Energy distribution, thermodynamic activities for friccohesity

Basic understanding of thermodynamics through adequate Friccohesity resonating energy transfer (FRET)

1. Basics of statistical thermodynamics
Mechanics: Study of position, velocity, force & energy
Classical Mechanics: Molecular Mechanics
Molecules are treated as rigid object or point, sphere, cube,...
Newton’s law of motion
Quantum Mechanics
Molecules are composed of electrons, nuclei, ...
Schrodinger’s equation  Wave function
Molecular boundaries: Geometries, stereoisomerism, space

2. Electric quadrupole: Orientational charge density
These dipoles affect alignment of MO

3. Statistical thermodynamics and mechanics
Thermodynamics study of relationships between macroscopic properties: V, P, T,
compressibility, KE = Statistical Mechanics
Statistical thermodynamics: Development of macroscopic properties as a consequence
of microscopic nature of system
Position & momenta of individual molecules as mechanical variables
Statistical thermodynamics is a link between microscopic properties & macroscopic
properties
microscopic
model is used
r
U
Statistical
Mechanics
mechanical variables
Thermodynamic Variables
P T V
RT
E
eqeK


Boltzmann distribution

4. Equilibrium macroscopic properties
Properties are consequence of average of individual molecules
Properties are invariant with time  Time average
Mechanical
Properties of
Individual
Molecules
position, velocity, energy, ...
Thermodynamic
Properties
temperature, pressure
internal energy, enthalpy,
Average
over
molecules
Average
over time
Statistical
thermodynamics
2 3 4
Quantum theory says -
•Each molecules can have only discrete values of energies
Evidence
•Black-body radiation
•Planck distribution
•Heat capacities
•Atomic & molecular spectra
•Wave-particle duality
Energy
Levels

5. Distribution of molecular States: Energy states
Configuration: At any instance, there may be n
o
molecules at e
0
, n
1
molecules at e
1
, n
2

molecules at e
2
, …
 {n
0
, n
1
, n
2
…} configurations
e
0
e
2
e
1
e
3
e
4
e
5
{ 3,2,2,1,0,0}
•Statistical
thermodynamics and
states of energy
•Determining the
equilibrium in energy
distribution depending
the states developed
in a process

6. Thermodynamics beta or coldness
•Thermodynamic
beta is known
as
 
coldness, is
reciprocal of
thermodynamic
temperature
 of a
system:
•T temperature,
k
B
 
Boltzmann
constant
•It was introduced
in 1971 as
"coldness
function" by
 Ingo
Müller, based on
reciprocal
temperature
function
•Thermodynamic
beta has unit
reciprocal to that
of energy (SI unit,
reciprocal
 J
-1

7. Partition functions of energy as a state of entropy for irreversible
systems
•Energy flows
downhill increasing
entropy
•Irreversible process is
reversed
•It cannot return both
the system and
surroundings to
original conditions
•Entropy increases
with increasing in
number of energy
levels

8. Macrostates and Microstates
•Physical system composed of N identical particles in a space of volume V
•N~10
23
is a thermodynamic limit having N → ∞, V → ∞
•However, the V/N ratio gives a finite value.
•Within N~10
23
thermodynamic limit, the extensive properties of system are directly
proportional to either N or V
•However, the intensive properties remain independent of these (N or V).
•If the particles under consideration are non-interacting, the total energy E could be
equal to sum of energies ε
i of particles, i.e.,
E = 
i n
i ε
i
where n
i is number of particles with energy ε
i.
•Also, the total number of particles is
N = 
i n
i
•Hence a macrostate can be defined as specification of actual values of the system
parameters N,V,E.
5 canonical states
contributing to
normal states of
benzene

9. n
u
m
b
e
r

o
f

m
i
c
r
o
s
t
a
t
e
s
macrostates
only one type
of lattice exists
crystalline substance
NaCl,
several types of microstates
amorphous substances like
dendrimers, supramolecules
a
b S = k
B
ln W
S = k
B
ln W
W= # microstates = 05
S =k
B ln 5
S=1.38065x10
-23
x2.303x0.69897 S = k
B
ln 1
S = 0
W= # microstates = 01
1.380650 × 10
−23
 
J/K

10. Entropy & probability of infinite numbers of microstates
Entropy depicts probability of infinite numbers of microstates in macrostate system
defined by Boltzmann distribution law
S = k
B
ln W (eqn1)
S entropy, k
B Boltzmann distribution constant, W is number of microstates of energy
S = f(disorder or randomness) on heating from absolute zero to another temperature
with entropy change as under
Several W of macrostate systems are developed when, T, dipole moment (Debye) or
chemical environment is changed For crystalline substance W = 1 because a population
of similar microstates are developed hence eqn. 1
S = k
B ln 1 & log 1 = 0 so the S = 0
Had there been irreversible oscillation then the S could have not been equal to 0 as the
many canonical states could have been existed: Irreversible system
Sor
0
0 dT
T
C
SSS
T
p
T 


11. Distinguishable particles
NN
N
N
NNN
N
w
i
i
i






!
!
!!!
!
321
Distinguishable particles and  depicts total distribution terms
wkSln
Degenerate States: The g
j
states have the same energy
NN
N
g
N
NNN
gggN
w
n
j
j
n
j j
N
j
NNN j









1
1321
321
!
!
!!!
!
321

12. 
r
A
r
B
d
AB
d
2
AB
t
cross
section
area

BA
AB
B
AB
AAB CC
Tk
dNZ
2/1
28











contact
point
collision, stronger
cohesive forces
no collision, no cohesive forces
and frictional or kinetic forces
high kinetic
energy
nonbonding
bonding
E
E
0
0
r, nm
r,nm
Thermokinetic: kinetics forming new bonding

13. Statistical thermodynamics of enzyme Catalysis
Lock & key
mechanism
Entropy & tentropy
play critical role ???
Entropy, tentropy & IMMFT
play critical role ???

14. Partition Function
Z
eg
NN
Zeg
eg
eg
NN
j
j
j
j
j
j
n
j
j
n
j
j
j
j















FunctionPartition
1
1
Putting the value of Z from equation 2 in equation 1
(1)
(2)
(2)

15. •At microscopic level, there is a large number of independent possibilities where the
microstate (N,V,E) of system can be formulated.
•With a non-interacting system, since the total energy E consists of a simple sum of
the N single-particle energies
•There is obviously be a large number of different ways where individual is chosen
to make the total energy equal to E.
•For a specific volume V of system composed of specific number (N) of particles,
there is a large number of different ways where the total energy E of system is
distributed among N particles constituting the system.
•Each of these (different) ways specifies a microstate (or, complexion) of the given
system.
•Thus, corresponding to a particular macrostate, we have many accessible
microstates.
•At any time, t , the macrostate of the system is equally likely to be in any one of its
microstates which are consistent with the constrained conditions of the given
system.
•The assumption forms the very basis of statistical mechanics and is known as
postulate of “equal a priori probabilities” for all accessible microstates of system
Macrostates and Microstates

16. •Random collection of the systems where each corresponds to the same macroscopic
thermodynamic state but has different microstructure, is an ensemble.
•The average over a period for a system is equivalent to average over ensemble at
one instant of time.
•The system of interest contains large number of molecules (N)
•The number of imagined elements form the ensemble at one time must be large (M)
Ensemble

17. •Microcanonical ensembles corresponds to isolated systems which exchange neither
energy nor mass with one another and which therefore keep the total number of
particles N, volume V and the total energy E as constant.
•Canonical ensembles correspond to closed isothermal systems which exchange
energy but not mass with one another and which therefore keep N, volume V and
temperature T as constant.
Types of Ensembles
Microcanonical
ensemble
Canonical ensemble

18. T
dq
ds
rev

T
dq
ds
Entropy is a state function defined by (per unit mass)
The second law defines entropy as a state function and permits following statements:
a) For a reversible process, the entropy of universe remains constant.
b) For an irreversible process, the entropy of universe increases
c)Is it due to distribution states like temperature, volume, energy.
Thus, a more general definition of entropy is
Second Law does not address anything specifically about the entropy of system, but
only that of the universe (system + surroundings).
Reversible process, entropy is represented by energy states of
compartments

19. When a substance passes from state 1 to state 2, the change in entropy is found by
integrating


2
1
12
T
dq
sss
rev
p
dp
R
T
dT
c
T
dq
ds
p

20. Calculate the change in entropy when 5 g of water at 0 C are raised to 100 C and then
converted to steam at that temperature. We will assume the latent heat of vaporization
is 2.253x106 J kg-1 at 100 C. (use extensive forms – capital letters – since mass is
involved.)
Step 1: Compute increase in entropy resulting from increasing the water temperature
from 0 to 100 C:


373
273
2733731
TdQSSS
rev
dQ
rev
= m(dq
rev
) = mc
w
dT where m is mass and c
w
is specific heat of water. It we
assume c
w
to be constant at 4.18x10
3
J kg
-1
K
-1
then



373
273
113
2733731 )1018.4)(005.0( TdTKkgJkgSSS
= 20.9 ln(373/273) = 6.58 J K
-1

21. Statistical thermodynamics exponential electronic changes: Cis or
natural fat non-cancerous while Trans non-bending, Cancerous
Both H atoms bind to the same side: Spacing of H in one side facilitates Cis fats to
bend
Bending allows Cis fats to bind with other chemicals & enzymes: Cis fats can’t pack
into crystal form at NTP
No bending occurs & don’t bind with enzymes type molecules: Trans fats straight &
can pack into crystal form & solidify at NTP: Cause cancer
Molecular shape is vital to its function, in similar way as shape of key is useful for lock
operation
Electronic energy distribution
New pattern of energy alignment develop
activities on dimensional distribution of
electron
Electronic energy distribution
Counterbalancing new pattern of energy
alignment on dimensional distribution of
electron as standing wave

22. Thermodynamics to kinetic: Cross section σ
AB

AB
A
B
radii
•N
A
 
Avogadro constant, σ
AB
 
reaction cross
section, k
B 
Boltzmann's constant
•Reactants A & B collide & their nuclei come
closer than a certain distance
•Area around A where it collides with an
approaching B molecule is cross section (σ
AB)
of reaction
•This area corresponds to a circle of radius σ
AB

as sum of their radii as a sphere
•The A molecule sweeps a volume πσ
2
AB.c
A per
sec as it moves ahead
•The c
A 
is an average velocity of particle & is
calculated as under
μ
AB = reduced
mass
 of reactants
AB
B
ABA
Tk
NZ


8

A
B
A
m
Tk
c

8



23. The water & ice have different densities
Density g/cm
3
or g/ml
Why does ice float on water?

4
2
0r
qq
F



Water density depends on its electrostatic forces
At low temperature, H atoms Oscillations ceased off
With void spaces that decreased density

24. Potential energy (ΔU), internal energy (ΔE), entropy (ΔS),
collisions (1/2 mv
2
), lattice disruption & interactions
liquid
gas
plasma
solids
ΔS
plasma > ΔS
gas > ΔS
liquid > ΔS
solid > ΔS > ΔS>
Which system is thermodynamically active & which is
kinetically active systems ????
wqdU 

25. Energy degenerate of solvent potential
Degeneration itself consumes energy








kT
EG
eP
kT
E
e
z
P


1









kT
E
gq
vi
i
viv
,
,
exp
 hE
vi 






2
1
,




















kT
h
gq
i
viv

2
1
exp
,
surface energy = mJcm
-2
Le Chatelier model
surface activities of cohesive forces
bulk or
internal
forces
Both the forces contribute in fluid dynamics

i
i
dnVdPSdTdG 

26. harmonic oscillator potential
dissociantion energy
e
l
e
c
t
r
o
n
i
c

e
n
e
r
g
y
internuclear distance r
r
e
=0
Morse
=1
=2
=3
=4
=5
=6
D
e
D
0
r
Geometrical distribution of molecular
forces coexisting between 2 molecules
Kinetic zone
Vibration
spectra
vibration
electron
energy
states
Anharmonic
oscillator
leading to
disrupt the
bonds
Reversible systems
2
nd

derivative

27. harmonic oscillator potential
dissociantion energy
e
l
e
c
t
r
o
n
i
c

e
n
e
r
g
y
internuclear distance r
r
e
=0
Morse
=1
=2
=3
=4
=5
=6
D
e
D
0
r
e
l
e
c
t
r
o
n
i
c

e
n
e
r
g
y
rere
=0
=1
=2
=3
=4
=5
=6
docked area
of impact
Geometrical distribution of
molecular forces coexisting
between 2 molecules
Kinetic
zone
Non-
kinetic
zone
Potential
enhance
Franck–Condon
principle
Vibration spectra vibration
electron energy states

28. •Proteins fold to lowest-energy fold in
micro sec to sec time scales. How can
they find right fold so fast?
•It is mathematically impossible for
protein folding to occur by randomly
trying every conformation until lowest
energy one is found (Levinthal’s
paradox)
•Search for minimum is not random
because direction toward native structure
is thermodynamically most favorable
Beginning of helix formation & collapse
Kinetic protein folding

29. E
0
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
E
9
E
10
0 0.10.20.30.40.50.6
H Cl
r
0
D
0
0
1000
2000
3000
4000
U
E, cm
-1
r, nm
different
vibrating
energy levels
HCl,anharmonic
oscillator vibrating at
energy level E
3
U = potential
energy
D
0= dissociation
energy
r
0 = bond length
E
3
E
3
E
7
E
7
Kinetics,
the bonds
are broken
new bonds
are formed
Reversible thermodynamics
with infinitesimal changes
rotational, vibrational &
translational motions
E = dq + pdV
Hooke's law
F= k.x
Thermodyna
mic motions
lead to
transition of
the either
phase or the
bond
compression elongation
x axis
ForceFvselongationxforahelicalspringasper
Hooke'slawandactualplotisindashedline.At
botton, the states of spring corresponding to some
pointsofplot,middleoneisinrelaxedstateasno
force is applied.
a
c
t
u
a
l
p
l
o
t
o
f
s
p
r
i
n
g

2

2
1
xkU
no force applied

30. Atomic and molecular transitions: Concept of tentropy
•Transition rule (SR) constrains possible transitions from one QS to another
•SRs are derived for electromagnetic transitions? in molecules, in atoms, in atomic
nuclei
•SR differs according to technique used to observe transition
•SR plays a role in chemical reactions as some reactions are spin forbidden
reactions ? Ultimately it is 
AO
which matters a lot for making 
MO
so change in
spin of AO leads to affect energy levels
•Thereby such intramolecular AOs constitute a concept of tentropy
•As each AO acts as set or array of one order while the another AO with
different AO energy acts as another set or array or canonical or partitioning
box. These differences cause disorder
•So reactions where spin state changes at least once from reactants to products
could be explained by SR
•In quantum mechanics a basis for a spectroscopic SR is value of transition moment
integral
d
2
*
1
 d
2
*
1 µ is transition moment operator

31. Transition moment integral: Zero
•Transition moment integral is zero if transition moment function, is
anti-symmetric/odd as y(x) = -y(-x) holds
•Symmetry of transition moment function is a direct product of
parities of its three components
•Symmetry characteristics of each component is obtained from
standard character format
•Rules for obtaining symmetries of a direct product is in texts on
character as-
21
*
Transition typeµ transforms as explainations
Electric dipolex, y, z Optical spectra
Electric quadrupolex
2
, y
2
, z
2
, xy, xz, yzConstraint x
2
 
+ y
2
 
+ z
2
 
= 0
Electric polarizabilityx
2
, y
2
, z
2
, xy, xz, yzRaman spectra
Magnetic dipoleR
x, R
y, R
z Optical spectra (weak)

32. Benzene symmetric
electric quadrupole
Q
Electronic or electric
charge is confined to
the centre of benzene
ring

33. Charge asymmetry and electric quadrupole
•The 
-

2
should have positive slope
•The 
+

2 should have negative slope with the same magnitude
•Integrated 
2 and 
-
may not bigger than 
+
and only the A
 affects
the quadrupole










 A
p
q
Electric
e
e
22
formation quadrupole 






NN
NN
eCh A as asymmetry arg

34. NMR single line or peak

35. Electron spin: Electron density or electric field or electric dipole
•Each orbital and each spin attains different energy states
•Energy state is prime criteria for succeeding a reaction
•So an interface between reaction and spin is evolved for spin
forbidden or allowed reaction concept
•Lone pair electrons also acquire different energy states
Singly Occupied Molecular Orbital
SOMO
LUMO
HOMO
Unstable

36. Electron spin: Electron density or electric field or electric dipole
•Each energy level influences a reaction
•So spin forbidden or allowed reaction concept seems valid as per
atomic orbital concept
•Since purpose of reaction is to influence electron spin of atom to
formulate MO so the spins of AOs are in focus
•Paired and unpaired electron states develop their own electric field or
domain
•Geometrical state and direction both seem to control the reactions
•Since HOMO and LUMO both are intra-atomic arrangements so they
also lead the reaction
Singly Occupied Molecular Orbital
SOMO
LUMO
HOMO
Unstable

37. Electron spin: Electron density or electric field or electric dipole
Both the position and direction of election are critical for a reaction
•For example physicochemical properties of CH
4
, H
2
C=CH
2
, HCCH
•Spatial electronic configurations as electron spin and orbital affect
•Singlet: Paired including HOMO and LUMO
•Singlet: Paired excited including HOMO and LUMO
•Triplet: SOMO-SOMO
•Thus the spin and orbital connected parameters decide a fate of reaction
and level of activation energy

38. Different PE states with several combinations
Minimum crossing points indicate reaction
occurs
26
Fe=1s
2
2s
2
2p
6
3s
2
3p
6
4s
2
3d
6
or 4s
2

39. Potential well and centre of confining energy of a reaction of MO
through spin and orbital

40. Transition state of 
MO

42. Availability AO for bonding with another atom: Hybridization

How do C atoms develop (-C-C-)
n bonds but Na does not ?
6
C=1s
2
2s
2
2p
2
or 1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
and

1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
form C-C bond in opposite
electron spin as per LJP ???. It also causes
1
H coupling in NMR
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
C
2
H
4
develops sharing of 2p
z
and 2p
z
2 electrons form 1 bond
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
C
2
H
2
develops sharing of 2p
z
and 2p
z
2 electrons form 1 bond
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1

as spinning
electrons come at
a closure distance CH
4
C=C sidewise overlap
Sp
2
hybridization
CC sidewise overlap
Sp hybridization
1
HNMR = 0.23 ppm
1
HNMR = 5.25 ppm
1
HNMR = 1.80 ppm

43. Central theme of MO: Affect reaction via Selection rule
•A molecule is viewed on a quantum mechanical level as a collection
of nuclei surrounded by delocalized MOs

AO
are summed to obtain molecular wave functions
•If 
AO reinforce each other, a bonding MO is formed (region of high
electron density exists between nuclei)
•If 
AO
cancel each other, an antibonding MO is formed (node of zero
electron density occurs between nuclei)
Amplitudes of wave functions
are added
Amplitudes of wave functions are
subtracted

44. (N
2 = NN), N= 1s
2
2s
2
2p
x
1
2p
y
1
2p
z
1
: No GW
•5 valence electrons
 in 2s
2
and 2p
3
orbitals, 3 p unpaired electrons
•It has highest
 electronegativities = 3.04 on Pauling scale, exceeding
by
 Cl
2 
=3.16, O
2 =3.44, F
2 
=3.98
•Its single-bond
 covalent radius = 0.071 nm < boron = 0.084, C =
0.076, > O
2
= 0.066
 and F
2
0.057
 nm
•1
st
three ionisation energies of N
2 = 1.402, 2.856, and 4.577
 MJ·mol
−1
,
and sum of 4
th
and 5
th
= 16.920
 MJ·mol
−1
.

How is it near H atom ?
•No radial nodes in 2p orbital is responsible for many of anomalous
properties of first row of p-block
•The 2p orbital is very small and is similar radius to 2s shell for
orbital hybridisation ???
•Very large electrostatic forces of attraction between nucleus and
valence electrons in 2s and 2p shells, resulting in very high
electronegativities
 ???? Contrary to H-C
C-H
NN

45. All nuclei molecule influence proper MOs
•Electronic energies: KE, interelectronic repulsions, internuclear
repulsions, and electron–nuclear attractions
•Nuclear potential is an average over electron configurations of sum of
electron–nuclear and internuclear electric potentials
•In molecular spectroscopy, ratios of periods of electronic, vibrational
and rotational energies are related to each other on scales in order of a
thousand
•BO is attached to approximation where energy components are
treated separately
•Nuclear spin energy is so small so it is normally omitted
nuclearrotationalelectronictotal
EEEE 
lvibrationa
E

46. Calculation of MOs for AB
n sandwich molecules
•If A atom lies in the same plane as B atoms, the shape is trigonal
planar
•If A atom lies above plane of B atoms, the shape is trigonal pyramidal
(pyramid with equilateral triangle)
•AB
3
like ClF
3
is T-shaped, a relatively unusual shape
•Atoms lie in one plane with two B¬A¬B angles of  90°, and a third
angle close to 180°
•Shapes of most AB
n
molecules is derived from just five basic
geometric arrangements
CCl
Cl
Cl
Cl
'A' similar atoms: Similar molecule
Central atom
Similar atoms

47. Determine electron domain geometry: Tentropic state
Step 1: Drawing Lewis structure for a molecule
Step 2: Counting number of electron domains on central atom
Step 3: Determining electron-domain geometry (EDG)
H – C  N:
:O:
:O – N = O:
–:
:
:
:
–
:F – Xe – F::
:
::
:
: :
EDG is based on number of electron domains around central atom
2 3
5
•EDG quickly hybridizes and optimizes the structure
•Hence a central atom remains highly tense ???
•Ultimately EDG decide the geometry and stability of molecules like
N
2
which does not respond to UV or h photons
•But CO
2
or H
2
O do so why is it so ??
Electrons repel each other

48. Electron domains and geometry
Number of
Electron Domains
Electron-Domain Geometry
2 Linear
3 Trigonal planar
4 Tetrahedral

49. Number of
Electron Domains
Electron-Domain Geometry
5 Trigonal bipyramidal
6 Octahedral
•Thus the molecules respond differently towards environments in
measurements of FTIR, NMR, UV, ESCA, HPLC, friccohesity
•This is the reason that the variable probes furnish different potential
of molecules ???

50. Step 4: Determine molecular geometry
Step 24 e¯ domains
NH
3

Electron-domain geometry and number of bonded atoms determine
molecular geometry
Step 1
:
H – N – H
–
H
Electron-domain
geometry tetrahedral
Step 3
Molecular geometry = Trigonal pyramidal

51. Molecular geometries derived from 5 electron-domain
geometries with definite PE
Linear
Bent
Trigonal planar
Bent
Trigonal
pyramidal
Tetrahedral
T-shaped
Seesaw
Trigonal
bipyramidal
Linear
T-shaped
Square planar
Square pyramidal
Octahedral
•Each geometry contains diversified spatial position of AOs
•Hence their PE and KE vary which decide their stability for making
their best use of each surface if as catalysts ??

52. Molecular geometry and polarity
+ -
H – F
H – F
Bond dipoles are vectors and therefore are additive
HF bond is polar and HF has a dipole moment ()
dipole moment > 0
dipole moment = 0
H
2O
CO
2
•Molecules
with more
than two
atoms
•Bond dipoles
are additive
since they are
vectors
Resultant dipole

53. All nuclei molecule influence proper MOs
•Nuclei in molecule influence MOs to require structure and symmetry
of a molecule
•All the times such treatment is not needed as more localized orbitals
act well
•In H
2
C=CH
2
, isolated double bonds, descriptions of localized π
orbitals are noted to distinguish energy levels
•Several important characteristics of MOs are prerequisite to illustrate
them
•Born-Oppenheimer approximation (BOA) a motion of atomic nuclei
and electrons in a molecule is separated
•It allows  of a molecule broken its electronic and nuclear
(vibrational, rotational) components
•Energy and  of average-size molecule is approximated i.e.
nuclear x 
electronictotal

54. H
AH
B
•How is  bonding and * antibonding? How do you know relative energy ordering
of these MOs?
•1s orbital depicts atomic wave function (
1s) and MO depicts wave function 
•Hence LCAOs is as -

2
predicts probability density function. Density functions for each MO are-
 = 
1
= 0.5 (
1sA
+ 
1sB
) * = 
2
= 0.5 (
1sA
- 
1sB
)
(
1
)
2
= 0.5 [(
1sA

1sA
) + 2(
1sA

1sB
) +(
1sB

1sB
)]
(
2
)
2
= 0.5 [(
1sA

1sA
) - 2(
1sA

1sB
) +(
1sB

1sB
)]
•Difference between two probability functions is in cross term (bold)
•Type and amount of overlap between two 1s atomic wave functions ((
1sA 
1sB) 
is overlap integral, S)
•In-phase overlap makes bonding orbitals and out-of-phase overlap makes
antibonding orbitals
Y
2
= (a + b)
2
Y
2
= a
2
+ b
2
+2ab
Y
2
= (a - b)
2
Y
2
= a
2
+ b
2
- 2ab

55. Bonding antibonding energy wise

56. Basics questions for  and 
2
How do C atoms develop (-C-C-)
n bonds but Na does not ?
6
C=1s
2
2s
2
2p
2
or 1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
and

1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
form C-C bond in opposite
electron spin as per LJP ???. It also causes
1
H coupling in NMR
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1
1s
2
2s
1
2p
x
1
2p
y
1
2p
z
1

2s
2
 
2s1
2
= 
2px
2
= 
2py
2
= 
2pz
2
•Which out of  and  bonds do develop 
2
of higher intensity and
why ?
•It defines role of cartesian coordinates as electron density which is eV is
distributed and not in control except geometrical optimization
•It generates a concept of conjugation similar to  electron conjugation
•How are h = c/ and h = p. prominent in Huckel MO approximation
•Similar dimension of 

for light photon as well as bonds 
as spinning
electrons come
at a closure
distance

57. H
AH
B
•How is  bonding and * antibonding? How do you know relative energy ordering
of these MOs?
•1s orbital depicts atomic wave function (
1s) and MO depicts wave function 
•Hence LCAOs is as -

2
predicts probability density function. Density functions for each MO are-
 = 
1
= 0.5 (
1sA
+ 
1sB
) * = 
2
= 0.5 (
1sA
- 
1sB
)
(
1
)
2
= 0.5 [(
1sA

1sA
) + 2(
1sA

1sB
) +(
1sB

1sB
)]
(
2
)
2
= 0.5 [(
1sA

1sA
) - 2(
1sA

1sB
) +(
1sB

1sB
)]
•Difference between two probability functions is in cross term (bold)
•Type and amount of overlap between two 1s atomic wave functions ((
1sA 
1sB) 
is overlap integral, S)
•In-phase overlap makes bonding orbitals and out-of-phase overlap makes
antibonding orbitals
Y
2
= (a + b)
2
Y
2
= a
2
+ b
2
+2ab
Y
2
= (a - b)
2
Y
2
= a
2
+ b
2
- 2ab

58. Bonding antibonding energy wise

59. The  bonding MO is more stable than atoms by themselves and *
antibonding MO so constructing MO diagram
H
E
n
e
r
g
y
HH
2
1s 1s

g
*
u
•To identify symmetry of
different MOs, subscripts g is
used (symmetric w.r.t. inversion
center)
•The u is for (anti-symmetric
w.r.t. inversion center) to labels
of each MO
•Electrons are then added to MO
diagram using Aufbau principle
Amount of stabilization of  MO (red arrow) is slightly less than amount
of destabilization of * MO (blue arrow) due to pairing of electrons
For H
2
, stabilization energy is 432 kJ/mol and bond order is 1
 MO
* MO

1,
2
02


orderbond

60. He with 1s AO form MO as He
2
in an identical way, except two
electrons in 1s AO on He
He
E
n
e
r
g
y
HeHe
2
1s 1s

g
*
u
Bond order in He
2
= (2-2)/2 = 0
•Hence He
2 does not exist
•But cation [He
2]
+
where one
electron in *MO is lost may
have bond order = (2-1)/2 = ½
•[He
2
]
+
cation is predicted to
exist
•Electron configuration for
cation is written in the same way
as we write those for atoms
except with MO labels replacing
AO labels-
[He
2]
+
= 
2
*
1
•MO predict about existence of
molecules
•It gives a clear picture of
electronic structure of any
hypothetical molecule that we
can imagine

62. Microstate at T = 0 for fermions
•Only one possible configuration (microstate) exists i.e. thermodynamic probability
W = 1
For W = 1, Boltzmann distribution S = klnW = 0
•Only electronic configurations obeying Pauli Principle are allowed
•Different arrangements of x electrons occupying y orbitals are not all equal in
energy due to different electron-electron repulsion energies in these microstates

 
smicrostate 45
!2! 252
! 52



x
x
D
t

 !! 2
! 2
0
0
ee
t
NNN
N
D


Determine microstates for nd
2
N
0 = total degenerate orbital in
subshell i.e. 5
N
e = number of electron in
configuration i.e. 2
N
0
= 5, N
e
= 2
WkSln

63. Electrons distributions in microstates for nd
1-10
D
t
values for d
10
is calculated as-
configurationd
1
d
2
d
3
d
4
d
5
d
6
d
7
d
8
d
9
d
10
microstates104512021025221012045101

 !! 2
! 2
0
0
ee
t
NNN
N
D



 
1
10)!10!-(10
10!
or
!10! 10102
! 52
nd
10



x
x
DFor
t

 
45,
!2! 252
! 52
,nd
2



x
x
DFor
t

 
10
(1)!1!
10!
or
!1! 112
! 52
,nd
1



x
x
DFor
t
•Density of states (DOS) is the number of available energy states per unit
energy per unit volume. units are J
-1
m
-3
or eV
-1
cm
-3

•It provides information on how energy states are distributed in a given solid denoted
as g(E) i.e. degenerate
Why only 1

64. Most probable distribution










n
j
j
n
j
jj
n
j
jj
n
j
j
n
j
jj
n
j j
N
j
NNNgNNw
NgNNw
w
N
g
Nww
j
111
11
1
lnln!lnln
!ln ln!lnln
0ln :Instead
!
! where0



































n
j j
j
j
n
j j
j
jjjj
n
j
jjj
N
g
Nw
N
N
NNgNw
NgNNw
1
1
1
ln)(ln
)1ln)(ln(ln
)1ln(ln!lnln



Common terms is
separated

65. Chemical potential
•Chemical potential is in middle of
gap
•Else it induces an imbalance
between conduction electrons than
vacant valence states.
equal
area!

 dxxx
x
x
dx
x
x
)/1)(/1(
lnln
2

66. H
i
g
h
e
r

P
E

w
i
t
h

s
t
r
o
n
g
e
r

C
F
S (entropy) J/K
dT
T
C
SSS
T
p
T 

0
0 Sor
dP
T
C
SSS
p
p
p 

0
0 Sor

67. Reorientations & molecular Collisions