Before we described the reverse
engineering methods for program tools, applications and etc. Here I have an idea to look at the
reverse engineering as tool to look inside the nature - the physical nature
where we live - quantum chemistry, quantum mechanics. As with other disciplines in
chemistry, computational chemistry uses tools to understand chemical reactions and
processes. Scientists use computer software to gain insight into chemical processes. The challenges for computational
chemistry are to characterize and predict the structure and stability of
chemical systems, to estimate energy differences between different states, and
to explain reaction pathways and mechanisms at the atomic level. Software tools
for computational chemistry are often based on empirical information.
Molecules consist of electrons and
nuclei. Most
applications of quantum chemistry separate the motion of the nuclei from the motion
of electrons (the Born-Oppenheimer approximation). The approximation results in
a model of nuclei moving on a potential energy surface, with electrons
adjusting instantly to changes in nuclear positions. Nuclear motion is
constrained by the interaction of nuclei and electrons. At any fixed positions
of the nuclei, potential energy is the sum of repulsions among the positively charged
nuclei and attractions arising from the electrons. Electrons are the “glue”
holding nuclei together.
Optimization the structure of the molecules
To calculate the properties of a
molecule, you need to generate a well-defined structure. A calculation often
requires a structure that represents a minimum on a potential energy surface.
Generating and Viewing Orbitals and Electronic Plots.
Orbital wave functions resulting
from semiempirical and ab initio quantum mechanical
calculations. It is interesting to view both the nodal properties and the
relative sizes of the wave functions. Orbital wave functions can provide
chemical insights.
Evaluating Chemical Pathways and Mechanisms
Calculating
single point properties and energies provides information about chemical
pathways and mechanisms.
MNDO and AM1
semi-empirical methods to calculate possible reaction pathways for the interaction of glycine
and cocaine.
To calculated
atomic charges and HOMO-LUMO (highest occupied and lowest unoccupied molecular
orbitals) electron densities for the two molecules. Using this information,
they proposed two possible reaction pathways and computed the heats of
formation for geometry-optimized structures.
Exploring Potential Energy Surfaces
Examine potential energy surfaces
with single point calculations, optimizations, and molecular dynamic
simulations. Energies and derivatives of energy, such as the forces on atoms,
are necessary to “construct” a potential energy surface. This concept gives the
molecular and quantum mechanics methods a unifying purpose: they generate the
energies and energy derivatives necessary to produce and examine potential
energy surfaces. The potential energy is always characterized by the body to
the source of power (force field). Both molecular and quantum mechanics
methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrödinger
equation (1) gives the wave functions and energies of a molecule.
HY = EY (1)
where
H is the molecular Hamiltonian,
Y is the wave function,
E is the energy.
The molecular Hamiltonian is
composed of these operators:
- the kinetic energy of the nuclei
(N) and electrons (E),
- nuclear-nuclear (NN) and
electron-electron repulsion (EE),
- and the attraction between nuclei
and electrons (NE)
(equation 1).
H(kinetic energy)N + (kinetic energy)E = + (repulsion)NN + (repulsion)EE + (attraction)NE (2)
Nuclei have many times more mass
than electrons. During a very small period of time when the movement of heavy
nuclei is negligible, electrons are moving so fast that their distribution is smooth.
This leads to the approximation that the electron distribution is dependent
only on the fixed positions of nuclei and not on their velocities. This
approximation allows two simplifications of the molecular Hamiltonian. The
nuclear kinetic energy term drops out of the molecular Hamiltonian. The nuclear
kinetic energy term drops out (equation 3).
H = (kinetic energy)E + (repulsion)NN + (repulsion)EE + (attraction)NE (3)
Since the nuclear-nuclear repulsion
is constant for a fixed configuration of atoms, this term also drops out. The
Hamiltonian is now purely electronic.
Helectronic = (kinetic energy)E + (repulsion)EE + (attraction)NE (4)
After solving the electronic
Schrödinger equation (equation 4), to calculate a potential energy surface, you
must add back nuclear-nuclear repulsions (equation 5).
HelectronicYelectronic = EelectronicYelectronic
VPES = Eelectronic + (repulsion)NN (5)
Generating the potential energy surface
(PES) using this equation requires solutions for many configurations of nuclei.
In molecular mechanics, the electronic energy is not evaluated explicitly. Instead,
these methods solve the potential energy surface by using a force field
equation. The force field equation represents electronic energy implicitly through
parameterization.
This means that the forces on the
atoms are zero for the structures found at A and C. The structure at C is more
stable than the structure at A. In fact, C is the global minimum for this
example. Geometry
optimizations seek minima such as A
and C.
Types of Calculations
There are three types of
calculations in HyperChem:
- single point,
- geometry optimization or
minimization,
- molecular dynamics.
Single Point
A single point calculation gives the
static properties of a molecule. The properties include potential energy, derivatives of the potential energy, electrostatic
potential, molecular orbital energies, and the coefficients of molecular
orbitals for ground or excited states. The input molecular structure
for a single point calculation usually reflects the coordinates of a stationary point on the potential energy
surface, typically a minimum or transition state. Use a geometry
optimization to prepare a molecule for a single point calculation. One type of
single point calculation, that of calculating vibrational properties, is
distinguished as a vibrations calculation in Hyper-Chem. A vibrations calculation predicts fundamental vibrational frequencies,
infrared absorption intensities, and normal modes for a geometry optimized
molecular structure. You can use a single point calculation that determines energies for ground
and excited states, using configuration interaction, to predict frequencies and
intensities of an electronic ultraviolet-visible spectrum.
Geometry Optimization
To carry out a geometry optimization
(minimization), HyperChem starts with a set of Cartesian coordinates for a
molecule and tries to find a new set of coordinates with a minimum potential
energy. The potential energy surface is very complex, even for a molecule
containing only a few dihedral angles. Since minimization calculations
cannot cross or penetrate potential energy barriers, the molecular structure
found during an optimization may be a local and not a global minimum. The minimum represents the potential energy closest to the starting
structure of a molecule. Researchers frequently use minimizations to generate a structure at a
stationary point for a subsequent single point calculation or to remove
excessive strain in a molecule, preparing it for a molecular dynamics
simulation.
Transition State Search
As mentioned earlier, a potential
energy surface may contain saddle points; that is, stationary points where
there are one or more directions in which the energy is at a maximum. A saddle
point with one negative eigenvalue (Собственное значение)
corresponds to a transition structure for a chemical reaction of changing
isomeric form. Transition structures also exist for reactions involving
separated species, for example, in a bimolecular reaction
A + B → C + D.
Activation energy, i.e., the energy of the transition structure relative to reactants,
can be observed experimentally. However, the only way that the geometries of
transition structures can be evaluated is from theory. Theory also can give
energetics and geometry parameters of short-lived reaction intermediates. Transition state search algorithms
rather climb up the potential energy surface, unlike geometry optimization
routines where an energy minimum is searched for. The characterization of even
a simple reaction potential surface may result in location of more than one
transition structure, and is likely to require many more individual
calculations than are necessary to obtain equilibrium geometries for either
reactant or product.
Calculated transition structures may
be very sensitive to the level of theory employed. Semi-empirical methods, since they are parametrized for energy
minimum structures, may be less appropriate for transition state searching than
ab initio methods are. Transition
structures are normally characterized by weak “partial” bonds, that is, being
broken or formed. In these cases UHF
calculations are necessary, and sometimes even the inclusion of electron correlation
effects.
Molecular Dynamics
A molecular dynamics simulation
samples phase space (the region defined by atomic positions and velocities) by
integrating numerically Newton’s equations of motion. Unlike single point and geometry
optimization calculations, molecular dynamics calculation account for thermal
motion. Molecules may contain enough thermal energy to cross potential
barriers. Molecular dynamics calculations provide information about possible
conformations, thermodynamic properties, and dynamic behavior of molecules.
Langevin Dynamics
Molecules in solution undergo
collisions with other molecules and experience frictional forces as they move
through the solvent. Using Langevin dynamics, you can model these effects and study the
dynamical behavior of a molecular system in a liquid environment. These
simulations can be much faster than molecular dynamics. Instead of explicitly
including solvent molecules, the Langevin method models their effect through
random forces applied to the molecule of interest to simulate collisions, and
frictional forces are added to model the energy losses associated with these
collisions.
Langevin dynamics simulations can be used to study the same kinds of
problems as molecular dynamics: time dependent properties of solvated systems
at non-zero temperatures. Because of the implicit treatment of the solvent, this method is
particularly wellsuited for studying large molecules in solution. It is
possible to decouple the time scales of molecular motion and focus on just the slow
modes associated with conformational changes, for example, or to follow the
rapid bond stretching motions leading to chemical reaction, while incorporating the
influence of molecular collisions. Langevin dynamics has been use to study
solvent effects on the structural dynamics of the active site of the enzyme
lysozyme, conformational equilibria and the dynamics of conformational transitions
in liquid alkanes, and temperature effects on a system of interacting quantum
harmonic oscillators.
Monte Carlo Simulations
The Monte Carlo method samples phase
space by generating random configurations from a Boltzmann distribution at a
given temperature. Averages computed from a properly equilibrated Monte Carlo
simulation correspond to thermodynamic ensemble averages. Thus, the Monte Carlo
method can be used to find average energies and equilibrium structural properties
of complex interacting systems. A sequence of successive
configurations from a Monte Carlo simulation constitutes a trajectory in phase
space; with HyperChem, this trajectory may be saved and played back in the same
way as a dynamics trajectory. With appropriate choices of setup parameters, the
Monte Carlo method may achieve equilibration more rapidly than molecular
dynamics. For some systems, then, Monte Carlo provides a more direct route to
equilibrium structural and thermodynamic properties. However, these
calculations can be quite long, depending upon the system studied. Monte Carlo
simulations provide an alternate approach to the generation of stable
conformations. As with HyperChem’s other simulation methods, the effects of
temperature changes and solvation are easily incorporated into the
calculations.
HyperChem uses two types of methods in calculations:
- molecular mechanics
- quantum mechanics.
The quantum mechanics methods implemented in HyperChem include:
- semi-empirical quantum mechanics
method
- ab initio quantum mechanics method.
Molecular Mechanics
Molecular mechanical force fields
use the equations of classical mechanics to describe the potential energy
surfaces and physical properties of molecules. A molecule is described as a collection of atoms
that interact with each other by simple analytical functions. This
description is called a force field. One component of a force field is the
energy arising from compression and stretching a bond. We try to find the potential energy.
The potential energy of a molecular system in a force field is the sum of individual components of the
potential, such as bond, angle, and van
der Waals potentials (equation 8). The energies of the individual bonding
components (bonds, angles, and dihedrals) are functions of the deviation of a
molecule from a hypothetical compound that has bonded interactions at minimum
values.
ETotal = term1 + term2 + ¼+ termn (8)
The absolute energy of a molecule in
molecular mechanics has no intrinsic physical meaning; ETotal values are useful only for
comparisons between molecules. Energies from single point calculations are
related to the enthalpies of the molecules. However, they are not enthalpies
because thermal motion and temperature dependent contributions are absent from
the energy terms (equation 8).
Unlike quantum mechanics, molecular
mechanics does not treat electrons explicitly. Molecular mechanics calculations
cannot describe bond formation, bond breaking, or systems in which electronic delocalization
or molecular orbital interactions play a major role in determining geometry or
properties. This discussion focuses on the individual components of a typical molecular
mechanics force field. It illustrates the mathematical functions used, why
those functions are chosen, and the circumstances under which the functions
become poor approximations.
Experiment with water
H2O- WATER (hydrogen oxide) H2O, they say. m. 18.016, just a steady Port.
hydrogen with oxygen. The liquid is odorless, taste and color.
The structure and physical properties of the molecule. Hydrogen and oxygen atoms in water
molecules are located in the corners of an isosceles triangle with the length
of the O-H 0.0957 nm; valence angle H-O-H 104,5 °; dipole moment of 6.17 *
10-30 * m Cl; molecular polarizability 1.45 * 10-3 Nm3; the average quadrupole
time - 1.87 * 10-41 m2 * Kl, the ionization energy of 12.6 eV, proton affinity
of 7.1 eV. When interaction. with other water molecules. atoms, ions and
molecules, including with other molecules in the water condenses. phases, these
parameters vary.
We make Minimizing the Energy of a
System. Molecular mechanics geometry
optimization; single point calculations; using reflection through a plane to
perform a symmetry transformation;
In this we minimize the energy of water
using the AMBER force field.
Energy minimization alters molecular geometry to lower
the energy of the system, and yields a more stable conformation. As the
minimization progresses, it searches for a molecular structure in which the
energy does not change with infinitesimal changes in geometry. This means that
the derivative of the energy with respect to all Cartesian coordinates, called
the gradient, is near zero. This is known as a stationary point on the
potential energy surface. If small changes in geometric parameters
raise the energy of the molecule, the conformation is relatively stable, and is
referred to as a minimum. If the energy lowers by small changes in one or more dimensions,
but not in all dimensions, it is a saddle point (точка перевала).
A molecular system can have many
minima. The one with the lowest energy is called the global minimum and the
rest are referred to as local minima. We find the global minimum energy. Before you build the chair structure
of any structure and perform a molecular mechanics optimization, you should
choose a molecular mechanics force field provided with HyperChem. A force field
contains atom types and parameters that must be assigned to the molecule before
you perform a molecular mechanics calculation. For this exercise, the AMBER force field.
The AMBER (Assisted Model Building
and Energy Refinement) is based on a force field developed for protein and
nucleic acid computations by members of the Peter Kollman research group at the
University of California, San Francisco. The original AMBER has become one of
the more widely used academic force fields and extensive work has gone into
developing it — resulting in a number of versions of the method and associated
parameters. Hyper-Chem gives results equivalent to Versions 2.0 and 3.0a of the
AMBER program distributed by the Kollman group and parameter sets for both
these versions are distributed with HyperChem. AMBER was first developed as a
united atom force field [S. J.Weiner et al., J. Am. Chem. Soc., 106, 765
(1984)] and later extended to include an all atom version [S. J. Weiner et al.,
J. Comp. Chem., 7, 230 (1986)]. HyperChem allows the user to switch back and
forth between the united atom and all atom force fields as well as to mix the
two force fields within the same molecule. Since the force field was developed
for macromolecules, there are few atom types and parameters for small organic
systems or inorganic systems, and most calculations on such systems with the
AMBER force field will fail from lack of parameters.
Set both the Electrostatic and van
der Waals 1-4 scale factors to 0.5.
These options determine the scaling
of nonbonded interactions for atoms that are separated by three bonds. The
AMBER parameters were derived with both scale factors set to 0.5 so you should
normally use these with the AMBER force field.
van der Waals
Intermolecular interactions,
interaction. molecules among themselves, do not result in tearing or the
formation of new chemical. ties. Intermolecular interaction determines the
difference between real gases from ideal, the existence of liquids and pier.
crystals. From intermolecular interactions depend on many others. structural,
spectroscopic, thermodynamic., The emergence of the concept of intermolecular interaction
associated with the name van der Waals forces, to explain the Holy-in real gases
and liquids offered in 1873, taking into account the intermolecular interaction
(see. Van der Waals equation) . Therefore, the intermolecular interaction
forces are often called van der Waals.
Polak-Ribiere
as the minimization algorithm. This algorithm is a good general-purpose
optimizer. Both Polak-Ribiere and Fletcher-Reeves
perform a series of one-dimensional searches, or cycles, in conjugate gradient directions.
Different algorithms might be appropriate in different circumstances.
RMS gradient
Use 0.1 for the RMS gradient condition,
and the default values for the other variables. The text boxes for RMS gradient and
maximum cycles let you set conditions for ending a calculation. When the
calculation reaches either of these criteria, the calculation ends.
The results:
Geometry optimization, MolecularMechanics,
molecule = (H2O).
PolakRibiere optimizer
Energy=0.071142 Gradient=2.295590
Converged=NO (1 cycles 4 points).
Energy=0.000015 Gradient=0.068458
Converged=NO (2 cycles 8 points).
Energy=0.000015 Gradient=0.068458
Converged=YES (2 cycles 8 points).
Bond=1.21803e-005 Angle=3.248e-006 Dihedral=0 Vdw=0 H-bond=0
Electrostatic=0.
Single Point, MolecularMechanics,
molecule = (H2O).
Total Energy=0.000015
Gradient=0.068466.
Bond=1.21835e-005 Angle=3.2483e-006 Dihedral=0 Vdw=0 H-bond=0
Electrostatic=0.
Vibrational Analysis, AbInitio,
molecule = (H2)).
AbInitio
Convergence limit = 0.0000100 Iteration limit = 50
Accelerate convergence = YES
The initial guess of the MO
coefficients is from eigenvectors of the core Hamiltonian.
Shell Types: S, S=P.
RHF Calculation:
Singlet state calculation
Number of electrons = 10
Number of Doubly-Occupied Levels = 5
Charge on the System = 0
Total Orbitals (Basis Functions) =
13
Primitive Gaussians = 21
Starting HyperGauss calculation with
13 basis functions and 21 primitive Gaussians.
2-electron Integral buffers will be
32000 words (double precision) long.
Two electron integrals will use a
cutoff of 1.00000e-010
Regular integral format is used.
2280 integrals have been produced.
Iteration = 1 Difference =
95.3496859822
Iteration = 2 Difference =
118.9651613722
Iteration = 3 Difference =
4.7871801718
Iteration = 4 Difference =
0.2399152043
Iteration = 5 Difference =
0.0099980717
Iteration = 6 Difference =
0.0008233598
Iteration = 7 Difference =
0.0000381018
Iteration = 8 Difference =
0.0000011855
ENERGIES AND GRADIENT
Total Energy = -47430.5805159 (kcal/mol)
Total Energy = -75.585431962 (a.u.)
Electronic Kinetic Energy =
47301.9415088 (kcal/mol)
Electronic Kinetic Energy =
75.380432680 (a.u.)
The Virial (-V/T) = 2.0027
eK, ee and eN Energy = -53065.5469682 (kcal/mol)
Nuclear Repulsion Energy =
5634.9664523 (kcal/mol)
RMS Gradient = 10.4734296 (kcal/mol/Ang)
Single Point
A single point calculation gives the
static properties of a molecule. The properties include potential energy, derivatives of the potential energy, electrostatic
potential, molecular orbital energies, and the coefficients of molecular
orbitals for ground or excited states. The input molecular structure
for a single point calculation usually reflects the coordinates of a stationary point on the potential energy
surface, typically a minimum or transition state.
Geometry optimization
The minimum represents the potential energy closest to the starting
structure of a molecule.
Ab initio (from the lat -. "From the beginning") - study the phenomenon of the natural laws without additional empirical assumptions or special models.
Ab initio (from the lat -. "From the beginning") - study the phenomenon of the natural laws without additional empirical assumptions or special models.
Ab initio in physics - the first solution of the problem of the fundamental principles
without additional empirical assumptions. Usually it refers to the direct
solution of the equations of quantum mechanics. Despite the name, this is often
done when any assumptions and simplifications. Such simplifications allow us to
calculate the system with a large number of atoms or atoms that has a larger
number of electrons. An example of this simplification is the use of
PAW-potentials.
Ab initio quantum chemistry methods are computational chemistry methods based on quantum
chemistry. The term ab initio
was first used in quantum chemistry by Robert Parr and coworkers, including
David Craig in a semiempirical study on the excited states of benzene. The all
methods based on to solve the Schrödinger equation. In the Hartree–Fock method
and the configuration interaction method, this approximation allows one to
treat the Schrödinger equation as a "simple" eigenvalue (собственное значение) equation of the electronic molecular Hamiltonian,
with a discrete set of solutions.
RHF calculation in computational physics and chemistry, the Hartree–Fock (HF) method
is a method of approximation for the determination of the wave function and the
energy of a quantum many-body system in a stationary state.
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