1. CHEMICAL
BONDING
B y
D r . B I N U S R E E J A Y A N
A S S I S T A N T P R O F E S S O R
D E P A R T M E N T O F C H E M I S T R Y
K S M D B C O L L E G E , S A S T H A M C O T T A , K O L L A M
2. SCHRODINGER EQUATION
• The fundamental equation of Quantum Mechanics.
𝐻𝜓=𝐸𝜓
• 𝐻 - 𝑡ℎ𝑒 𝐻𝑎𝑚𝑖𝑙𝑡𝑜𝑛𝑖𝑎𝑛 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 ( 𝑎𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟),
• 𝜓 - 𝑤𝑎𝑣𝑒− 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑟 𝑒𝑖𝑔𝑒𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑟 𝑠𝑡𝑎𝑡𝑒 function
• E - Total energy of the system
𝑯• mathematical command
• The total energy operator or quantum mechanical operator of energy.
• Sum of kinetic energy operator ( 𝑻) and potential energy operator ( 𝑽).
𝐻𝜓=( 𝑻 + 𝑽 )=𝐸𝜓
𝑻 ≠0;
𝑽 may or may not be zero
Cap indicates An operator
3. Schrödinger equation for a single particle of mass m with zero potential
energy
• Moving in one dimension (x direction).
• 𝐻𝜓 = ( 𝑻 + 𝑽 )𝜓 =𝐸𝜓
• Given, 𝑽 =0.
• 𝐻𝜓 = 𝑻x 𝜓 =𝐸𝜓
−ℎ2
8𝜋2 𝑚
.
𝜕2 𝜓
𝜕𝑥2 =𝐸𝜓
𝑻x=
−ℎ2
8𝜋2 𝑚
.
𝜕2
𝜕𝑥2
• Moving in two dimension (xy plane)
• 𝐻𝜓 = ( 𝑻 + 𝑽 )𝜓 =𝐸𝜓
• Given, 𝑽 =0.
• 𝐻𝜓 = ( 𝑻x + 𝑻y )𝜓 =𝐸𝜓
•
−ℎ2
8𝜋2 𝑚
(
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2)𝜓 =𝐸𝜓
• Moving in three dimension (xyz space).
• 𝐻𝜓 = ( 𝑻 + 𝑽 )𝜓 =𝐸𝜓
• Given, 𝑽 =0.
• 𝐻𝜓 = ( 𝑻x + 𝑻y + 𝑻z )𝜓 =𝐸𝜓
−ℎ2
8𝜋2 𝑚
(
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2 +
𝜕2
𝜕𝑧2)𝜓 =𝐸𝜓
4. Writing Schrodinger equation for
chemical system
• For a given chemical system with “n” nuclei and “e” electrons, the total number of terms
(kinetic energy and potential energy terms)
𝑁𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝐻 𝒕𝒆𝒓𝒎𝒔=
(𝒏+𝒆)(𝒏+𝒆+𝟏)
2
(𝑛 + 𝑒)(𝑛 + 𝑒 + 1)
2
Kinetic energy terms Potential energy terms
𝑛 + 𝑒
Electron-nuclear attraction
𝑛𝑒
Nuclear- Nuclear repulsion
𝑛(𝑛 − 1)
2
Electron electron repulsion
𝑒(𝑒 − 1)
2
5. EXAMPLES….
System n e Total number of
terms in
Hamiltonian
H 1 1 3
He+ 1 1 3
He 1 2 6
He2 2 4 1
C 1 6 28
CH4 5 10 120
6. Set Up SWE
• 𝐻, 𝑡h 𝑒𝐻𝑎𝑚𝑖𝑙𝑡𝑜𝑛𝑖𝑎𝑛𝑜𝑝𝑒𝑟𝑎𝑡or: The total energy operator or quantum mechanical operator of
energy.
• Sum of kinetic energy operator ( 𝑻) and potential energy operator ( 𝑽).
𝐻𝜓=( 𝑻 + 𝑽 ) 𝜓 =𝐸𝜓
𝑻 =
−ℏ 𝟐
𝟐𝒎
𝛁 𝟐
𝑽= attraction between e-n + repulsion between e-e + repulsion between n-n
𝑽 =
𝒒 𝟏 𝒒 𝟐
𝒓
𝐻=
−ℏ 𝟐
𝟐𝒎
𝛁 𝟐+
𝒒 𝟏 𝒒 𝟐
𝒓
r
𝒒 𝟏
𝒒 𝟐
8. APPROXIMATION METHODS
Why approximation methods are needed???
Complete solution of Schrodinger equation is possible only for a few systems….
1. Electron moving under a constant potential field.
2. Rigid Rotor
3. Harmonic Oscillator
4. Hydrogen atom ----- Real sytem
SWE exactly soluble
True 𝛹 & E
All properties
10. Approximation methods….
• Variational Approximations From The (Rayleigh-ritz) Variational Principle
• Time-independent Perturbation Theory For Schr¨odinger Eigenvalue Problem
• Self Consistent Field [SCF] Method…
• Time-dependent Perturbation Theory And Fermi’s Golden Rule
• Semiclassical Approximation And The WKB Method.
• Bohr-sommerfeld Approximation For Excited States
• Partial Wave Approximation In Potential Scattering……..
11. VARIATION METHOD…
• Applicable to system whose 𝜓 can be guessed.
• Guessed 𝜓 ---- Trial 𝜓
• Adjustable parameters ----- Variation parameters
• Trial 𝜓 gives Ea
Based on Eckart’s Theorem / Variation Theorem
𝑬 𝒂 ≥ 𝑬 𝟎
𝐻 𝜓= E 𝜓
𝐸 =
𝜓∗
𝐻 𝜓𝑑𝜏
𝜓𝜓∗ 𝑑𝜏
𝐸 =
< 𝜓∗/H/𝜓 >
< 𝜓/𝜓∗ >
𝐸0 =
𝜓0
∗
𝐻 𝜓0 𝑑𝜏
𝜓0 𝜓0
∗
𝑑𝜏
𝐸 𝑎 =
𝜓∗ 𝐻 𝜓𝑑𝜏
𝜓𝜓∗ 𝑑𝜏
12. Variation Theorem – Definition
The energy evaluated by trial wavefunction should be always greater than or equal to true
energy of the system in its ground state.
OR
The variation theorem says that no approximate wave function can have lower energy than the
exact ground state energy of the system
OR
If the ground state function 𝜓0 𝑓𝑜r a system is approximated by a trial wave function 𝜓, has
the same boundary conditions of the problem, then the calculated energy (Ea) must be greater
than or equal to the actual ground state energy.
𝐸0 =
𝜓0
∗
𝐻 𝜓0 𝑑𝜏
𝜓0 𝜓0
∗
𝑑𝜏
𝐸 𝑎 =
𝜓∗ 𝐻 𝜓𝑑𝜏
𝜓𝜓∗ 𝑑𝜏
𝑬 𝒂 ≥ 𝑬 𝟎
𝑖𝑓, 𝜓0 ≠ 𝜓
