Objectives of this course
- Presentation of basic knowledge about the computational
methods of theoretical chemistry
- In particular about their reliability, the range of applicability
and expected accuracy in solving problems of structural
chemistry, spectroscopy, thermochemistry, and chemical
reactivity
- The course is proposed for student interested in applications
of theoretical chemistry rather than in its further development.
It is assumed that students know quantum chemistry at the level
of the III Semester Course “Introduction to quantum chemistry’’
The objectives of theoretical chemistry
- Prediction of properties of single molecules, in particular:
- molecular structure (geometry – bond lengths, angles)
- molecular charge distribution (dipole, quadrupole
moments)
- energetics: bond dissociation energies, conformation
energies, barriers, activation energies, reaction energies
- spectra (rotational, vibrational, electronic, NMR, EPR,...
- electric and magnetic properties of molecules:
polarizability, magnetic susceptibility
The objectives of theoretical chemistry, continued
- Prediction of properties of molecular aggregates, supramolecular
and macroscopic systems, in particular:
- intermolecular interactions
- thermodynamic properties and functions (like entropy)
and chemical equilibrium constants
- properties of liquids and solids
- relaxation processes
- characteristics of phase transitions
- rates of chemical reactions in the gas, liquid and solid phase
- mechanisms of catalytic reactions
Parts of theoretical chemistry
- Quantum Chemistry
- electronic structure theory - Born-Oppenheimer approximation
and the concept of the Potential Energy Surface (PES) or curve
- theory of nuclear (rovibrational) dynamics in molecules
- theory of molecular collisions and reactions
- theory of nonadiabatic processes
- Statistical thermodynamics and mechanics
- analytic methods (classical and quantum)
- computer simulation methods - Monte Carlo methods
(classical and quantum) and classical molecular dynamics
Quantum Mechanics
- non-relativistic (Schrodinger-Coulomb equation)
- relativisitc (Dirac-Coulomb equation)
- quantum field theory (Quantum ElectroDynamics, QED)
Example of achievable accuracy – dissociation energy (in 1/cm)
of the chemical bond
hydrogen
36118.7978(2)
36118.2659(3)
36118.0695(9)
36118.0696(4)
deuterium
36749.0910(2)
36748.5634(3)
36748.3633(9)
36748.3629(6)
theory
Schrodinger-Coulomb
relativistic
QED
experiment
Born-Oppenheimer approximation for diatomic molecules (PEC)
Electronic Schrodinger equation
Nuclear Schrodinger equation
- rotations - J quantum number (rigid rotor model)
- oscilations – v quantum number (harmonic oscillator model)
Potential V(R) for nuclear motion in a diatomic molecule
Harmonic oscilator potential
Wave functions of the harmonic oscillator
Effect of zero-point vibrations - ZPE
Dissociation energy of a diatomic molecule: A-B  A + B
E(A) + E(B)
ZPE
E(AB) (lowest point)
Two definitions:
Electronic binding energy (well depth): De = E(A) + E(B) - E(AB@Req)
Dissociation energy:
D0 = E(A) + E(B) - [E(AB@Req + ZPE]
= De - ZPE
Born-Oppenheimer approximation for polyatomics (PES)
Electronic Schrodinger equation
Nuclear Schrodinger equation
- rotations - J quantum number (rigid rotor model)
- oscilations – v quantum numbers (harmonic oscillator model)
- tunelling motions – for floppy molecules (ammonia moleucle)
Three-atom molecule
H2O
N=3
# of deg. freed. = 3N-6 = 3
r2
H1
O
r1
H2
Stationary points on PEC or PES
Minima andmaxima in 1-D
f(x)
minimum: f’(x0)=0
f”(x0)>0
maximum: f’(x0)=0
f”(x0)<0
example: f = ax2 + bx + c
f’ = 2ax + b
f” = 2a
a > 0 parabola - minimum; a<0 parabola - maximum
(inflection points – less interesting)
Similarly for PES’s – functions in 3N-6 dimensions:
PES = E(q1, q2, q3, …, q3N-6(5) )
In a stationary point:
E
0
qi
Derivative of energy - gradient
To locate stationary points on PES we must find points, where all
gradients vanish.
To distinguish minima and maxima ona has to compute the matrix
of the second derivatives – the Hessian

  2 E
 2
 2q1
  E
q q
2
1
 2
  E
q3q1
  2 E

q42q1
  E

qnq1
2E
q1q2
2E
q22
2E
q3q2
2E
q4q2
2E
qnq2
2E
q1q3
2E
q2q3
2E
q32
2E
q4q3
2E
qnq3
2E
q1q4
2E
q2q4
2E
q3q4
2E
q42
2E
qnq4
 2 E 

q1qn 
 2 E 
q2qn 
 2 E 

q3qn 
 2 E 

q4qn 
 2 E 
qn2 

n = 3N-6(5)
  2 E
 2
 2q1
  E
q q
2 1
 2
Hessian=   E
q3q1
 ..
 .
 2
  E

qnq1
2E
q1q2
2E
q22
2E
q3q2
..
.
2E
qnq2
2E
 2 E 
…

q1q3
q1qn 
 2 E …  2 E 
q2q3
q2qn 

 2 E …  2 E 
q32
q3qn 
.. . .
 2 E 
.
.

q4qn 
 2 E …  2 E 
qnq3
qn2 

Hessian is diagonalized and we look at its eigenvalues
When all are positive we have a minimum

Hessian diagonalized!
 2 E
 2
Q1
 0


New coordinates  0


 0

 0


0
0
0
2E
Q22
0
0
2E
Q32
0
0
0
0
0
0
2E
Q32

0 

0 


0 


0 

2
 E 
Qn2 

Eigenvalues of the Hessian
Criteria

Minimum:
Saddle points:
Maximum:
All eigenvalues of
All eigenvalues of the
All eigenvalues of
the Hessian are
Hessian are positive
the Hessian are
positive
except for one
negatiove
Minimum on PES – equilibrium geometry
Saddle point on PES - transition state (a pass between two minima),
reaction barrier, barier separating konformers
Equilibrium geometry = locate minimum on PES
Transition state geometry = locate a saddle point on PES
Energy Profile = calculate cross-section of PES along one coordinate
How is potential energy minimized (minimum located at the PES)?
We know that in a minimum the first derivatives of energy (the gradient) is zero
Start from an input structure (a point on PES)
 evaluate gradient at this point (a vector)
 go in the direction of the steepest descent (given by the gradient
vector) as long as the energy decreases.
 when the energy stops to decrease compute the gradient again
and repeat the procedure
 when the gradient reaches zero you are at the minimum (optimized
structure and the equilibrium energy)