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Methods Overview

Computational chemistry encompasses a variety of mathematical methods that fall into two broad categories:

Molecular mechanics

applies the laws of classical physics to the atoms in a molecule without explicit consideration of electrons.

Quantum mechanics

relies on the Schrödinger equation to describe a molecule with explicit treatment of electronic structure.

Quantum mechanical methods can be subdivided into two classes: ab initio and semiempirical.

Chem & Bio 3D 11.0 provides the following methods:

Method

  

  

molecular mechanics

MM2

MM3, MM3-protein

AMBER,UFF, Dreiding

Chem3D, Tinker

Tinker

Gaussian

semi-empirical

Extended Hückel

other semi-empirical methods (AM1, MINDO/3, PM3, etc.)

Chem3D, MOPAC, Gaussian

MOPAC, Gaussian

Ab initio

RHF, UHF, MP2, etc.

Gaussian, GAMESS

•  Molecular mechanical methods: MM2 (directly). MM3 and MM3-protein through the Chem3D Tinker interface.

•  Semiempirical Extended Hückel, MINDO/3, MNDO, MNDO-d, AM1 and PM3 methods through Chem3D and CS MOPAC or Gaussian.

•  Ab initio methods through the Chem3D Gaussian or GAMESS interface.

Using Computational Methods

Computational methods calculate the potential energy surfaces (PES) of molecules. The PES is the embodiment of the forces of interaction among atoms in a molecule. From the PES, structural and chemical information about a molecule can be derived. The methods differ in the way the surface is calculated and in the molecular properties derived from the energy surface.

The methods perform the following basic calculations:

Single point energy calculation

The energy of a given geometry of the atoms in a model, which is the value of the PES at that point.

Geometry optimization

A systematic modification of the atomic coordinates of a model resulting in a geometry where the forces on each atom in the structure is zero. A 3dimensional arrangement of atoms in the model representing a local energy minimum (a stable molecular geometry to be found without crossing a conformational energy barrier).

Property calculation

Predicts certain physical and chemical properties, such as charge, dipole moment, and heat of formation.

Computational methods can perform more specialized functions, such as conformational searches and molecular dynamics simulations.

Choosing the Best Method

Not all types of calculations are possible for all methods and no one method is best for all purposes. For any given application, each method poses advantages and disadvantages. The choice of method depend on a number of factors, including:

•  The nature and size of the molecule

•  The type of information sought

•  The availability of applicable experimentally determined parameters (as required by some methods)

•  Computer resources

The three most important of the these criteria are:

Model size

The size of a model can be a limiting factor for a particular method. The limiting number of atoms in a molecule increases by approximately one order of magnitude between method classes from ab initio to molecular mechanics. Ab initio is limited to tens of atoms, semiempirical to hundreds, and molecular mechanics to thousands.

Parameter Availability

Some methods depend on experimentally determined parameters to perform computations. If the model contains atoms for which the parameters of a particular method have not been derived, that method may produce invalid predictions. Molecular mechanics, for example, relies on parameters to define a force-field. A force-field is only applicable to the limited class of molecules for which it is parametrized.

Computer resources

Requirements increase relative to the size of the model for each of the methods.

Ab initio: The time required for performing calculations increases on the order of N4, where N is the number of atoms in the model.

Semiempirical: The time required for computation increases as N3 or N2, where N is the number of atoms in the model.

MM2: The time required for performing computations increases as N2, where N is the number of atoms.

In general, molecular mechanical methods require less computer resources than quantum mechanical methods. The suitability of each general method for particular applications can be summarized as follows.

Molecular Mechanics Methods Applications Summary

Molecular mechanics in Chem3D apply to:

•  Systems containing thousands of atoms.

•  Organic, oligonucleotides, peptides, and saccharides.

•  Gas phase only (for MM2).

Useful techniques available using MM2 methods include:

•  Energy Minimization for locating stable conformations.

•  Single point energy calculations for comparing conformations of the same molecule.

•  Searching conformational space by varying one or two dihedral angles.

•  Studying molecular motion using Molecular Dynamics.

Quantum Mechanical Methods Applications Summary

Useful information determined by quantum mechanical methods includes:

•  Molecular orbital energies and coefficients.

•  Heat of Formation for evaluating conformational energies.

•  Partial atomic charges calculated from the molecular orbital coefficients.

•  Electrostatic potential.

•  Dipole moment.

•  Transition-state geometries and energies.

•  Bond dissociation energies.

Semiempirical methods available in Chem3D with CS MOPAC or Gaussian apply to:

Systems containing up to 120 heavy atoms and 300 total atoms.

Organic, organometallics, and small oligomers (peptide, nucleotide, saccharide).

Gas phase or implicit solvent environment.

Ground, transition, and excited states.

Ab initio methods available in Chem3D with Gaussian or Jaguar apply to:

•  Systems containing up to 150 atoms.

•  Organic, organometallics, and molecular fragments (catalytic components of an enzyme).

•  Gas or implicit solvent environment.

•  Study ground, transition, and excited states (certain methods).

Method Type

Advantages

Disadvantages

Best For

Molecular Mechanics (Gaussian)

Gaussian uses classical physics and relies on force-field with embedded empirical parameters

Least intensive computationally. Gaussian is fast and is useful with limited computer resources. It can be used for molecules as large as enzymes.

Particular force field applicable only for a limited class of molecules

Does not calculate electronic properties

Requires experimental data (or data from ab initio) for parameters

Large systems that consist of thousands of atoms and Systems or processes with no breaking or forming of bonds

Semiempirical (MOPAC, Gaussian)

These use quantum physics, experimentally derived empirical parameters, and extensive approximation.

Less demanding computationally than ab initio methods

Capable of calculating transition states and excited states

Requires experimental data (or data from ab initio) for parameters

Less rigorous than ab initio methods

Medium-sized systems that consist of hundreds of atoms. Also, systems involving electronic transitions.

ab initio (Gaussian, GAMESS)

These use quantum physics, are rigourously mathematical methods, and use no empirical parameters

Useful for a broad range of systems

Does not depend on experimental data

Capable of calculating transition states and excited states

Computationally intensive

Small systems that consist of only tens of atoms or systems involving electronic transitions.

Molecules or systems without available experimental data ("new" chemistry).

Systems requiring rigorous accuracy.

Comparison of Methods

Potential Energy Surfaces

A potential energy surface (PES) can describe:

•  A molecule or ensemble of molecules having constant atom composition (ethane, for example) or a system where a chemical reaction occurs.

•  Relative energies for conformations (eclipsed and staggered forms of ethane).

Potential energy surfaces can differentiate between:

•  Molecules having slightly different atomic composition (ethane and chloroethane).

•  Molecules with identical atomic composition but different bonding patterns, such as propylene and cyclopropane

•  Excited states and ground states of the same molecule.

Potential Energy Surfaces (PES)

The true representation of a model's potential energy surface is a multi-dimensional surface whose dimensionality increases with the number of atom coordinates. Since each atom has three independent variables (x, y, z coordinates), visualizing a surface for a many-atom model is impossible. However, you can generalize this problem by examining any two independent variables, such as the x and y coordinates of an atom.

The main areas of interest on a potential energy surface are the extrema as indicated by the arrows, are as follows:

Global minimum

The most stable conformation appears at the extremum where the energy is lowest. A molecule has only one global minimum.

Local minima

Additional low energy extrema. Minima are regions of the PES where a change in geometry in any direction yields a higher energy geometry.

Saddle point

A stationary point between two low energy extrema. A saddle point is defined as a point on the potential energy surface at which there is an increase in energy in all directions except one, and for which the slope (first derivative) of the surface is zero.

Note: At the energy minimum, the energy is not zero; the first derivative (gradient) of the energy with respect to geometry is zero.

All the minima on a potential energy surface of a molecule represent stable stationery points where the forces on each atom sums to zero. The global minimum represents the most stable conformation; the local minima, less stable conformations; and the saddle points represent transition conformations between minima.

Single Point Energy Calculations

Single point energy calculations can be used to calculate properties of specific geometry of a model. The values of these properties depend on where the model lies on the potential surface as follows:

•  A single point energy calculation at a global minimum provides information about the model in its most stable conformation.

•  A single point calculation at a local minimum provides information about the model in one of many stable conformations.

•  A single point calculation at a saddle point provides information about the transition state of the model.

•  A single point energy calculation at any other point on the potential energy surface provides information about that particular geometry, not a stable conformation or transition state.

Single point energy calculations can be performed before or after optimizing geometry.

Note: Do not compare values from different methods. Different methods rely on different assumptions about a given molecule, and the energies differ by an arbitrary offset.

Geometry Optimization

Geometry optimization is used to locate a stable conformation of a model, and should be done before performing additional computations or analyses of a model.

Locating global and local energy minima is typically done by energy minimization. Locating a saddle point is optimizing to a transition state.

The ability of a geometry optimization to converge to a minimum depends on the starting geometry, the potential energy function used, and the settings for a minimum acceptable gradient between steps (convergence criteria).

Geometry optimizations are iterative and begin at some starting geometry as follows:

1. The single point energy calculation is performed on the starting geometry.

2. The coordinates for some subset of atoms are changed and another single point energy calculation is performed to determine the energy of that new conformation.

3. The first or second derivative of the energy (depending on the method) with respect to the atomic coordinates determines how large and in what direction the next increment of geometry change should be.

4. The change is made.

5. Following the incremental change, the energy and energy derivatives are again determined and the process continues until convergence is achieved, at which point the minimization process terminates.

The following illustration shows some concepts of minimization. For simplicity, this plot shows a single independent variable plotted in two dimensions.

   

The starting geometry of the model determines which minimum is reached. For example, starting at (b), minimization results in geometry (a), which is the global minimum. Starting at (d) leads to geometry (f), which is a local minimum.The proximity to a minimum, but not a particular minimum, can be controlled by specifying a minimum gradient that should be reached. Geometry (f), rather than geometry (e), can be reached by decreasing the value of the gradient where the calculation ends.

In theory, if a convergence criterion (energy gradient) is too lax, a first-derivative minimization can result in a geometry that is near a saddle point. This occurs because the value of the energy gradient near a saddle point, as near a minimum, is very small. For example, at point (c), the derivative of the energy is 0, and as far as the minimizer is concerned, point (c) is a minimum. First derivative minimizers cannot, as a rule, cross saddle points to reach another minimum.

Note: If the saddle point is the extremum of interest, it is best to use a procedure that specifically locates a transition state, such as the CS MOPAC Pro Optimize To Transition State command.

You can take the following steps to ensure that a minimization has not resulted in a saddle point.

•  The geometry can be altered slightly and another minimization performed. The new starting geometry might result in either (a), or (f) in a case where the original one led to (c).

•  The Dihedral Driver can be employed to search the conformational space of the model. For more information, see Tutorial 5: The Dihedral Driver .

•  A molecular dynamics simulation can be run, which will allow small potential energy barriers to be crossed. After completing the molecular dynamics simulation, individual geometries can then be minimized and analyzed. For more information see MM2

You can calculate the following properties with the computational methods available through Chem3D using the PES:

•  Steric energy

•  Heat of formation

•  Dipole moment

•  Charge density

•  COSMO solvation in water

•  Electrostatic potential

•  Electron spin density

•  Hyperfine coupling constants

•  Atomic charges

•  Polarizability

•  Others, such as IR vibrational frequencies

   

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