## Computational Chemistry

# Computational Chemistry

In 1929, shortly after the emergence of quantum mechanics, Paul Dirac made his famous statement that in principle the physical laws necessary to understand all of chemistry were at that point known—the only difficulty was that their application to chemical systems generally led to equations that were too difficult to solve. Consequently, at that time quantum principles could be rigorously applied only to simple atoms and molecules, such as H, He, H_{2}^{+}, and H_{2}.

During the 1930s the first approximate **quantum mechanical** methods for molecules were developed, leading to some success in modeling electronic behavior in many-electron molecules. For example, Eric Hückel developed a rudimentary molecular orbital model for the behavior of electrons in organic polyenes. During the late 1930s and early 1940s the first electronic digital computers were developed, and after World War II their use significantly expanded the application of classical and quantum mechanical principles to chemical problems.

A measure of the progress that computer technology spawned was the awarding of the Nobel Prize in chemistry in 1966 to Robert Mulliken for his creation of molecular orbital theory and its use in the calculation of the electronic structure of molecules. In his acceptance speech Mulliken announced the emergence of computational chemistry as a recognized specialty within the field of chemistry: "In conclusion, I would like to emphasize strongly my belief that the era of computing chemists, when hundreds if not thousands of chemists will go to the computing machine instead of the laboratory for increasingly many facets of chemical information, is already at hand."

The 1998 Nobel Prize in chemistry, awarded to Walter Kohn (who developed the **density-functional theory** ) and John Pople (who developed the use of computational methods in quantum chemistry) for their contributions to the advancement of computational chemistry, provided further evidence that the field had become a mature, reliable, and essential method of scientific inquiry. Today a vast literature exists, and computational chemistry has become an essential subject in the education of chemists at both the graduate and undergraduate levels.

The technical breadth of computational chemistry and the interdisciplinary character of its applications make the formulation of a concise definition challenging. Computational chemistry might be broadly defined as the attempt to model chemical and biochemical phenomena (structure, properties, reactivity, etc.) via computer-implementation of the theoretical principles of classical and quantum mechanical physics. Chemists and biochemists are interested in a wide array of molecules, including simple inorganic molecules, organic species of intermediate complexity, **transition metal** complexes with **heavy metal** ions, polymers, and biological macro-molecules, such as proteins and nucleic acids. In their studies of these diverse species they have a wide range of computational tools at their command. Therefore, assessing the strengths and weaknesses of these tools, and choosing the most appropriate method for the task at hand is a serious challenge for the computational chemist.

## Computational Quantum Chemistry

At the most fundamental level chemical phenomena are determined by the behaviors of **valence** electrons, which in turn are governed by the laws of quantum mechanics. Thus, a "first principles" or *ab initio* approach to chemistry would require solving Schrödinger's equation for the chemical system under study. Unfortunately, Schrödinger's equation cannot be solved exactly for molecules or multielectron atoms, so it became necessary to develop a variety of mathematical methods that made approximate computer solutions of the equation possible.

**The Hartree–Fock (HF) approximation.** The HF method is based on the Born–Oppenheimer and orbital approximations. Under the Born–Oppenheimer approximation the **nuclear** and electronic degrees of freedom of a molecule are decoupled, and the nuclei are held fixed while the electronic contribution to the energy is calculated. In the orbital approximation

the electrons occupy individual spin-orbitals, and as a consequence the *N* -electron Schrödinger equation is transformed into *N* one-electron equations. Both approximations facilitate computation, and the HF method proceeds by selecting a trial wave function (a molecular orbital formed as a linear combination of **atomic orbitals** , LCAO-MO) containing adjustable parameters, and subsequently solving a set of *N* coupled **integro-differential** equations through an iterative (self-consistent field) procedure.

**Post-Hartree–Fock (PHF) methods.** HF calculations for small to intermediate sized molecules generally yield reliable geometries, but fail to various degrees in predicting other important molecular properties. This is due to the **electron correlation error** introduced by the orbital approximation. PHF implementations introduce electron correlation into the calculation either by Møller–Plesset perturbation methods (MP2 and MP4), or by using wave functions based on many electron configurations (configuration interaction), rather than the **single Slater determinant** wave functions used in the HF method. These methods yield excellent results, but are computationally expensive (and good quantitative agreement with experiment comes at a price). As the level of theoretical rigor increases, the ability to interpret the results in terms of traditional chemical concepts decreases. For this reason, many chemists are willing to sacrifice quantitative agreement for qualitative, conceptual understanding.

**Semiempirical quantum mechanics.** The computational effort in *ab initio* calculations increases as the fourth power of the **size of the basis set** , and, therefore, its application to large molecules is expensive in terms of time and computer resources. Consequently, semiempirical methods treating only the valence electrons, in which some integrals are ignored or replaced by empirically based parameters, have been developed. The various semiempirical parameterizations now in use (MNDO, AM1, PM3, etc.) have greatly increased the molecular size that is accessible to quantitative modeling methods and also the accuracy of the results.

**Density functional theory (DFT).** DFT is an alternative to the HF method, in which the fundamental role is played by the electron density ψ^{2}, rather than the wave function. The basis for this method is a proof by Pierre Hohenberg and Kohn that all physical properties of a molecule are completely determined by its electron density. The computational savings that DFT offers come from the fact that the wave function of an *n* -electron molecule depends on 3*n* spatial coordinates, whereas the electron density depends on just three spatial coordinates. Consequently, DFT calculations generally scale as the third power of the size of the basis set, rather than the fourth power of the HF methods.

## Classical Computational Methods

Many chemical and biochemical systems of interest are too large for analysis with quantum mechanical methods, either *ab initio* or semiempirical. However, some of their properties may be modeled by classical or semi-classical methods. Classical computational methods do not provide electronic structure information.

**Molecular mechanics (MM).** MM is a nonquantum mechanical method of computing molecular structures and properties that treats a molecule as a flexible collection of atoms held together by chemical bonds. The method minimizes the molecular potential energy, which is generally calculated classically in terms of internal degrees of freedom such as bond lengths, bond angles, dihedral angles, and electrostatic and van der Waals nonbonding interactions. The MM minimization depends on an empirically-based parameterization scheme and is able to handle molecules with thousands of atoms. The MM method has been highly developed, especially by researchers in the pharmaceutical industry.

**Molecular dynamics (MD).** Within the MD method of computation, atomic and molecular trajectories are generated by the numerical integration of Newton's laws of motion. This requires specification of initial conditions and a knowledge of the forces acting on all constituents, which can be obtained either algebraically or numerically from a previously calculated potential energy surface. MD has important application in a variety of chemical disciplines, but is especially useful in biochemistry in the study of protein-folding, in probing alternative minimal energy states of macromolecules, and in other areas, such as enzyme-substrate docking.

**Monte Carlo simulations (MC).** MC computational methods are used to solve a wide variety of problems in mathematics and the natural sciences, including the evaluation of integrals, the solution of differential equations, and the modeling of physical phenomena. Unlike MD, which is **deterministic** , MC (as its name suggests) is based on the generation of random changes in the variables of a system, followed by reliance on some criterion for deciding whether the changes lead to a valid or significant new state of the system. For example, MC can be used to calculate bulk thermodynamic quantities by generating random changes in the positions of the atoms in an ensemble of molecules. The energy of the new arrangement is then calculated and its importance evaluated using the Boltzmann factor, ê(-E/kT). If this new arrangement passes the test, it is included (properly weighted, statistically) in the **manifold of ensemble states** that are used to calculate the average thermodynamic properties of the system being modeled.

**Quantum mechanics/molecular mechanics hybrid method (QM/MM).**

MM is applicable to macromolecules such as enzymes, but cannot model the bond-breaking and bond-making that occurs within enzyme-substrate complexes at active sites. The solution is provided by a QM/MM hybrid approach, in which QM is used to model the active site and MM models are used for the rest of the enzyme structure.

Advances in computer technology and improved algorithm efficiency have greatly increased the size and range of chemical systems to which these computational methodologies can be applied. In addition, the increased availability of commercial software with efficient user interfaces for preparing input and analyzing output has made computational methods accessible to a rapidly growing number of chemists.

see also Molecular Orbital Theory; Quantum Chemistry; Theoretical Chemistry.

*Frank Rioux*

## Bibliography

Cramer, Christopher J. (2002). *Essentials of Computational Chemistry: Theories and Models.* New York: Wiley.

Hehre, Warren J.; Radom, Leo; Schleyer, Paul V. R., et al. (1986). *Ab Initio Molecular Orbital Theory.* New York: Wiley.

Leach, Andrew R. (2001). *Molecular Modelling: Principles and Applications*, 2nd edition. New York: Prentice Hall.

Lipkowitz, Kenny B., and Boyd, Donald B., eds. (1990). *Reviews in Computational Chemistry.* New York: VCH.

Schleyer, Paul V. R.; Allinger, N. L.; and Clark, T., et al., eds. (1998). *Encyclopedia of Computational Chemistry.* New York: Wiley.

### Internet Resources

WWW Computational Chemistry Resources. Available from <http://www.chem.swin.edu.au/chem_ref.html>.

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