METHODS AND APPLICATIONS OF COMPUTATIONAL CHEMISTRY

What is potential energy of a system ?

Energy stored in the position or arrangement of a system, relative to other systems eg particles or its surrounding eg external fields

What is the purpose of computational chemistry

Determining and being able to predict qualitative ideas about chemical process and comparing with experiments using PES

PES relates molecular geometry to energy, which is the driving force for reactions, so we will define, calculate and explore PES

Why do we use internal coordinate instead of simple Cartesian coordinate

Cartesian coordinates change upon bond translation and rotation even though energy doesn’t

Internal coordinates for all the structural properties we measure, by using their vectors to specify molecular geometry

Allows computer to be aware

Equation of internal energy

U=KE+PE

What does the Born-Oppeinhemer approximation state?

It states that the nucleus is so large that the electrons ‘see’ thr nuclei as stationery, but us help up solve maths for electron behaviour

What is a Potential energy surface ?

A plot of the PE with respect to all thr molecular coordinates which represent a molecular structure

X-axis is inter nuclear bond distance (R)/A

Y-axis is PE

From this plot you can form the derivative curve of dE/dR

What is the reaction path?

The route lowest in energy / least resistance of R → TS → P, made up by reaction coordinates

Molecular geometreis increased in quantity as the number of atoms increase, how do we determine molecular geometrías we are interested on a PES

Stable geometries (minimas on graphs ) - structures we cannot spontaneously change to lower their energy; energy increases if we change internal coordinates

Transition states (saddle points on graphs) - highest point in the lowest oath connecting two stage geometries; energy decreases if we change internal coordinates

Both are stationary points on the PES

What are the internal coordinates?

Any other coordinate not part of the reaction path, orthogonal to the PE axis

Cartesian coordinate define a single point on the PES, what do they represent and be converted into?

A molecular structure, once they are known ghey can be converted into atom position vectors, bond lengths, bond angles and dihedral angles

What are internal coordinates ?

Coordinates which if changed they would e,as to a change in energy to the molecule

Internal angle for a non-linear molecule?

3N-6

Internal angle for a linear molecule?

3N-5

What are normal coordinates?

Unique set of intern, coordinates related to normal modes of vibrations eg reaction coordinates

What is a rotamer?

Same molecule, different conformation; dihedral angles changes In order to find most favourable conformer

Position vectors in terms of coordinates for bond vector/length AB^(→)

$$=r_{A}-r_{B}$$

Position vectors in terms of coordinates for magnitude of bond length AB^(→) or BA^(→)

Modulus of vector vector hub

$$=\left\vert BA\cdot BA\right\vert$$

Position vectors in terms of coordinates for bond angle between BA^(→) and BC^(→)

$$BA\cdot BC=\vert BA\vert\vert BC\vert\cos\theta$$

Position vectors in terms of coordinates for dihedral angle ABC ^(→)

$$\cos\left(\tau\right)=\frac{a\cdot b}{\left\vert a\right\vert\left\vert b\right\vert}$$

Sign if angle $\tau$ is $\frac{a\cdot b}{\left\vert a\cdot b\right\vert}\cdot\frac{BC}{\left\vert BC\right\vert}$

What are stationery points defined as on the PES with repesct to the internal coordinates R

Defined by the first derivative

Gradient will always be 0

How are the stationery points: minimas and saddle points differentiated on a PES

By the second derivate

Minimas - always be positive (due to upward curvature in all directions)

Saddle point - for one coordinate X will be negative (due to downward curvature in one direction, upwards for others )

What does the collinear approach for PES, for a three atom system, what affects the gradient of energy along a bond coordinate?

The derivative of energy, depend on what is kept constant

$\left(\frac{\partial E}{\partial R_{AB}}\right)_{\theta,R_{BC}}=0$ and $\left(\frac{\partial E}{\partial R_{BC}}\right)_{\theta,R_{AB}}=0$

Potential energy curve of a harmonic curve for a diatomic molecule?

$$PE=\frac12k\left(R-R_{e}\right)^2$$

Equations for steepest descent optimisation

$$R^{\left(new\right)}=R^{\left(m\right)}+\Delta R$$

And

$$\Delta R=-\frac{\partial E}{\partial R}$$

Where R(m) is the randomly chosen coordinate

R(m) and $\Delta R$ are vectors

Equations for Newton-Raphson optimisation, if the curve is harmonic

$$R_{e}=R-k^{-1}\cdot\frac{\partial E}{\partial R}$$

Using, $E\left(R\right)=\frac{k}{2}\left(R-R_{e}\right)^2$

1st and inverse of 2nd derivative must be worked out

Equations for Newton-Raphson optimisation, if the curve is anharmonic

$$\Delta R=R^2-R^1=-\frac{g^{\left(1\right)}}{k^{\left(1\right)}}$$

For every new R we must ca,curate the new gradient and curvature

Force acting on each atom, wtr to coordinate changes equation