Roll number 9311

Applied inorganic chemistry

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.