Roll moments, optical absorption spectra and electron and

            Roll number 9311

 

              Applied inorganic chemistry

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             Submitted To              Dr. Jawwad Saif

 

     Submitted By              Sana Munir

 

MSc.  Applied Chemistry

 

First Semester
(Morning)

 

 

 

 

              Quantum Mechanics in Molecular
Bond Theories

Quantum mechanics (QM;
also acknowledged as quantum physics or quantum theory), is a fundamental
concepts in physics which defines nature at the minutest gauges of energy
levels of atoms and the sub-atomic particles. In other words, Quantum mechanics
is the science of the very small. It describes the behavior of matter and its
influences with energy at the level of atoms and sub-atomic particles. While
Matter is any material that takes up mass and takes up space by having volume.
It comprises atoms and everything made up of atoms, but not other energy
phenomena or waves such as light or sound.

 

The principal
approximate methods are considered in molecular quantum mechanics in valence
bond theory and molecular theory. Valence bond theory originated in the Heitler
and London and was developed extensively by Pauling. Molecular orbital theory
has its origin in the early research work in band spectroscopy of diatomic
molecules and has been widely used to describe many aspects of molecular
structure and diverse molecular properties such as electronic dipole moments,
optical absorption spectra and electron and nuclear magnetic resonance. Among
those in the original works were Hund, Mulliken, Lennard-Jones and Slater. We
are concerned here in exclusively with molecular orbital theory, and
particularly with the theories and problems are encountered in carrying out the
calculations of molecular orbitals of large molecules.

Molecular orbital
theory is the method of determining molecular structure in which electrons are
not assigned to individual bonds between atoms, but are treated as moving under
the influence of the nuclei in the whole molecule. The spatial and energetic
properties of electrons within atoms are fixed by quantum mechanics to form
orbitals that contain these electrons. While atomic orbitals contain electrons
ascribed to the single atom, molecular orbitals which surround number of atoms
in a molecule contain valence electrons between atoms. Molecular orbital theory
which proposed in the early twentieth century revolutionized the study of
bonding by approximating the position of bonding electrons_the molecular
orbitals_as linear combination of atomic orbitals (LCAO).

Molecular orbital
theory provides a precise description of molecular electronic structure only
for one- electron molecules, but for many-electron molecules it provides a
sufficiently good approximate description to be generally useful. The full
analytical calculation of the molecular orbital for most system of interest may
be reduced to a purely mathematical problem, the central feature of which is
the calculation and diagonalization of an effective interaction energy matrix
of the system.

Semi-empirical Methods:

Experimental data on
atoms and prototype molecular systems are used to estimate values for
quantities entering into the calculations as parameters, and for this reason
the procedures are widely known as Semi-empirical methods.

Approximate molecular
orbital theory can be approached from two principally different points of view:

Huckel and Extended
Huckul Method:

Select appropriate
values for the elements of interaction energy matrix from essentially empirical
considerations,and is characteristics of the so-called Huckel and extended
Huckel methods.

Approximate Self
Consistent Field Theory:

The other approach
depends clearly on the mathematical formalism, and involves introducing
approximations for the atomic and molecular integrals entering the expression for
the elements of the energy interaction matrix. The later approach is referred
to as approximate self consistent field theory.

The Schrodinger
Equation:

According to classical
mechanics, the energy E of the system of interacting particles is the sum of the
kinetic energy contribution T and a potential energy system V

T+V=E

Schrodinger that the
proper way to describe the wave character of particles was to replace the
kinetic-and potential energy function T, V with linear operators^T V and set up
a wave equation of the form

{T + V}* = E*

The solution of this
equation is called wave function>£, would explain the spatial motion of all
the particles of the system moving in the field of force specified by the
potential energy operator V.

In single electron
systems, such as the hydrogen atom, the problem is basically to explain the
motion of the electron in the columbic force field of the nucleus. In this case
the quantum mechanical potential energy and classical potential energy function
are identical and for an electron moving in the field of nucleus o f charge Ze.

 V = -ZeV”1

Where is the distance
of the electron from the nucleus and is the unit of electronic charge. With the
coordinate system centered on the atomic nucleus, one need consider only the
kinetic energy of the electron. Schrodinger prescription essential that the classical
kinetic energy expression for the single particle T=t^

Planck’s m are the
momentum and the mass of the particle correspondingly, be replaced by the
linear differential operator, where h is Planck’s constant m the electronic
mass and d2 is Cartesian coordinates. Thus the Schrodinger equation for the
hydrogen atom takes the form

  ‘ k% v2 ~ ^ r ) *(1) = E*Q) (L 7 8ir2m)

In this one electron
system, the wave function SE^1 contain only the coordinates of the single
electron and the 1 is parentheses signifies.

F(A) linear operator M
considered the functions f and y obeys the equation

M (f + n) = Mf + Mi

M(Cf) = CMf where c is
the constant

In dealing with the
equation of quantum mechanics, it is appropriate to introduce the new units
which are suitable to atomic dimension and which remove one of the constants
from the wave function.

These are referred to
as atomic units. The atomic unit of length is defined as the radius of the
first orbit and quantity in the original Bohr Theory of the hydrogen atom. It
is often referred to as the Bohr radius. The atomic unit of electronic charge
is the protonic charge,

e = 4.80298 X 1010 esu.

The atomic unit of
energy is the energy of interaction of two units of charge separated by one
Bohr radius and is called the Hartee. The atomic unit of mass is the electron
mass

m = 9.0191 X 10~28g

The Schrodinger
equation for a larger system consisting of a set of interacting electrons and
nuclei is formulated in a similar manner. This first requires specification of
the full Hamiltonian for the system. The Hamiltonian is again the sum of
kinetic energy operators for the nuclei and for the electrons together with the
potential energy terms representing the various columbic interactions. These
are repulsive for electron-electron and nucleus-nucleus pairs but electrons
between electrons and nuclei. If there are N nuclei and n electrons the
many-particle Hamiltonian operator   is:

 

 

Here MA is the mass of
nucleus A; m and e is the electronic mass and charge, respectively; ZAe is the
charge of nucleus A; and r is the distance between particles I and j. Summation
involving indices a and b are over atomic nuclei and those involving p and q
are over electrons.

The Schrodinger
equation for the entire system is thus:

   

Where   is now a complete wave function for all the
particles in the molecule and E is the total energy of the system. Since each
particle is described by three Cartesian coordinates, this is the partial
differential oq notation in SN + 3n variables.

The Orbital
Approximation:

The orbital approach to
estimated solutions of many-electrons Schrodinger equation is an attempt to
construct a satisfactory approximate many-electron wave function from a
combination of functions, each dependent upon the coordinates of one electron
only. For an electron system, the easiest way to do this is to correlate the
electrons with n one electron functions  
and write the total wave function  
as the product of the one electron functions  

Such one electron
functions   are called orbital and the
product function as such is known as a Hartee product. The probability density
function   is just the product of
one-electron possibility densities   .
From elementary possibility theory, this situation only arises the events
related with each of the probabilities  
which take place independently to one another. Thus, the physical model
involves the approximation of various electron wave functions by product of
orbital’s in an independent electron order model.

Electron Spin:

 An orbital which gives a complete condition of
the spatial distribution of an electron, but it is still incomplete that it
does not gives specify the state of electron spin. In addition to spatial
motion, the orbital angular momentum is reflected; an electron that possesses
an additional fundamental angular momentum is identified with an electron spin.
The representation of spin angular momentum is by the vector operators and has
components Sx, Sy and Sz which convince the basic communication relations characteristics
of general angular momentum operators. The spin operators are communicated with
the general Hamiltonian operators, which have no spin, coordinates, so that it
is hopeful to increase simplification by making approximate wave functions
which are the Eigen functions of suitable angular momentum operators. The
components Sx, Sy and Sz are converting with the spin squared operator S² but
not with each other. Thus, the most hope for this is a function which is all
together an Eigen state of S² and one of the components of s, usually taken
randomly as Sz.

Electronic
configurations and Electronic states:

Having some of the
general features of the orbital for approach to approximate solutions of the
Schrodinger equation, now consider a manner in which the electronic structure
of the system is described by the obtaining orbital. For a molecule with 2n
electrons, results of the solution of Schrodinger equation in the orbital
approximation in molecular 2n spin orbital which associated with the distinct
energy of orbital. In a wave function  of
spin restricted orbital, a given spatial orbital may be associated with both
the spin of an electron and an electron of P spin, with the energies of the
orbital of the two resulting spin of the 
orbital’s, being of course; degenerate. For the ground state of the 2n
electron system, the n spatial orbital’s will be occupied it is possible to say
that this system have an electronic configuration   

Electronic
configuration is represented by the energy level of orbitals. The diagram of
the orbital energy level is for a four electron system. A configuration in
which occupied orbital’s having the maximum of two electrons is known as a
closed shell configuration.

If the number of
electrons is odd, 2n+1, the electronic configuration of the ground state will
be   and is represented by the orbital
energy level diagram. Open shell configuration is a type of configuration and
is the characteristics of free radicals.

 

Figure1: Orbital
energy-level diagram for ground electronic configuration of (a) Closed-shell
and (b) Open shell system.

Hartee-Fock Equation
for Molecular Orbital’s:

The established of the
proper form of many electron wave functions for close shell as a single
determinant of spin orbital’s and developed a suitable expression for the
energy of electrons, we continue the details of the real determination of the
spatial orbital’s   for a closed shell
system if there is no limitation in these functions then we can assume
differential equations for the optimum force of the orbital of electrons by
tempting to the variational method. Firstly Fock proposed these differential
equations based on earlier work by Hartee and are known as the Hartee-Fock
equations.

According to the
principal of variation, the adjustment of an approximate many electron wave
function in the following equation to the energy of the lower state,

 

Then the correct
solutions of the many electron wave equations will be approached. The best
orbital of molecules, therefore; obtained by changing all the contributing one
electron functions   in the determinant
until the energy achieves its minimum values. This will not, give the accurate
many electrons   for a closed shell
system, but somewhat the closest possible approach is obtained in the form of a
single determinant of orbitals. Such orbital’s are referred to as self
consistent, or Hartee-Fock molecular orbital’s.

Orbitals and
Probability Density:

Value of the wave
function si (?) at a specified point in space x, y, z, is relative to the
amplitude of the electron matter wave at that point. However, many wave
functions are complex function containing I= ??1I and the amplitude of the
matter wave has no real physical significance.

The square of the wave
function ?2 is a little more useful. This is because the square is a wave
functional of proportional to the possibility of locating an electron in a
particular volume of gap within an atom. The function ?2 is often called the
probability density.

 For an electron the probability density can be
visualized in the number of different ways. For example, ?2 can be represented
by a graph in which unreliable strength of color is used to show the relative
opportunity for the presence of an electron in a given region in space. The
greater the possibility of an electron in a particular volume, then the density
of the color is higher in that region. The image below shows the probability
distribution for the spherical 1s, 2s, 3s orbital’s.

 

 

 

 

Shapes of Atomic
Orbitals:

The shape of s orbitals
is spherical. The distance r from the nucleus is the most important reason of
an electron’s probability distribution. However, the other type of orbitals
such as p, d and f orbital are in the angular position of electrons relative to
the nucleus is also a factor of the probability density.

 

 With the exception of the d orbital it
describes the clover shape four possible orientations because d orbital almost
look like a p orbital’s whose shaped is like dumbbells which are oriented along
one of the axis x, y, z comma y and comma z. The d orbital cannot go around the
middle. It’s not sufficient value to describe the f orbitals.

Applications of Quantum
Mechanics:

Fluorescence and
Phosphorescence:

Fluorescence means the
light is emitted by s substance it absorbed. It is a form of photoluminescence.
In most cases, the substance emitted the light has longer wavelength, and have
lower energy, than the radiation to be absorbed. However when the
electromagnetic radiation absorbed is powerful, it is possible for an electron
to absorbed two photons; the absorption of these two photons leads to the
emission of radiations which have a shorter wavelength than the radiation
absorbed. The substance emitted the radiation, may also have the same
wavelength as the absorbed radiation termed “resonance fluorescence”.

 

When an electron of a
molecule or an atom back to its ground state by the emission of  photon of light after excited to a higher
excited state by some kind of energy fluorescence occur. The most prominent
examples of fluorescence are when the radiation is absorbed in the ultraviolet
region of the spectrum, and thus it is undetectable to the human eye, and
emitted light is in the visible region.

The one form of
photoluminescence is the phosphorescence which is related to fluorescence.
Phosphorescence is different from the fluorescence, because radiations absorb
by a phosphorescent material do not emit readily. The electrons excite to the
higher state, is occurred by the change of spin state. Once a spin state is
different, electrons cannot back into the ground state readily because the
re-emission involves the quantum mechanically transition of forbidden energy.
In certain materials, radiations absorbed very slowly in these transitions that
may be re-emitted at a lower concentration for up to several hours after the
original excitation.

 

Fluorescence and
Phosphorescence: Energy scheme used to explain the difference between
fluorescence and phosphorescence

 Phosphorescent materials have many examples
that are paint, the shine in the dark toys and clock dials that glow after
being charged for some time by the intense light such as in many normal reading
or light of the room. Normally the shining within minutes slowly fades out in a
dark room.

Lasers:

A laser is a machine
which emits lights of single wavelength which is based on the motivated
emission of photons during the process of optical application.

The principles of laser
operation are based on quantum mechanics (except the free electron lasers,
which can be explained only by a classical electrodynamics). When an electron
is excited from a lower energy state to a higher energy state it will not
reside that way everlastingly. In an excited state an electron may decay to a
vacant state of lower energy according to a particular time constant
characterizing that change. Without external influence, when such an electron
decays then the photon is emitted and this process is called “spontaneous
emission”. The period related with the emitted photon is random. In the
material if many electrons are present in an excited state then the nature of
radiation is monochromatic, but if the individual photons which have no common
phase relationship and would begin in the random direction. This is the method
of fluorescence and thermal emission.

Holography:

Holography is the
visual technique that enables the man to make the three dimensions images. It
involves the use of the intrusion, diffraction, power of the light copy and
appropriate lighting of the recording. If the image is changed the location and
path of the presentation system is changed precisely the same manner as if the
object will still there, thus the image will be three dimensional.

X-rays:

The one form of an
electromagnetic radiation is x-rays and the range of wavelength of x-rays is
0.01 to 10 nanometers.

X-radiations (composed
of x-rays) are a form of electromagnetic radiation. The range of wavelength is
0.01 to 10 nanometers, which have 
frequencies in the range of 30 pet hertz to 30 exahertz (3•1016 Hz to
3•1019 Hz) and have energies in the range from 100eV to 100 keV.

X-rays can be generated
from a tube called x-ray tube, a emptiness of tube uses high energy to
accelerate the electrons released by a hot cathode to a high rate. The
electorns of high velocity  collide with
an anode which is made of metal for creating the x-rays.

A specific resource of
x-rays which is largely use in study in synchrotron emission which is produced
by accelerator of particles. The exclusive features are x-rays output which
has  orders of magnitude better than the
x-ray tubes, broad spectra of  x-ray,
brilliant collimation and linear polarization.

Quantum Mechanical View
of Atom:

 The basic unit of matter is called atom that contains
nucleus surrounding by negative charges electrons. The nucleus of the atoms
contains a mixture of protons which have 
positive charges and neutrons which have no charges and called as
neutral. The nucleus is bounded with electrons 
by electromagnetic force. Atoms are tiny particles having diameters of
the tenths of a nanometer and have small masses relative to the volume which is
implied by these proportions.

 

           

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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