Quantum Mechanics and ChemicalBonding Submitted to:Dr. Jawad Saif Submitted by:Muhammad Ahsan RasoolRoll No: 16444  Department of Applied ChemistryBS (Hons) Applied Chemistry 5thSemester (Evening) Government College UniversityFaisalabad    Quantum mechanics:Mechanicsis the science of motion, It is a branch of physical science which deals withthe behaviour of the substances or matter under the action of different forces.

The development ofquantum theory began 1900 in order to rationalize the basis for theunderstanding of microscopic particles as atoms, molecules and elementaryparticles. It is developed out of classical mechanics and Maxwell’selectromagnetic theory. So the electrons are chemically most important partsof  atoms . so we have to discusselectrons also in proper ways. Soon classical mechanic was disapproved and newmodel with quantum mechanics was published to overcome for the new observationsthat did not suitable for the classical mechanics.Quantum mechanics has itsown significance as it enables to calculate energy levels and other propertiesof atoms and molecules. It also helps to understand the periodic variations indifferent properties as ionization potential and electron affinity etc. It alsohelps to understand the nature of bond in different molecules.

The energylevels bond length bond angles can also be calculated for many molecules. Withthe concepts of quantum mechanics it can be predict that whether molecules hasdipole moment or not.Classical mechanics hasdeals with the motion of large particles and in this technique matter isconsidered as made up of particles, each of particles has its own mass. Themotion of these particles is usually explained by the Newton’s laws of motion.First Law of Motion:Thislaw states that an object is at rest and remains to its rest position and anobject is at motion and remains its motion as long as no outside force acts onit.Second Law of Motion:ThisLaw states that if an unbalanced force acts on a object, then that object willaccelerate in the direction of force and the amount of acceleration will beinversely proportional to the mass of that object and directly proportional tothe force.Third Law of Motion:Thislaw states that for every action there is an equal but in opposite directionreaction.Newton’s Law is mostfamiliar law here force is a vector quantity,, for a single object of mass ‘m’Newton’s second law is expressed asIt can be written as Classical Mechanics hascertain limitations some of following are given as,·        It could notexplain the motion of bodies travelling with the speed of light.

·        It could notexplain atomic spectra which are observed with different atoms.·        It also could notexplain the emission of the radiation emitted from a hot mass or from the blackbody radiation.·        Rutherford showed thatatom has all of its positive charge in nucleus with the electrons surroundingit, but it was not observed with the laws of classical mechanics.Significance of Quantum Mechanics:v  Significance of Quantum mechanics is explained with followingpoints,v  Classical mechanics could not explain the study of microscopicparticles while Quantum mechanics explains the study of microscopic particles.v  Classical mechanic is not based of Schrodinger wave equationwhile quantum mechanics is usually based on Schrodinger wave equations.v  Classical mechanics could not explain the dual nature ofparticles while quantum mechanics do.

v  Quantum mechanics measure everything but not with 100% accuracy.v  Quantum mechanics can be used to know the gross systemproperties where classical mechanics is valid through the effort involved maybe greater than those on the classical bases. This idea is known as”Correspondence principle”.

v  In quantum mechanics usually small units are used as ,nm, mm, cm.v  It can easily explain the relationship between energy andmatter.v  It can utilize the data of classical mechanics and give aninsight into reaction mechanics.v  Quantum mechanics is usually can discuss many problem of bothphysical and medical science.Postulates of Quantum Mechanics:There are number ofquantum mechanics postulates as they have been needed.

1.      Thestate of quantum is fully described by a wave function  or ?(r, t) that is the function of coordinatesof particles and time. It contains all the information about the system.2.      Thewave function ?(x, t) and its first and second derivation    and  must be continuous finite and single valuedfor all values of x.

3.      Everyphysical property A of the system can be characterized in quantum mechanics bya linear operator  or Hermitian operator. This operatorsatisfies the following condition for any pair of functions and which describesphysical states of system.4.      Theonly possible value which a measurement of the property A can yield is theeigen values of the equation.Where  is the operator corresponding with theobservable.5.

Averagevalue of the property A associated with the operator  is given by=Where ? is the normalize wave function for the state.Or        Where? is the system, state function.6.      Thewave function of a system changes with time according to time dependentSchrodinger wave equation.

-These postulates cannot be proved or derived; we can treat thesepostulates in the same light as the acceptance of Newton’s second law ofmotion. This classical law is accepted without proof on the strength of itsagreement with experimental results.    Derivation of Magnetic Quantum number:The ? equation isgenerally written as,OrIt is a second order differential equation.Above equation is ofsame form as the wave equation for the particle in a box. In term of sine and sine,its solution isIn order fo a wavefunction to be acceptable, it should be of well-behaved class.

One of the suchrequirement of such a function is that it must be single valued. In order to met thisrestriction, the function  should have same value for ? = 0 as it has for? = 2, since these areidentical angular positions.Very often in thetreatment of H-atom the solution of   equation is generally written in the form ofexponential function.Where C is constant and is an imaginary number.

Substituting the valueof C into equation of Where m = 0,  Must be single valued, and also therefore  must same values after any number of wholerevolutions. It is clear that the form of  function depends on the value of the integerm. In effect, m serves to quantize the eigenvalues A in  equation such that . The integer m iscalled magnitude quantum number. It is concerned with the behaviour ofelectrons in atom when it is placed in a magnetic field.This quantum numberdetermines the orientation of the orbitals, i.e the number of different ways inwhich a given orbitals (s, p, d, f) in the presence of magnetic field, can bearranged into the space along x, y and z-axis.Derivation of Azimuthal Quantum Number:According to the ?-equation.

) + ? = 0It is convenient to define the new independent variable z asSo that Comparing equations adby adding and subtracting we have final relation asWhere is known as azimuthalquantum number  and m in  meant to indicate that the forms of associatedLegendre Polynomial, which are functions of z, depends on the value of positiveintegral  and on the absolute value of  of the integer m, which have already definedin last sections. The value of  is restricted to 0, 1, 2, 3,… and magneticquantum number is restricted to It is used to determinethe shape of the orbital, whether the cloud is spherical dumb-bell shaped hassome more complicated shape.Derivation of Principal Quantum Numbers:Principal quantumnumber usually represents the average size of the electron cloud, as theaverage distance of electron from nucleus. It is also used to determine theenergy of electron in a given orbital. It is represented by the integer n.As radial equation is given as But By substituting thevalues and equations add reduction of equations we have equation given as This is a second order differential equation and has a generalsolutionBy differentiation and substituting the value of ‘a’ we getAnd for ‘E’This equation is sameas the Bohr’s expression, In this equation n is the principal quantum number.

It has values, 1, 2, 3, ….. It is obvious that Schrodinger wave equation givesthe same results as the Bohr’s theory for the energy of the electron in thefirst orbit of H atom.Treatment of Hydrogen Molecule With Respect To QuantumMechanics:We know that covalentbond is formed by sharing of electrons which usually hold two of nucleitogether. The hydrogen molecule  is the simplest molecule is which there is anelectron pair bond. The are two basic quantum mechanical treatments forHydrogen molecule given as,·        Valence-Bond Theory·        Molecular-Orbital TheoryMolecular orbital areoften constructed as linear combination of atomic orbitals and the electronsare fed into the related orbitals with due regards to the Pauli’s principle.

Valence Bond Theory:The valence bond methodis based on the familiar chemical ideas of resonance and resonance structures.This method is favoured by the some chemists as it gives more pictorial view ofbonding and is closely related to the classical structural theory of organicchemistry. It is based on assumption that half-filled atomic orbitals ofcombining atoms interact to show newer orbitals of the large size. These neworbitals are usually responsible to make the system to show its stability.

Allof others orbitals on atoms remain undisturbed. The half-filled bondingorbitals merge into one another to form stable orbitals. In other words theatoms contribute their valancy to form bond with their other atoms so that themolecules thus formed consist of atomic cores and bonds are formed betweenthese cores.This method was used byHeitler and London to calculate the energy of  molecules and thus the strength of a covalentelectron pair bond is known. They applied quantum theory to the problem of thestructures of homo-nuclear hydrogen molecule.

SWE can be written to describethe behaviour of the electrons in a molecule. Solutions ofthese equations can be obtained by using approximate techniques. The Heitlerand London treatment is called Valence Bond Method.This molecule composedof two protons and two electrons has been found spectroscopically to have inits lowest state a binding energy of  and an internuclear distance of  when the energy of two separated atom is takenas zero.

In term of distance the potential energy of molecule follow as  When the protons andelectron get near to each other then there are different forces of interaction.Sum of which are attractive and other repulsive. The distance between twoprotons is. Similarly the distancebetween two electron is . The potential energyof the system is already known. The force of attraction contributes nativevalue to potential energy and force of repulsion contributes positive value.The four negative terms related to the electrons 1 and 2 for the proton ‘a’ and ‘b’ the positive term to repulsion between proton and proton andelectron-electron.The Hamiltonian operator is Where M is mass of electrons and  and  referred to the coordinated to the twoelectron.

Similarly the wave equation for hydrogen molecule is First of all considerthe hydrogen molecule at large values of. The system willconsist of two normal hydrogen atom each involving a proton and electron.Suppose under these condition the nucleus ‘a’ has associated with its electron1 and nucleus b has electron 2.For the first atom wedesignate the IS orbitals wave function by  for the second  and the wave function of the two atoms by   According to Heitler and London the wavefunction for a system of two atoms can be represented by the product of twoindividual wave functions and is denoted by .                         But it is also possiblethat the nucleus ‘a’ may have associated with it electron 2 and nucleus ‘b’with electron 1.                 The wave function  of the molecule is given by the linearcombination of two products.

On bond formationexchange of electron takes place and the overall wave function chosen byHeitler and London consists of equal mixture of  and .+2The term involving  represent enhancement of electron density inoverlap region. Therefore the strength o bond in VB theory can be measure ofaccumulation of electron density in the inter-nuclear region. The VB approachtreats the bond as purely covalent and over estimates the covalent nature ofthe molecule. In the formation of ? each electron is linked with a particularnucleus and the bonding results when the electrons are brought near each other.It is common sense that the linkage of two electrons leads to the formation ofa bond. The Heitler London approach represents the quantum mechanical equalentof the electron pair covalent bond postulated by G.N.

Lewis.Furthermore both theelectrons in  molecule ground state must occupy the same ISorbital the Pauli’s exclusion principle demands that there spins must beopposite i.e for 1 the spin must be and for the other. Consequently electronpairing to form covalent bond occurs only between electrons of opposite spins. Postulates for VB method:·        The atoms which unite to form a moleculecompletely retain their identity.

·        The formation of a double bond is due tooverlap of atomic orbitals if the two atoms each having one unpaired electronscome together the atomic orbitals according to these unpaired electrons overlaptheir electronic wave interact and the spin of the two electrons is getmutually neutralized resulting the formation of a covalent bond which is localizedbetween the two atoms. If the electron in atomic orbital is of parallel spin nobond formation will take place.·        If the atomic orbitals possess more thanone unpaired electrons formation of more than one covalent bond is possible.For example  there are three covalent bonds in it.·        Valence electrons which are alreadypaired cannot make covalent bond unless they are unpaired. Unpairing ofelectrons require energy however energy release by the bond formation usuallycompensate the unpairing energy.·        The strength of covalent bond roughlydepends on the extent of atomic orbital overlap. Greater the extent of overlapstronger is the covalent bond.

Molecular Orbital Theory:The molecular orbitaltheory considers a molecule as a whole one unit. According to its concept theatomic orbital of atoms which are combing they overlap to form new orbitalswhich are called as molecular orbitals. It represents the properties of wholemolecule. The molecular orbital surround two or more nuclei. Two atomicorbitals when overlaps the form two types of molecular orbitals which differfrom each other on the basis of energy.The molecular orbitalhaving low amount of energy is usually known is bonding orbital while anothermolecular orbital having high amount of energy is known as anti-bodingmolecular orbital. Bonding molecular orbital is usually represented by sigma  while anti bonding molecular orbital isrepresented by sigma star.

The filling ofelectrons in molecular orbital theory is usually done by three principles whichare given as·        Aufbau principle·        Pauli’s exclusion principle·        Hund’s ruleTo explain Molecularorbital theory in Hydrogen atom take a look that each of hydrogen atom has oneelectron in its valence shell, both of these electron are present in s-orbital.They overlap and both of electrons are now present in bonding orbital whileanti-bonding orbital is missing.Let us consider theformation of simple homo-nuclear diatomic molecule such as hydrogen molecule inwhich two similar atoms are joined by an electron pair. Although the atoms areidentical but they are separated by  and. Each hydrogen atomhas single electron in its valence shell.

The effective overlap of wavefunction will take place when·        The orbital have identical energy·        The orbital overlap to a specific extent·        Orbital have same symmetryWave functions are given as??The two possible wave functions for bonding and anti-bonding aregiven asAnd So far we have studiedthe MOT for the s-orbital as in case of Hydrogen. Other type of molecularbonding occurs between two of p-orbitals which overlap to form Molecularorbitals. As we know there are three type of p atomic orbitals which areusually directed along three axis as x, y and z. For the formation of molecularorbitals from p-orbitals there is two type of overlapping as ·        Head on Overlap:·        Sideways Overlap:Head on Overlap:In head on overlapthere are both of p orbitals which are along x-axis. The combination of atomicorbital give rise to  which is bonding orbital and  which is anti-bonding molecular orbital. Bothof these orbitals lie on x-axis.Sideways Overlap:When the axis of twop-orbitals is along y and z axis then they will be parallel to each other, theyinteract to form molecular orbitals which are parallel to each other.The bonding molecularorbitals  and  usually have zero electron density on thenuclear axis which are called as nodal plane.

The electron density is equallydistributed above and below the nodal plane. On the other hand anti-bondingmolecular orbitals have low of electron density in the inter-nuclear region.There are different of atoms or molecules showing MOT properties. Diatomicmolecules show the property of MOT.

Like He, . The electronicconfiguration of He-atom is . The 1s orbital ofHe-atoms combine to form one bonding  and one anti-bonding 1s orbitals which isshown in diagram.In above MOT diagrameach of He-atom contributes two electrons, two electrons enter in bondingorbital and two enters in anti-bonding orbital so the bond order is equal tozero asAnd thus  molecule is not formed.For  the systematic diagram pf MOT are given as;

x

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