Theoretical chemistry
Theoretical chemistry is the application of non-experimental (usually mathematical or computer simulation technology ) 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 computational implementation through programs usually in Fortran or C. The computer programs developed are then in the context of computational chemistry used not only by representatives of theoretical chemistry, but also from other areas of chemistry to assist in solving chemical problems. Subordinate new approaches to interpretation of results are explored.
Ab initio methods
Methods posed by the Schrödinger equation or its relativistic extension ( Dirac equation ), included as a parameter only fundamental constants, and are therefore referred to as ab initio methods. These methods are scientifically sound best, 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 be solved analytically only for some trivial and otherwise Einelektronensysteme approximate solutions are needed, which quickly take with increasing scope of the system, however, a too high computational power to complete. Computer programs for ab initio calculation of chemical structures, for example, GAUSSIAN and GAMESS.
The Schrödinger equation is an eigenvalue equation ( partial differential equation) and has the following form:
.
The time-independent case, it consists of the so-called Hamiltonian H of the wave function and the total energy E of the system,
Applies. The (known) Hamiltonian here describes the kinetic and potential energies of the participating particles (electrons and nuclei ). It acts on the (unknown ) wave function. The amount of square, that is interpreted as the probability density of the particles of the system involved. If it is known, all the properties of a system can be calculated relatively simply as the expected value over the property of the respective associated operator.
Born- Oppenheimer approximation
Due to the large mass difference between electrons and atomic nuclei movement of the nuclei can be removed in most cases a very good approximation, which, after further separation of the translation of the entire system, a core Schrödinger equation results that the vibration and rotation of the system, for example a molecule which describes. What remains is the electronic Schrödinger equation can be solved for fixed atomic core layers pointwise. Any resulting (electronic) energy to the nuclear Schrödinger equation with.
Qualitatively, this means that the electrons move in the potential of the atomic nuclei, and adjust instantaneously to changes in the core geometry. Its limit, this approximation is, where a small change in the geometry of the core is connected to a large change in the electronic structure. Such situations are occasionally found in certain geometries of four-atom molecules or even larger, particularly if they are located in an electronically excited state.
The Born- Oppenheimer approximation allows only the idea that molecules possess an equilibrium geometry to swing their atoms then. Mathematically, the Born- Oppenheimer approximation, that in the electronic Schrödinger equation, the term for the kinetic energy of the nuclei is set equal to zero and the term for the potential energy of the nucleus-nucleus interaction to a parameter is defined by Coulomb's law and the selected arrangement of the atomic nuclei is determined.
The solution of the Schrödinger equation overall, which includes the kinetic energy of the nuclei is the smallest system, the hydrogen molecule practically possible. In two steps, first the electronic Schrödinger equation is solved instead of selected core geometries, then searched based on these data, an analytical approximate solution of the energy surface and then inserted into the nuclear Schrödinger equation. It must be said that the nuclear Schrödinger equation is comparatively rarely solved because you have to be limited due to time a few degrees of freedom of the system ( a system of N atoms has 3N -6 vibrational degrees of freedom, linearly arranged molecules 3N -5) and it is also possible is to calculate quantities such as vibrational frequencies of molecules in a different way to a good approximation (see below for calculation of physical properties )
Hartree- Fock method
In Hartree- Fock calculation refers to a ab initio method of theoretical chemistry to calculate properties of many-electron systems that are no longer analytically solvable approximation. Named Douglas Rayner Hartree this are and Vladimir Aleksandrovich Fock. In this method, the wave function is approximated recognized as a determinant of Einelektronenfunktionen ( the so-called orbitals ), which in turn are recognized generally as a linear combination of several (typically atom -centered ) basis functions with unknown coefficients. The solution of the Hartree -Fock equation is ultimately attributed 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 as opposed to the solution of differential equations of several variables. In so-called open- shell systems can appear to respect the symmetry rather than a determinant also linear combinations of determinants with predetermined by symmetry coefficients.
Multi-configuration SCF wave functions
In some cases it is not sufficient (but respectively with a plurality of symmetry determined by fixed coefficients ) to be set, the wave function as a determinant in order to detect the system qualitatively. Instead, the relevant determinants must be identified and whose coefficients are optimized together with the orbitals. Such an approach is often required in the description of electronically excited states. The specific selection of these determinants, however, designed to be difficult and also depends to some extent on the considered geometry of the molecule. Therefore often not individual determinants, but particularly relevant orbitals are considered, for example, the (energetic ) highest occupied and lowest unoccupied HOMO and LUMO. Furthermore, all of the determinants are considered that occupied by replacing the ( viewed ) can be produced by the orbitals of the unoccupied. This approach is referred to as Complete Active Space Self Consistent Field ( CASSCF ). Due to the exponentially growing number of determinants to be considered the maximum number of such orbitals considered to be limited to about 12-16.
Correlated calculations
The accuracy of the Hartree -Fock or MCSCF / CASSCF solutions is usually not high enough, so that it then usually performed a correlated calculation, or the Hartree -Fock in solving. MCSCF equation arising unoccupied orbitals are used. The number corresponds to the calculated orbitals namely the number of basis functions used, and is significantly larger than the number of occupied orbitals of the Hartree-Fock wave functions, or MCSCF are usually. In principle, the wave function is a linear combination of determinants recognized at the Hartree -Fock wave function as the leading determinant (large coefficient) in case of correlated methods. Other determinants are formed by occupied orbitals are replaced by unoccupied orbitals (so-called suggestions ).
In the correlated methods are in single- reference case, typically either the perturbation theory ( Moller- Plesset approach), Configuration Interaction or Coupled Cluster (CC ) used approaches in multi- reference methods, either the Multi- Reference Configuration Interaction MRCI method or the multi- reference perturbation theory.
In general, only one-and two specialist suggestions of the reference wave function for all methods regarding considered, with due to the nature of the Coupled Cluster approach, certain classes of higher excitations are taken into account here. When Coupled Cluster approach, this is referred to as CCSD ( S for Single, Double D for the number of excitations in the approach). The Coupled Cluster approach was developed by Hermann Fritz Coester and Kümmel in the late 1950s in nuclear physics and applied from the 1960s, in quantum chemistry ( Jiři Čížek, Josef Paldus ).
In the Configuration Interaction method of the reference wave function (SCF or MCSCF / CASSCF ) are first with respect to, for example, generates all the inputs and double excitations by one or two occupied orbitals are replaced by the corresponding number of unoccupied orbitals. The CI wave function is taken as a linear combination of all of these determinants and the corresponding ( CI ) coefficients of the determinants determined so that the resulting energy minimum (possibly negative). Usually, only one-and two specialist suggestions will be considered. A special case is the so-called full- CI, in which all sorts of suggestions are generated. Full -CI calculations are, however, associated with such a large computational cost, so that they can be performed only in small systems as a benchmark. The MRCI (SD) method is considered to be very accurate for both the properties as well as for the absolute energy of the ground state energy differences and to electronically excited states. Single Reference CI methods, in particular CI (SD) are against it because of the lack of size consistency as inaccurate. Size consistency for example, would mean that two hydrogen molecules, which are located in a very large distance from each other, the same total energy supply as twice the energy that is calculated for a single hydrogen molecule. Due to the lack of three-and four-fold excitations in the first case, the result in the second case, however, is much more negative. While this is true in principle, for MRCI (SD ), but the multi- reference approach compensates to a large part of the error.
In perturbation theory the Hamiltonian is split as a sum of an unperturbed operator and a perturbation operator, which disorder should be " small". The eigenvalue solutions of the unperturbed operator are known. When correlated perturbation theory is used as one operator who has the Hartree -Fock or MCSCF / CASSCF wave functions for the solution. then calculated as the difference from the real Hamiltonian of the system. In the single- reference case, the MP2 and MP4 ( SDQ) are frequently used methodology in multiple-reference case, the so-called CASPT2 method ( with a CASSCF wave function as the reference wave function ), rarely the CASPT3 method.
In the solution of the disturbance equation shows that the disturbed portion further splits into a wave function of the first order, second order, etc., with the Gesamtwellenfuktion is the sum of the unperturbed and perturbed wave functions of the different. The computational complexity is added with each coming correction significantly higher. However, it is not necessarily that the series converges to the exact result, ie that it is not guaranteed that the calculated wave function and the associated energy / properties are getting better with increasing effort. In fact, in some cases oscillations around the exact value or a divergence of results are observed.
In the coupled-cluster approach, the wave function is represented as. This guarantees the one hand, the size consistency of the method and results on the other hand to the fact that certain higher excitation types are also recognized. Coupled cluster calculations of the CCSD ( T) type are considered very accurate. The wave function here is not in closed form, so that the properties should be calculated in other ways, but for which there are appropriate management strategies.
Configuration Interaction variational method, that is, the calculated power is always higher than the exact energy. However, this does not apply to the perturbation theory or the Coupled Cluster approach. However, perturbation theory and coupled cluster approach is in contrast to the CI method size consistent. Size consistency means that the energy that results practically non- interacting with each other in calculating a Super Systems of two due to the large distance chosen ( like) molecules, must be equal to twice the calculated energy of a single molecule. Due to this lack of CI method hardly CI calculations on ( single- reference ) Hartree -Fock calculations are now placed more, whereas MRCI (SD) calculations are very accurate. Full -CI calculations, in which all possible in the orbital space suggestions regarding the SCF or MCSCF function are taken into account, are a special case both variational and size- consistent, but also the most expensive by far.
The effort in performing correlated methods does not increase linearly with the size of the molecule, but is usually used in the methods and between, where N is a measure of the size of the molecule (such as the number of basis functions). This can be attributed to the fact that the orbitals delocalised, i.e. more or less spread over the whole molecule. The resulting at the Hartree - Fock calculation orbitals can, however, locate relatively well with different methods. Correlation methods that use these localized orbitals, promise a significant reduction of the above scaling behavior with the molecular size and is being intensely investigated. Core problem here is that truly local orbitals must be " trimmed ", which means that local orbitals not strictly orthogonal to each other, the integral of two such orbitals of zero is a little bit different.
Semi-empirical methods
The so-called semi-empirical methods in general, the matrix form of the Hartree- Fock equation can be simplified by certain variables ( integrals) in the matrix equation either neglected, replaced by experimentally determined values or adjusted on a training set. Such a training set is usually made (experimental or to a high computational methods determined ) variables such as bond lengths, dipole moments, etc., of a number of molecules that are to be reproduced as well as possible by varying the free parameters. Semi-empirical methods can handle several 100 atoms systems with (at least ).
Density Functional
In the density functional theory ( DFT) one makes use of the fact that the description of the ground state, the electron density is independent of the number of electrons required as a function of only three spatial variables; would qualify as a further variable to the spin density. However, here there is the problem that the necessary terms ( operators) are not all exactly known in the equations to be solved, but in some cases approximations are necessary. Current density functionals achieve the accuracy of simple correlated ab initio methods (such as the perturbation theory of second order ) and can be used up to about 1000 atoms for systems. Frequently used DFT calculations for geometry optimization of molecules.
Force fields
In the so-called force field methods are accessed, however, back to a classic mode of conception, according to which the atoms are connected in molecules by small springs with spring constant specific to each other, which also describe the change of bond and torsion angles. This method is especially suitable for very large ( bio) molecules, which can not be solved using other methods, and is primarily for the geometry optimization. However, spring constant corresponding to a plurality of possible combinations atom (three for bond angles, torsion angles for four to two bonds ) are determined. (Partial ) charges on atoms and their interaction with each other are also considered.
A frequently encountered even among experts of theoretical chemistry prejudice states that the description of bond breakage is inherently problematic or even impossible with force field methods. This prejudice is neither content nor historically accurate. It is based on two commonly used but arbitrary and unrealistic model assumptions: (1) The modeling of binding by a spring ( harmonic oscillator ); this can be replaced by a more realistic model with a spring that can tear at intense stretching also ( anharmonic oscillator, for example, Morse oscillator ); ( 2) The simplifying assumption that atoms do not change the type and number of their nearest neighbor atoms (definition of so-called atomic types); This assumption alone can drop compensation. The frequently put forward, additional objection that one could not classically mechanically, but only quantum mechanically describe a bond breaking, because of this essential correct description of the electron is true, but at the same time irrelevant. In a force field of the type referred to here there is no explicit description of electrons and no classical mechanics; it is only an approximation to the results of a quantum-mechanical treatment of the electrons. If one uses the mentioned more realistic modeling, we obtain a reactive force field. Some of the much-quoted first work on classical molecular dynamics (see below) already in use reactive force fields. This came by later, wide use of some non-reactive force fields somewhat forgotten.
Non-reactive force field methods can answer many conformational issues; with reactive force fields can also describe chemical reactions. Force fields are used for classical molecular dynamics.
Calculation of physical properties
The physical properties of a system such as its dipole moment is assigned a quantum mechanical operator. With knowledge of the wave function of the property, the dipole moment can be calculated as the expected value on the operator, so as. Also, the property may be determined as a single or multiple derivative of the electronic energy of the system according to certain variables which depend on the physical property. The latter method can also be used, when the wave function is not explicitly known ( for example when coupled cluster approach), and is not limited to the equilibrium geometry in contrast to the first method.
Geometry optimization
Since the solution of the Schrödinger equation only pointwise, ie for discrete geometries is possible and a sufficiently accurate solution for a geometry is already connected with a high computational complexity, a sub-branch of theoretical chemistry involved in the preparation of the algorithms with which excellent geometries with the least possible computational effort can be found. Excellent geometries are, for example, the equilibrium geometry ( energy minimum ), as well as chemical reactions of the transition state on the reaction coordinate as a saddle point. The energy difference between reactants and transition state determines the activation energy of the reaction, the energy difference between reactants and products of the reaction energy. Methods are frequently used in which the energy is calculated at a point next to and its first derivative and the second derivative is estimated.
Simulation of chemical reactions
For the simulation of chemical reactions an analytical representation of the involved potential energy surface (s) in the relevant range of possible geometries of the system under consideration is to generally (eg molecule ) is necessary, ie an analytic function, the energy of the system depending on its geometry reproduces. To this end, for each face calculates the corresponding energy at certain excellent geometries and determined on the basis of which an approximate analytical representation of the surface, for which there are different approaches. As the number of internal degrees of freedom of a system consisting of N atoms, 3N -6 is ( linear molecules 3N -5), a complete potential energy surface, that is one which takes account of all degrees of freedom of the system, calculated for three to four atom molecules be. For larger systems must have a selection of relevant geometry parameters (that is, as a rule, certain bond lengths, angles or torsion angles ) are taken, the values of the remaining Geometrieparamter to the excellent geometries are optimized for energy efficiency. After the surfaces are present in analytical form, the core Schrödinger equation can be solved and thus the progress of the chemical reaction on the computer can be simulated. Here, too, there are different approaches.
Qualitative explanation schemes
Especially in the early days of theoretical chemistry a number of explanatory schemes have been prepared with the help of various aspects could be explained qualitatively. One example is the so-called VSEPR theory, with the help of which the geometry of simple molecules can be predicted with a central atom. But even more recently, new concepts have been developed, such as the electron localization function ( ELF) or the topological concept of 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). Places with low pair density are associated with a high locality of an electron and topologically with a chemical bond. ELF can be practically calculated at the HF and DFT levels. Bader method forming of the first and second derivatives of the electron density in accordance with the spatial coordinates of a link with intuitive expectations as chemical bonds.