CHM2S1-A Introduction to Quantum Mechanics Dr R. L. Johnston

II: Quantum Mechanics of Atoms and Molecules

5. Electronic Structures of Atoms

5.1 The hydrogen atom and hydrogenic ions.

5.2 Quantum numbers. 5.3 Orbital angular momentum. 5.4 Atomic orbitals.5.5 Electron spin.5.6 Properties of atomic orbitals. 5.7 Many-electron atoms.

 

THE UNIVERSITYOF BIRMINGHAM

6. Bonding in Molecules

6.1 The Born-Oppenheimer approximation.

6.2 Potential energy curves.

6.3 Molecular orbital (MO) theory.

6.4 MO diagrams.

6.5 MOs for 2nd row diatomic molecules.

6.6 Molecular electronic configurations.

6.7 Bond order.

6.8 Paramagnetic molecules.

6.9 Heteronuclear diatomic molecules.

5. Electronic Structures of Atoms

5.1 The Hydrogen Atom and Hydrogenic Ions

• The series of atoms/ions H, He+, Li2+, Be3+ …

all have: 1 electron (charge –e)

nucleus (charge +Ze)

• To determine the electronic wavefunction ( = e) and allowed electronic energy levels (E), we must set up and solve the Schrödinger Equation for a single electron in an atom.

• Consider 3-D motion of the electron (-e) relative to the nucleus (+Ze):

Hamiltonian Operator

Kinetic Energy where

Potential Energy where

(electrostatic attraction between electron and nucleus).

VTH ˆˆˆ

22

2

T̂ e

Ne

Ne mmm

mm

r4

ZerVV

0

2

ˆ

+ZemN

-eme

2

2

2

2

2

22

zyx

222 zyxr

r

• We must solve the 2nd order differential equation:

• Because of the spherical symmetry of the atom, it is convenient to describe the position of the electron in spherical polar coordinates (r,,), rather than Cartesians (x,y,z).

• Due to this spherical symmetry, the wavefunction can be separated into a product of radial and angular components:

(r,,) = R(r).Y(,)

• Imposing boundary conditions 3 quantum numbers (n,,m)

n,,m(r,,) = Rn,(r).Y,m(,)

EψψrVψ2

22

radial wavefunction

angularwavefunction(spherical harmonic)

5.2 Quantum Numbers

n,,m(r,,) = Rn,(r).Y,m(,)

depends on 3 quantum numbers (n,,m).

1. Principal Quantum Number (n)

• Positive integer n = 1, 2, 3 …• For H and 1-e ions, the electron energy depends only on n

(this is the same result as from the Bohr model).• For many-electron atoms E depends on n and .

2

2

20

2

4

nn

Z

h8

eE

2. Orbital Angular Momentum Quantum Number ()

• For a given value of n, can take the integer values:

= 0, 1, 2 … (n-1)

e.g. n = 1 = 0n = 2 = 0, 1n = 3 = 0, 1, 2

3. Orbital Magnetic Quantum Number (m)

• For a given value of , m can take the integer values:

m = 0, 1, 2 …

e.g. = 0 m = 0

= 1 m = 0, 1

• There are (2+1) allowed values of m.

5.3 Orbital Angular Momentum

• Q.Nos. and m arise from the angular part of the wavefunction and they relate to the angular momentum of the electron due to its motion around the nucleus (orbital angular momentum).

– determines magnitude of angular momentum vector (J)

• Because is quantized, so is |J|:

1J

0 1 2 3|J| 0 2 6 12

• The orientation of the angular momentum vector (J) is also quantized – as it depends on m.

• The component of angular momentum along a reference direction (e.g. the z-axis):

Jz = m

• e.g. for = 2 (m = 0, 1, 2)

angular momentum vector J

has magnitude |J| = 6 and

5 allowed orientations:

Jz = 0, , 2.

m

5.4 Atomic Orbitals

• The wavefunctions (n,,m) describing an electron in an atom are known as atomic orbitals.

• Each atomic orbital (1-e wavefunction) is uniquely defined by 3 quantum numbers (n,,m).

• The orbital () gives the spatial distribution of the electron (via the

Born interpretation of ||2).

• Orbitals are often drawn as 3-D surfaces which enclose approx. 90% of the probability of finding the electron.

Electronic Shells, Sub-shells and Orbitals

• All orbitals with the same value of n together constitute an electronic shell.

• For each n, orbitals with the same value of together constitute an electronic sub-shell.

• Each sub-shell consists of (2+1) orbitals, each with a different m value.

n 1 2 3 4Shell K L M N

0 1 2 3Sub-shell s p d f

No. orbitals 1 3 5 7

• Possible combinations of shells and sub-shells.

• Total number of orbitals in shell n is n2.

n

0 1 2 3 …

1 1s2 2s 2p3 3s 3p 3d4 4s 4p 4d 4f…

Electronic Shells, Sub-shells and Orbitals

n=3

l=0

l=1

l=2 ml =2 ml =1 ml =0 ml =1 ml =2 3d orbitals

ml =1 ml =0 ml =1 3p orbitals

ml =0 3s orbital

shell sub-shells orbitals

Atomic Orbital Energy Diagram for H

• Orbital energies depend only on the

principal quantum number n.

• For a given n value (shell), all sub-shells

() and orbitals (m) have the same energy

– i.e. they are degenerate.

Quantum Numbers and Atomic Orbitals

• Quantum theory postulates that electrons in atoms are fixed in regions of space corresponding to atomic orbitals. The probability of finding an electron at a particular point in space (r) is proportional to |(r)|2 (Born Interpretation).

• Orbitals differ from each other in size, shape and orientation:

– the orbital size is defined by the principle quantum number n

– the type of orbital (it’s shape) is defined by the angular momentum quantum number

– the orientation of the orbital is defined by the orbital magnetic quantum number m

Shapes of Atomic Orbitals

•s orbitals have l = 0 and ml = 0

d x2 y

2 dz2 dxy dxz dyz

•d orbitals have l = 2 and ml = 2, 1, 0 ,1, 2

pz px py

•p orbitals have l = 1 and ml = 1 , 0, 1

5.5 Electron Spin

• Stern and Gerlach (1921) observed that a beam of Ag atoms is split into 2 beams by an inhomogeneous magnetic field.

• Dirac (1930) introduced relativistic effects into Quantum Mechanics and showed that to completely describe the state of an electron we must specify:

1. The orbital (n,,m)

2. The spin state of the electron

• Electron spin is characterised by 2 quantum numbers:

– spin angular momentum q. no. s ( = ½ for all electrons)

– spin magnetic q. no. ms ( = ½).

• Spin angular momentum (S) has magnitude:

• The projection of S on the z-axis:

Sz = ms = ½

• There are two possible electron spin states:

ms = +½ “spin up”

ms = -½ “spin down”

• Spin is an intrinsic property of the electron and is not connected with orbital motion.

• Complete specification of an electron in an atom requires 4 quantum numbers (n,,m,ms) – as s is fixed.

2

31S

ss

5.6 Properties of Atomic Orbitals

1. Shape – determined by the angular wavefunction Y,m(,)

1s orbital (n = 1, = 0, |J| = 0)

• Normalized wavefunction:

• Has no angular dependence(spherically symmetric, depends only on r).

• All s orbitals are spherically symmetrical.

2s orbital ns orbital

• As n, orbitals expand – average radius r

0ar23

0

ea

11s1

+

2rer2s2 nrerns f

2p orbitals (n = 2, = 1, |J| = 2)

• There are 3 degenerate orbitals, with 3 different m values (0,1).

• e.g. m = 0 Jz = m = 0 2pz orbital (pointing along z-axis)

• m = 1 Jz = 2px, 2py orbitals

02ar25

0z ercos

a

1

24

1p2

++

+

--

-

2. Nodal Properties

a) Angular Nodes• s orbitals – have no angular nodes (spherically symmetric)• p orbitals – have one angular node (a nodal plane, where = 0)• No. of angular nodes = (s = 0, p = 1, d = 2 …)

b) Radial Nodes• s orbitals – have a maximum in at nucleus (r = 0)

• For other orbitals (p, d, f …), = 0 at nucleus due to angular nodes.

• All orbitals decay exponentially at large r values.• Radial nodes – nodal surfaces where = 0, with changing sign

either side of the node.• No. of radial nodes = n1 1s 0

2s 1 2p 03s 2 3p 1 3d 0

• Total number of nodes (angular + radial) = n1

Angular and Radial Nodes in

Angular Nodes Radial Nodes

-

+ = 0

2pz orbital

+

-

3s orbital

= 0

+

3. Radial Distribution Functions

• Electron density at point (r) in space:

• Probability of finding electron at point r:

• Radial Distribution Function – probability density of finding electron at distance r from nucleus:

• P(r)dr – probability of finding electron in a shell of thickness dr at distance r from the nucleus.

• For non-spherical orbitals (p, d, f …), use:

2rr

dP2

rr

22 rr4P r

22 rRrP r

Example: H(1s)

• Peak in RDF at r = a0 (the Bohr radius for n = 1).

• The delocalized electron is represented by a wave which has maximum probability at r = a0.

r

4r22

a0

Hydrogenic Ions

• 1e ions with nuclear charge Z.

• Max. in 1s RDF at

• As Z, orbitals contract (rmax).

• Orbital Energies

• As Z, orbitals more tightly bound (En – more negative).

Z

ar 0max

2

2

nn

ZE

5.7 Many-Electron Atoms

• For atoms with 2 or more electrons, the Schrödinger Eqn. must include KE terms for all electrons and PE terms for all e-e and e-n interactions.

• Example: He (2 electrons)

2 terms in

3 terms in

S.E.

where

Z = +2e

-e(1)r1

r2

r12

-e(2)

E

r

e

r

e

r

e

mm ee 120

2

20

2

10

222

221

2

44

2

4

2

22

2

2

2

2

2

22

iiii

zyx

T̂V̂

• For N electrons: N = (r1,r2,r3 … rN)

• For > 1 e, S.E. cannot be solved analytically – though good numerical solutions may be obtained from computer calculations.

Approximations must be made.

The Orbital Approximation

• The total wavefunction (N) for the N-electron atom is approximated by the product of N 1-e orbitals similar to those of hydrogenic ions:

N = (r1,r2,r3 … rN) = (r1)(r2)(r3)…(rN)

Orbital Energies for Many-Electron Atoms

• For H and hydrogenic ions, E depends only on n.– e.g. 3s, 3p and 3d orbitals are degenerate.

• For many-electron atoms, E depends on n and .

• In shell n, En as i.e. E(ns) < E(np) < E(nd) …

• Why ?

Effective Nuclear Charge

• An electron at distance r from nucleus experiences an effective nuclear charge:

Zeff < Z

(Z = actual charge on nucleus = atomic number)

• Electrons inside a sphere of radius r repel the electron and shield (screen) it from the nucleus:

Zeff = Z

( = shielding constant).

Properties of

• For large R, behaves like 2 independent H(1s) AOs.

• For small R, there is significant overlap between the AOs. • Destructive interference between atomic

wavefunctions (AOs) A and B depletion

of electron density between the nuclei

(decrease of and 2).

• The depletion of electron density between

the nuclei leads to decreased e-n attraction:

E() > E(A,B)

is an antibonding MO.

is cylindrically symmetrical about the internuclear (A-B) axis

labelled as a MO (or sometimes *, as it is antibonding).

A B

R

A -B

+

• Electrons in orbitals with low n (core orbitals) have greater electron density (higher probability of finding the electron) close to the nucleus experience higher Zeff lower energy.

• Electrons in orbitals with same n, but

lower , have peaks in RDF which are

closer to the nucleus better at shielding

other electrons and better at penetrating

shielding:

Zeff ns > np > nd …

E ns < np < nd …

• Orbitals with same n and are still

degenerate

e.g. E(2px) = E(2py) = E(2pz)

Electronic Configurations

• To determine the electronic configuration of an atom (how the electrons are distributed among the atomic orbitals), we need to know:

– The available orbitals and their relative energies.– The number of electrons.– The rules for filling the orbitals.

Rules for Filling Orbitals

1. The Pauli Exclusion Principle

No two electrons in a particular atom or ion can have the same values of all 4 quantum numbers (n,,m,ms).

No more than 2e can occupy a given orbital

If 2e are in the same orbital (same n,,m), they must have opposite (= paired) spins.

H 1s1

He 1s2

Li 1s22s1

X

2. The Aufbau (Building-up) Principle

• Add available electrons into orbitals – starting from the lowest in energy, put maximum of 2e in each orbital.

• General order of orbital occupation:

1s < 2s < 2p < 3s < 3p < 4s ~ 3d < 4p < 5s < 4d …

• Usually completely fill each sub-shell () before starting to fill another.

– Exceptions – e.g. in the 1st row transition metals:

Cr [Ar]4s13d5

Cu [Ar]4s13d10

The Aufbau Principle

n

The 3d and 4s orbitals are close in energy for the first transition metalseries.

For heavier atoms (higher atomic number Z), E(3d) < E(4s).

As Z, the sub-shells of the inner (lower n) orbitals become approx. degenerate

Variation of Orbital Energy with Z

3. Spatial Separation

• Electrons occupy different orbitals (m) of a given sub-shell () before starting to pair electrons.

e.g. N [He]2s22p3 = [He]2s22px12py

12pz1

• Electrons in different orbitals on average are further apart lower e-e repulsion lower E.

4. Hund’s Rule (of Maximum Multiplicity)

• Provided rules 1 and 2 are satisfied, an atom in its ground state adopts the configuration with the maximum number of unpaired (= parallel) spins.

e.g. C [He]2s22p2 = [He]2s22px12py

1

(equivalent to … 2px12pz

1 and 2py12pz

1).

Parallel Spins Antiparallel Spins

• Electrons with parallel spins () tend to avoid each other better than those with antiparallel spins () reduced e-e repulsion lower E.

• This is known as spin correlation.

6. Bonding in Molecules

Exact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H2

+.

Full Hamiltonian operator for H2+ :

e

HA HBRAB

rA rB

AB0

2

BA0

22e

e

22B

2A

p

2

R4

e

r

1

r

1

4

e

m2m2H

ˆ

nuclear K.E. e-n attraction n-n repulsionelectron K.E.

KE PE

6.1 The Born-Oppenheimer Approximation

• Nuclei are much heavier (thousands of times) than electrons.

they move much more slowly.

• In the Born-Oppenheimer approximation, nuclei are treated as being stationary.

• Consider motion of electrons relative to fixed nuclei.

• Total wavefunction () is split into the product of electronic and nuclear wavefunctions:

= e.n

Example 1: H2+

• Fix RAB ( R).• Schrödinger Equation for electronic

motion:

• Note: although the electronic SE is solved for fixed RAB (= R), the solutions (e,Ee) depend on the value of R.

e

HA HBR

rA rB

R4

e

r4

e

r4

e

m2H

0

2

B0

2

A0

22e

e

2

e

ˆ

electron K.E. e-n attraction n-n repulsion(constant)

2e

2

2e

2

2e

22e

zyx

eeee EH ˆ

Example 2: H2

• Electronic Hamiltonian:

• SE for molecules containing other elements (atomic no. Z):

– e-n attraction terms e2 Ze2 – n-n repulsion terms +e2 +Z2e2.

120

2

2B2A1B1A0

222

21

e

2

r

1

R

1

4

e

r

1

r

1

r

1

r

1

4

e

m2H

ˆ

electron K.E. e-n attraction n-n repulsion

2i

2

2i

2

2i

22i

zyx

e-e repulsion

e1

HA HBR

r1A r1B

e2

r2Br2A

r12

6.2 Potential Energy Curves

• For a diatomic molecule, solving the electronic SE for different fixed positions of the nuclei (i.e. fixed inter-nuclear distances, R) gives the molecular potential energy curve V(R).

• Re = equilibrium bond length

• De = PE well depth = dissociation energy.

• As R , V(R) 0 (dissociation limit).

• This can be extended to larger molecules:– e.g. for a general triatomic molecule we get a potential energy

surface V(R1,R2,). R1 R2

V(R)

RRe

-De

0

• How can we determine and E for an electron in a molecule?• SE can be solved exactly for (within the B-O approximation) but it is

complicated and SE cannot be solved exactly for > 1e.• We need to make more approximations Molecular Orbital theory.

6.3 Molecular Orbital Theory

• Electrons in molecules have spatial distributions which are described by 1e wavefunctions called molecular orbitals (MOs) – analogous to atomic orbitals (AOs).

• Let represent a MO and an AO.

• MOs are spatially delocalised over the molecule.

• Probability of finding electron at point r in space in MO :

P(r) = | (r)|2d

Linear Combination of Atomic Orbitals (LCAO) Approximation

• Construct MOs () as linear combination of AOs ():

• ci = coefficients (numbers) = contribution of ith AO to the MO.

• N AOs N MOs.

• Justification– When electron is close to one nucleus (A) it experiences an electrostatic

(Coulomb) attraction that is greater than that to B. MO wavefunction () close to A, resembles an atomic orbital centred

on A (A).

N

1iiic

Example: H2+ and H2

• MOs formed as linear combinations of H(1s) AOs.

• 2 AOs (A,B) 2 MOs (+,)

+ = N+(A + B) in-phase (bonding)

= N(A B) out-of-phase (antibonding)

A

A

B

B

R

Properties of +

• For large R, + behaves like 2 independent H(1s) AOs.

• For small R, there is significant overlap between the AOs.

• Constructive interference between atomic

wavefunctions (AOs) A and B build-up

of electron density between the nuclei

(increase of and 2).

R

A B

R

A B

+

• The accumulation of electron density between the nuclei leads to increased e-n attraction (electrons interact strongly with both nuclei):

E(+) < E(A,B)

+ is a bonding MO.

Covalent bonding – due to sharing of electrons.

+ is cylindrically symmetrical about the internuclear (A-B) axis

labelled as a MO:

+

side view

+ +

cross section

Properties of

• For large R, behaves like 2 independent H(1s) AOs.

• For small R, there is significant overlap between the AOs.

• Destructive interference between atomic

wavefunctions (AOs) A and B deplete

of electron density between the nuclei

(decrease of and 2).-B

A B

R

A

node

• The depletion of electron density between the nuclei leads to decreased e-n attraction.

E() > E(A,B)

is an antibonding MO.

is cylindrically symmetrical about the internuclear (A-B) axis

labelled as a * MO (* denotes antibonding character):

+

nodal plane

Normalization of the MO Wavefunction

• What are the normalization constants (N+ and N)?

+ = N+(A + B)

= N(A B)

• Normalization condition:

where = + or

• e.g. for +

atomic orbitals are normalized:

define overlap integral between orbitals A and B:

normalization constant:

1dτdτ 2 *

1dτdτ 2BA

2 N

1d2dd BA2B

2A

2 N

1dd 2B

2A

dS BAAB

ABS12

1

N

• Similarly, for we get:

• Since 0 < SAB < 1, this means that N > N+ .

• Similar arguments can be used to show that the antibonding orbital

() is raised in energy by more than the bonding orbital (+) is lowered in energy when a bond is formed.

• Note: if we ignore overlap, then N+ = N = 1/2

ABS12

1

N

6.4 Molecular Orbital Diagrams

Example 1. H2+

• Ground state configuration: (+)1

1e in bonding orbital bound state.

• Excited state configuration: ()1

1e in antibonding orbital unbound state.

• Energies defined relative to dissociation

H2+ H + H+

A=H(1sA) B=H(1sB)

+ = 1

= 2*

Energy

V(R)

Re

-De

0

+

R

Example 2. H2

• Ground state configuration: (+)2

2e in bonding orbital bound state.

• H2 has a shorter stronger bond than H2+ (more bonding electrons).

• Note: De(H2) < 2De(H2+) due to e-e repulsion.

A=H(1sA) B=H(1sB)

+ = 1

= 2*

Energy

Re/pm De/kJ mol-1

H2+ 106 255

H2 74.1 430

Example 3. He2

• Ground state configuration: (+)2()2

No net covalent bonding (bonding and a-b contributions cancel out).

Only weak dispersion forces hold He atoms together (see Intermolecular Forces lectures).

• He2+ has the configuration (+)2()1 and does have net covalent

bonding.

A=He(1sA) B=He(1sB)

+ = 1

= 2*

Energy

6.5 MOs for 2nd Row Diatomic Molecules

• Valence AOs = 2s, 2px, 2py, 2pz

• Core AOs = 1s (not involved in bonding)

• Linear combinations of 2s orbitals:

= N (A(2s) B (2s)) 1(2s) and 2 *(2s) (as for H2)

• Some combinations are not allowed – zero net overlap = “orthogonal orbitals”:

+ +

A(2s) + B (2px)

+

+

A(2px) + B (2pz)

• The 2p orbitals interact to give -type (0 angular nodes with respect

to the molecular axis) and -type (1 angular node) MOs, which can

be bonding (,) or antibonding (*,*).

3

4*

1

2*

2p 2p

4*

1

2*

3

2pz-2pz overlap > overlap 3-4* splitting > 1-2*.

6.6 Molecular Electronic Configurations

• Follow same rules as for atomic electronic configurations (Aufbau principle, Hund’s rule etc.).

• Note: the ordering of MOs can vary – e.g. the 3(2p) and 1(2p) MOs are sometimes reversed:

3 < 1 (O2, F2)3 > 1 (B2, C2, N2)

Due to 2s-2p mixing (hybridization) whichraises 3 and lowers2*.

As Z, the 2s-2p separationincreases, so s-p mixing is weaker.

6.7 Bond Order

• The strength of a covalent bond is the net outcome of occupying bonding and antibonding orbitals.

• Bond Order b = ½(NB N*)

NB = number of electrons in bonding MOs

N* = number of electrons in antibonding MOs

• ExamplesNB N* b

H2+ 1 0 0.5

H2 2 0 1

He2 2 2 0

N2 8 2 3

O2 8 4 2

F2 8 6 1

F2+ 8 5 1.5

6.8 Paramagnetic Molecules

• Even with even numbers of electrons, certain molecules are paramagnetic (i.e. they have unpaired electron spins).

• e.g. O2– Ground state electronic configuration: (1)2 (2*)2 (3)2 (1)4 (2*)2

– There are 2 electrons in the antibonding pair of 2* orbitals.

– From Hund’s rule – the lowest energy configuration has the most unpaired spins = 2.

– The magnetic effects of these 2 electrons do not cancel out.

3

4*

1

2*

2p 2p

6.9 Heteronuclear Diatomic Molecules

• Generally AOs of different atoms have different energies – depending on relative electronegativities of the atoms.

• The MO closest in energy to anAO has more character(greater LCAO coefficient) ofthat AO.

bonding and antibonding orbitalsusually have opposite characters.

• e.g. HF

= 0.19H(1s)+0.98F(2pz)

* = 0.98H(1s)0.19F(2pz)

(1s+2p)

*(1s-2p)

HF

non-bondingF(2px,2py)AOs.