Famous Opinions of QM

Famous Opinions of QM“A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it.”

(Max Planck, 1920)

“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?‘” 

(Albert Einstein, 1954)

“Those who are not shocked when they first come across quantum physics cannot possibly have understood it.”

(Niels Bohr, 1958)  

Famous Opinions of QMThe one great dilemma that nails us…day and night is the wave-particle dilemma.

(Erwin Schrodinger, 1959)  

I think I can safely say that nobody understands quantum mechanics.

(Richard Feynman, 1965)  

Particles and Waves• EM radiation can behave as either a wave or a

particle depending on the situation• “Light has properties that have no analogy at the

macroscopic level, and thus, we have to combine two different ideas to describe its behavior.”

Wavefunctions• A wavefunction is a probability

amplitude. The “square” of a wavefunction gives the probability density…the likelihood of finding the particle in region of space.

• The wavefunctions and kinetic energies available to a quantum particle are quantized if the particle is subject to a constraining potential.

• We can determine the wavefunctions and KEs available to our system by considering the field of force (the PE) our system is subject to.

The Hamiltonian• Erwin Schrodinger developed a mathematical

formalism that incorporates the wave nature of matter.

• H, the “Hamiltonian,” is a special kind of function that gives the energy of a quantum state, which is described by the wavefunction, Y.

• H contains a KE part and a PE part:

• By solving the Schrodinger equation (below) with a known Hamiltonian, we can determine the wavefunctions and energies for quantum states.H E

2 2 2 2

2 2 2 2ˆ ( , , )

8h d d dH V x y zm dx dy dz

H-atom wavefunctions• In the H atom, we are interested in describing the

regions in space where it is likely we will find the electron, relative to the nucleus…we want the wavefunction for the electron.

• We can model the attraction of the H atom’s single electron to its single proton with a “Coulombic” potential curve:

e-

P+r

r0

V (r) e2

r• The V(r) potential becomes part of the

Hamiltonian for the electron.

H-atom wavefunctions (cont.)

• The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates.

p+

e-

r = interparticle distance (0 ≤ r ≤ )

q = angle from “xy plane” (/2 ≤ ≤ - /2)

= rotation in “xy plane” (0 ≤ ≤ 2)

H-atom wavefunctions (cont.)

• Then the Schrodinger equation for the hydrogen atom becomes:

H E

2 22

2 2 2 2

2

1 1sin8 sin sin

h rr r r

Ze Er

3-dimensional KE operator in spherical polar coordinates

Radial Coulombic PE operator

H-atom wavefunctions (cont.)

• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized:

2

218

220

4

2

2

10178.28 n

ZJxh

menZEn

n is the principle quantum number, and can have integer numbers ranging from 1 to infinity.

The higher n, the greater the distance between the nucleus (+) and the electron (-).

H-atom wavefunctions (cont.)

• In solving the Schrodinger Equation, two other quantum numbers become evident:

l…the orbital angular momentum quantum number. Ranges in value from 0 to (n - 1 ).

ml … the “z component” of orbital angular momentum. Ranges in value from - l to 0 to +l .

• We can characterize the hydrogen-atom orbitals using the quantum numbers: n, l , ml

Orbitals and Quantum Numbers

• Naming the electron orbitals is done as follows– n is simply referred to by the quantum

number– l (0…n… - 1) is given a letter value as

follows:• 0 = s• 1 = p• 2 = d• 3 = f- ml (- l …0…l ) is usually “dropped”

For example: for n = 3, l = 2 “3d orbital”

Quantum Mechanical Model

The Bohr model is deterministic…uses fixed “orbits” around a central nucleus to describe electron structure of atoms. The QM model is probabilistic…uses probabilities to describe electron structure.A probabilistic electron structure is much more difficult to visualize. HOWEVER, the electronic energy levels are still quantized.

Deterministic vs. Probabilistic

• In the Bohr model, you can always find the electron in an atom, just like you can always find the moon as it orbits the earth.

• You can always determine the relative location of the nucleus and electron in Bohr’s model.

• This is because the electron follows a particular orbit around the nucleus.

• In the QM model, the electron does not travel along a particular path around the nucleus.

• You can never determine the electron’s exact location…you can only find where it is likely to be.

• The Bohr orbit is replaced by orbital which describes a volume of space in which the electron is likely to be found.

Quantum Numbers and Orbitals (cont.)

• Table 7.1: Quantum Numbers and Orbitals

n 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

Incr

easin

g En

ergy

Orbital Shapes (cont.)• Example: Write down the orbitals associated with n = 4.

Ans: n = 4 l = 0 to (n - 1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f

4s (1 ml sublevel)4p (3 ml sublevels)4d (5 ml sublevels4f (7 ml sublevels)

Which of the following sets of quantum numbers (n, l, m) is not allowed?

A. (3, 2, 2).

B. (0, 0, 0).

C. (1, 0, 0).

D. (2, 1, 0).

Electron OrbitalsOrbitals represent a probability space where an electron is likely to be found.All atoms have all orbitals, but many of them are not occupied. The shapes of the orbitals are determined mathematically...they are not intuitive.

s orbitals

p orbitals

d orbitals

Electron Orbital Shapes• The “1s” wavefunction has no angular

dependence (i.e., it only depends on the distance from the nucleus).

1s 1Zao

32

eZa0r

1Zao

32

e

*Probability =

Probability is spherical

Electron Orbital Shapes (cont)s (l = 0) orbitals

as n increases, orbitals demonstrate n - 1 nodes.

Node: an area of space where the electron CAN’T be, ever, no matter how much it wants to.

Aside: What’s a node??• Remember the guitar-string standing wave

analogy?• A standing wave is a motion

in which translation of the wave does not occur.

• In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs.

Electron Orbital Shapes (cont.)

not spherical, but lobed.labeled with respect to orientation along x, y, and z.

2p (l = 1) orbitals

2px 2py 2pz

2pz

14 2

Zao

32

e

2 cos

Electron Orbital Shapes (cont.)

• more nodes as compared to 2p (expected.).

• still can be represented by a “dumbbell” contour.

3pz

281

Zao

32

6 2 e 3 cos

3p (l = 1) orbitals

Orbital Shapes (cont.)3d (l = 2) orbitalslabeled as dxz, dyz, dxy, dx2-y2 and dz2.

Orbital Shapes (cont.)

We will not show the exceedingly complex probability distributions associated with f orbitals.

4f (l = 3) orbitals

Electron Orbital Energies in the H-atom

• energy increases as 1/n2

• orbitals of same n, but different l are considered to be of equal energy (“degenerate”).• the “ground” or lowest energy orbital is the 1s.

Orbital Summary• Orbital E increases with n.

– At higher n, the electron is farther away from the nucleus...this is a higher energy configuration.

• Orbital size increases with n. – There is a larger area of space where you are likely to find

an electron at higher E’s. • Orbital shape is the same no matter the value of n.

– 3s looks like 1s, except it’s bigger and has more nodes. Same for p, d, f, etc.

• Number of nodes in an orbital goes as n - 1. – 1s has zero nodes, 2s has one node, 3s has two nodes...– 2px, 2py, 2pz each have one node, 3px, 3py, 3pz each have

two nodes– the 3d orbitals each have two nodes, 4d have three, etc.– Note that number of nodes indicates relative energy!

• All atoms have all orbitals…but in an unexcited atom, only those closest to the nucleus will be occupied by electrons

Orbital Quiz• The shape of a given type of orbital

changes as n increases.

• The number of types of orbitals in a given energy level is the same as the value of n.

• The hydrogen atom has a 3s orbital.

• The number of lobes on a p-orbital increases as n increases. That is, a 3p orbital has more lobes than a 2p orbital.

• The electron path is indicated by the surface of the orbital.

T

F

TF

F

Electron Spin• Experiments demonstrated

the need for one more quantum number.

• Specifically, some particles (electrons in particular) demonstrated inherent angular momentum…

• Basically, this means that electrons have two ways of interacting with an applied magnetic field.

Interpretation: the electron is a bundle of “spinning” charge

“spin up”

“spin down”

Electron Spin (cont.)• The new quantum

number is ms (analogous to ml).

• For the electron, ms has two values:

+1/2 and -1/2

ms = 1/2

ms = -1/2

Pauli Exclusion PrincipleDefn: No two electrons may

occupy the same quantum state simultaneously.

In other words: electrons are very territorial. They don’t like other electrons horning in.

In practice, this means that only two electrons may occupy a given orbital, and they must have opposite spin.

Quantum Number Summary• n: principal quantum number

– index of size and energy of electron orbital– can have any integral value: 1, 2, 3, 4, …

• l: angular momentum quantum number– related to the shape of the orbitals– can have integral values 0 … n - 1

• ml: magnetic quantum number– related to orbital orientation (relative to the other l-

level orbitals)– can have integral values –l … 0 … +l

• ms: electron spin quantum number– related to the “magnetic moment” of the electron– can have half-integral values –1/2 or +1/2

Polyelectronic Atoms• For polyelectronic atoms, a direct solution of

the Schrodinger Eq. is not possible.

• When we construct polyelectronic atoms, we use the hydrogen-atom orbital nomenclature to discuss in which orbitals the electrons reside.

• This is an approximation (and it is surprising how well it actually works).

2 22

2 2 2 2

2

1 1sin8 sin sin

h rr r r

Ze Er

No solution for polyelectronic atoms!!

The Aufbau Principle• When placing electrons into orbitals in the

construction of polyelectronic atoms, we use the Aufbau Principle.

• This principle states that in addition to adding protons and neutrons to the nucleus, one simply adds electrons to the hydrogen-like atomic orbitals

• Pauli exclusion principle: No two electrons may have the same quantum numbers. Therefore, only two electrons can reside in an orbital (differentiated by ms).

Orbital Energies

1s

2s

3s

4s

2p

3p

3d

Ener

gy

H has only one electron, so all of the sublevels in a given principal level have the same energy...they are degenerate.

In many-electron atoms, a given electron is simultaneously attracted to the nucleus and repelled by other electrons, causing the energies of the sublevels to change relative to H.

When we put electrons in orbitals, we fill them in order of increasing energy, not n.

Let’s fill some orbitals

RULES• Orbitals are filled starting

from the lowest energy.• The two electrons in an

orbital must have opposite spin.

• Example: Hydrogen (Z = 1)

1s 2s 2p• Example: Helium (Z = 2)

1s 2s 2p

1s1

1s2

Let’s fill some more orbitals• Lithium (Z = 3)

1s 2s 2p

1s 2s 2p

• Berillium (Z = 4)

• Boron (Z = 5)

1s 2s 2p

1s22s1

1s22s2

1s22s22p1

Filling Orbitals (cont.)• Carbon (Z = 6)

1s 2s 2pHund’s Rule: Lowest energy configuration is the one

in which the maximum number of unpaired electrons are distributed amongst a set of degenerate orbitals.

1s22s22p2

REVISED RULES• Orbitals are filled starting from the lowest energy.• The two electrons in an orbital must have

opposite spin.• Hund’s Rule: the orbitals in degenerate series

(such as 2p in the example above) must each have an electron before any of them can have two.

Filling Orbitals (cont.)• Carbon (Z = 6)

1s 2s 2p

1s22s22p2

• Nitrogen (Z = 7)

1s 2s 2p

1s22s22p3

Filling Orbitals (cont.)• Oxygen (Z = 8)

1s 2s 2p

1s 2s 2p

• Fluorine (Z = 9)

1s22s22p4

1s22s22p5

1s 2s 2p

• Neon (Z = 10)

1s22s22p6

full

Filling Orbitals (cont.)• Sodium (Z = 11)

3s1s22s22p63s1

Ne

[Ne]3s1

1s 2s 2p

• Compare to Neon (Ne) (Z = 10)1s22s22p6

full

1s 2s 2p

3s 3p

Filling Orbitals (cont.)• Sodium (Z = 11)

3s1s22s22p63s1

3s 3p

• Phosphorus (P) (Z = 17)[Ne]

3s23p3

Ne

[Ne]3s1

Ne

3s 3p

• Argon (Z = 18)[Ne]

3s23p6Ne

Filling Orbitals (cont.)We now have the orbital configurations for the first 18 elements.

Elements in same column have the same # of valence electrons!Valence Electrons: The total number of s

and p electrons in the highest occupied energy level.

The Aufbau Principal (cont.)• Similar to Sodium, we begin the next row of the periodic table by adding electrons to the 4s orbital.

• Why not 3d before 4s?• 3d is closer to the nucleus

• 4s allows for closer approach; therefore, is energetically preferred.

Back to Filling Orbitals• Elements Z=19 and Z= 20:

Z= 19, Potassium:

Z= 20, Calcium:

4s 4p

Ar

4s 4p

Ar

1s22s22p63s23p64s1 = [Ar]4s1

1s22s22p63s23p64s2 = [Ar]4s2

Filling Orbitals (cont.)• Elements Z=21 to Z=30 have occupied d orbitals:

Z= 21, Scandium:

Z= 30, Zinc:

4sAr

4s 4p

Ar3d

4p

3d

1s22s22p63s23p64s23d1 = [Ar] 4s23d1

1s22s22p63s23p64s23d10 = [Ar] 4s23d10

The Aufbau Principal (cont.)• Elements Z=19 and Z= 20:

Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1 Z= 20, Calcium: 1s22s22p63s23p64s2 = [Ar]4s2

• Elements Z = 21 to Z = 30 have occupied d orbitals:

Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1

Z= 30, Zinc: 1s22s22p63s23p64s23d10 = [Ar] 4s23d10

Z = 24, Chromium: [Ar] 4s13d5 exception

What if you forget the orbital-filling order?

1. Write down the orbitals for each n on separate lines.

2. Arrows drawn as shown will give you the order in which the orbitals should be filled.

Note that this scheme fills 4s before 3d, as expected.

Polyelectronic Atoms

+ e-

“Screening”: The presence of other electrons means a given electron does not feel the attraction of the nucleus as strongly as it would in hydrogen.

“Penetration”: Orbitals that have some probability density close to the nucleus will be energetically favored over orbitals that do not.

Periodic Table This orbital filling scheme gives rise to the modern periodic table.

Periodic Table After Lanthanum ([Xe]6s25d1), we start filling 4f.

Periodic Table After Actinium ([Rn]7s26d1), we start filling 5f.

Periodic Table Row headings correspond to the highest occupied energy level for any element in that period.

“Valence” only refers to s and p electrons in the highest occupied energy level.

Periodic Table Column headings give total number of valence electrons for any element in that group.

What is the electron configuration for the indicated element?

A. 1s22s22p63s23p64s23d3

C. 1s22s22p63s23p64s23d2

B. 1s22s22p63s23p64s24d3

D. 1s22s22p73s23p64s23d2

Valence Electrons• The total number of s and p electrons in the

highest occupied energy level.• As we’ll see, all the “action” happens at the

valence electrons.• Elements in the same group (column) in the

periodic table have the same number of valence electrons.

• This means elements in the same group tend to have similar chemical properties.

Valence Electrons (cont.)Chemists use Lewis dot symbols to indicate the number of valence electrons in an atom. The valence electrons are drawn as dots around the atomic symbol, with orbital occupancy indicated...that is, electrons that occupy the same orbital appear as paired dots.HOWEVER, we will encounter situations where it is more convenient to spread the dots out around the element symbol.

C C

Which of the following electron configurations represents N?

A. 1s21p32s2

B. 1s22s22p23s1

C. 1s22s32p2

D. 1s22s22p2

But we wrote the electron configuration of N as 1s22s22p3 the other day!

The electron configuration 1s22s22p33s1 represents an excited state of N.

Excited States

An Excited State of Nitrogen (Z = 7)

1s 2s 2p

1s22s22p23s1

Ground State of Nitrogen (Z = 7)

1s 2s 2p

1s22s22p3

3s

hν

• By putting the “right” amount (a quantized amount) of energy into an atom, we can move electrons from a low energy orbital to a higher-energy orbital…

• Such an electron is said to be in an excited state.

Periodic Trends• The valence electron structure of atoms

can be used to explain various properties of atoms.

• In general, properties correlate down a group of elements.

• A warning: such discussions are by nature very generalized…exceptions do occur.

Ionization Energy• The energy required to remove an electron

from a gaseous atom or ion.

• The electron is completely “removed” from the atom.

Z+

Z-Z+

(Z-1)-

e-Energy

Ionization Energy (IE)• Ionization is an endothermic process...we

must put energy in to separate the negatively-charged electron from the positively-charged nucleus.

• The greater the propensity for an atom to “hold on” to its electrons, the higher the ionization energy will be.

IE + X X+ + e-

• Generally done using photons, with energy measured in eV (1 eV = 1.6 x 10-19 J).

Ionization Examples• Removal of valence versus core

electronsNa(g) Na+(g) + e- IE1 = 5.14 eV

Na+(g) Na2+(g) + e-IE2 = 47.3 eV

[Ne]3s1 [Ne]

[Ne] 1s22s22p5

(removing “valence” electron)

(removing “core” electron)• Takes significantly more energy to

remove a core electron…. core electron configurations tend to be energetically stable.

WOW!

Ionization Examples• We can perform multiple ionizations of

valence electrons:

Al(g) Al+(g) + e- I1 = 580 kJ/mol first

Al+(g) Al2+(g) + e- I2 = 1815 kJ/mol second

Al2+(g) Al3+(g) + e- I3 = 2740 kJ/molthird

Al3+(g) Al4+(g) + e- I4 = 11,600 kJ/molfourthWOW!

Ionization• First Ionization Energies:

Column 1

Column 8

First Ionization E Trends

Increases from left to right across a period.

Decreases down a group.

Reason: increasing Z+ (the number of protons in the nucleus) which attracts the valence electron Reason: increasing distance between electron and nucleus

Ease of Ionization

Which reaction represents the ionization of F?

A. 1s22s22p5

B. 1s22s22p5

1s22s22p6

1s22s22p43s1

C. 1s22s22p5 1s22s22p4

D. 1s22s22p5 1s22s12p6

Electron AffinityElectron Affinity: the energy change associated with the addition of an electron to a gaseous atom.

+ZZ-

+Z(Z+1)-

e-

Energy

• We will stick with our thermodynamic definition, with energy released being a negative quantity.

Electron Affinity• Some elements have very high electron

affinity:

• Group 7 (the halogens) and Group 6 (O and S specifically).

Wow!

Electron Affinity• Some elements have essentially no electron

affinity:

• Orbital configurations can explain both observations.

N?

Electron Affinity• Why is EA so great for the halogens?

F(g) + e- F-(g) EA = -327.8 kJ/mol1s22s22p5 1s22s22p6 [Ne]

• Why is EA so poor for nitrogen?

N(g) + e- N-(g) EA > 0 (unstable)1s22s22p3 1s22s22p4

(e- must go into occupied orbital)

Electron Affinity• How do these arguments do for O?

O(g) + e- O-(g) EA = -140 kJ/mol1s22s22p4 1s22s22p5

• What about the second EA for O?

O-(g) + e- O2-(g) EA > 0 (unstable)1s22s22p5 1s22s22p6

[Ne] configuration, but electron repulsion is just too great.

Bigger Z+ overcomes e- repulsion.

Atomic Radii• Atomic Radii are defined as the covalent radii, and are obtained by taking 1/2 the distance of a bond:

r = atomic radius

Atomic Radii• Decrease across a row due to increase in Z+

• Increase down column due to population of orbitals of greater n.

Ionization and Atomic Radii

Ease of Ionization

Which atom would you expect to have the lowest ionization energy?

A. 1s22s22p3

B. 1s22s22p63s23p5

C. 1s22s22p63s23p64s2

D. 1s22s22p63s23p64s23d104p65s1

• Metals … Tend to lose electrons. – Good conductors of heat, electricity; malleable

solids• Non-metals … Tend to gain electrons.

– Poor conductors; not malleable• Metalloids … Can lose or gain electrons.

– Both metallic and nonmetallic properties

1A

2A 3A 4A 5A 6A 7A

8A

Metals

Non-metals

We can partition the periodic table into general types of elements.

What about Hydrogen?• Hydrogen (H). Valence: 1s1

• H is not really a metal or a halogen, although it has some properties of both– forms compounds in the same ratio as the

alkali metals, but the bonding mechanism is different

– can gain an electron to form H- like a halogen• Elemental H is found as a diatomic gas: H2• Under 3 million atm of pressure, H exhibits

metallic properties…

H

Metallic Hydrogen??Under the millions of atm of pressure exerted in a diamond-anvil cell, elemental H changes from a gas to an opaque solid that conducts electricity. The covalently-bonded diatomics turn into a network of protons in a “sea” of electrons.

This is just a photograph of a diamond anvil cell…that’s not metallic H in there.

Chemical Bonds• In broad terms, a chemical bond is a term used

to characterize an interaction between two atoms that results in a reduction in the energy of the system relative to the isolated atoms.

• The degree of energy reduction or “stabilization” is given by the energy required to break the bond (known as the “bond energy”)

Chemical Bonds (cont.)

Stabilization