The document discusses methods and procedures for energy minimization in computational chemistry, focusing on geometry optimization to achieve stable molecular conformers. Various techniques including the Newton-Raphson, steepest descent, and conjugate gradient methods are outlined, along with practical applications and tools available in software for molecular mechanics and conformational analysis. It also covers concepts like local and global energy minima, Monte Carlo methods, and genetic algorithms for conformational exploration.

1. Methods and procedure
Of
ENERGY MINIMIZATION
SHIKHAD. POPALI
HARSHPAL SINGHWAHI
GURUNANAK COLLEGEOF PHARMACY

2. Introduction
When a molecule is built in a computational chemistry software package, the initial
geometry does not necessarily correspond to one of the stable conformers.
energy minimization is usually carried out to determine a stable conformer this same
process also is commonly referred to as geometry optimization.
Energy minimization is a numerical procedure for finding a minimum on the
energy surface starting from a higher energy initial structure, labeled "1" as illustrated
in Figure. During energy minimization, the geometry is changed in a stepwise fashion
so that the energy of the molecule is reduced, from steps 2 to 3 to 4 as shown in
Figure. After a number of steps, a local or global minimum on the potential energy
surface is reached..

3. Most energy minimization methods proceed by determining the energy and the
slope of the function at point 1. If the slope is positive, it is an indication that the
coordinate is too large, If the slope is negative, then the coordinate is too small. The
numerical minimization technique then adjusts the coordinate; if the slope is
positive, the value of the coordinate is reduced as shown by point 2. The energy
and the slope are again calculated for point 2. If the slope is zero, a minimum has
been reached. If the slope is still positive, then the coordinate is reduced further, as
shown for point 3, until a minimum is obtained.
X new = X old +
correction
X new refers to the value of the geometry at the next step
X old refers to the geometry at the current step
correction is some adjustment made to the geometry.
Equation to adjust the geometry to reach the minimum energy

4. Methods of Energy Minimization
Newton-Raphson Method
The Newton-Raphson method is the most computationally expensive per
step of all the methods utilized to perform energy minimization. It is based
on a Taylor series expansion of the potential energy surface at the current
geometry. The equation for updating the geometry is
X new = X old - E1(X old)
E2(X old)
The correction term depends on both the first derivative (also called the slope
or gradient) of the potential energy surface at the current geometry and also on
the second derivative (otherwise known as the curvature). It is the necessity of
calculating these derivatives at each step that makes the method very expensive
per step, especially for a multidimensional potential energy surface where there
are many directions in which to calculate the gradients and curvatures. However,
the Newton-Raphson method usually requires the fewest steps to reach the
minimum.

5. Steepest Descent Method
Rather than requiring the calculation of numerous second
derivatives, the steepest descent method relies on an
In this method, the second derivative is assumed to be a constant.
Therefore, the equation to update the geometry becomes
X new = X old - γ E1(X old)
where γ is a constant, because of the approximation, it is not as efficient
and so more steps are generally required to find the minimum.

6. Conjugate Gradient Method
In the Conjugate Gradient method, the first portion of the search takes
place in the direction of the largest gradient, just as in the Steepest
Descent method. However, to avoid some of the oscillating back and
forth the conjugate gradient method mixes in a little of the previous
direction in the next search. This allows the method to move rapidly to
the minimum. The equations for the conjugate gradient method in two
or more dimensions are more complex than those of the other two
methods, so they will not be given here.
Procedure of Energy Minimization
Continued from the next slide

7. Here as an example we are drawing the structure of DB75 [2,5-bis(4-
amidinophenyl)furan], also known as furamidine, is an analog of pentamidine.
“Optimize Geometry” option which can be located from “Extensions” tab as
shown in Figure, allows the user to optimize the structure of the molecule to
its stable conformation.

8. The molecule makes several changes in its atom position through rotation and calculates
energy in every position. This process is repeated many times to find the position with
lowest energy.
Optimized Structure of DB 75
From the “Extensions” tab, user can select “Molecular Mechanics” which allows the user
to select other parameters to calculate energy according to their choice as shown in
Figure 4.

9. Different options in “Force Field” can be selected by clicking on “Setup Force Field”
which displays different types of force fields, algorithms associated with it, number
of steps and Convergence as shown in Figure.
From the “Molecular Mechanics”, user can select “Calculates Energy” option which
allows the user to calculate energy of the molecule.

10. User can view other properties of the builder molecules by clicking on the “View” tab ->
“Properties”

11. The “Molecular Properties” allows the user to view the IUPAC name, molecular
weight, chemical formula, energy, estimated dipole moment, number of atoms and
bond
from the “Extensions” tab, click "Molecular Mechanics" from where user can
select “Conformer Search” which allows the user to find the number of
conformers by different types of rotations with the molecule which is shown
in Figure

12. The different options in the conformer search are displayed which allows the
user to give parameters according to their choice.

13. Conformational Analysis
1. Local and Global energy minima
If the 3D structure created initially is on the energy curve at the position
energy minimization will stop when it reaches the first stable conformation it
encounters—a local energy minimum, At this point, variations in structure
result in low-energy changes and so the minimization will stop. In order to
the saddle to the more stable conformation, structural variations would have
be carried out which increase the strain energy of the structure and these will
rejected by the program. The minimization program has no way of knowing
there is a more stable conformation (a global energy minimum ) beyond the
energy saddle.

14. 2. Molecular Dynamics
Steric energy 3.053kcal/mole Steric energy 2.180 kcal/mole
It is stable by 0.9 kcal/mole

15. 3. Stepwise bond rotation
It is to generate different conformations by automatically rotating
every single bond by a set number of degrees. For example, 12
different conformations of butane were generated by automatically
rotating the central bond in 30° steps. The steric energy of each
conformation was calculated and graphed, revealing that the most
stable conformation was the fully staggered one, whereas the least
stable conformation was the eclipsed one. In this operation, energy
minimization is not carried out on each structure because the aim is to
identify both stable and unstable conformations. The number of
conformations generated will depend on the number of rotatable
bonds present and the set amount of rotation. For example, a
with three rotatable bonds could be analyzed for conformations
resulting from 10° increments at each bond to generate 46,656
conformations. With four rotatable bonds, 30° increments would
generate 20,736 conformations.

16. 4. Monte Carlo and the Metropolis method
The Monte Carlo method of conformational analysis
introduces a bias towards stable conformations such that
more processing time is spent on these—a process known as
importance sampling.
Different conformations are generated by carrying out
random bond rotations. This is quite different from molecular
dynamics, where atoms are shifted in space. As each
conformation is generated, it is energy minimized to give a
stable conformation, and its steric energy is calculated and
compared with the previous structure. If the steric energy of
the new conformation is lower (more stable), it is accepted
used as the starting structure for the next conformation. If the
steric energy is higher, it may be accepted or rejected
depending on a probability formula which takes into account
both the energy of the new conformation and the
‘temperature’ of the system.

17. The Metropolis method (also known as simulated annealing
)
It is an approach which can be used to increase the
of finding the global minimum. It involves a number of cycles
where the Monte Carlo algorithm is run at different
temperatures. In the first cycle, a high temperature is set (T1)
and a set of structurally diverse conformations is generated.
The most stable conformation is then used as the starting
structure for the next run where the temperature is set at a
lower value. This process is repeated several times with the
probability equation becoming more ‘choosy’ about which
structures are accepted. This slowly ‘focuses’ the search on a
particular area of conformational space which can be searched
more rigorously. In this way, there is more chance of finding
the global minimum, but there is still no guarantee of success.

19. 5. Genetic and evolutionary algorithms
First of all, the conformation of a molecule has to be represented in a
manner which will allow an evolutionary process of mutation and selection
to take place. Quite simply, the torsion angles for the rotatable bonds in
molecule are stored as a sequence of numbers. This sequence
to a ‘chromosome’, where each ‘gene’ signifies a torsion angle. An initial
population of ‘chromosomes’ representing different conformations is
created by randomly choosing values for the different torsion angles. The
stability of each conformation is then calculated by molecular mechanics.
The next stage in genetic algorithms is to create a new population of
chromosomes or conformations. First of all, sets of ‘parents’ are chosen
from the initial population. This is a random process, but a statistical bias is
built into the selection process such that the most stable conformations
chosen as parents. This means that a particularly stable conformation can
involved in several ‘relationships’. The new population of conformations is
now generated. The chromosomes from each parent undergo a
or recombination process to generate new chromosomes where each
chromosome has torsion angles contributed each parent.

21. Structure comparison and overlays
2D overlay of cocaine and procaine.
Overlay of cocaine and procaine using Chem3D.