**Computational chemistry** is a branch of

chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

that uses principles of

computer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

to assist in solving chemical problems. It uses the results of

theoretical chemistryTheoretical chemistry seeks to provide theories that explain chemical observations. Often, it uses mathematical and computational methods that, at times, require advanced knowledge. Quantum chemistry, the application of quantum mechanics to the understanding of valency, is a major component of...

, incorporated into efficient

computer programA computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

s, to calculate the structures and properties of

moleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s and solids. Its necessity arises from the well-known fact that apart from relatively recent results concerning the hydrogen molecular ion (see references therein for more details), the quantum

n-body problemThe n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...

cannot be solved analytically, much less in closed form. While its results normally complement the information obtained by chemical

experimentAn experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

s, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials.

Examples of such properties are structure (i.e. the expected positions of the constituent atoms), absolute and

relativeIn physics, interaction energy is the contribution to the total energy that is caused by an interaction between the objects being considered.The interaction energy usually depends on the relative position of the objects...

(interaction)

energiesIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

,

electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

ic

charge distributionThe linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

s,

dipoleIn physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

s and higher multipole moments, vibrational frequencies, reactivity or other

spectroscopicSpectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

quantities, and

cross sectionsA cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

for

collisionIn mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

with other particles.

The methods employed cover both static and dynamic situations. In all cases the computer time and other resources (such as memory and disk space) increase rapidly with the size of the system being studied. That system can be a single molecule, a group of molecules, or a solid. Computational chemistry methods range from highly accurate to very approximate; highly accurate methods are typically feasible only for small systems.

*Ab initio* methodsAb initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...

are based entirely on theory from first principles. Other (typically less accurate) methods are called empirical or semi-empirical because they employ experimental results, often from acceptable models of atoms or related molecules, to approximate some elements of the underlying theory.

Both

*ab initio* and semi-empirical approaches involve approximations. These range from simplified forms of the first-principles equations that are easier or faster to solve, to approximations limiting the size of the system (for example,

periodic boundary conditionsIn mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...

), to fundamental approximations to the underlying equations that are required to achieve any solution to them at all. For example, most

*ab initio* calculations make the Born–Oppenheimer approximation, which greatly simplifies the underlying

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

by freezing the nuclei in place during the calculation. In principle,

*ab initio* methodsAb initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...

eventually converge to the exact solution of the underlying equations as the number of approximations is reduced. In practice, however, it is impossible to eliminate all approximations, and residual error inevitably remains. The goal of computational chemistry is to minimize this residual error while keeping the calculations tractable.

In some cases, the details of electronic structure are less important than the long-time

phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

behavior of molecules. This is the case in conformational studies of proteins and protein-ligand binding thermodynamics. Classical approximations to the potential energy surface are employed, as they are computationally less intensive than electronic calculations, to enable longer simulations of

molecular dynamicsMolecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...

. Furthermore,

cheminformaticsCheminformatics is the use of computer and informational techniques, applied to a range of problems in the field of chemistry. These in silico techniques are used in pharmaceutical companies in the process of drug discovery...

uses even more empirical (and computationally cheaper) methods like

machine learningMachine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...

based on physicochemical properties. One typical problem in cheminformatics is to predict the binding affinity of drug molecules to a given target.

## History

Building on the founding discoveries and theories in the

history of quantum mechanicsThe history of quantum mechanics, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859-1860 winter statement of the black body radiation problem by Gustav...

, the first theoretical calculations in chemistry were those of

Walter HeitlerWalter Heinrich Heitler was a German physicist who made contributions to quantum electrodynamics and quantum field theory...

and

Fritz LondonFritz Wolfgang London was a German theoretical physicist. His fundamental contributions to the theories of chemical bonding and of intermolecular forces are today considered classic and are discussed in standard textbooks of physical chemistry.With his brother Heinz, he made a significant...

in 1927. The books that were influential in the early development of computational quantum chemistry include

Linus PaulingLinus Carl Pauling was an American chemist, biochemist, peace activist, author, and educator. He was one of the most influential chemists in history and ranks among the most important scientists of the 20th century...

and

E. Bright Wilson's 1935

*Introduction to Quantum Mechanics – with Applications to Chemistry*,

EyringHenry Eyring was a Mexican-born American theoretical chemist whose primary contribution was in the study of chemical reaction rates and intermediates....

, Walter and Kimball's 1944

*Quantum Chemistry*, Heitler's 1945

*Elementary Wave Mechanics – with Applications to Quantum Chemistry*, and later

CoulsonCharles Alfred Coulson FRS was an applied mathematician, theoretical chemist and religious author.His major scientific work was as a pioneer of the application of the quantum theory of valency to problems of molecular structure, dynamics and reactivity...

's 1952 textbook

*Valence*, each of which served as primary references for chemists in the decades to follow.

With the development of efficient

computerA computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

technology in the 1940s, the solutions of elaborate

wave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...

s for complex

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

ic systems began to be a realizable objective. In the early 1950s, the first semi-empirical atomic orbital calculations were carried out. Theoretical chemists became extensive users of the early digital computers. A very detailed account of such use in the United Kingdom is given by Smith and Sutcliffe. The first

*ab initio* Hartree–Fock calculations on diatomic molecules were carried out in 1956 at MIT, using a

basis setA basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...

of Slater orbitals. For diatomic molecules, a systematic study using a minimum basis set and the first calculation with a larger basis set were published by Ransil and Nesbet respectively in 1960. The first polyatomic calculations using Gaussian orbitals were carried out in the late 1950s. The first

configuration interactionConfiguration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...

calculations were carried out in Cambridge on the

EDSACElectronic Delay Storage Automatic Calculator was an early British computer. The machine, having been inspired by John von Neumann's seminal First Draft of a Report on the EDVAC, was constructed by Maurice Wilkes and his team at the University of Cambridge Mathematical Laboratory in England...

computer in the 1950s using

Gaussian orbitalIn computational chemistry and molecular physics, Gaussian orbitals are functions used as atomic orbitals in the LCAO method for the computation of electron orbitals in molecules and numerous properties that depend on these.- Rationale :The principal reason for the use of Gaussian basis functions...

s by Boys and coworkers. By 1971, when a bibliography of

*ab initio* calculations was published, the largest molecules included were

naphthaleneNaphthalene is an organic compound with formula . It is a white crystalline solid with a characteristic odor that is detectable at concentrations as low as 0.08 ppm by mass. As an aromatic hydrocarbon, naphthalene's structure consists of a fused pair of benzene rings...

and

azuleneAzulene is an organic compound and an isomer of naphthalene. Whereas naphthalene is colourless, azulene is dark blue. Its name is derived from the Spanish word azul, meaning "blue"...

. Abstracts of many earlier developments in

*ab initio* theory have been published by Schaefer.

In 1964,

Hückel methodThe Hückel method or Hückel molecular orbital method proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene,...

calculations (using a simple linear combination of atomic orbitals (LCAO) method for the determination of electron energies of molecular orbitals of π electrons in conjugated hydrocarbon systems) of molecules ranging in complexity from butadiene and

benzeneBenzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....

to

ovaleneOvalene is a polycyclic aromatic hydrocarbon with the formula C32H14, which consists of ten peri-fused six-membered rings. It is very similar to coronene....

, were generated on computers at Berkeley and Oxford. These empirical methods were replaced in the 1960s by semi-empirical methods such as

CNDOCNDO is the abbreviation for Complete Neglect of Differential Overlap. Although CNDO is based on quantum chemistry, it is more specifically one of the first semi-empirical quantum chemistry methods. It uses two approximations:...

.

In the early 1970s, efficient

*ab initio* computer programs such as ATMOL,

GAUSSIANGaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...

, IBMOL, and POLYAYTOM, began to be used to speed up

*ab initio* calculations of molecular orbitals. Of these four programs, only GAUSSIAN, now massively expanded, is still in use, but many other programs are now in use. At the same time, the methods of

molecular mechanicsMolecular mechanics uses Newtonian mechanics to model molecular systems. The potential energy of all systems in molecular mechanics is calculated using force fields...

, such as

MM2In the context of molecular modeling, a force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles . Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical...

, were developed, primarily by

Norman AllingerNorman "Lou" Allinger is an American computational chemist and Distinguished Research Professor Emeritus of Chemistry at the University of Georgia in Athens....

.

One of the first mentions of the term "computational chemistry" can be found in the 1970 book

*Computers and Their Role in the Physical Sciences* by Sidney Fernbach and Abraham Haskell Taub, where they state "It seems, therefore, that 'computational chemistry' can finally be more and more of a reality." During the 1970s, widely different methods began to be seen as part of a new emerging discipline of

*computational chemistry*. The

*Journal of Computational Chemistry*The Journal of Computational Chemistry is a peer-reviewed scientific journal published since 1980 by John Wiley & Sons. It covers research, contemporary developments in theory and methodology, and applications in all areas of computational chemistry, including ab initio quantum chemistry methods...

was first published in 1980.

## Fields of application

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. In theoretical chemistry, chemists, physicists and mathematicians develop

algorithmIn mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

s and computer programs to predict atomic and molecular properties and reaction paths for

chemical reactionA chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...

s. 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 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, or more accurate quantum chemical methods, to find stationary points on the energy surface as the position of the nuclei is varied.
- Storing and searching for data on chemical entities (see chemical database
A chemical database is a database specifically designed to store chemical information. This information is about chemical and crystal structures, spectra, reactions and syntheses, and thermophysical data.- Chemical structures :...

s).
- Identifying correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

s between chemical structureA chemical structure includes molecular geometry, electronic structure and crystal structure of molecules. Molecular geometry refers to the spatial arrangement of atoms in a molecule and the chemical bonds that hold the atoms together. Molecular geometry can range from the very simple, such as...

s 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
Drug design, also sometimes referred to as rational drug design or structure-based drug design, is the inventive process of finding new medications based on the knowledge of the biological target...

and catalysisCatalysis is the change in rate of a chemical reaction due to the participation of a substance called a catalyst. Unlike other reagents that participate in the chemical reaction, a catalyst is not consumed by the reaction itself. A catalyst may participate in multiple chemical transformations....

).

## Accuracy

The words

*exact* and

*perfect* do not appear here, as very few aspects of chemistry can be computed exactly. However, almost every aspect of chemistry can be described in a qualitative or approximate quantitative computational scheme.

Molecules consist of nuclei and electrons, so the methods of

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

apply. Computational chemists often attempt to solve the non-relativistic

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, with relativistic corrections added, although some progress has been made in solving the fully relativistic

Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...

. In principle, it is possible to solve the Schrödinger equation in either its time-dependent or time-independent form, as appropriate for the problem in hand; in practice, this 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.

Accuracy can always be improved with greater computational cost. Significant errors can present themselves in

ab initioab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...

models comprising many electrons, due to the computational expense of full relativistic-inclusive methods. This complicates the study of molecules interacting with high atomic mass unit atoms, such as transitional metals and their catalytic properties. Present algorithms in computational chemistry can routinely calculate the properties of molecules that contain up to about 40 electrons with sufficient accuracy. Errors for energies can be less than a few kJ/mol. For geometries, bond lengths can be predicted within a few picometres and bond angles within 0.5 degrees. The treatment of larger molecules that contain a few dozen electrons is computationally tractable by approximate methods such as

density functional theoryDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...

(DFT).

There is some dispute within the field whether or not 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 by

classical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

methods that employ what are called

molecular mechanicsMolecular mechanics uses Newtonian mechanics to model molecular systems. The potential energy of all systems in molecular mechanics is calculated using force fields...

. In QM/MM methods, small portions of large complexes are treated quantum mechanically (QM), and the remainder is treated approximately (MM).

## Methods

A single molecular formula can represent a number of molecular isomers. Each isomer is a local minimum on the energy surface (called the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

) created from the total energy (i.e., the electronic energy, plus the 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 structureThe transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest energy along this reaction coordinate. At this point, assuming a perfectly irreversible reaction, colliding reactant molecules will always...

and the coordinate is the

reaction coordinateIn chemistry, a reaction coordinate is an abstract one-dimensional coordinate which represents progress along a reaction pathway. It is usually a geometric parameter that changes during the conversion of one or more molecular entities....

. This process of determining stationary points is called geometry optimization.

The determination of molecular structure by geometry optimization became routine only after 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 estimated. More importantly, it allows for the characterization of stationary points. The frequencies are related to the eigenvalues of the Hessian matrix, which contains second derivatives. 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 (i.e., an imaginary frequency), then the stationary point is a transition structure. If more than one eigenvalue is negative, then the stationary point is a more complex one, and is usually of little interest. When one of these is found, it is necessary to move the search away from it if the experimenter is looking solely 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 by making use of the Born–Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation. This leads to the evaluation of the total energy as a sum of the electronic energy at fixed nuclei positions and the repulsion energy of the nuclei. A notable exception are certain approaches called

direct quantum chemistryDirect quantum chemistry covers a set of quantum chemical methods not using the Born–Oppenheimer representation. Direct quantum chemistry considers the motion of the nuclei and the electrons on the same time scales...

, 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 relative total energies can be compared using molecular mechanics. The ways of determining the total energy to predict molecular structures are:

### *Ab initio* methods

The programs used in computational chemistry are based on many different

quantum-chemicalQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

methods that solve the molecular

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

associated with the

molecular HamiltonianIn atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...

. 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*Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...

. 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 within the mathematical and/or physical approximations made).

The simplest type of

*ab initio* electronic structure calculation is the Hartree–Fock (HF) scheme, an extension of

molecular orbital theoryIn chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...

, in which the correlated 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 towards 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 correlationElectronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...

. 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 choose a

basis setA basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...

. This is a set of functions, usually centered on the different atoms in the molecule, which are used to expand the molecular orbitals with the

LCAOA linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...

ansatzAnsatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...

. 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

configurationsMulti-configurational self-consistent field is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate...

need to be used. Here, the coefficients of the configurations and the coefficients of the basis functions are optimized together.

The total molecular energy can be evaluated as a function of the

molecular geometryMolecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...

; in other words, the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

. Such a surface can be used for reaction dynamics. The stationary points of the surface lead to predictions of different

isomerIn chemistry, isomers are compounds with the same molecular formula but different structural formulas. Isomers do not necessarily share similar properties, unless they also have the same functional groups. There are many different classes of isomers, like stereoisomers, enantiomers, geometrical...

s and the

transition structureTransition state theory explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium between reactants and activated transition state complexes....

s for conversion between isomers, but these can be determined without a full knowledge of the complete surface.

A particularly important objective, called computational

thermochemistryThermochemistry is the study of the energy and heat associated with chemical reactions and/or physical transformations. A reaction may release or absorb energy, and a phase change may do the same, such as in melting and boiling. Thermochemistry focuses on these energy changes, particularly on the...

, is to calculate thermochemical quantities such as the

enthalpy of formationThe standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states...

to chemical accuracy. Chemical accuracy is the accuracy required to make realistic chemical predictions and is generally considered to be 1 kcal/mol or 4 kJ/mol. To reach that accuracy in an economic way it is necessary to use a series of post-Hartree–Fock methods and combine the results. These methods are called

quantum chemistry composite methodsQuantum chemistry composite methods are computational chemistry methods that aim for high accuracy by combining the results of several calculations. They combine methods with a high level of theory and a small basis set with methods that employ lower levels of theory with larger basis sets...

.

### Density functional methods

Density functional theory (DFT) methods are often considered to be

*ab initio methods*Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...

for determining the molecular electronic structure, even though many of the most common

functionalIn mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...

s use parameters derived from empirical data, or from more complex calculations. In DFT, the total energy is expressed in terms of the total one-

electron densityIn quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either...

rather than the wave function. In this type of calculation, there is an approximate

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

and an approximate expression for the total electron density. DFT methods can be very accurate for little computational cost. Some methods combine the density functional exchange functional with the Hartree–Fock exchange term and are known as

hybrid functionalHybrid functionals are a class of approximations to the exchange-correlation energy functional in density functional theory that incorporate a portion of exact exchange from Hartree-Fock theory with exchange and correlation from other sources...

methods.

### Semi-empirical and empirical methods

Semi-empirical

quantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

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

HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

is not explicitly included. For π-electron systems, this was the

Hückel methodThe Hückel method or Hückel molecular orbital method proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of pi electrons in conjugated hydrocarbon systems, such as ethene,...

proposed by

Erich HückelErich Armand Arthur Joseph Hückel was a German physicist and physical chemist. He is known for two major contributions:*The Debye–Hückel theory of electrolytic solutions...

, and for all valence electron systems, the

extended Hückel methodThe extended Hückel method is a semiempirical quantum chemistry method, developed by Roald Hoffmann since 1963. It is based on the Hückel method but, while the original Hückel method only considers pi orbitals, the extended method also includes the sigma orbitals.The extended Hückel method can be...

proposed by

Roald HoffmannRoald Hoffmann is an American theoretical chemist who won the 1981 Nobel Prize in Chemistry. He currently teaches at Cornell University in Ithaca, New York.-Escape from the Holocaust:...

.

### Molecular mechanics

In many cases, large molecular systems can be modeled successfully while avoiding quantum mechanical calculations entirely.

Molecular mechanicsMolecular mechanics uses Newtonian mechanics to model molecular systems. The potential energy of all systems in molecular mechanics is calculated using force fields...

simulations, for example, use a single classical expression for the energy of a compound, for instance the

harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

. All constants appearing in the equations must be obtained beforehand from experimental data or

*ab initio* calculations.

The database of compounds used for parameterization, i.e., the resulting set of parameters and functions is called the

force fieldIn the context of molecular modeling, a force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles . Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical...

, 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.

These methods can be applied to proteins and other large biological molecules, and allow studies of the approach and interaction (docking) of potential drug molecules (e.g.

http://www.bio-balance.com/JMGM_article.pdf and

http://www.bio-balance.com/GPCR_Activation.pdf).

### Methods for solids

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 zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...

. 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

nuclearMolecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...

variables are

separatedIn mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....

(within the Born–Oppenheimer representation), in the time-dependent approach, the

wave packetIn physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...

corresponding to the nuclear

degrees of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

is propagated via the

time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...

operator (physics)In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

associated to the time-dependent

Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

(for the full

molecular HamiltonianIn atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...

). In the

complementaryIn physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...

energy-dependent approach, the time-independent Schrödinger equation is solved using the

scattering theoryIn mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

formalism. The potential representing the interatomic interaction is given by the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

s. In general, the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

s are coupled via the

vibronic couplingIn theoretical chemistry, the vibronic coupling terms, , are proportional to the interaction between electronic and nuclear motions of molecules. The term "vibronic" originates from the concatenation of the terms "vibrational" and "electronic"...

terms.

The most popular methods for propagating the

wave packetIn physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...

associated to the

molecular geometryMolecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...

are:

- the split operator technique,
- the Chebyshev (real) polynomial
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...

,
- the multi-configuration time-dependent Hartree
Multi-configuration time-dependent Hartree is a general algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles. MCTDH can thus determine the quantal motion of the nuclei of a molecular system evolving on one or...

method (MCTDH),
- the semiclassical method.

### Molecular dynamics

Molecular dynamics (MD) use either

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

,

Newton's laws of motionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

or a mixed model to examine the time-dependent behavior of systems, including vibrations or Brownian motion and reactions. MD combined with

density functional theoryDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...

leads to hybrid models.

## Interpreting molecular wave functions

The

atoms in moleculesThe atoms in molecules or atoms-in-molecules or quantum theory of atoms in molecules approach is a quantum chemical model that characterizes the chemical bonding of a system based on the topology of the quantum charge density...

model developed by

Richard BaderRichard F. W. Bader is a quantum chemist, noted for his work on the Atoms in Molecules approach. This approach explains chemistry, particularly established chemical models such as the Lewis structure, in terms of the topology of the electron density function in three-dimensional space.He was born...

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 pairA Lewis electron pair is a pair of electrons with opposite spins located in a molecule. The pair of electrons can comprise either a covalent bond, or a lone pair, localized in a mostly non-bonding molecular orbital....

s and the

valence bond modelIn chemistry, valence bond theory is one of two basic theories, along with molecular orbital theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds...

. Bader has demonstrated that these empirically useful models are connected with the

topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

of the quantum charge density. This method improves on the use of Mulliken population analysis.

## Software packages

There are many self-sufficient software packages used by computational chemists. Some include many methods covering a wide range, while others concentrating on a very specific range or even a single method. Details of most of them can be found in:

- Biomolecular modelling programs: proteins, nucleic acid.
- Molecular mechanics programs.
- Quantum chemistry and solid state physics software supporting several methods.
- Molecular design software
Molecular design software is software for molecular modeling, that provides special support for developing molecular models de novo.In contrast to the usual molecular modeling programs such as the molecular dynamics and quantum chemistry programs, such software directly supports the aspects related...

- Semi-empirical
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...

programs.
- Valence bond
VB computer programs for modern valence bond calculations:-* CRUNCH is by Gordon A. Gallup and his group.* GAMESS includes calculation of VB wave functions by the TURTLE code, due to J.H. van Lenthe....

programs.

## See also

## Other references

- C. J. Cramer
*Essentials of Computational Chemistry*, John Wiley & Sons (2002).
- T. Clark
*A Handbook of Computational Chemistry*, Wiley, New York (1985).
- R. Dronskowski
*Computational Chemistry of Solid State Materials*, Wiley-VCH (2005).
- F. Jensen
*Introduction to Computational Chemistry*, John Wiley & Sons (1999).
- K.I. Ramachandran, G Deepa and Krishnan Namboori. P.K.
*Computational Chemistry and Molecular Modeling Principles and applications* Springer-Verlag GmbH ISBN 978-3-540-77302-3.
- D. Rogers
*Computational Chemistry Using the PC, 3rd Edition*, John Wiley & Sons (2003).
- P. v. R. Schleyer (Editor-in-Chief).
*Encyclopedia of Computational Chemistry*. Wiley, **1998**. ISBN 0-471-96588-X.
- D. Sherrill. Notes on Quantum Mechanics and Computational Chemistry http://vergil.chemistry.gatech.edu/notes/.
- J. Simons
*An introduction to Theoretical Chemistry*, Cambridge (2003) ISBN 978-0521530477.
- A. Szabo, N.S. Ostlund,
*Modern Quantum Chemistry*, McGraw-Hill (1982).
- D. Young
*Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems*, John Wiley & Sons (2001).
- D. Young's Introduction to Computational Chemistry.

## External links