Theoretical chemistry

Theoretical chemistry is the application of non-experimental (usually mathematical or computer simulation ) methods to explain or predict chemical phenomena . Therefore, it deals primarily with the development or further development of methods by which the chemical and physical properties of matter can be calculated, as well as their computer technical implementation through programs mostly in Fortran or C . The developed computer programs are then used within the framework of computer chemistry not only by representatives of theoretical chemistry, but also by other areas of chemistry to support the solution of chemical problems. New approaches to the interpretation of results are also researched.

Ab initio methods

Methods based on the Schrödinger equation or its relativistic extension ( Dirac equation ) contain only natural constants as parameters and are therefore referred to as ab initio methods. These methods are scientifically best founded, but can only be applied to relatively small systems with relatively few atoms if quantitatively accurate results are desired. This is due to the fact that the Schrödinger or Dirac equation can only be solved analytically for some trivial one-electron systems and otherwise approximate solutions are necessary, which, however, quickly take up too much computing power with increasing system size. Computer programs for the ab initio calculation of chemical structures are for example GAUSSIAN and GAMESS .

The Schrödinger equation is an eigenvalue equation (partial differential equation) and has the following form:

{\ displaystyle {\ text {Operator}} \ cdot {\ text {Eigenfunction}} = {\ text {Eigenvalue}} \ cdot {\ text {Eigenfunction}}}

.

In the time-independent case, this consists of the so-called Hamilton operator Ĥ, the wave function and the total energy E of the system, where ${\ displaystyle \ Psi}$

{\ displaystyle {\ hat {H}} \ Psi = E \ Psi}

applies. The (well-known) Hamilton operator describes the kinetic and potential energies of the particles involved ( electrons and atomic nuclei ). It acts on the (unknown) wave function . The absolute square of , is interpreted as the probability density of the particles involved in the system. If it is known, all properties of a system can be calculated relatively easily as expected values using the operator assigned to the respective property. ${\ displaystyle \ Psi}$ ${\ displaystyle \ Psi}$ ${\ displaystyle \ Psi \ Psi ^ {*}}$ ${\ displaystyle \ Psi}$

Born-Oppenheimer approximation

Due to the large difference in mass between electrons and atomic nuclei, the movement of the atomic nuclei can usually be separated in a very good approximation, whereby after further separation of the translation of the overall system, a Kernschrödinger equation results, which describes the oscillation and rotation of the system, for example a molecule . What remains is the electronic Schrödinger equation, which can be solved point by point for fixed atomic nuclei. The resulting (electronic) energies flow into the Kernschrödinger equation.

Qualitatively, this means that the electrons move in the potential of the atomic nucleus and instantly adjust to changes in the nucleus geometry. This approximation finds its limit where a small change in the core geometry is associated with a large change in the electronic structure. Such situations are occasionally found in certain geometries of tetratomic or even larger molecules, especially when these are in an electronically excited state.

The Born-Oppenheimer approximation enables the idea that molecules have an equilibrium geometry around which their atoms then oscillate. Mathematically, the Born-Oppenheimer approximation means that in the electronic Schrödinger equation the term for the kinetic energy of the nuclei is set to zero and the term for the potential energy of the nucleus-nucleus interaction becomes a parameter that is determined by Coulomb's law and the chosen arrangement of the atomic nuclei is determined.

The solution of the overall Schrödinger equation, which also includes the kinetic energy of the nuclei, is only practically possible for the smallest system, the hydrogen molecule. Instead, the electronic Schrödinger equation is first solved in two steps on selected core geometries, then an analytical approximate solution of the energy hypersurface is sought based on this data and this is then inserted into the Kernschrödinger equation. It has to be said that the Kernschrödinger equation is solved comparatively seldom, because for time reasons one has to limit oneself to a few degrees of freedom of the system (a system of N atoms has 3N-6 degrees of freedom, linearly arranged molecules 3N-5) and there are also possibilities gives to calculate quantities such as the oscillation frequencies of molecules in other ways in good approximation (see below under calculation of physical properties)

Hartree-Fock method

The Hartree-Fock calculation is an ab initio method of theoretical chemistry to approximately calculate properties of multi-electron systems that can no longer be analytically solved. This is named after Douglas Rayner Hartree and Wladimir Alexandrowitsch Fock . With this method, the wave function is applied approximately as a determinant of one-electron functions (the so-called orbitals ), whereby these in turn are usually applied as a linear combination of so-called (usually atom-centered) basis functions with unknown coefficients. The expected energy value of this determinant wave function is minimized using Ritz's principle of variation , with the coefficients of the basis functions acting as the variation parameters. The solution of the Hartree-Fock equation is ultimately reduced to the calculation of integrals over the basis functions and the diagonalization of a matrix. These arithmetic operations can be solved very efficiently with computers in contrast to the solution of differential equations of several variables. In so-called open shell systems, instead of a determinant, linear combinations of determinants with coefficients specified by symmetry can also occur in order to respect symmetry.

In the Hartree-Fock method, the electrons move in the mean field of the other electrons. By using the averaged potential, however, the electron correlation , i.e. the exact interaction of the electrons with one another, is neglected. Therefore, the Hartree-Fock energy never reaches the exact value, even if an infinitely large basis set were used (so-called Hartree-Fock limit).

Multi-configuration SCF wave functions

In some cases it is not sufficient to use the wave function as one determinant (or with several but with fixed coefficients determined by symmetry) in order to capture the system qualitatively correctly. One speaks of so-called multi-reference systems. Instead, the relevant determinants must be identified which are necessary for a qualitatively correct description of the system and whose coefficients must be optimized together with the orbitals. Such a procedure is often necessary when describing electronically excited states, since here a determinant often cannot provide a qualitatively correct description. However, there are also molecules whose ground state can only be described qualitatively correctly by several determinants, e.g. B. biradical systems. The targeted selection of these determinants turns out to be difficult and also depends to a certain extent on the geometry of the molecule under consideration. Therefore, often not individual determinants, but first of all the particularly relevant orbitals are considered, for example the (energetically) highest occupied and lowest unoccupied, in the simplest case HOMO and LUMO . All determinants are then taken into account that can be generated by replacing the (considered) occupied orbitals with those that are not occupied. This approach is called Complete Active Space Self Consistent Field (CASSCF). Due to the exponentially growing number of determinants to be taken into account, the maximum number of orbitals to be taken into account is limited to approx. 12-16.

Correlated calculations

The accuracy of the Hartree-Fock or MCSCF / CASSCF solutions is usually not high enough, so that a correlated calculation is then usually carried out. MCSCF equation resulting unoccupied orbitals are used. The number of calculated orbitals corresponds to the number of basic functions used and is usually significantly greater than the number of occupied orbitals that are part of the Hartree-Fock or MCSCF wave function. In principle, in correlated methods, the wave function is applied as a linear combination of determinants with the Hartree-Fock wave function as the leading determinant (large coefficient). Further determinants are formed by replacing occupied orbitals with unoccupied orbitals (so-called excitations).

In the case of correlated methods, in the single-reference case either the perturbation theory ( Moller-Plesset approach ), configuration interaction or coupled cluster (CC) approaches are used, in multi-reference methods either the multi-reference configuration interaction MRCI method, the multi-reference method Reference perturbation theory or multi-reference coupled cluster approaches.

As a rule, only single and double excitations with regard to the reference wave function are taken into account in all methods, whereby, due to the nature of the coupled cluster approach, certain classes of higher excitations are also taken into account. In the coupled cluster approach, this is referred to as CCSD (S for single, D for double according to the number of excitations in the approach). The coupled cluster approach was developed by Hermann Kümmel and Fritz Coester in nuclear physics at the end of the 1950s and applied in quantum chemistry from the 1960s ( Jiři Čížek , Josef Paldus ).

In the configuration interaction method, first of all, with regard to the reference wave function (SCF or MCSCF / CASSCF), e.g. B. all single and double excitations are generated by replacing one or two occupied orbitals with the corresponding number of unoccupied orbitals. The CI wave function is applied as a linear combination of all these determinants and the corresponding (CI) coefficients of the determinants are determined so that the resulting energy is minimal (as negative as possible). Usually only single and double excitations are taken into account. A special case is the so-called full CI, in which all possible suggestions are generated. Full-CI calculations are so complex that they can only be performed as a benchmark for small systems. The MRCI (SD) method is considered to be very precise both for the properties and for the absolute energy of the ground state and energy differences to electronically excited states. Single reference CI methods, especially CI (SD), on the other hand, are considered imprecise due to the lack of size consistency. For example, size consistency would mean that two hydrogen molecules that are very far apart from each other deliver the same total energy as twice the energy calculated for a single hydrogen molecule. Due to the lack of triple and quadruple excitations in the first case, the result in the second case is much more negative. In principle, this also applies to MRCI (SD), but the multi-reference approach compensates for a large part of the error.

In the case of perturbation calculations, the Hamilton operator is split up as the sum of an undisturbed operator and a perturbation operator , whereby the perturbation should be "small". The eigenvalue solutions of the undisturbed operator are known. In the case of correlated perturbation calculations, the operator who has the Hartree-Fock or the MCSCF / CASSCF wave functions for the solution is used. then results as the difference to the real Hamilton operator of the system. The MP2 or MP4 (SDQ) methodology is often used in the single reference case, the so-called CASPT2 method (with a CASSCF wave function as the reference wave function) in the multi -reference case , and more rarely the CASPT3 method. ${\ displaystyle {\ hat {H}}}$ ${\ displaystyle {\ hat {H}} ^ {0}}$ ${\ displaystyle {\ hat {H}} ^ {1}}$ ${\ displaystyle {\ hat {H}} ^ {0}}$ ${\ displaystyle {\ hat {H}} ^ {0}}$ ${\ displaystyle {\ hat {H}} ^ {1}}$

When solving the perturbation equation, the result is that the perturbed portion is further split into a wave function of the first order, second order, etc., the total wave function being the sum of the undisturbed and the various perturbed wave functions. The computational effort increases significantly with each additional correction. It is not necessary, however, that the series converges to the exact result; This means that there is no guarantee that the calculated wave function and the associated energy / properties will always get better with increasing effort. In fact, oscillations around the exact value or a divergence of the results are sometimes observed.

In the coupled cluster approach, the wave function is represented as. On the one hand, this guarantees the size consistency of the method and, on the other hand, means that certain higher types of excitation are also recorded. Coupled cluster calculations of the CCSD (T) type are considered to be very accurate. The wave function is not available here in a closed form, so that the properties have to be calculated in a different way, for which there are corresponding procedures. ${\ displaystyle \ psi = \ exp (T) \ psi (0)}$

The Configuration Interaction method is variational, i.e. the calculated energy is always higher than the exact energy. However, this does not apply to perturbation theory or the coupled cluster approach. However, in contrast to the CI method, perturbation theory and the coupled cluster approach are consistent in size. Size consistency means that the energy resulting from the calculation of a supersystem of two (identical) molecules that practically do not interact with one another due to the large selected distance must be equal to twice the calculated energy of a single molecule. Due to this shortcoming of the CI method, nowadays hardly any CI calculations are based on (single reference) Hartree-Fock calculations, whereas MRCI (SD) calculations are considered to be very precise. Full-CI calculations, in which all possible suggestions in the orbital space regarding the SCF or MCSCF function are taken into account, are, as a special case, both variational and consistent in size, but by far the most expensive.

The effort involved in performing correlated methods does not increase linearly with the size of the molecule, but rather lies between and with the methods commonly used , where N is a measure of the size of the molecule (e.g. the number of basis functions). This can be attributed to the delocalizing of the orbitals, i.e. H. are more or less spread over the whole molecule. However, the orbitals resulting from the Hartree-Fock calculation can be localized relatively well using various methods. Correlation methods using these localized orbitals promise a significant reduction in the above scaling behavior with molecular size and are being intensively researched. The core problem here is that really local orbitals have to be "cut", which means that local orbitals are not strictly orthogonal to one another, so the integral over two such orbitals is slightly different from zero. ${\ displaystyle N ^ {5}}$ ${\ displaystyle N ^ {7}}$

In addition to the molecular orbital-based methods discussed here, correlated calculations can also be represented using valence structure theory (VB theory) (e.g. VBSCF, BOVB). This has the advantage that the results are easy to interpret chemically.

Semi-empirical methods

The so-called semi-empirical methods generally introduce approximations at the level of the Hartree-Fock matrix by either neglecting certain quantities (integrals) in the matrix equation or replacing them with parameters. The parameters are either defined by experimentally determined values or fitted to a training set. Such a training set usually consists of quantities (determined experimentally or with very good calculation methods) such as bond lengths, dipole moments etc. of a series of molecules that are to be reproduced as well as possible by varying the free parameters. Semiempirical methods can treat systems with (at least) several hundred atoms. Since the empirical parameters can implicitly take into account correlation effects, semiempirical methods should not only be faster but also more accurate than Hartree-Fock calculations. However, the expected accuracy depends heavily on how similar the molecule to be calculated is to the training set. Typically a minimal basis set with Slater-like basis functions is used and only the valence electrons are explicitly taken into account. Therefore, the modeling of the nucleus-nucleus repulsion, which must also parametrically consider the interaction of the electrons of the inner shells, is of great importance. The semi-empirical model is determined by the approximations used (i.e. which integrals are neglected) and how the remaining integrals are evaluated / parameterized. The three most common models are complete neglect of differential overlap (CNDO), intermediate neglect of differential overlap (INDO) and neglect of differential diatomic overlap (NDDO).

The simplest semiempirical approach is the Hückel approximation , since it does not calculate any integrals. However, it is only applicable to π-electron systems. The theory was later extended to σ systems (Extended Hückel Theory, EHT).

Density functional

The density functional theory (DFT) makes use of the fact that the description of the ground state requires the electron density, independent of the number of electrons, as a function of only three position variables; if necessary, the spin density is added as a further variable . The basis is the Hohenberg-Kohn theorem . The total energy of the system is principally linked to the density via a functional . However, there is the problem that the exact functional is not known for most applications and approximations are therefore necessary. In practice, therefore, the choice of a suitable approximated functional is crucial for accuracy. Most modern DFT methods are based on the Kohn-Sham approach, which is based on the use of a Slater determinant. Today's density functionals achieve the accuracy of simple correlated ab initio methods (such as second-order perturbation theory) and can be used for systems with up to approx. 1000 atoms. DFT calculations are often used to optimize the geometry of molecules. The systematic improvement is less pronounced than with ab initio methods, but with the so-called Jacob's Ladder a kind of hierarchy of DFT methods has been established. With regard to the application of quantum chemical methods to chemical problems, the density functional theory is of considerable importance. It is now widely used in industrial and academic research. This was recognized not least by the awarding of the Nobel Prize for Chemistry to Walter Kohn in 1998.

Force fields

With the so-called force field methods, on the other hand, one falls back on a classic way of thinking, according to which the atoms in molecules are connected to one another by small springs with certain spring constants, which also describe the change in binding and torsion angles (ball-spring model) Generally speaking, force fields are a parameterization of the potential energy . The fact that a distinction is made between different configurations and even conformations in chemistry shows that the atomic nuclei do not have very significant quantum properties in relation to chemical problems. This is used in molecular mechanical or force field methods. These methods are particularly suitable for very large (bio) molecules that cannot be mastered with other methods and are mainly used to optimize their geometry or for dynamic simulations. However, corresponding parameters (spring constants) must be determined for a large number of possible atom combinations (two for bonds, three for bond angles, four for torsion angles). (Partial) charges on atoms and their electrostatic interaction with each other are also taken into account. The various force field methods differ in the functional form of the interaction and the way in which the parameters are determined.

A common prejudice among experts in theoretical chemistry is that the description of bond breaks using force field methods is inherently problematic or even impossible. This prejudice is neither substantively nor historically correct. It is based on two frequently used but arbitrary and unrealistic model assumptions: (1) The modeling of a bond by a spring (harmonic oscillator); this can be replaced by a more realistic modeling with a spring, which can also tear if it is stretched more strongly (anharmonic oscillator, e.g. Morse oscillator); (2) The simplifying assumption that atoms do not change the type and number of their nearest neighbors (definition of so-called atom types); this assumption can be dropped without replacement. The additional objection that is often raised that a bond break cannot be described in a classical mechanical way, but only in a quantum mechanical way, because of the essential correct description of the electrons, is correct, but at the same time irrelevant. In a force field of the kind referred to here, there is no explicit description of electrons and also no classical mechanics; it is only an approximation function for the results of a quantum mechanical treatment of the electrons. If the more realistic modeling mentioned is used, a reactive force field is obtained. Some of the much-cited first papers on classical molecular dynamics (see below) already used reactive force fields. This was somewhat forgotten due to the later, widespread use of some non-reactive force fields.

Non-reactive force field methods can answer many conformational questions; chemical reactions can also be described with reactive force fields. Force fields are used, for example, in classical molecular dynamics .

Calculation of physical properties

A quantum chemical operator is assigned to the physical properties of a system such as its dipole moment . If the wave function is known , the property can be calculated as an expected value using the operator, i.e. the dipole moment as . In addition, the property can also be determined as a single or multiple derivative of the electronic energy of the system according to certain quantities that depend on the physical property. The latter method can also be used if the wave function is not explicitly known ( e.g. with the coupled cluster approach ) and, in contrast to the first method, is not limited to the equilibrium geometry. ${\ displaystyle \ mu}$ ${\ displaystyle \ Psi}$ ${\ displaystyle \ int \ Psi ^ {*} \ mu \ Psi \ mathrm {d} \ tau}$

Geometry optimization

Since the solution of the Schrödinger equation within the Born-Oppenheimer approximation is only possible point by point, i.e. for discrete geometries, and a sufficiently precise solution for a geometry is already associated with a high computational effort, a sub-branch of theoretical chemistry deals with the establishment of algorithms with which excellent geometries can be found with as little computational effort as possible. Excellent geometries are, for example, the equilibrium geometry (energetic minimum) and, in chemical reactions, the transition state as a saddle point on the reaction coordinate. The energy difference between educts and transition state determines the activation energy of the reaction, the energy difference between educts and products determines the reaction energy. Methods are often used in which, in addition to the energy at a point, its first derivative is calculated and the second derivative is estimated.

Simulation of chemical reactions

To simulate chemical reactions, an analytical representation of the energy hypersurface (s) involved in the relevant area of the possible geometries of the system under consideration (e.g. molecule) is generally necessary, i.e. an analytical function that determines the energy of the system depending on its geometry reproduces. For this purpose, the associated energy is calculated for each surface on certain distinguished geometries and, based on this, an approximate analytical representation of the surface is determined, for which there are different approaches. Since the number of internal degrees of freedom of a system consisting of N atoms is 3N-6 (linear molecules 3N-5), a complete energy hypersurface, i.e. one that takes all degrees of freedom of the system into account, can only be calculated for three to a maximum of four-atom molecules become. In the case of larger systems, a selection of the relevant geometry parameters (i.e. usually certain bond lengths, angles or torsion angles) must be made, with the values of the remaining geometry parameters being energetically optimized for the geometries awarded. After the surfaces are in analytical form, the Kernschrödinger equation can be solved and the progress of the chemical reaction can be simulated on the computer. There are also various approaches to this. An illustrative example was z. B. the simulation of the Miller-Urey experiment using ab initio nanoreactors.

Qualitative explanatory schemes

Especially in the early days of theoretical chemistry, a number of explanatory schemes were set up with the help of which various aspects could be explained qualitatively. One example is the so-called VSEPR theory , which can be used to predict the geometry of simple molecules with a central atom. But new concepts have also been developed recently, such as the electron localization function (ELF) or the topological concept by Richard Bader (atoms in molecules 1990). ELF is a method to make chemical bonds visible. It is based on the pair density of two electrons with the same spin (same spin pair probability density). Locations with a low pair density are associated with a high locality of an electron and topologically with a chemical bond. ELF can be practically calculated at HF and DFT level. The Bader method uses the first and second derivative of the electron density according to the spatial coordinates to create a connection with intuitive ideas such as chemical bonds.

literature

Markus Reiher, Paulo A. Netz: What is the meaning of theoretical concepts in chemistry? In: Chemistry in Our Time . tape 33 , no. 3 , 1999, p. 177-185 , doi : 10.1002 / ciuz.19990330312 .
MP Allen, DJ Tildesly: Computer Simulation of Liquids . Oxford University Press, 1989, ISBN 0-19-855645-4
PW Atkins, RS Friedman: Molecular Quantum Mechanics . 4th ed., Oxford University Press, Oxford 2004, ISBN 0-19-927498-3 .
CJ Cramer Essentials of Computational Chemistry , Wiley, Chichester 2002, ISBN 0-471-48552-7 .
F. Jensen: Introduction to Computational Chemistry . John Wiley & Sons, Chichester 1999, ISBN 978-0-471-98425-2 .
W. Kutzelnigg: Introduction to Theoretical Chemistry . Wiley-VCH, Weinheim 2002, ISBN 3-527-30609-9 .
AR Leach Molecular Modeling. Principles and Applications . 2nd ed., Pearson Prentice Hall, Harlow 2001, ISBN 0-582-38210-6 .
E. Lewars: Computational Chemistry. Introduction to the Theory and Applications of Molecular and Quantum Mechanics. Kluwer Academic Publishers, New York, Boston, Dordrecht, London, Moscow, 2004. ISBN 1-4020-7285-6 . E-book ISBN 0-306-48391-2
J. Reinhold: quantum theory of molecules . 3rd edition, Teubner, 2006, ISBN 3-8351-0037-8 .
A. Szabo, NS Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . McGraw-Hill, 1989, ISBN 0-07-062739-8

Individual evidence

↑ M. Born, R. Oppenheimer: To the quantum theory of molecules . In: Annals of Physics . tape 389 , no. 20 , 1927, ISSN 1521-3889 , pp. 457-484 , doi : 10.1002 / andp.19273892002 .

↑ Björn O. Roos, Peter R. Taylor, Per EM Sigbahn: A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach . In: Chemical Physics . tape 48 , no. 2 , May 15, 1980, ISSN 0301-0104 , p. 157-173 , doi : 10.1016 / 0301-0104 (80) 80045-0 .

^ Chr. Møller, MS Plesset: Note on an Approximation Treatment for Many-Electron Systems . In: Physical Review . tape 46 , no. 7 , October 1, 1934, p. 618-622 , doi : 10.1103 / PhysRev.46.618 .

↑ Jiří Čížek: On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell ‐ Type Expansion Using Quantum ‐ Field Theoretical Methods . In: The Journal of Chemical Physics . tape 45 , no. 11 , December 1, 1966, ISSN 0021-9606 , p. 4256-4266 , doi : 10.1063 / 1.1727484 ( scitation.org [accessed October 11, 2019]).

↑ Kerstin. Andersson, Per Aake. Malmqvist, Bjoern O. Roos, Andrzej J. Sadlej, Krzysztof. Wolinski: Second-order perturbation theory with a CASSCF reference function . In: The Journal of Physical Chemistry . tape 94 , no. July 14 , 1990, ISSN 0022-3654 , pp. 5483-5488 , doi : 10.1021 / j100377a012 .

↑ JH van Lenthe, GG Balint-Kurti: The valence-bond self-consistent field method (VB-SCF): Theory and test calculations . In: The Journal of Chemical Physics . tape 78 , no. 9 , May 1, 1983, ISSN 0021-9606 , p. 5699-5713 , doi : 10.1063 / 1.445451 .

↑ Philippe C. Hiberty, Sason Shaik: Breathing-orbital valence bond method - a modern valence bond method did includes dynamic correlation . In: Theoretical Chemistry Accounts . tape 108 , no. 5 , November 1, 2002, ISSN 1432-2234 , p. 255-272 , doi : 10.1007 / s00214-002-0364-8 .

^ Sason S. Shaik, Philippe C. Hiberty: A Chemist's Guide to Valence Bond Theory . John Wiley & Sons, 2007, ISBN 978-0-470-19258-0 .

↑ ^a ^b Walter Thiel: Semiempirical quantum chemical methods . In: Wiley Interdisciplinary Reviews: Computational Molecular Science . tape 4 , no. 2 , March 1, 2014, ISSN 1759-0884 , p. 145–157 , doi : 10.1002 / wcms.1161 .

↑ P. Hohenberg, W. Kohn: Inhomogeneous Electron Gas . In: Physical Review . tape 136 , 3B, November 9, 1964, pp. B864-B871 , doi : 10.1103 / PhysRev.136.B864 .

^ W. Kohn, LJ Sham: Self-Consistent Equations Including Exchange and Correlation Effects . In: Physical Review . tape 140 , 4A, November 15, 1965, pp. A1133 – A1138 , doi : 10.1103 / PhysRev.140.A1133 .

↑ John P. Perdew, Karla Schmidt: Jacob's ladder of density functional approximations for the exchange-correlation energy . In: AIP Conference Proceedings . tape 577 , no. 1 , July 6, 2001, ISSN 0094-243X , p. 1-20 , doi : 10.1063 / 1.1390175 .

↑ Axel D. Becke: Perspective: Fifty years of density-functional theory in chemical physics . In: The Journal of Chemical Physics . tape 140 , no. 18 , April 1, 2014, ISSN 0021-9606 , p. 18A301 , doi : 10.1063 / 1.4869598 .

^ Norman L. Allinger: Force Fields: A Brief Introduction . In: Encyclopedia of Computational Chemistry . American Cancer Society, 2002, ISBN 978-0-470-84501-1 , doi : 10.1002 / 0470845015.cfa007s .

↑ Lee-Ping Wang, Alexey Titov, Robert McGibbon, Fang Liu, Vijay S. Pande: Discovering chemistry with an ab initio nanoreactor . In: Nature Chemistry . tape 6 , no. December 12 , 2014, ISSN 1755-4349 , p. 1044-1048 , doi : 10.1038 / nchem.2099 ( nature.com [accessed December 31, 2019]).

^ AD Becke, KE Edgecombe: A simple measure of electron localization in atomic and molecular systems . In: The Journal of Chemical Physics . tape 92 , no. 9 , May 1, 1990, ISSN 0021-9606 , pp. 5397-5403 , doi : 10.1063 / 1.458517 .

^ Richard FW Bader: A quantum theory of molecular structure and its applications . In: Chemical Reviews . tape 91 , no. 5 , July 1, 1991, ISSN 0009-2665 , pp. 893-928 , doi : 10.1021 / cr00005a013 .

Web links

Wiktionary: theoretical chemistry - explanations of meanings, word origins, synonyms, translations

lectures4you.de: Compilation of freely accessible online teaching material on theoretical chemistry

[1] M. Born, R. Oppenheimer: To the quantum theory of molecules . In: Annals of Physics . tape 389 , no. 20 , 1927, ISSN 1521-3889 , pp. 457-484 , doi : 10.1002 / andp.19273892002 .

[2] Björn O. Roos, Peter R. Taylor, Per EM Sigbahn: A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach . In: Chemical Physics . tape 48 , no. 2 , May 15, 1980, ISSN 0301-0104 , p. 157-173 , doi : 10.1016 / 0301-0104 (80) 80045-0 .

[3] Chr. Møller, MS Plesset: Note on an Approximation Treatment for Many-Electron Systems . In: Physical Review . tape 46 , no. 7 , October 1, 1934, p. 618-622 , doi : 10.1103 / PhysRev.46.618 .

[4] Jiří Čížek: On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell ‐ Type Expansion Using Quantum ‐ Field Theoretical Methods . In: The Journal of Chemical Physics . tape 45 , no. 11 , December 1, 1966, ISSN 0021-9606 , p. 4256-4266 , doi : 10.1063 / 1.1727484 ( scitation.org [accessed October 11, 2019]).