Molecular Modelling

Question 1

What is molecular modelling?

Answer 1

> Molecular modelling = molecular mechanics = force field calculations.

> Unlike quantum mechanics (accurate but slow), this approach does not solve the electronic Schrodinger equation).

> It is an empirical approach, atom-atom interactions are described using some suitable mathematical expressions and electron-electron and electron-nuclei interactions are ignored → this approach is much FASTER than quantum mechanics (and gives results that are almost as good).

Question 2

What are we looking for when we model molecular systems?

Answer 2

> We are looking for the MINIMUM ENERGY of the molecule.

Question 3

Why are we allowed to ignore electron-electron interactions?

Answer 3

> BORN-OPPENHEIMER APPROXIMATION: nuclei are much heavier than atoms, therefore, electrons move on a much faster timescale. → We can consider nuclei to be stationary and electrons to be moving in the constant potential energy field of the nuclei.

> Can also postulate that electrons move so fast that there is an average distribution of electrons around each nucleus (molecular mechanics).

Question 4

How are diatomic molecules modelled>

Answer 4

> Harmonic approximation: the molecule is represent s 2 balls connect by a spring.

> Energy is calculated using Hooke’s law. The structure with the best geometry is the lowest energy structure.

Question 5

What is Hooke’s Law and what are its limits?

Answer 5

E= 1/2 k(x-x0)2

where k=force constant
x0= the ideal interatomic distance (equilibrium bond length).

> With Hooke’s law, the molecule never dissociates. (even with the atoms at infinite separation, the molecule remains bound).

> The harmonic potential (i.e. Hooke’s law) can be used only for intermolecular distances of x0 ± 10%.

Question 6

What is steric energy?

Answer 6

> Steric energy is a measure of strain in the molecules’ bond lengths (and angles, torsional angles etc.).

Question 7

How do you calculate and interpret the force of a molecule?

Answer 7

F= -∂E/∂x = -k(x-x0)

> If x=xo then F=0 → the net force = 0, atoms are in their ideal positions and there is no movement.

> If x0→ the force drives the atom to the right towards x0.

> If x>x0 the F<0 → the force drives the atom to the left towards x0.

Question 8

What is meant by the term “minimisation of energy”?

Answer 8

> Minimisation of energy is the procedure of searching for the best (minimum energy) structure with force F=0.

Question 9

What is the Lennard-Jones potential?

Answer 9

E = 4 Ɛ[ (σ/r)12 - (σ/6)6]

> It describes both the equilibrium bond length and dissociation.

Question 10

What is a force field?

Answer 10

> Force field= combination of equations and their parameters used to describe interatomic interactions.

Question 11

What are the advantages and disadvantages of quantum mechanics (QM)?

Answer 11

> Advantages:
→ rigorous description of electron-electron and electron- nuclei interactions.
→ from 1st principals (does not require experimental input).
→ Can cope with new unexpected structures.
→ only method to use if interested in electronic properties.

> Disadvantages:
→ Slow
→ Solving Schrodinger equation requires approximations which lower QM accuracy.

Question 12

What are the advantages and disadvantages of molecular mechanics (MM)?

Answer 12

> Advantages:
→ very fast
→ applicable to large systems
→ allows dynamics of molecules to be studied (evolution of the structure over time).
→If properly parameterised it can be as accurate as QM.

> Disadvantages:
→ parameters of the force field need to be properly chosen (hard).
→ QM of experiment needs to be done first.
→Non-transferable, cannot cope with new unknown structures.
→ Applicable for describing atoms but not electronic properties.

Question 13

When can QM be used?

Answer 13

>QM should be used to describe electronic properties of:
→ molecular orbitals
→ electronic states
→ light absorption
→ charge transfer
→ bond rearrangements (mechanisms)

Question 14

When can MM be used?

Answer 14

→ large molecules especially biomolecules
→ dynamics of molecules
→ molecules in solution
→ some inorganic solids (e.g. Si)

Question 15

When can both methods be used?

Answer 15

→ Small/medium molecules

→ some inorganic solids

Question 16

The energy of a molecular system is given by a sum of intramolecular and intermolecular interactions, what are these interactions?

Answer 16

> Intramolecular = all ‘chemical’ interactions between atoms within 1 molecule. Depends only on the geometry of each molecule.

> Intermolecular = All interactions between atoms of different molecules: non-bonded and electrostatic.

Question 17

What is the Valence force field?

Answer 17

> Molecules are not fixed, they undergo bond stretching, bond bending (change of angle) and torsion (change of dihedral angle) - a typical valence force field includes all of these terms.

> E= Estretch + Ebend + Etorsion

Question 18

How is the stretching term dealt with?

Answer 18

> Stretching - changing the bond length

> Most simple form for the bond stretching term is the harmonic oscillator (Hooke’s law).

> For a molecule containing many atoms, the stretch term is the sum of stretch contributions over all bonds between the atoms.

> Can also use the Morse potential: Estretch(rij)= Dij[1-e^-aij(rij-rij0)]2

where Dij = dissociation energy (binding strength) and aij= strength of the potential

(describes dissociation better than quadratic but is less easy computationally)

> Can also use a Cubic potential:
Estretch (rij) = 1/2 kij (rij-rij0)2 (1-2(rij-rij0))

> or the Quartic potential:
Estretch (rij) = 1/2 k1 (ij)(rij-rij0)2 + 1/2 k2(ij) (rij-rij0)4

> These expressions are better at describing the anharmonicity of bonds (important for the calculations of vibrational frequencies).

Note: the HARMONIC POTENTIAL = most widely used because it is mathematically the simplest.

Question 19

How do we know if there is a bond between 2 atoms?

Answer 19

> Check the distance, there is a cut off distance (rij ± 10-20%) beyond which atoms are considered to be not chemically bound and the stretching interaction does not apply.

Question 20

How is the bending term dealt with?

Answer 20

> Bending term= changing the bond angle

> Ebend (θ ijk) - 1/2 kijk(θijk - θijk0)2

where θijk0 = the natural (ideal) angle between atoms ijk.

> For a molecule containing many atoms, the bending term is the sum of the contributions over all angles between all bonded atoms.

> Again, more complex forms can be used such as quartic or cosine:

Ebend (θijk) - Kijk/2sin^2θijk(cosθ ijk-θcosijk0)^2

Question 21

How is the torsional term dealt with?

Answer 21

> Torsional term= changing the dihedral angle.

> Harmonic potential:
Etorsion (χijkl) = 1/2 kijkl (χijkl-χijkl0)^2

This expression implies that there is 1 preferred value for a torsional (dihedral) angle (suitable for improper torsions but not suitable for proper torsions).

Question 22

What are proper and improper torsions?

Answer 22

> Proper torsion= not restricted and a full 360° rotation around the j0k bond is possible (e.g. rotation around the c-c bond in ethane).

> Improper torsion= the values taken by the dihedral angles are restricted (e.g. in benzene c-c-c-c ~0, atom are all in the same plane).

Question 23

What expression can be used to describe the periodicity of a torsional angle?

Answer 23

Etorsion (χ)= 1/2Vijkl [1-cos(nχ-ϒ)]

where:
n=periodicity of torsion
Vijlk= force constant (the height difference between the maxima and minima in the torsional potential)
ϒ= the phase factor which determines where the torsional potential passes through a minimum.

> For ethane, all minima have equal depths (not always the case). A more general expression that accounts for several minima of unequal depths:

Etorsion (χ)= 1/2 ΣVn [1-cosχ-ϒ)]
n=a number from 1 to some value (depending on the force field).

Question 24

How do different types of force fields differ?

Answer 24

> Force fields differ in:
→choice of equations
→choice of atom types (united or all atom)
→choice of parameters

Question 25

What is the parameter fitting procedure?

Answer 25

(i) Choose (experimental or computational) reliable values of molecules’ properties.
(ii) Choose some initial values of force field parameters.
(iii) Use these values to calculate the target properties.
(iv) Are the desired properties reproduced accurately?
NO → Update force field paramerters
YES→ Parameter fitting complete → RUN CALCULATIONS

Question 26

Is it possible to use a force field optimised for one set of molecules to calculate a different class of molecules?

Answer 26

> Not necessarily, it depends on whether the force field is all purpose or specialised.

Question 27

Describe the potential energy curve for a 1D case.

Answer 27

> E vs r (distance) goes through a minimum (typically interested in the value of rmin and the binding energy).

Question 28

Describe the features of the potential energy surface (PES) of a polyatomic molecule.

Answer 28

> horizontal axis= internal coordinates (bonds, angles, torsion).

> The number of internal coordinates = number of degrees of freedom in vib. spec = 3N-6.

> 3 mains features of a PES:
→ Maxima: rarely of interest
→ Minima: global or local. Global minimum = the deepest minimum and is the most stable structure of the molecule.
→ Saddle Points: these are the minima in some direction (along some internal coordinates) but maxima in other directions. The most interesting saddle point= a transition state ( a minimum in all directions except one).

Question 29

How do we know that we have reached an energy minimum?

Answer 29

> criterion of success = any small change in geometry should increase the energy or forces should be ~ equal to 0. or the change in energy should be smaller than a set value.

> analysis of vib. frequencies tells us if the structure is a minimum or a saddle point.

Question 30

What are gradient based methods of finding an energy minimum?

Answer 30

> The potential energy E(qi) of the system as a function of all the internal coordinates qi, is given by all the terms in the force field.

> Gradient of energy (slope) = 1st derivative of energy wrt all internal coordinates.

gradE(q) = dE(q)/dq

→ gradient tells us how fast the energy decreases and what is the direction of the fastest decrease.
→In the 1D case, gradE(q) = -force
→ Polyatomic molecules: gradE (q) is a column of values dE/dqi = force on all atoms.