3s

3p1, 3p0, 3p-1

3d2, 3d1, 3d0, 3d-1, 3d-2

3d(xy), 3d(xz), 3d(yz),

3d(x2-y2), 3dz2

Wavefunction of H

n = 4 l = 0 m = 0

n = 4 l = 1 m = 1,0, -1

n = 4 l = 2 m = 2,1,0, -1,-2

4s

4p1, 4p0, 4p-1

4d2, 4d1, 4d0, 4d-1, 4d-2

n = 4 l = 3 m= 3,2,1,0, -1,-2,-3

4f3, 4f2, 4f1, 4f0, 4f-1, 4f-2, 4f-3

Heisenberg Uncertainty Principle

p

Observation

p ?

Measurement effects state of system.

There is a limit imposed on the degree of certainty to which you can know the position (r) and momentum (p) of a particle

r r1

r2r ?

DpDr ≥ h/4pDp – error in p

Dr – error in r

Exercise

An electron is traveling between 0.11000 C and 0.11500 C. What is the smallest error in the position you can expect? What is the error in position if it were a proton?

Need error in momentum Dp?

We know that Dv = 0.00500 C

Therefore Dp = m Dv

Dp = (9.109*10-31 kg)*(0.00500 * 2.998*108 m/s)

Dp = 1.36*10-24 kg m/s

For an electron

Recall that DpDr ≥ h/4p

Dr ≥ h/(4pDp)

Dr ≥ (6.626*10-34 Js)/[(4*3.14159)*(1.36*10-24 kg m/s)]

Dr ≥ 3.88*10-11 m

Dr ≥ (6.626*10-34 Js)/[(4*3.14159)*(2.51*10-21 kg m/s)]

For an proton

Dp = (1.674 × 10-27 kg)*(0.00500 * 2.998*108 m/s)

Dp = 2.51*10-21 kg m/s

Dp = m Dv

Dr ≥ 2.10*10-14 m

Probability DistributionA particle position and momentum cannot be known exactly

Therefore a particle is characterized by a probability distribution function

The probability distribution is determined by the wavefunction:

P α r2Y2(n,l,m,s;r)Y(x) P(x)

Hydrogen Orbitals The hydrogen orbital are determined from the wavefunctions

Ex) 1s a Y2( 1, 0, 0, r ) 2s a Y2( 2, 0, 0; r )

S orbitals - are spherical, i.e. they are identical in all directions

The probability distribution can be graphed as a function of the radius as P(r) = r2 Y2

radial probability density plot

Notice the shell structure as n increases

Notice the nodes in the wavefunction

For 1s 0 nodes

For 2s 1 nodes

For 3s 2 nodes

P OrbitalsThe p orbitals are constructed from the hydrogen wavefunctions with n > 1, and l = 1.

Y(2,1,1) and Y(2,1,-1), are complex values, and are combined to make them real valued.

The resulting functions are aligned along the x and y axis. The remaining function Y(2,1,0) is aligned along z axis.

i.e. Y(2,1,1), Y(2,1,0), Y(2,1,-1)

P OrbitalsThese dumbbell shaped orbital are referred to as the p (polar) orbitals, which are labeled according to their orientation, 2px, 2py, 2pz

The number of nodes increases with n as n-1, i.e 1 for 2p

Note that when the nodal plane is crossed the orbital changes sign

The orbitals are plotted as the boundary enclosing total of 90% probability

Px Py Pz

D OrbitalsThe d orbitals are constructed from the hydrogen wavefunctions with n > 2, and l = 2.

Y(3,2,2) and Y(3,2,-2) a well as Y(3,2,1) and Y(3,2,-1), are combined to make them real valued functions.

The four corresponding distribution functions are have four lobes in the xy, xz and yz plane. (in between the axes)

i.e. Y(3,2,2), Y(3,2,1), Y(3,2,0), Y(3,2,-1), Y(3,2,-2),

D Orbitals

A fourth orbital exists in the xy plane aligned on the axes, the other fits between the axes.

The remaining fifth orbital , dz2, resembles a Pz orbital with a donut like shape in the xy plane (z2-x2 and z2-y2 are superimposed)

Sign changes when nodal plane (cone) is crossed

There are 2 nodes 3d

F Orbitals

Constructed from seven H wavefunctions to make them real valued

Composed of 8 lobes

There are 3 nodes for 4f

The Orbitals of the Hydrogen Atom

0 nodes

1 node

2 nodes

Radial nodes

1 planar node 2 planar nodes

ConceptsProperties of waves (wavelength, frequency, amplitude, speed)

Electromagnetic spectrum, speed of light

Planck’s equation and Planck’s constant

Wave-particle duality (for light, electrons, etc.)

Atomic line spectra and relevant calculations

Ground vs. excited states

Heisenberg uncertainty principle

Bohr and Schrödinger models of the atom

Quantum numbers (n, l, ml)

Shells (n), subshells (s,p,d,f) and orbitals

Different kinds of atomic orbitals (s, p, d, f) and nodes