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Computational chemistry

2007 Schools Wikipedia Selection. Related subjects: General Chemistry

   Computational chemistry is a branch of chemistry that uses the results
   of theoretical chemistry incorporated into efficient computer programs
   to calculate the structures and properties of molecules and solids,
   applying these programs to real chemical problems. Examples of such
   properties are structure (i.e. the expected positions of the
   constituent atoms), energy and interaction energy, charges, dipoles and
   higher multipole moments, vibrational frequencies, reactivity or other
   spectroscopic quantitities, and cross sections for collision with other
   particles. The term computational chemistry is also sometimes used to
   cover any of the areas of science that overlap between computer science
   and chemistry. Electronic configuration theory is the largest
   subdiscipline of computational chemistry.

Introduction

   The term theoretical chemistry may be defined as a mathematical
   description of chemistry, whereas computational chemistry is usually
   used when a mathematical method is sufficiently well developed that it
   can be automated for implementation on a computer. Note that the words
   exact and perfect do not appear here, as very few aspects of chemistry
   can be computed exactly. Almost every aspect of chemistry, however, can
   be described in a qualitative or approximate quantitative computational
   scheme.

   Molecules consist of nuclei and electrons, so the methods of quantum
   mechanics apply. Computational chemists often attempt to solve the
   non-relativistic Schrödinger equation, with relativistic corrections
   added, although some progress has been made in solving the fully
   relativistic Schrödinger equation. It is, in principle, possible to
   solve the Schrödinger equation, in either its time-dependent form or
   time-independent form as appropriate for the problem in hand, but this
   in practice is not possible except for very small systems. Therefore, a
   great number of approximate methods strive to achieve the best
   trade-off between accuracy and computational cost. Present
   computational chemistry can routinely and very accurately calculate the
   properties of molecules that contain no more than 10-40 electrons. The
   treatment of larger molecules that contain a few dozen electrons is
   computationally tractable by approximate methods such as density
   functional theory (DFT). There is some dispute within the field whether
   the latter methods are sufficient to describe complex chemical
   reactions, such as those in biochemistry. Large molecules can be
   studied by semi-empirical approximate methods. Even larger molecules
   are treated with classical mechanics in methods called molecular
   mechanics.

   In theoretical chemistry, chemists, physicists and mathematicians
   develop algorithms and computer programs to predict atomic and
   molecular properties and reaction paths for chemical reactions.
   Computational chemists, in contrast, may simply apply existing computer
   programs and methodologies to specific chemical questions. There are
   two different aspects to computational chemistry:
     * Computational studies can be carried out in order to find a
       starting point for a laboratory synthesis, or to assist in
       understanding experimental data, such as the position and source of
       spectroscopic peaks.
     * Computational studies can be used to predict the possibility of so
       far entirely unknown molecules or to explore reaction mechanisms
       that are not readily studied by experimental means.

   Thus computational chemistry can assist the experimental chemist or it
   can challenge the experimental chemist to find entirely new chemical
   objects.

   Several major areas may be distinguished within computational
   chemistry:
     * The prediction of the molecular structure of molecules by the use
       of the simulation of forces to find stationary points on the energy
       hypersurface as the position of the nuclei is varied.
     * Storing and searching for data on chemical entities (see chemical
       databases).
     * Identifying correlations between chemical structures and properties
       (see QSPR and QSAR).
     * Computational approaches to help in the efficient synthesis of
       compounds.
     * Computational approaches to design molecules that interact in
       specific ways with other molecules (e.g. drug design).

Molecular structure

   A given molecular formula can represent a number of molecular isomers.
   Each isomer is a local minimum on the energy surface (called the
   potential energy surface) created from the total energy (electronic
   energy plus repulsion energy between the nuclei) as a function of the
   coordinates of all the nuclei. A stationary point is a geometry such
   that the derivative of the energy with respect to all displacements of
   the nuclei is zero. A local (energy) minimum is a stationary point
   where all such displacements lead to an increase in energy. The local
   minimum that is lowest is called the global minimum and corresponds to
   the most stable isomer. If there is one particular coordinate change
   that leads to a decrease in the total energy in both directions, the
   stationary point is a transition structure and the coordinate is the
   reaction coordinate. This process of determining stationary points is
   called geometry optimisation.

   The determination of molecular structure by geometry optimisation
   became routine only when efficient methods for calculating the first
   derivatives of the energy with respect to all atomic coordinates became
   available. Evaluation of the related second derivatives allows the
   prediction of vibrational frequencies if harmonic motion is assumed. In
   some ways more importantly it allows the characterisation of stationary
   points. The frequencies are related to the eigenvalues of the matrix of
   second derivatives (the Hessian matrix). If the eigenvalues are all
   positive, then the frequencies are all real and the stationary point is
   a local minimum. If one eigenvalue is negative (an imaginary
   frequency), the stationary point is a transition structure. If more
   than one eigenvalue is negative the stationary point is a more complex
   one, and usually of little interest. When found, it is necessary to
   move the search away from it, if we are looking for local minima and
   transition structures.

   The total energy is determined by approximate solutions of the
   time-dependent Schrödinger equation, usually with no relativistic terms
   included, and making use of the Born-Oppenheimer approximation which,
   based on the much higher velocity of the electrons in comparison with
   the nuclei, allows the separation of electronic and nuclear motions,
   and simplifies the Schrödinger equation. This leads to evaluating the
   total energy as a sum of the electronic energy at fixed nuclei
   positions plus the repulsion energy of the nuclei. A notable exception
   are certain approaches called direct quantum chemistry, which treat
   electrons and nuclei on a common footing. Density functional methods
   and semi-empirical methods are variants on the major theme. For very
   large systems the total energy is determined using molecular mechanics.
   The ways of determing the total energy to predict molecular structures
   are:-

Ab initio methods

   The programs used in computational chemistry are based on many
   different quantum-chemical methods that solve the molecular Schrödinger
   equation associated with the molecular Hamiltonian. Methods that do not
   include any empirical or semi-empirical parameters in their equations -
   being derived directly from theoretical principles, with no inclusion
   of experimental data - are called ab initio methods. This does not
   imply that the solution is an exact one; they are all approximate
   quantum mechanical calculations. It means that a particular
   approximation is rigorously defined on first principles (quantum
   theory) and then solved within an error margin that is qualitatively
   known beforehand. If numerical iterative methods have to be employed,
   the aim is to iterate until full machine accuracy is obtained (the best
   that is possible with a finite word length on the computer, and whithin
   the mathematical and/or physical approximations made).
   Diagram illustrating various ab initio electronic structure methods in
   terms of energy.
   Enlarge
   Diagram illustrating various ab initio electronic structure methods in
   terms of energy.

   The simplest type of ab initio electronic structure calculation is the
   Hartree-Fock (HF) scheme, in which the Coulombic electron-electron
   repulsion is not specifically taken into account. Only its average
   effect is included in the calculation. As the basis set size is
   increased the energy and wave function tend to a limit called the
   Hartree-Fock limit. Many types of calculations, known as
   post-Hartree-Fock methods, begin with a Hartree-Fock calculation and
   subsequently correct for electron-electron repulsion, referred to also
   as electronic correlation. As these methods are pushed to the limit,
   they approach the exact solution of the non-relativistic Schrödinger
   equation. In order to obtain exact agreement with experiment, it is
   necessary to include relativistic and spin orbit terms, both of which
   are only really important for heavy atoms. In all of these approaches,
   in addition to the choice of method, it is necessary to chose a basis
   set. This is set of functions, usually centred on the different atoms
   in the molecule, which are used to expand the molecular orbitals with
   the LCAO ansatz. Ab initio methods need to define a level of theory
   (the method) and a basis set.

   The Hartree-Fock wave function is a single configuration or
   determinant. In some cases, particularly for bond breaking processes,
   this is quite inadequate and several configurations need to be used.
   Here the coefficients of the configurations and the coefficients of the
   basis functions are optimised together.

   The total molecular energy can be evaluated as a function of the
   molecular geometry, in other words the potential energy surface.

Example: Is Si[2]H[2] like acetylene (C[2]H[2])?

   A series of ab initio studies of Si[2]H[2] shows clearly the power of
   ab initio computational chemistry. They go back over 20 years, and most
   of the main conclusions were reached by 1995. The methods used were
   mostly post-Hartree-Fock, particularly Configuration interaction (CI)
   and Coupled cluster (CC). Initially the question was whether Si[2]H[2]
   had the same structure as ethyne (acetylene), C[2]H[2]. Slowly (because
   this started before geometry optimization was widespread), it became
   clear that linear Si[2]H[2] was a transition structure between two
   equivalent trans-bent structures and that it was rather high in energy.
   The ground state was predicted to be a four-membered ring bent into a
   'butterfly' structure with hydrogen atoms bridged between the two
   silicon atoms. Interest then moved to look at whether structures
   equivalent to vinylidene - Si=SiH[2] - existed. This structure is
   predicted to be a local minimum, i. e. an isomer of Si[2]H[2], lying
   higher in energy than the ground state but below the energy of the
   trans-bent isomer. Then surprisingly a new isomer was predicted by
   Brenda Colegrove in Henry F. Schaefer, III's group. This prediction was
   so surprising that it needed extensive calculations to confirm it. It
   requires post Hartree-Fock methods to obtain a local minimum for this
   structure. It does not exist on the Hartree-Fock energy hypersurface.
   The new isomer is a planar structure with one bridging hydrogen atom
   and one terminal hydrogen atom, cis to the bridging atom. Its energy is
   above the ground state but below that of the other isomers. Similar
   results were later obtained for Ge[2]H[2] and SiGeH[2] . More
   interestingly, similar results were obtained for Al[2]H[2] (and then
   Ga[2]H[2] and AlGaH[2]) which have two electrons less than the Group 14
   molecules. The only difference is that the four-membered ring ground
   state is planar and not bent. The cis-mono-bridged and vinylidene-like
   isomers are present. Experimental work on these molecules is not easy,
   but matrix isolation spectroscopy of the products of the reaction of
   hydrogen atoms and silicon and aluminium surfaces has found the ground
   state ring structures and the cis-mono-bridged structures for Si[2]H[2]
   and Al[2]H[2]. Theoretical predictions of the vibrational frequencies
   were crucial in understanding the experimental observations of the
   spectra of a mixture of compounds. This may appear to be an obscure
   area of chemistry, but the differences between carbon and silicon
   chemistry is always a lively question, as are the differences between
   group 13 and group 14 (mainly the B and C differences). The silicon and
   germanium compounds were the subject of a Journal of Chemical Education
   article.

Density Functional methods

   Density functional theory (DFT) methods are often considered to be ab
   initio methods for determining the molecular electronic structure, even
   though many of the most common functionals use parameters derived from
   empirical data, or from more complex calculations. This means that they
   could also be called semi-empirical methods. It is best to treat them
   as a class on their own. In DFT, the total energy is expressed in terms
   of the total electron density rather than the wave function. In this
   type of calculation, there is an approximate Hamiltonian and an
   approximate expression for the total electron density. DFT methods can
   be very accurate for little computational cost. The drawback is, that
   unlike ab initio methods, there is no systematic way to improve the
   methods by improving the form of the functional.

Semi-empirical and empirical methods

   Semi-empirical quantum chemistry methods are based on the Hartree-Fock
   formalism, but make many approximations and obtain some parameters from
   empirical data. They are very important in computational chemistry for
   treating large molecules where the full Hartree-Fock method without the
   approximations is too expensive. The use of empirical parameters
   appears to allow some inclusion of correlation effects into the
   methods.

   Semi-empirical methods follow what are often called empirical methods
   where the two-electron part of the Hamiltonian is not explicitly
   included. For π-electron systems, this was the Hückel method proposed
   by Erich Hückel, and for all valence electron systems, the Extended
   Hückel method proposed by Roald Hoffmann.

Molecular mechanics

   In many cases, large molecular systems can be modelled successfully
   while avoiding quantum mechanical calculations entirely. Molecular
   mechanics simulations, for example, use a single classical expression
   for the energy of a compound, for instance the harmonic oscillator. All
   constants appearing in the equations must be obtained beforehand from
   experimental data or ab initio calculations.

   The database of compounds used for parameterization - (the resulting
   set of parameters and functions is called the force field) - is crucial
   to the success of molecular mechanics calculations. A force field
   parameterized against a specific class of molecules, for instance
   proteins, would be expected to only have any relevance when describing
   other molecules of the same class.

Interpreting molecular wave functions

   The Atoms in Molecules model developed by Richard Bader was developed
   in order to effectively link the quantum mechanical picture of a
   molecule, as an electronic wavefunction, to chemically useful older
   models such as the theory of Lewis pairs and the valence bond model.
   Bader has demonstrated that these empirically useful models are
   connected with the topology of the quantum charge density. This method
   improves on the use of Mulliken charges.

Computational chemical methods in solid state physics

   Computational chemical methods can be applied to solid state physics
   problems. The electronic structure of a crystal is in general described
   by a band structure, which defines the energies of electron orbitals
   for each point in the Brillouin zone. Ab initio and semi-empirical
   calculations yield orbital energies, therefore they can be applied to
   band structure calculations. Since it is time-consuming to calculate
   the energy for a molecule, it is even more time-consuming to calculate
   them for the entire list of points in the Brillouin zone.

Chemical dynamics

   Once the electronic and nuclear variables are separated (within the
   Born-Oppenheimer representation), in the time-dependent approach, the
   wave packet corresponding to the nuclear degrees of freedom is
   propagated via the time evolution operator (physics) associated to the
   time-dependent Schrödinger equation (for the full molecular
   Hamiltonian). In the complementary energy-dependent approach, the
   time-independent Schrödinger equation is solved using the scattering
   theory formalism. The potential respresenting the interatomic
   interaction is given by the potential energy surfaces. In general, the
   potential energy surfaces are coupled via the vibronic coupling terms.

   The most popular methods for propagating the wave packet associated to
   the molecular geometry are
     * the split operator technique,
     * the Multi-Configuration Time-Dependent Hartree method (MCTDH),
     * the semiclassical method.

   Molecular dynamics (MD) examines (using Newton's laws of motion) the
   time-dependent behaviour of systems, including vibrations or Brownian
   motion, using a classical mechanical description. MD combined with
   density functional theory leads to the Car-Parrinello method.

Software packages

   A number of self-sufficient software packages include many
   quantum-chemical methods, and in some cases molecular mechanics
   methods. The following table illustrates the capabilities of the most
   versatile software packages that show an entry in two or more columns
   of the table. There are separate lists for specialized programs, such
   as:-
     * Density functional theory programs
     * Semi-empirical programs
     * Solid state system programs with periodic boundary conditions.
     * Valence Bond programs

   Package Molecular Mechanics Semi-Empirical Hartree-Fock
   Post-Hartree-Fock methods Density Functional Theory
   ACES N N Y Y N
   CADPAC N N Y Y Y
   COLUMBUS N N Y Y N
   DALTON N N Y Y Y
   GAUSSIAN Y Y Y Y Y
   GAMESS (UK) N Y Y Y Y
   GAMESS (US) N Y Y Y Y
   JAGUAR Y N Y Y Y
   MOLCAS Y Y Y Y Y
   MOLPRO N N Y Y Y
   MPQC N N Y Y Y
   NWChem Y N Y Y Y
   PLATO Y N N N Y
   PQS Y Y Y Y Y
   PSI N N Y Y N
   TURBOMOLE N N Y Y Y
   Q-Chem Y N Y Y Y

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