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

- 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

The objectives of theoretical chemistry, continued

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 deuterium theory 36118.7978(2) 36749.0910(2) Schrodinger-Coulomb 36118.2659(3) 36748.5634(3) relativistic 36118.0695(9) 36748.3633(9) QED 36118.0696(4) 36748.3629(6) experiment

Born-Oppenheimer approximation for diatomic molecules (PEC)

Electronic Schrodinger equation

- rotations - J quantum number (rigid rotor model)

- oscilations – v quantum number (harmonic oscillator model)

Nuclear Schrodinger equation

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)

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

ZPE

Born-Oppenheimer approximation for polyatomics (PES)

Electronic Schrodinger equation

- rotations - J quantum number (rigid rotor model)

- oscilations – v quantum numbers (harmonic oscillator model)

- tunelling motions – for floppy molecules (ammonia moleucle)

Nuclear Schrodinger equation

Three-atom molecule

H2O N=3 # of deg. freed. = 3N-6 = 3O

H1 H2

r1r2

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:

Eqi

0 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

2E

q12

2E

q1q2

2E

q1q3

2E

q1q4

2E

q1qn2E

q2q1

2Eq2

2

2Eq2q3

2Eq2q4

2Eq2qn

2E

q3q1

2E

q3q2

2E

q32

2E

q3q4

2E

q3qn2E

q4q1

2Eq4q2

2Eq4q3

2Eq4

2

2Eq4qn

2E

qnq1

2E

qnq2

2E

qnq3

2E

qnq4

2E

qn2

2E

q12

2E

q1q2

2E

q1q3

2E

q1qn2E

q2q1

2E

q22

2E

q2q3

2E

q2qn2E

q3q1

2E

q3q2

2E

q32

2E

q3qn2E

q4qn2E

qnq1

2E

qnq2

2E

qnq3

2E

qn2

…

…

…

. . ....

......

…

Hessian=

n = 3N-6(5)

Hessian is diagonalized and we look at its eigenvaluesWhen all are positive we have a minimum

2E

Q12

0 0 0 0

02EQ2

2 0 0 0

0 02E

Q32

0 0

0 0 02EQ3

2 0

0 0 02E

Qn2

Eigenvalues of the Hessian

Criteria

Minimum:All eigenvalues of

the Hessian are

positive

Maximum:

All eigenvalues of

the Hessian are

negatiove

Saddle points:

All eigenvalues of the

Hessian are positive

except for one

Hessian diagonalized!

New coordinates

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 PESEnergy 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)