Computational chemistry
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Computational chemistry is a branch of theoretical chemistry whose major goals are to create efficient mathematical approximations and computer programs that calculate the properties of molecules (such as total energy, dipole and quadrupole moment, vibrational frequencies, reactivity and other diverse spectroscopic quantitities and cross sections for collision of molecules with diverse atomic or subatomic projectiles) and to apply these programs to concrete chemical objects. The term is also sometimes used to cover the areas of overlap between computer science and 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 and has been described in a qualitative or approximate quantitative computational scheme.
It is, in principle, possible to use one very accurate method and apply it to all molecules. Although such methods are well-known and available in many programs, the computational cost of their use grows factorially (even faster than exponentially) with the number of electrons. 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 molecules that contain a few dozen electrons is computationally tractable by approximate methods such as DFT. There is some dispute within the field whether the latter methods are sufficient to describe complex chemical reactions, such as those in biochemistry.
In theoretical chemistry, chemists and physicists together 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 approaches in doing this:
- Computational studies can be carried out in order to find a starting point for a laboratory synthesis
- Computational studies can be used to explore reaction mechanisms and explain observations of laboratory reactions
Several major areas may be distinguished within computational chemistry:
- The computational representation of atoms and molecules
- Storing and searching for data on chemical entities (see chemical databases)
- Identifying correlations between chemical structures and properties (see QSPR and QSAR)
- Theoretical elucidation of structures based on the simulation of forces
- 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)
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 empirical or semi-empirical parameters in their equations - are derived directly from theoretical principles, with no inclusion of experimental data - are generally called ab initio methods. Most of the time this is referring to approximate quantum mechanical calculations. The approximations made in these cases, however, are usually mathematical in nature, such as using a simpler functional form or getting an approximate solution for a complicated differential equation.
Most ab initio methods use the Born-Oppenheimer representation, allowing the separation of electronic and nuclear motion, and simplifying the Schrödinger equation. A notable exception are certain approaches called direct quantum chemistry, which treat electons and nuclei on a common footing. The calculation usually proceeds in two steps:
- determination of the electronic structure,
- determination of the chemical dynamics.
Electronic structure
The electronic structure is determined by solving the time-independent Schrödinger equation associated with the electronic molecular Hamiltonian. The molecular geometry is considered as an adiabatic parameter. Usually the basis set (which is usually built from the LCAO ansatz) used to solve the Schrödinger equation is not complete and does not span the Hilbert space associated with ionization and scattering processes (see continuous spectrum for more details). This approximation allows one to treat the Schrödinger equation as a "simple" eigenvalue equation of the electronic molecular Hamiltonian with a discrete set of solutions.
The obtained eigenvalues are functions of the molecular geometry which are called potential energy surfaces.
The most common type of ab initio electronic structure calculation is called a Hartree-Fock (HF) calculation, in which the Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect is included in the calculation. This is a variational calculation, therefore the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree-Fock limit. Many types of calculations begin with a HF calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation. Møller-Plesset perturbation theory (MP) and Coupled cluster (CC) are examples of such methods.
A method that avoids making the variational overestimation of HF in the first place is Quantum Monte Carlo (QMC), in its variational, diffusion, and Green's functions flavors. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. Such calculations can be very time consuming, but they are probably the most accurate methods known today.
Density Functional Theory (DFT) methods are often considered to be ab initio methods for determining the molecular electronic structure, even many of the most common functionals usually use parameters derived from empirical data, or more complex calculations. 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.
Ab initio electronic structure methods have the advantage that they can be made to converge to the exact solution, when all approximations are sufficiently small in magnitude. The convergence, however, is usually not monotonic, and sometimes the smallest calculation gives the best result for some properties. The bad side of ab initio methods is their cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales as N4 (N being the number of basis functions) – a calculation twice as big takes 16 times as long to complete – and correlated calculations often scale much less favorably (correlated DFT calculations being the most efficient of this lot).
Computational chemical methods can also 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 semiempirical 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.
The most popular classes of ab initio electronic structure methods:
- Hartree-Fock
- Møller-Plesset perturbation theory
- Multi-configurational self-consistent field (MCSCF)
- Configuration interaction
- Multi-Reference Configuration Interaction
- Coupled cluster
- Quadratic configuration interaction
- Quantum Monte Carlo
- Density functional theory
- Generalized Valence Bond (GVB)
- Modern valence bond theory
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.
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 examines (using Newton's laws of motion) the time-dependent behaviour of systems, including vibrations or Brownian motion, most often with a classical mechanical description as well. Combined with density functional theory methods molecular dynamics is called Car-Parrinello method.
Semiempirical methods
Electronic structure
Within the framework of Hartree-Fock calculations, some pieces of information (such as two-elecron integrals) are sometimes approximated or completely omitted. In order to correct for this loss, semiempirical methods are parametrized, that is their results are fitted by a set of parameters, normally in such a way as to produce results that best agree with experimental data, but sometimes to agree with ab initio results.
Semiempirical methods followed 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 Huckel method proposed by Erich Hückel and for all valence electron systems, the Extended Huckel method proposed by Roald Hoffmann.
Semiempirical calculations are much faster than their ab initio counterparts. Their results, however, can be very wrong if the molecule being computed is not similar enough to the molecules in the database used to parametrize the method.
Semiempirical calculations have been most successful in the description of organic chemistry, where only a few elements are used extensively and molecules are of moderate size.
As with empirical methods, we can distinguish methods that are:-
- restricted to pi-electrons, and those
- restricted to valence electrons,
the latter being by far the largest group of methods.
Molecular mechanics
In many cases, large molecular systems can be modelled succesfully 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 proteins.
Software packages
A number of software packages that are self-sufficient and include many quantum-chemical methods are available. The following is a table illustrating the capabilities of various software packages, (corrections to table entries requested). There is a separate list of Valence Bond Programs.
| Package | Molecular Mechanics | Semi-Empirical | Hartree-Fock | Post-Hartree-Fock methods | Density Functional Theory | Periodic |
| ACES | N | N | Y | Y | N | N |
| ADF | N | N | N | N | Y | Y |
| AMPAC | N | Y | N | N | N | N |
| CADPAC | N | N | Y | Y | Y | N |
| COLUMBUS | N | N | Y | Y | N | N |
| CRYSTAL | N | N | Y | N | Y | Y |
| DALTON | N | N | Y | Y | Y | N |
| GAUSSIAN | Y | Y | Y | Y | Y | Y |
| GAMESS | N | Y | Y | Y | Y | N |
| JAGUAR | Y | N | Y | Y | Y | N |
| MOLCAS | Y | Y | Y | Y | Y | N |
| MOLPRO | N | N | Y | Y | Y | N |
| MOPAC | N | Y | N | N | N | Y |
| MPQC | N | N | Y | Y | Y | N |
| NWChem | Y | N | Y | Y | Y | Y |
| PLATO | Y | N | N | N | Y | Y |
| PQS | Y | Y | Y | Y | Y | N |
| PSI | N | N | Y | Y | N | N |
| TURBOMOLE | N | N | Y | Y | Y | N |
| Q-Chem | N | N | Y | Y | Y | N |