Inorganic ChemistryInorganic Chemistry

Bonding and Coordination Chemistry

Books to followInorganic Chemistry by Shriver & Atkins

Physical Chemistry: Atkins

Bonding in s,p,d systems: Molecular orbitals of diatomics, Bonding in s,p,d systems: Molecular orbitals of diatomics, d-orbital splitting in crystal field (Od-orbital splitting in crystal field (Ohh, T, Tdd).).

Oxidation reduction: Metal Oxidation states, redox Oxidation reduction: Metal Oxidation states, redox

potential, diagrammatic presentation of potential data.potential, diagrammatic presentation of potential data.

Chemistry of Metals: Coordination compounds (Ligands & Chemistry of Metals: Coordination compounds (Ligands &

Chelate effect), Metal carbonyls – preparation stability andChelate effect), Metal carbonyls – preparation stability and

application.application.

Wilkinson’s catalyst – alkene hydrogenationWilkinson’s catalyst – alkene hydrogenationHemoglobin, myoglobin & oxygen transportHemoglobin, myoglobin & oxygen transport

CHEMICAL BONDING:A QUANTUM LOOK

H2 // Na+Cl- // C60

Failure of Classical Mechanics• Total energy, E = ½ mv2 + V(x)

• p = mv ( p = momentum )• E = p2/2m + V(x) ……… . . Eq.1

“ A moving ball I know it all ”

• Newton’s second law is a relation between the acceleration d2x/dt2 of a particle and the force F(x) it experiences.

• Therefore, v = p/m• Or, p• = F(x)

“ Hit the ball hard, it will move fastHit it soft, it will move slow”

• Continuous variation of energy is possible.

Macroscopic World: “Classical Mechanics - the God”

A young Max Planck was to

give a lecture on radiant heat.

When he arrived he inquired

as to the room number for the

Planck lecture. He was told,

"You are much too young to be

attending the lecture of the

esteemed professor Planck."

Max Planck E = h1900 German physicist

“Each electromagnetic oscillator is limited to discrete values and cannot be varied arbitrarily”

Plank had applied energy quantization to the oscillators in the blackbody but had considered the electromagnetic radiation to be wave.

PHOTOELECTRIC EFFECT

When UV light is shone on a metal plate in a vacuum, it emits charged particles (Hertz 1887), which were later shown to be electrons by J.J. Thomson (1899).

As intensity of light increases, force increases, so KE of ejected electrons should increase.

Electrons should be emitted whatever the frequency ν of the light.

Classical expectations

Hertz J.J. Thomson

I

Vacuum chamber

Metal plate

Collecting plate

Ammeter

Potentiostat

Light, frequency ν

Maximum KE of ejected electrons is independent of intensity, but dependent on ν

For ν<ν0 (i.e. for frequencies below a cut-off frequency) no electrons are emitted

Actual results:

Photoelectric Effect.

(i) No electrons are ejected, regardless of the intensity of the radiation, unless its frequency exceeds a threshold value characteristic of the metal.

(ii) The kinetic energy of the electron increases linearly with the frequency of the incident radiation but is independent of the intensity of the radiation.

(iii) Even at low intensities, electrons are ejected immediately if the frequency is above the threshold.

Major objections to the Rutherford-Bohr model • We are able to define the position and velocity of each electron precisely.

• In principle we can follow the motion of each individual electron precisely like planet.

• Neither is valid.

Werner HeisenbergHeisenberg's name will always be associated with his theory of quantum mechanics, published in 1925, when he was only 23 years.

• It is impossible to specify the exact position and momentum of a particle simultaneously.

• Uncertainty Principle. x p h/4 where h is Plank’s

Constant, a fundamental constant with the value 6.62610-34 J s.

Einstein

• KE 1/2mv2 = h- is the work function

• h is the energy of the incident light.

• Light can be thought of as a bunch of particles which have energy E = h. The light particles are called photons.

h = ½ mv2 +

If light can behave as particles,why not particles

behave as wave?

Louis de BroglieThe Nobel Prize in Physics 1929 French physicist (1892-1987)

• E = mc2 = h • mc2 = h• p = h / { since = c/} = h/p = h/mv

• This is known as wave particle duality

Wave Particle Duality

Light and matter exhibit Light and matter exhibit wave-particle dualitywave-particle duality

Relation between wave and particle propertiesRelation between wave and particle propertiesgiven by the given by the de Broglie relationsde Broglie relations

Photoelectric effectPhotoelectric effect

Flaws of classical mechanicsFlaws of classical mechanics

Heisenberg uncertainty principleHeisenberg uncertainty principle limits limitssimultaneous knowledge of conjugate variablessimultaneous knowledge of conjugate variables

The state of a system in classical mechanics is defined by specifying all the forces acting and all the position and velocity of the particles.

Wave equation?Schrödinger Equation.

• Energy Levels

• Most significant feature of the Quantum Mechanics: Limits the energies to discrete values.

• Quantization.

1887-1961

For every dynamical system, there exists a wave function Ψ

that is a continuous, square-integrable, single-valued function

of the coordinates of all the particles and of time, and from

which all possible predictions about the physical properties of

the system can be obtained.

The wave function

If we know the wavefunction we know everything it is possible to know.

Square-integrable means that the normalization integral is finite

Time period = T, Velocity = v, v = /T, Frequency, = 1/T, v =

Derivation of wave equation

If the wave is moving to the right with velocity ‘v’ at time ‘t’

x

y

y(x,t) = A sin 2/(x-vt)

= v/

A wave eqn.is born

• y = A sin 2(x/v - t)

• Differentiating y W.R.T x, keeping t constant

2y/x2 + (4y = 0

• In three dimension the wave equation becomes:

x2 + y2 + z2 + (4

· It can be written as 2

· We have = h/mv

2m2v2h

· E = T + V or T = (E-V) (E = total energy)

· V = Potential energy, T = Kinetic energy

· T = 1/2 mv2 = m2v2/2m

· m2v2 = 2m(E-V)

2m/ h(E - V)

• This can be rearranged as

h2/m) 2 + V}· H h2/m)2 + V) Hamiltonian operator

2y/x2 + (4y = 0

{(-h2/82m)(2/x2 + 2/y2 + 2/z2) + V} = E

How to write Hamiltonian for different systems?

{(-h2/82m)2 + V} = E

• Hydrogen atom:

• KE = ½ m (vx2 + vy

2 + vz2)

• PE = -e2/r, (r = distance between the electron and the nucleus.)

• H = {(-h2/82m) 2 –e2/r}

2 + (82 m/h2)(E+e2/r) = 0

• If the effective nuclear charge is Ze

• H = {(-h2/82m )2 –Ze2/r}

r

+Ze

-e

H2+ Molecule

e (x,y,z)

ra rb

A RAB B

the wave function depends on the coordinates of the two nuclei, represented by RA and RB, and of the single electron, represented by

r1.

• PE = V = -e2/ra – e2/rb + e2/Rab

• H = (-h2/82m)2 + ( – e2/ra - e2/rb + e2/Rab)

• The Wave equation is

2 + (82 m/h2) (E+ e2/ra + e2/rb – e2/Rab) = 0

e (x,y,z)

ra

rbA Rab

B

H2+

{(-h2/82m)2 + V} = E

Born-Oppenheimer approximation

V = -e2/40[1/ra+1/rb-1/Rab]

• V = -2e2/r1 – 2e2/r2 + e2/r12

• H = (-h2/82m) (12 + 2

2) + V

• The Wave equation is

• (12 + 2

2 ) + (82 m/h2)(E-V) = 0

e1 (x1, y1, z1)

r12 r1

e2 (x2, y2, z2) Nucleus (+2e)

He Atom

r2

{(-h2/82m)2 + V} = E

• PE = V = ?

• H = (-h2/82m)(12 + 2

2) + V

• The Wave equation is

• (12 + 2

2 ) + (82 m/h2)(E-V) = 0

Rab

ra1 rb2

A B

H2

e1 (x1, y1, z1) r12 e2 (x2, y2, z2)

ra2

rb1

V = -e2/40[1/ra1+1/rb1 + 1/ra2 +1/rb2 -1/r12 -1/Rab]

attractive potential energy Electron-electron repulsion

Internuclear repulsion

An electron moving along x-axis in a field V(x)

V=0

a

x =0 x =a

Particle in a box

• Applying Boundary conditions: = 0 at x = 0 C = 0

= D sin kx

d2 /dx2 + 82 m/h2 (E-V) = 0

Assume V=0 between x=0 &

x=a

Also = 0 at x = 0 & ad2/dx2 + [82mE/h2] = 0

V=0

a

x =0 x =a

d2/dx2 + k2 = 0 where k2 = 82mE/h2

Solution is: = C cos kx + D sin kx

= D sin kx • Applying Boundary Condition: = 0 at x = a, D sin ka = 0

• sin ka = 0 or ka = n,

• k = n/a

• n = 0, 1, 2, 3, 4 . . .n = D sin (n/a)x

• k2 = 82m/h2[E] or E = k2h2/ 82m

• E = n2 h2/ 8ma2 k2= n2 2/a2

• n = 0 not acceptable: n = 0 at all x

• Lowest kinetic Energy = E0 = h2/8ma2

V=0

a

x =0 x =a

An Electron in One Dimensional Box

n = D sin (n/a)x

• En = n2 h2/ 8ma2

• n = 1, 2, 3, . . .

• E = h2/8ma2 , n=1

• E = 4h2/8ma2 , n=2

• E = 9h2/8ma2 , n=3

Energy is quantized

V = V = a

x = 0 x = a

Characteristics of Wave Function: What Prof. Born Said

• Heisenberg’s Uncertainty principle: We can

never know exactly where the particle is.

• Our knowledge of the position of a particle can never be absolute.

• In Classical mechanics, square of wave

amplitude is a measure of radiation intensity

• In a similar way, 2 or * may be related

to density or appropriately the probability of

finding the electron in the space.

The wave function is the probability amplitude

*2

Probability density

The sign of the wave function has not direct physical significance: the positive and negative regions of this wave function both corresponds to the same probability distribution. Positive and negative regions of the wave function may corresponds to a high probability of finding a particle in a region.

• Let (x, y, z) be the probability function,

d = 1 Let (x, y, z) be the solution of the wave equation

for the wave function of an electron. Then we may anticipate that (x, y, z) 2 (x, y, z)

• choosing a constant in such a way that is converted to = (x, y, z) = 2 (x, y, z) 2 d = 1

Characteristics of Wave Function: What Prof. Born Said

The total probability of finding the particle is 1. Forcing this condition on the wave function is called normalization.

2 d = 1 Normalized wave function

• If is complex then replace 2 by *

• If the function is not normalized, it can be done by multiplication of the wave function by a constant N such that

• N2 2 d = 1

• N is termed as Normalization Constant

Acceptable wave functions

• For electron bound to an atom/molecule, the wave function must be every where finite, and it must vanish in the boundaries

• Single valued

• Continuous

• Gradient (d/dr) must be continuous

• *d is finite, so that can be normalized

• Stationary States

• E = Eigen Value ; is Eigen Function

The wave equation has infinite number of solutions, all of which do not corresponds to any physical or chemical reality.

Need for Effective Approximate Method of Solving the Wave Equation

• Born Oppenheimer Principle.

• How can we get the most suitable approximate wave function?

• How can we use this approximate wave function to calculate energy E?

Operators“For every dynamical variables there is a corresponding operator”

Energy, momentum, angular momentum and position coordinates

Operators Symbols for mathematical operation

Eigen values

The permissible values that a dynamical variable may have are those given by = a- eigen function of the operator that corresponds to the observable whose permissible values are a -operator

- wave functiona - eigen value

If performing the operation on the wave function yields original function multiplied by a constant, then is an eigen function of the operator

= a

= e2x and the operator = d/dx

Operating on the function with the operator

d /dx = 2e2x = constant.e2x

e2x is an eigen function of the operator

For a given system, there may be various possible values.

As most of the properties may vary, we desire to determine the average or expectation value.We know

= aMultiply both side of the equation by *

* = *aTo get the sum of the probability over all space * d = *a da – constant, not affected by the order of operation

Removing ‘a’ from the integral and solving for ‘a’

a = * d/ * d

cannot be removed from the integral.

a = < >/ < >

Variation Method: Quick way to get E

• H = E H = E = E • If is complex,

• E = *H d/ * d• E= H / ……(4)

• Bra-Ket notation

• Given any , E can be calculated.

• If the wave function is not known, we can begin by educated guess and use Variation Theorem.

1 E1

2 E2

What does E = H / tell us ?

“If a trial wave function is used to calculate the energy, the value calculated is never less than the true energy” – Variation Theorem.

1 E1

2 E2

The Variation Theorem tells that

• E1 , E2 Eg, Eg true energy of the ground state

• IF, E1 E2,

• Then E2 and 2 is better approximation to the energy

and corresponding wave function 2 to the true wave

function

• We can chose a whole family of wave function at the same time, like trial function with one or more variable parameters C1, C2, C3,….

• Then E is function of C1, C2, C3 …….etc.

• C1, C2, C3 …. etc. are such that E is minimized with respect to them.

• We will utilize this method in explaining chemical bonding.

Variation Method: The First Few Steps

Chemical Bonding

• Two existing theories,

• Molecular Orbital Theory (MOT)

• Valence Bond Theory (VBT)

Molecular Orbital Theory

• MOT starts with the idea that the quantum mechanical principles applied to atoms may be applied equally well to the molecules.

H-CC-H

MOT: We can write the following principles

Describe Each electron in a molecule by a certain wave function - Molecular Orbital (MO).

Each is defined by certain quantum numbers, which govern its energy and its shape.

Each is associated with a definite energy value.

Each electron has a spin, ± ½ and labeled by its spin quantum number ms.

When building the molecule- Aufbau Principle (Building Principle) - Pauli Exclusion Principle.

Simplest possible molecule:H2

+ : 2 nuclei and 1 electron.

• Let the two nuclei be labeled as A and B & wave functions as A & B.

• Since the complete MO has characteristics separately possessed by A and B,

= CAA + CBB

• or = N(A + B)

= CB/CA, and N - normalization constant

This method is known as Linear Combination of Atomic Orbitals or LCAO

A and B are same atomic orbitals except for

their different origin.

• By symmetry A and B must appear with

equal weight and we can therefore write

• 2 = 1, or = ±1

• Therefore, the two allowed MO’s are

= A± B

For A+ B we can now calculate the energy

• From Variation Theorem we can write the energy function as

• E = A+B H A+B/A+B A+B

A+B H A+B = A H A +

• B H B +

• A H B +

• B H A

• = A H A + B H B +2AH B

Looking at the numerator: E = A+B H A+B/A+B A+B

= A H A + B H B + 2AH B

Numerator = 2EA + 2

ground state energy of a hydrogen atom. let us call this as EA

A H B = B H A =

= resonance integral

Physical Chemistry class test answer scripts will be shown to the students on 3rd March (Tuesday) at 5:30 pm in

Room C-306: Sections 11 and 12

• A+B A+B = A A +

• B B +

• A B +

• B A

• = A A + B B + 2A B

Looking at the denominator:E = A+B H A+B/A+B A+B

= A A + B B + 2A B

A and B are normalized,

so A A = B B = 1

A B = B A = S, S = Overlap integral.

Denominator = 2(1 + S)

E+ = (EA + )/ (1 + S)

Also E- = (EA - )/ (1 – S)

Summing Up . . . E = A+B H A+B/A+B A+B

Numerator = 2EA + 2

Denominator = 2(1 + S)

S is very small Neglect S

E± = EA ±

Energy level diagram

EA -

EA +

BA

Rules for linear combinationRules for linear combination

1. Atomic orbitals must be roughly of the same energy.

2. The orbital must overlap one another as much as possible- atoms must be close enough for effective overlap.

3. In order to produce bonding and antibonding MOs, either the symmetry of two atomic orbital must remain unchanged when rotated about the internuclear line or both atomic orbitals must change symmetry in identical manner.

Linear combination of atomic orbitalsLinear combination of atomic orbitals

Rules for the use of MOsRules for the use of MOs * When two AOs mix, two MOs will be produced * Each orbital can have a total of two electrons (Pauli principle) * Lowest energy orbitals are filled first (Aufbau principle) * Unpaired electrons have parallel spin (Hund’s rule)

Bond order = ½ (bonding electrons – antibonding electrons)

A B

A B

AB = N(cA A + cBB)

Linear Combination of Atomic Orbitals (LCAO)Linear Combination of Atomic Orbitals (LCAO)

2AB = (cA

2 A

2 + 2cAcB A B + cB2 B

2)

Overlap integral

The wave function for the molecular orbitals can be approximated The wave function for the molecular orbitals can be approximated by taking linear combinations of atomic orbitals.by taking linear combinations of atomic orbitals.

Probability density

c – extent to which each AOcontributes to the MO

cA = cB = 1

+. +. . .+

bondingg

Amplitudes of wave functions added

g = N [A + B]

Constructive interferenceConstructive interference

2AB = (cA

2 A

2 + 2cAcB A B + cB2 B

2)

electron density on original atoms,electron density on original atoms,

density between atomsdensity between atoms

The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei.

The energy of the molecule is lower

Nuclei are shielded from each other

Amplitudes of wave functions

subtracted.

Destructive interferenceDestructive interferenceNodal plane perpendicular to the Nodal plane perpendicular to the H-H bond axis (en density = 0) H-H bond axis (en density = 0) Energy of the en in this orbital is Energy of the en in this orbital is higher.higher.

+. -. ..

node

antibondingu = N [A - B]

cA = +1, cB = -1 u

+ -

A-B

The electron is excluded from internuclear region The electron is excluded from internuclear region destabilizing destabilizing

AntibondingAntibonding

When 2 atomic When 2 atomic orbitalsorbitals combine there are 2 combine there are 2resultant resultant orbitalsorbitals..

low energy bonding orbitallow energy bonding orbital

high energy high energy antibondingantibonding orbital orbital1sb 1sa

1s

*

E1s

MolecularMolecular orbitalsorbitals

EgEg. s . s orbitalsorbitals

Molecular potential energy curve shows the variation of the molecular energy with internuclear separation.

Looking at the Energy Profile• Bonding orbital

• called 1s orbital

• s electron

• The energy of 1s orbital

decreases as R decreases

• However at small separation, repulsion becomes large

• There is a minimum in potential energy curve

11.4 eV

109 nm

HH22

Location ofBonding orbital4.5 eV

LCAO of n A.O n M.O.

The overlap integral

dS BA*

The extent to which two atomic orbitals on different atom The extent to which two atomic orbitals on different atom overlaps : the overlap integraloverlaps : the overlap integral

S > 0 Bonding S < 0 anti

S = 0 nonbondingBond strength depends on the Bond strength depends on the

degree of overlapdegree of overlap

Homonuclear Diatomics

• MOs may be classified according to:

(i) Their symmetry around the molecular axis.

(ii) Their bonding and antibonding character.

1s 1s* 2s 2s* 2p y(2p) = z(2p)

y*(2p) z*(2p)2p*.

dx2-dy2 and dxy

Cl4Re ReCl42-

A

Bg- identical under inversion

u- not identical

Place labels Place labels gg or or uu in this diagram in this diagram

g

g

u

u

First period diatomic moleculesFirst period diatomic molecules

1s2HE

nerg

yHH2

1s 1s

g

u*

Bond order = ½ (bonding electrons – antibonding electrons)

Bond order: 1

1s2, *1s2 HeE

nerg

yHeHe2

1s 1s

g

u*

Molecular Orbital theory is powerful because it allows us to predict whether molecules should exist or not and it gives us a clear picture of the of the electronic structure of any hypothetical molecule that we can imagine.

Diatomic molecules: The bonding in He2

Bond order: 0

Second period diatomic moleculesSecond period diatomic molecules

1s2, *1s2, 2s2

Bond order: 1

Li

Ene

rgy

LiLi2

1s 1s

1g

1u*

2s 2s

2g

2u*

1s2, *1s2, 2s2, *2s2

Bond order: 0

Be

Ene

rgy

BeBe2

1s 1s

1g

1u*

2s 2s

2g

2u*

Diatomic molecules: Homonuclear Molecules of the Second Period

SimplifiedSimplified

SimplifiedSimplified

Diamagnetic??

2g

2u*

3g

1u

1g*

3u*

MO diagram for BMO diagram for B22

Li : 200 kJ/molF: 2500 kJ/mol

Same symmetry, energy mix-the one with higher energy moves higher and the one with lower energy moves lower

2g

2u*

3g

1u

1g*

3u*

B BB2

2s 2s

2g

2u*

2p

2p

3g

3u*

1u

1g*

(px,py)

HOMO

LUMO

MO diagram for BMO diagram for B22

Paramagnetic

1g

1u

1g

1g

1u

1g

C2

DiamagneticParamagnetic ?X

1g

1u

1g

1g

1u

1g

Li2 to N2O2 and F2

General MO diagrams

Orbital mixing Li2 to N2

Bond lengths in diatomic molecules

Filling bonding orbitals

Filling antibonding orbitals

SummarySummaryFrom a basis set of N atomic orbitals, N molecular orbitals are From a basis set of N atomic orbitals, N molecular orbitals are constructed. In Period 2, N=8.constructed. In Period 2, N=8.

The eight orbitals can be classified by symmetry into two sets: 4 The eight orbitals can be classified by symmetry into two sets: 4 and 4 and 4 orbitals. orbitals.

The four The four orbitals from one doubly degenerate pair of bonding orbitals from one doubly degenerate pair of bonding orbitals and one doubly degenerate pair of antibonding orbitals.orbitals and one doubly degenerate pair of antibonding orbitals.

The four The four orbitals span a range of energies, one being strongly orbitals span a range of energies, one being strongly bonding and another strongly antibonding, with the remaining bonding and another strongly antibonding, with the remaining two two orbitals lying between these extremes. orbitals lying between these extremes.

To establish the actual location of the energy levels, it is necessary To establish the actual location of the energy levels, it is necessary to use absorption spectroscopy or photoelectron spectroscopy.to use absorption spectroscopy or photoelectron spectroscopy.

Distance between b-MO and AO

Heteronuclear Diatomics….

The energy level diagram is not symmetrical. The bonding MOs are

closer to the atomic orbitals which are lower in energy.

The antibonding MOs are closer to those higher in energy.

c – extent to which each atomicorbitals contribute to MO

If cAcB the MO is composed principally of A

HHFF

HF

1s 12s, 2p 7

=c1 H1s + c2 F2s + c3 F2pz

Largely nonbonding

2px and 2py

Polar

12 2214