A base set in theory and chemical computation is a set of functions (called base functions) used to represent electronic wave functions in the Hartree-Fock method or density-functional theory to convert partial differential equations from model to algebraic equations suitable for efficient implementation of the computer.
The use of base sets is equivalent to the use of approximate resolution of identity. The state of a single particle (molecular orbital) is then expressed as a linear combination of the base function.
The base set can consist of atomic orbitals (resulting in a linear combination of atomic orbital approaches), which are the usual choices in quantum chemical communities, or field waves normally used in the solid state community. Several types of atomic orbitals may be used: Gaussian-type orbitals, Slater-type orbitals, or numerical atomic orbitals. Of the three, Gaussian type orbitals are the most commonly used, as they enable the efficient implementation of the Post-Hartree-Fock method.
Video Basis set (chemistry)
Introduction
In modern computational chemistry, quantum chemical calculations are performed using limited basic functions. When a finite base is extended to a complete set of functions (unlimited), the calculations using such a set of bases are said to be close to the specified complete border limit (CBS). In this article, basic functions and atomic orbitals are sometimes used interchangeably, although it should be noted that their basic functions are usually not true atomic orbitals, since many of the basic functions are used. to describe the polarization effect in the molecule.
In the basic set, the wave function is represented as a vector, a component corresponding to the basic function coefficients in linear expansion. One-electron operator corresponds to the matrix, (ranked two tensors), in this base, whereas the two-electron operator is ranked four tensor.
When molecular calculations are performed, it is common to use a base consisting of atomic orbitals, centered on each nucleus within a molecule (a linear combination of atomic orbital analsis). The best physically motivated base set is a Slater-type orbital (STO), which is a solution to the Schrödinger equation of hydrogen-like atoms, and decays exponentially away from the nucleus. While hydrogen-like atoms lack the interaction of many electrons, it can be shown that Hartree-Fock's molecular orbital and density-functional theory also exhibit exponential decay. Furthermore, STO-S types also meet Kato's peak conditions in the nucleus, meaning that they can accurately represent electron density near the nucleus.
However, computing the integral with STO is computationally difficult and it was later realized by Frank Boys that STO can be approximated as a linear combination of gaussian-type orbital (GTOs) instead. Since the product of two GTOs can be written as linear combinations of GTO, integral to Gaussian basic functions can be written in a closed form, leading to large computational savings (see John Pople).
Dozens of sets of Gaussian orbital bases have been published in the literature. Base sets usually come in an increasing size hierarchy, providing a controlled way to get a more accurate, yet higher cost solution.
The smallest set of bases is called the minimal base set . A minimal set of bases is one in which, on every atom in a molecule, a single base function is used for every orbital in the Hartree-Fock calculation of free atoms. For atoms like lithium, the basic functions of type p are also added to the basic functions associated with the 1s and 2s orbital of the free atom, since lithium also has a bound state of 1s2p. For example, every atom in the second period of the periodic system (Li - Ne) will have a set of five base functions (two functions and three p functions).
The minimal base set is almost appropriate for gas phase atoms. At the next level, additional functions are added to describe the polarization of electron densities of atoms in the molecule. This is called polarization function . For example, while the minimal base set for hydrogen is a function that is close to a 1s atomic orbitals, a simple set of polarization bases usually has two functions s and one p (consisting of three basic functions: px, py and pz). This adds flexibility to the established base, effectively enabling molecular orbitals involving hydrogen atoms to become more asymmetric about the hydrogen nuclei. It is very important to model chemical bonds, because bonds are often polarized. Similarly, the d-type function can be added to the specified base with the valence p orbitals, and the -f function to the base specified by the d-type orbitals, and so on.
Another common addition to the base set is the addition of diffuse functions . This is a basic Gaussian function extended with small exponents, which gives flexibility to the "tail" part of the atomic orbitals, away from the nucleus. The diffuse base function is important to describe the anion or dipole moment, but they can also be important for accurate modeling of intra- and intermolecular bonds.
Maps Basis set (chemistry)
Set minimum base
The most common minimal base set is STO-nG, where n is an integer. This n value represents the number of Gaussian primitive functions consisting of a single base function. In this basic set, the same number of Gaussian primitives consists of core or valence orbitals. The minimal set of bases usually gives insufficient rough results for research-quality publications, but is much cheaper than their larger counterparts. The commonly used basic sets of this type are:
- STO-3G
- STO-4G
- STO-6G
- STO-3G * - STO-3G polarized version
There are several other minimum base sets that have been used such as the MidiX base set.
Double valence basics set
During most molecule bonds, it is a valence electron that mainly takes part in bonding. In recognition of this fact, it is common to represent a valence orbital with more than one basic function (each which in turn can consist of a linear combination of fixed primitive Gaussian functions). The set base where there are several basic functions corresponding to each valence valence orbitals are called double valence, triple, quadruple-zeta, etc., base sets (zeta,?, Commonly used to represent exponents of STO basic functions). Because different orbitals of the split have different spatial extents, this combination allows the electron density to adjust its spatial range according to a particular molecular environment. In contrast, the minimal set of bases does not have the flexibility to adapt to different molecular environments.
Pople base set
Notations for the split-valence base set emerging from the John Pople group are typically X-YZg . In this case, X represents the number of primitive Gaussians consisting of each basic function of the basic atomic orbitals. The Y and Z show that the valence orbitals consist of two basic functions respectively, the first consisting of a linear combination of the primitive Gaussian Y function, the other consists of a linear combination of Z the primitive Gaussian function. In this case, the presence of two numbers after the hyphen implies that the base set is the set base of the split-valence double-zeta . Split-valence triple-and quadruple-zeta base sets are also used, denoted as X-YZWg , X-YZWVg , etc. Here is a list of commonly used split base valence sets of this type:
- 3-21G
- 3-21G * - Polarization function at heavy atom
- 3-21G ** - Polarization function on heavy and hydrogen atoms
- 3-21 G - Diffuse function on heavy atoms
- 3-21 G - Diffuse function on heavy and hydrogen atoms
- 3-21 G * - Polarization and diffuse functions on heavy atoms
- 3-21 G ** - Polarization functions on heavy and hydrogen atoms, and diffuse functions on heavy atoms
- 4-21G
- 4-31G
- 6-21G
- 6-31G
- 6-31G *
- 6-31 G *
- 6-31G (3df, 3pd)
- 6-311G
- 6-311G *
- 6-311 G *
The basic circuit 6-31G * (defined for atoms H to Zn) is a polarized double-zeta valence base that adds to 6-31G sets six d -types of the Cartesian-Gaussian polarization function at each Li atom through Ca and ten f -the Gaussian Cartesian polarization polarization function of each Sc atom through Zn.
The base set of people is somewhat out of date, since consistent series of correlations or consistent polarizations usually produce better results with similar resources. Also note that some sets of base pople have grave deficiencies that can lead to incorrect results.
Consistent set of correlation-based
One of the most commonly used basic sets is that developed by Dunning and co-workers, as they are designed to convert Post-Hartree-Fock calculations systematically to establish a complete baseline using empirical extrapolation techniques.
For the first and second line atoms, the base set is cc-pVNZ where N = D, T, Q, 5.6,... (D = double, T = triples, etc.). The 'cc-p', short for 'consistent-polarized correlation' and 'V' shows them a basic set of valences only. They include larger shells of polarization functions (connecting) ( d , f , g , etc.). Recently the 'consistent set of correlated-polarized bases' has been widely used and is the current state of the art for correlation or post-Hartree-Fock calculations. The example is:
- cc-pVDZ - Two-zeta
- cc-pVTZ - Triple-zeta
- cc-pVQZ - Quadruple-zeta
- cc-pV5Z - Quintuple-zeta, etc.
- aug-cc-pVDZ, etc. - The augmented version of the previous base is set with additional diffuse functions.
- cc-pCVDZ - Double-zeta with core correlation
For period-3 atom (Al-Ar), additional functions are needed; this is the base cc-pV (N d) Z. Even larger atoms can use a pseudopotential base set, cc-pVNZ-PP, or relativistic regulated Douglas-Kroll base, cc-pVNZ-DK.
While the usual basic Dunning circuit is for valence calculation only, the set can be supplemented by further functions that describe the correlation of the electron nucleus. This core-valence set (cc-pCVXZ) can be used to approach the right solutions to all-electron problems, and they are necessary for accurate geometric calculations and nuclear properties.
The weighted core-valence set (cc-pwCVXZ) is also recently recommended. The weighted set aims to capture core-valence correlations, while ignoring most corelation cores, to produce accurate geometries at a cost smaller than the set of cc-pCVXZ.
Diffuse functions can also be added to describe anions and long-term interactions such as Van der Waals style, or to perform electronic excited calculations, calculation of electric field properties. A recipe for building additional supplementary functions exists; as many as five augmented functions have been used in the second calculation of hyperpolarizability in the literature. Due to this rigorous base set construction, extrapolation can be done for almost any energetic property. However, care must be taken when extrapolating the energy difference as individual energy components converge at different levels: Hartree-Fock energy fused exponentially, whereas the correlation energy merely integrates polynomialically.
To understand how to get a number of functions, use the cc-pVDZ base for H: There are two orbital s ( L = 0) i> L = 1) orbital that has 3 components along the z -axis ( m L = -1,0,1 )
y and p z . So, five total spatial orbital. Notice that each orbital can withstand two opposite spin electrons.
For example, Ar [1s, 2s, 2p, 3s, 3p] has 3 s orbital (L = 0) and 2 sets of orbital p (L = 1). Using cc-pVDZ, orbital is [1s, 2s, 2p, 3s, 3s ', 3p, 3p', 3d '] (where' represents an additional in polarization orbital), with 4 orbital, 3 sets of orbital p and 1 set orbital d.
Polarization of consistent set base
Recent density-functional theory has become widely used in computational chemistry. However, the consistent-correlation basis described above is less than optimal for density-functional theory, since consistent-correlation sets have been designed for Post-Hartree-Fock, whereas density-functional theory shows faster convergence set basis rather than wave function method.
Adopting a methodology similar to a consistent series of correlations, Frank Jensen introduces a consistent polarization (pc-n) base set as a way to quickly match functional theory density calculations to a complete baseline. Like the Dunning set, pc-n sets can be combined with basic extrapolation techniques to get the CBS value.
Set pc-n can be added with diffuse function to get set augpc-n.
The base of Karlsruhe
The basic set of Karlsruhe comes in various flavors
- def2-SV (P) - Separate valence with polarization functions on heavy (not hydrogen) atoms
- def2-SVP - Split valence polarization
- def2-SVPD - Split valence polarization with diffuse function
- def2-TZVP - Polarization of three-zeta valence
- def2-TZVPD - Polarization of triple-zeta valence with diffuse function
- def2-TZVPP - Triple-zeta validation with two sets of polarization functions
- def2-TZVPPD - Valence triple-zeta with two sets of polarization functions and a set of diffus functions
- def2-QZVP - Polarization quadruple-zeta valence
- def2-QZVPD - Polarization of quadruple-zeta valence with diffuse function
- def2-QZVPP - Valid quadruple-zeta with two sets of polarization functions
- def2-QZVPPD - Valid quadruple-zeta with two sets of polarization functions and a set of diffus functions
Completely optimized base base
Gaussian-type orbital base sets are usually optimized to reproduce the lowest possible energy for systems used to train base sets. However, energy convergence does not imply the convergence of other properties, such as nuclear magnetic shields, dipole moments, or electron momentum densities, which investigate different aspects of electronic wave function.
Manninen and Vaara have proposed the completeness of the basic-optimized sets, where exponents are obtained by maximizing the completeness profile of an electron rather than minimizing energy. Complenetessally optimized base sets are a way to easily approach the complete baseline of each property at all levels of theory, and the procedure is simple to automate.
Completely optimized base sets are tailored to specific properties. In this way, the flexibility of the base set can be focused on the computing demands of the selected property, usually resulting in a much faster convergence to the complete set of base limits than can be achieved with an optimized set of energy bases.
Plane-based waveform
In addition to the localized base set, the base set of aircraft waves can also be used in quantum-chemical simulations. Typically, the basic choice of ground plane waves is based on cutoff energy. The plane wave in the simulation cell that fits under the energy criterion is then included in the calculation. This base set is popular in calculations involving three-dimensional periodic boundary conditions.
The main advantage of the plane-wave base is that it is guaranteed to meet in a subtle, monotonic way to the target wave function. In contrast, when a localized set of bases is used, monotonic convergence to a set boundary may be difficult because of a problem with excessive completeness: in large base sets, functions of different atoms begin to look alike, and many eigenvalues ​​from overlapping matrices approach zero.
In addition, certain integrals and operations are much easier to program and run with the wave-plane base function than with their local counterparts. For example, a diagonal kinetic energy operator in a reciprocal space. Integration over real space operators can be performed efficiently using fast Fourier transforms. The Fourier Transform properties allow a vector to represent the total energy gradient with respect to the plane-wave coefficient calculated by computational effort scaling as NPW * ln (NPW) where NPW is the number of plane waves. When this property is combined with a separate pseudopotential Kleinman-Bylander type and a pre-existing conjugate gradient solution technique, a dynamic simulation of periodic problems containing hundreds of atoms is possible.
In practice, the set of the plane-wave base is often used in combination with 'effective core potential' or pseudopotensial, so a field wave is used only to illustrate the density of the valence charge. This is because the nuclear electrons tend to be concentrated very close to the nucleus of the atom, resulting in a large wave function and density gradient near the nucleus that is not easily explained by a fixed plane-wave base unless a very high energy cutoff, and therefore a small wavelength, is used. The combined method of a fixed-plane base with pseudopotential core is often abbreviated as a calculation of PSPW .
Furthermore, since all functions in the base are mutually orthogonal and unrelated to a particular atom, the base of the wave-plane set does not indicate a basic-set superposition error. However, the base-base base set depends on the size of the simulated cell, which makes it difficult to optimize cell size.
Due to the assumption of periodic boundary conditions, the set of the plane-wave base is less suitable for gas phase calculations than for local set sets. Large vacuum areas need to be added to all sides of the gas phase molecule to avoid interaction with molecules and periodic copies. However, plane waves use the same accuracy to describe the vacuum region as the region in which the molecule is, meaning that getting a limit that is completely noninteracting can be computationally expensive.
Basic real space set
Analogue with the base wave of the set, where the base function is the eigen function of the momentum operator, there is a base set whose function is the eigen function of the position operator, ie, pointing to a uniform mesh in the real space. The actual implementation may use a finite difference, or interpolate with the sinc function (a.k.a. Lagrange function) or wavelet.
The Sinc functions form an orthonormal, analytical, and complete set of bases. Convergence to the basic set of complete set is systematic and relatively simple. Similarly for the plane wave base, the accuracy of the sinc base set is controlled by the energy cutoff criteria.
In the case of wavelets, it is possible to make adaptive mesh, so that more points are used close to the core. Wavelets rely on the use of localized functions that enable the development of linear scale methods.
See also
- Basic sets superposition errors
- Angular momentum
- The atomic orbitals
- Molecular orbitals
- List of quantum chemistry and solid state physics software
References
Source of the article : Wikipedia
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