Quantum Chemistry
Introduction:
Classical Mechanics Was Invented By Sir Isaac Newton To Describe And Predict The Motions Of Objects Such As The Planets As They Move About The Sun. Although Classical Mechanics Was A Great Success When Applied To Objects Much Larger Than Atoms, It Was A Complete Failure When Applied To Atoms And Molecules. It Was Superseded By Quantum Mechanics, Which Has Enjoyed Great Success In Explaining And Predicting Atomic And Molecular Properties. However, Quantum Mechanics Was Built Upon Classical Mechanics, And Someone Has Said That If Classical Mechanics Had Not Been Discovered Prior To Quantum Mechanics, It Would Have Had To Be Invented In Order To Construct Quantum Mechanics.
The Old Quantum Theory:
The Atomic Nature Of Matter:
All Of Ordinary Matter Is Made Up Of Atoms. Although The Idea Of Atoms Was Introduced By The Greek Philosopher Democritus, It Became Part Of Chemistry Only After 1803, When Dalton Proposed His Atomic Theory. This Theory Asserts That Each Element Has Its Own Characteristic Type Of Atoms, And That Atoms Combine As Units To Produce Compounds. Dalton’s Theory Did Not Include Any Information About The Structure Of Atoms And How They Could Bind To Other Atoms To Make Molecules. Dalton Even Proposed That Atoms Might Have Little Hooks That Could Snag Similar Hooks On Other Molecules. In The 1870s Thomson Showed That The “cathode Rays” Emitted By Negatively Charged Metals In A Vacuum Consisted Of Negatively Charged Particles, Now Known As Electrons. Thomson Pictured An Atom As Containing Stationary Electrons Imbedded In A Positive Matrix, Like Raisins In An English Plum Pudding. However, Rutherford Discovered In 1911 That An Atom Contains A Very Small Positive Nucleus, So That The Electrons Must Orbit The Nucleus. The Charge On The Electron Was Measured By Millikan In A Series Of Experiments Carried Out Between 1908 And 1917. Moseley Discovered The Concept Of Atomic Number In 1913, Showing That Each Element Was Characterized By The Charge On Its Nucleus. The Atomic Nucleus Can Be Regarded As Being made Up Of Protons And Neutrons. Protons And Neutrons Now Appear To Be Made Up Of Quarks, And The “superstring” Theory Proposes That Quarks, Electrons, And Other Fundamental Particles Consist Of Vibrations Of Tiny Strings. For Most Chemical Purposes, It Is Sufficient To Regard An Atom As Consisting Of A Positive Nucleus And A Number Of Negative Electrons.
Blackbody Radiation: The Failure Of Classical Theory:
Any Object Will Radiate Energy In The Form Of Electromagnetic Radiation Purely As A Consequence Of Its Temperature. The Red Glow Of An Electric Heater And The Bright White Light Of The Tungsten Fi Lament In An Incandescent Light Bulb Are Familiar Examples. This Radiation Is Referred To As blackbody Radiation.
The Physical Properties Of Blackbody Radiation Depend Only On The Temperature Of The Object, Not On Its Composition. If We Measured The Intensity Of Blackbody Radiation Versus The Wavelength Emitted At Different Temperatures, We Would Obtain A Series Of Curves Similar To The Ones Shown In Figure 1. Experiments At The End Of The Nineteenth Century By Josef Stefan And Wilhelm Wien Led To Two Important Empirical Laws Of Blackbody Radiation, Now Named The Stefan-Boltzmann Law And Wien’s Law. (An Empirical Law Is One That Is Formulated Purely On The Basis Of Experimental Data.)
- Stefan-Boltzmann Law: The Total Intensity Of Blackbody Radiation Emitted By An Object (obtained By Integrating The Curves In Figure 1 Over All Wavelengths) Is Proportional To The Fourth Power Of The Absolute Temperature (that Is, The Temperature In Kelvins)
(total Radiant Emittance) = σT4
Where The Stefan–Boltzmann Constant σ has The Experimental Value σ =5.67051 × 10−8 Jm−2 S−1 K−4 = 5.67051 × 10−8Wm−2 K−4 Where The Joule (J) Is The Unit Of Energy And The Watt (W, Equal To Joules Per Second) Is The Unit Of Power And T Is The Absolute Temperature. The Law Was Used By Stefan To Estimate The Surface Temperature Of The Stars And Sun.
Figure 1. The Quantized Energies Of An Oscillator As Postulated By Planck. The Horizontal Line Segments Are Plotted At The Heights Of The Assumed Energy Values, 0, hν, 3hν, 4hν, 5hν, 6hν, 7hν, Etc.
Wien’s Law: The Wavelength Of Maximum Intensity (λmax) Is Inversely Proportional To The Absolute Temperature:
λmaxT = 2.90 X 10-3 M. K
Since, λmax=hxc/4.965 x T x KB where H, C And KB Is Plank Constant Velocity Of Light And Boltzmann Constant Respectively.
Einstein’s Theory Of The Photoelectric Effect:
In 1905, Only Fi Ve Years After Planck Presented His Quantum Theory, Albert Einstein Used The Theory To Explain The photoelectric Effect—a Phenomenon In Which electrons are Ejected From The Surface Of Certain Metals Exposed To Electromagnetic Radiation.
Experimentally, The Photoelectric Effect Is Characterized By Three Primary Observations:
- The Number Of Electrons Ejected Is Proportional To The Intensity Of The Light.
- No Electrons Can Be Ejected If The Frequency Of The Light Is Lower Than A Certain threshold Frequency, which Depends Upon The Identity Of The Metal.
- The Kinetic Energy Of The Ejected Electrons Is Proportional To The Difference Between The Frequency Of The Incident Light And The Threshold Frequency.
The Photoelectric Effect Could Not Be Explained By The Wave Theory Of Light. In The Wave Theory The Energy Of A Light Wave Is Proportional To The Square Of The Amplitude (intensity) Of The Light Wave, Not Its Frequency. This Contradicts The Second Observation Of The Photoelectric Effect. Building On Planck’s Hypothesis, Einstein Was Able To Explain The Photoelectric Effect By Assuming That Light Consisted Of Particles (light Quanta) Of Energy h, Where is The Frequency Of The Light. These Particles Of Light Are Called photons. Electrons Are Held In A Metal By Attractive Forces, And So Removing Them From The Metal Requires Light Of A Sufficiently High Frequency (which Corresponds To A Sufficiently High Energy) To Break Them Free. We Can Think Of Electromagnetic Radiation (light) Striking The Metal As A Collision Between Photons And Electrons. According To The Law Of Conservation Of Energy, We Have Energy Input Equal To Energy Output.
If exceeds The Threshold Frequency, Einstein’s Theory Predicts:
Where,∅, Called Work Function Of The Metal, Is Similar To Ionization Energy And 1/2 Mv2 Is Kinetic Energy Of Emitted Electron. The Work Function Measures How Strongly The Electrons Are Held In The Metal. The Threshold Frequency Is The Smallest Frequency For Which Equation 1 Has A Solution. This Occurs When The Kinetic Energy Of The Electron Is Zero, In Which Case Equation 1 Gives
hvthreshold = ∅
Therefore,
The Kinetic Energy Of Electron Is Also Equal To = -e X Vs, Where Vs Is The Stopping Potential.
Consequently, Einstein’s Theory Predicts That The Kinetic Energy Of The Ejected Electron Is Proportional To The Difference Between The Incident And Threshold Frequencies, As Required.
Figure 2: shows A Plot Of The Kinetic Energy Of Ejected Electrons Versus The Frequency Of Applied Electromagnetic Radiation.
The Wave–Particle Duality Of Light:
Since Light Exhibits A Particle-like Nature In Some Experiments And Wave-like Properties In Other Experiments, We Say That It Exhibits A Wave–particle Duality. We Cannot Give A Simple Answer To The Question: “What Is Light Really Like?”We Use The Wave Description When It Explains A Particular Experiment, And Use The Particle Description When It Explains Another Experiment.
The Bohr Model Was An Early Attempt To Formulate A Quantum Theory Of Matter:
The Work Of Planck And Einstein Showed That The Energy Of Electromagnetic Radiation At A Given Frequency () Is Quantized In Units Of H. The Extension Of This Quantum Hypothesis To Matter Paved The Way For The Solution Of Yet Another Nineteenth-century Mystery In Physics: The Emission Spectra Of Atoms.
Emission Spectra Of Atoms: Evidence Of The Energy Quantization Of Matter:
Ever Since The Seventeenth Century, Chemists And Physicists Have Studied The Characteristics Of emission Spectra, which Are Either Continuous Or Line Spectra Of The Radiation Emitted By Substances. The Emission Spectrum Of A Substance Can Be Seen By Energizing A Sample Of Material Either With Thermal Energy (heating) Or With Some Other Form Of Energy (such As A High-voltage Electrical Discharge). A “red-hot” Or “white-hot” Iron Bar Freshly Removed From A High-temperature
source Produces A Characteristic Glow. This Visible Glow Is The Portion Of Its Emission Spectrum That Is Sensed By Eye. The Warmth Of The Same Iron Bar Represents The Infrared Region Of Its Emission Spectrum.
The Emission Spectra Of The Sun And Of A Heated Solid Are Both Continuous; That Is, All Wavelengths Of Visible Light Are Represented In The Spectra. The Emission Spectra Of Atoms In The Gas Phase, On The Other Hand, Do Not Show A Continuous Spread Of Wavelengths From Red To Violet; Rather, The Atoms Produce Bright Lines In Different Parts Of The Visible Spectrum. These line Spectra are Light Emissions At Specific Wavelengths.
The Emission Spectrum Of Hydrogen:
At The End Of The Nineteenth Century, Physicists Began Exploring The Emission Spectra Of Atoms In Quantitative Detail. Of Particular Interest, Because Of The Simplicity And Importance Of The First Element, Was The Emission Spectrum Of Hydrogen .The Swedish Physicist Johannes Rydberg Analyzed The Existing Experimental Data And Formulated The Following Equation For The Frequencies Of The Lines In The Hydrogen Emission Spectrum:
where n1 And n2 Are Two Positive Integers And RH Is A Constant Known As Rydberg’s Constant, Equal To 109677.581 Cm−1 If The Wavelengths Are Measured In A Vacuum. Classical Physics Was Unable To Explain This Relationship.
The First Five Spectral Series In The Emission Spectrum Of Hydrogen
The Wave-Particle Duality Of Matter—The De Broglie Hypothesis:
Niels Bohr Received The 1922 Nobel Prize In Physics For His Hydrogen Atom Theory, Based On An Assumption Of Quantization Of Angular Momentum. In 1923 A Graduate Student At The University Of Paris Named Prince Louis De Broglie Was Trying To Find A Physical Justification For Bohr’s Hypothesis Of Quantization. In Classical Physics, One Thing That Is Quantized Is The Wavelength Of A Standing Wave. De Broglie Sought A Way To Relate This To Bohr’s Theory And Came Up With The Idea That A Moving Particle Such As An Electron Is Accompanied By A “fictitious Wave.”
According To Einstein’s Theory Of Relativity, A Particle Of Energy E has A Mass m such That
E = Mc2 --------------------------------1
where C Is The Speed Of Light. If We Apply This To A Photon And Use The Planck–Einstein Relation, For The Energy And If We Replace Mc By The Momentum P, The Eq. 1 Becomes
Where λ is The Wavelength Of The Photon And h is Planck’s Constant. De Broglie Deduced That The Velocity Of The Wave Accompanying A Particle Was The Same As The Velocity Of The Particle.
We Omit De Broglie’s Argument, Which Is More Complicated. The Quantization Assumption Of Bohr’s Theory Arises Naturally From Eq. 3 If One Assumes That The Circumference Of A Circular Electron Orbit In A Hydrogen Atom Is Equal To An Integral Number Of Wavelengths. This Assumption Means That The Wave Repeats Itself With The Same Phase (with Crests In The Same Positions) On Each Trip Around The Orbit. For A Circular Orbit
In Terms Of Kinetic Energy
If A Charged Particle Carrying Charge q is Associated Through A Potential Difference V Volts, Then Kinetic Energy Ek = qV .Therefore, De-Broglie Wavelength For Charged Particle Of Charge Q And Accelerated Through A potential Difference Of V Volt Is
When A Material Particle Like Neutron Is In Thermal Equilibrium At Temperature T, Then They Possess Maxwell Distribution Of Velocities And So The Kinetic Energy Of Most Of Material Is Given By
Ek = kT
Where k is Boltzmann’s Constant k = 1.38 x10-23
The Heisenberg Uncertainty Principle
One Of The Assumptions Of Classical Physics Is That The Dynamical Variables (positions And Momenta) Of A Particle In Motion Have Well-defined, Precise Values. However, The Concept Of A Precise Position Becomes Ill-defined When We Try To Describe A Particle As A Wavelike Object. A Wave Is An Object That Is Extended Over Some Region Of Space. To Describe The Problem Of Trying To Locate A Subatomic Particle That Behaves Like A Wave, Werner Heisenberg Formulated What Is Now Known As The Heisenberg Uncertainty Principle: It Is Impossible To Know Simultaneously Both The Momentum P (defined As Mass Time’s Velocity) And The Position Of A Particle With Certainty. Stated Mathematically We Have,
Where are The Uncertainties In Measuring The Position And Momentum, Respectively. Thus, If We Measure The Momentum Of A Particle More Precisely (that Is, If We Make a Small Quantity), Our Knowledge Of The Position Will Become Correspondingly Less Precise (that Is, will Become Larger). Similarly, If The Position Of The Particle Is Known More Precisely, Then Its Momentum Must Be Known Less Precisely.
This Inverse Relationship Arises Because The Position Of A Wavelike Particle Is Determined By The Region Of Space Occupied By The Wave, But The Momentum, Through The De Broglie Relationship, Is Related To The Wavelength Of The Wave.
The Time Independent Schrödinger Wave Equation
In Classical Mechanics, The State Of A Particle Is Defined Uniquely By Its Position And Momentum. If You Know Both Of These Quantities, Then You Can Predict The Future Motion Of The Particle Based On The Forces Acting Upon It. According To Heisenberg’s Uncertainty Principle, Though, This Sort Of Knowledge Is Unavailable For A Quantum Particle, As The Position And Momentum Cannot Be Simultaneously Specified. Schrödinger postulated That The Complete Information About The State Of A Quantum Particle Was contained In A Function (x), Called The wave Function, which Is A Function Of The Position Of The Particle (given By x for A One-dimensional System). One Of The Most Important Properties Of Wavelike Objects Is The Ability To Exhibit Constructive And Destructive Interference. For This To Be Possible, The Wave Function Must Be Able To Take On Positive or negative Values. By Analogy With The Laws Of Optics, Schrödinger Proposed That The Wave Function For A Particle Of Mass m in One Dimension Is The Solution To The Equation
we Abbreviate The Schrodinger Equation In The Form:
HΨ = E Ψ Where H Is Hamiltonian Operator
The Time Dependent Schrödinger Wave Equation
HΨ=ih DΨ/dt Where i is The Imaginary Unit, Defined To Equal The Square Root Of −1
The Particle In A One-Dimensional Box: A Simple Model
In This Problem A Particle Of Mass Is Placed In A One-dimensional Box Of Length l Particle Is Free To Move. Box Has Infinite Walls.
It Is Assumed That The Potential Energy Of Particle Is Zero Everywhere Inside The Box I.e.
V(x)=O.
The Time-independent One-dimension Schrodinger Equation Will Be
H = E Ψ (1)
Where H Is Hamiltonian Operator:
That Is = 0 Outside The Box. This Means That The Particle Cannot Exist Outside The Region O L. Within The Box, The Schrodinger Equation For The Motion Of The Particle Takes The Form.
A General Solution Of Equation (2) Is Given By.
Ψ = A Sin ( α x) + B Cos ( α x) -------------------(6)
Where A And B Are Constant.
Boundary Conditions:
(1) Ψ (x)=0 At X=0
Ψ = A Sin ( α 0) + B Cos ( α 0)
The Above Expression Will Be True Only When B = 0, Thus The Wavefunction As Given By Equation (4) Becomes.
Ψ = A Sin ( α 0) ------------------(7)
(2) Ψ(x) = 0 At X = ????, Then
0= A Sin ( α l)
The Above Expression Will Be True Only Where α l Is An Integral Multiple Of π That Is,
α l = N π ---------------(8)
When N Can Have Only Integral Values Of L, 2, 3,..... And Is Known As Quantum Number. A Value Of N = 0 Is Eliminated Since It Leads To α= 0 I.e. Ψ (x)=0 Everywhere Within The Box. Substituting A From Equation (6) In Equation (5) We Get.
Energy Of A Particle In 1-D Box Is Quantized.
So Energy Difference Between The Two Successive Energy Levels Will Be
E = En+1 - En
Energy Level Diagram In 1-D Box:
Graph Of Wave Functions And Probability:
Normalization Of Ψ
Since The Total Probability Of Finding The Particle Within The Box Is 1, Therefore, According To Born’s Interpretation Of The Wave Function , The Normalization Of Ψ Required That
Particle In A Symmetrical 1-D Box From -l To +l:
Consider A Free Particle Is Confined To Move In 1D Box Of Length -l To +l.
V = 0 Inside The Box
V = Else Where
Schrodinger Wave Equation For 1-D Box Is
Possible Solutions:
Particle In Two-Dimensional Box:
The Mathematical Form Of Potential For 2-D Box:
V (x, Y) = 0 0 < X < ?x, 0 < Y y = 0 Otherwise,
For 2-D Box Wave Function Ψ Will Dependent Upon Two Independent Variable X And Y, And
[ Ψ (x, Y) = X (x). Y (y)] Is The Multiplication Of Both Function.
Schrodinger Wave Equation For Free Particle For 2-D Box:
Degeneracy:
Sometimes, There Are Several Solutions With The Exact Same Energy. Such Solutions Are Called ‘degenerate’
Degeneracy In 2-D Box
Zero-point Energy:
nx=1 Ny=1 ⇒ E11 =h2/m?2
Particle In Three-Dimensional Box
Now We Will Discuss The Motion Of Particle Of Mass M In A Three-dimensional Box. As In One Dimensional Box In Three-dimensional Box Also, The Potential Energy Is Zero With In The Box And Infinite Outside The Box.
So Three Dimensional Schrodinger Equation.
Where The Function Ψ Will Depend Upon Three Independent Variable X, Y, Z To Solve The Above Equation We Write The Function Ψ As The Product Of Three Wave Function.
Schrodinger Wave Equation For A Particle In Three Dimension Box.
The Term α2 Inthe Above Equation Is A Constant Quantity. Hence The Sum Of The Three Terms On The Left-hand Side Of Equation (4) Must Also Be A Constnat Quantity. If We Change The Value Of X (or Y Or Z) Keeping The Other Two Variables Constants Even Then The Above Constancy Has To Be Satisfied. This Is Possible Only When Each Term Is Independent Of The Other Terms And Each Is Equal To A Constant Quantity So That The Sum Of Three Constants Is Equal To α2.
Now We Have Three Separate Equations To Be Solved Each Of Them Has A Form Of One-domensional Box. Thus, The Normalized Wave Function Of A Three-dimensional Box Is
Degeneracy Of 3-D Box:
The Schrodinger Equation And De BroglieWaves
The Particle In A Box Model Provides An Illustration Of The Fact That The Energy Eigenfunction
represents De Broglie Waves. In The Case Of Zero Potential Energy
Where We Use The Definition Of The Momentum, p = Mv. From The De Broglie Wavelength Formulae
The Energy Of A De Broglie Wave Is Inversely Proportional To The Square Of Its Wavelength. The Wavelength Of The Wave Function Is The Value Of x such That The Argument Of The Sine Function Equal To 2π
As The Value Of n increases, The Energy Increases, The Wavelength Decreases And The Number Of Nodes Increases. It Is An Important General Fact That A Wave Function With More Nodes Corresponds To A Higher Energy.
| Harmonic Oscillator |
One-Dimensional Simple Harmonic Oscillator
A Diatomic Vibrating Molecule Can Be Represented By A Simple Model, The So-called Simple Harmonic Oscillate (S.H. 0.). The Force Acting On The Molecule Is Given By F = -kx, Where Xis The Displacement From Theequihbriumll Position And K Is A Constant Called The Force Constant. The Potential Energy V (x) Of This Molecule Is Given By
Equation (1) Is The Equation Of A Parabola. Thus, If We Plot Potential Energy Of A Particle Executing Simple Harmonic Oscillations As A Function Of Displacement From The Equilibrium Position, We Get A Curve As Shown In Figure.
This Energy Is Called The Zero Point Energy Of The Oscillator. Classical Mechanics Predicts That The Zero-point Energy Of The Oscillator Is Zero Whereas Quantum Mechanics Predicts That The Zero-point Energy Is Non-zero. The Occurrence Of The Zero-point Energy Is Consistent With The Heisenberg Uncertainty Principle.
Graph Of Wave Function And Probability:
Degeneracy For 2-D Harmonic Oscillator:
Degeneracy For 3-D Harmonic Oscillator:
Virial Theorem:
Simple Harmonic Oscillator's Formula
With The Formula Given Below We Can Solvethe Problems Of The Simple Harmonic Oscillator
Hydrogen Atom & Rigid Rotor
Quantization Of Electronic Energy:
Necessity Of Replacing Bohr’s Theory:
The Mathematical Framework Of The Bohr’s Theory Was Based On The Basic Assumption Of Quantization Of Orbital Angular Momentum Of The Electron. It Was Seen Earlier That This Theory Led To The Quantization Of Electronic Energies Which Formed The Basic For Explaining The Experimental Spectra Of Hydrogen like Species Such As H, He+, Li2+ And Be3+. However, This Theory Was Not Entirely Satisfactory As It Failed To Provide An Interpretation Of Relative Line Intensities In The Hydrogen Spectrum And Also And Also Failed Completely When It Was Applied To Explain The Energies And Spectra Of More Complex Atoms In This Section, We Consider The Application Of Schrodinger’s Wave Theory To One-electron Atom, In The Subsequent Section, It Will Be Shown How The Principles Of This Theory Can Be Applied, In A More Approximate Way, To Many Electron Atoms.
Setting Of Schrodinger Equation:
The Time Independent Form Of Schrodinger Equation Is
HOP ψtotal = Etotal ψtotal ……(1)
Where HOP Is The Hamiltonian Operator, Etotal is The Total Nonrelativistic Energy And ψtotal is The Wave Function For The Total System. Since The Hydrogen Like System Contains Two Particles, Namely, Nucleus And Electron, It Is Obvious That The Wave Function ψtotal depends On The Six Coordinate Variables, Three For The Electron (xe, Ye, Ze) And Three For The Nucleus (xn, Yn, Zn), Both Sets Of Coordinate Refer To The Common Origin. The Hamiltonian Operator Consists Of Two Terms, Viz, The Kinetic And Potential Energy Terms. The Kinetic Energy Operator Will Contain Two Terms, One For The Electron And One For The Nucleus. Thus, We Have
Equation (3) Can Be Broken Into Two Simpler Equations, One Involving The Free Movement Of The Centre Of Mass Of The Atom In Space And The Other Involving The Relative Motion Of The Electron With Respect To The Nucleus Within The Atom.
The Two Equations Are:
Equation (4) Is Simply The Schrodinger Equation For A Free Particle Of Mass (mc+ Mn), Etrans Is The Trapslational Kinetic Energy Associated With The Free Movement Of The Centre Of Mass Of The Atom Through Space. Equation (5) Is The Shrodinger Equation Which Represents The System In Which An Electron Of Reduced Mass μ Is Revolving Around The Stationary Nucleus Of Positive Charge Z At A Distance Of R. The Behaviour Of This Electron Can Be Described By The Function ψe And E Is The Corresponding Energy Of The Electron. The Allowed Values Of Electronic Energies Can Be Obtained By Solving Equation (5) And Equation (4) which Describes The Motion Of The Centre Of Mass Is Of The Same Form As That Of The Particle In A Three Dimensional Box.
Schrodinger Equation In Terms Of Spherical Polar Coordinates:
The Solution Of Schrodinger Equation Becomes Very Much Simplified If The Equation Is Expressed In The Coordinate System That Reflects The Symmetry Of The System. In The Present Case The Potential Energy Fieldis Spherical Symmetry (V Depends Only On r), And Thus, It Is Convenient To Transform The Schrodinger Equation (5) Into The Spherical Polar Coordinates R,ø and ? By Using The Relations
Splitting Of Schrodinger Equation:
Rearranging The Above Expression, We Get
Since The Operator In Equation (11) Consists Of Two Terms. One Depending On ? And The Other φ On , We Can Write The Wave Function Y?φ as:
Three Split Expressions Of The Schrodinger Equation:
Thus, The Schrodinger Equation (6) For The Hydrogen Like Species Can Be Separated Into Three Equations.
These Are:
- Equation Involving Only R:
We Now Consider The Acceptable Solutions Of Equations (16), (17) And (18)
Solutions Of ? Dependent Equation:
Equation (17) Has Already Been Solved In Connection With The Rigid Rotator System. Its Solution Is
These Solutions Are Obtained Provided The Following Conditions Are Satisfied.
? = 0, 1, 2, 3, .
m = 0, ± 1, ± 2, , ± ?
The Quantum Number f is Called The Azimuthal Quantum Number Or The Subsidiary Quantum Number And It Represents The Quantization Of The Square Of Total Angular Momentum According To The Equation
Solution Of dependent Equation:
With The Help Of A Suitable Transformation Of Independent Variable And From The Forms Of Solutions As The Variable Approaches Zero And Infinity, It Is Possible To Write In The Following More Familiar Form, Known As Associated Laguerre Equation.
The Solution Of Equation (25), As Determined By The Power Series Method, Is The Associated Laguerre Polynomial Of Degree (k-j) And Order J, And Is Given By
In Order That The Associated Laguerre Polynomial Is The Acceptable Solution Of Equation (25) The Following Quantum Restriction Has To Be Satisfied.
k = 1,2,3,……, …..(34)
Acceptable Solutions, As Usual, Means That The Wave Function L(ρ) Should Be A Well-behaved Function And Should Vanish As ρ → ∞ . The Latter Condition Requires That The Polynomial Must Be Restricted To A Finite Number Of Terms.
This Quantum Restriction Leads To The Fact That λ, Which Is Equal To K - ?( Equation (29)) Must Also Be An Integer.
Let λ Be Written As N, So That
k = N+ l ... (35)
Expression Of The Normalized R Functions:
Thus, The Function R As Given By Equation (26) Is:
Allowed Values Of N And Its Relation With l:
The Constant N Is Reffered To As The Principal Quantum Number. If The Associated Laguerre Polynomial (equation 32) Is Not To Vanish, We Must Have
Expressions For The Function Rn, ? :
The Exact Form Of The Function Rn, ? (r) For The Given Values Of N And ? can Be Determined From Equations (37), (32) And (33).
These Equations Are:
Virial Theorem For Hydrogen Atom:
Potential:
Where, E = Total Energy
T = Kinetic Energy
V =potential Energy
V = -2T (Virial Theorem For Hydrogen Atom)
Degeneracy In H-atom:
(1) If A Spin Is Not Included Degeneracy = n2 (where, N Is Principle Quantum Number)
(2) If Spin Is Included Degeneracy = 2n2
Hartree:
Hartree's Method Of Self-consistent Fields: An Important Method For Obtaining A Suitable Central Field Is Due To Hartree. In This Method, A Suitable Function ( Ri ) Is First Chosen With Property Close To That Given By Equation
The Single Particle States ψi Are Then Obtained By Solving The Schrodinger Equation. Now, The Part Of The Potential Due To All Electrons Other Than The Jth Is Calculated By Assuming That Each Of These Electrons Yields A Charge Distnbution As Given By Their Wavefunctions Already Worked Out.
Consider The Jth electron , J ≠ i. The Average Charge In A Volume Element Dτj Due To This Electron Is E Dτj | Ψ(rj)|2 and The Potential Energy Of Interaction With The Ith electron Is
where The First Term Is Due To The Nucleus.
With This Potential, We Can Repeat The Whole Process, I.e. Calculate The Wavefunctions (in General, Numerically), Etc. The Process Is Repeated Until Two Successive Reactions Yield The Same V¯(ri) to The Desired Accuracy. In Other Words, The Electron Supposed To Move In A Potential Which They Themselves Generate.
This Is Why The Method Is Called ‘self-consistent’.
The Numeric Calculations Necessary Are Possible Only With High Speed Computers And That Too With A Great Deal Of Technical Expertise.
We Shall, However, Illustrate The Method With An Elementary Calculation For Helium. Hartree's Method Is Designed For Heavier Atoms Than Helium But The Purpose Here Is To Clarify The General Technique. We Have Used A Small Programmable Calculator For This Purpose And Would Strongly Advise The Student To Try This Calculation Himself.
Applications To Helium Atom:
We Start By Assuming A Potential
V0 (r) = -2/r …(4)
This Is The Potential Either Electron Moves In If The Other Were Absent. [ In Practice, It Would Save Considerable Labour To Start With A More Realistic Potential In Accordance With Equation (1). The Radial Function For The Ground State Is
|
Figure: The Normalized Wave Function U(r) Corresponding To Eigenvalue E ≈ 0.82 |
Adding The Term –Z/r1 [See Equation (3)], We Obtain
The Suffix On V‾1 Indicating That We Have Now Worked Out The First Interation. Note That This Function Already Has The Correct Limiting Behavior For R1 << And R1 >>1. We Can Solve The Radial Equation Numerically To Obtain The Ground State Wavefunction. The Normalized Wavefunction U (r), Corresponding To An Eigenvalue E ≈ 0.82 Is Plotted In Figure Above. Here U(r) = RR(r) Is The More Convenient Function To Use. The Radial Equation Is Then
Since For The 's’ Wave Functions Only The l = 0 Terms Survive On Integration Over Angles, We Have
Each Of The Integrals In Equations (15), (16) And (17) Has To Be Evaluated Numerically.
For The Wave Function In Figure Above. We Obtain,
(t) = 1.29
(v) = -3.20 …..(19)
And (v12) = 0.97
The Total Energy Of Helium Atom (ground State) Is Not Twice The Sum Of The Above Values. We Must Remember That The Electrostatic Energy Of The Two Electrons Must Be Added Just Once. Thus,
E0 = 2((t) + (v)) + (V12) -2.85 ….(20)
Which Turns Out To Be -77.6 EV; Somewhat Larger Than The Observed Value Of -79 EV.
The Above Calculation, Chosen For Its Simplicity, Misses Some Of The Subtler Points. For Example, The Two Electron In The Ground State Are Both In The 1s State So That The Self-consistent Potential For Them Is The Same. This, However, Is Not True In General. For The Potential Of One Of The Electrons Due To Other,
showing Explicitly That V1 ( R1) And V2 ( R2) Are Different Functions. (Hence The Suffixes On V1 And V2).
One Can Use These To Calculate The Energies As Before.
Since The Effective Hamiltonian For The Various Electrons Are Different, The Resulting Solutions Of The Wave Equations For The Electrons Will Not, In General, Be Orthogonal. Moreover, The Wave Functions Used Will Have To Satisfy The Exclusion Principle. These Features Are Incorporated In The Hartree-Fock Method.
The Complete Hydrogen Like Atomic Wave Functions Expressed As Real Functions For N = 1, 2, And 3. The Quantity Z Is The Atomic Number Of The Nucleus And = Zr / 0 Where A0 Is The Bohr Radius.