Basic set (chemistry)

In theoretical and computational chemistry , a basic set is a set of functions (the so-called basic functions) that is used to represent the electronic wave function in Hartree-Fock methods or density functional theory in order to convert the partial differential equations of the model into algebraic equations, which are useful for efficient implementations on a computer.

The use of basic sentences is equivalent to the use of an approximate solution of the identity. The single-particle states (molecular orbitals) are then expressed in the form of linear combinations of the basis functions.

The basic set can either be composed of atomic orbitals (which leads to the approach of the LCAO method ) or plane-waves . The former approach is primarily used by quantum chemists, while the latter is typically used by solid-state researchers. In principle, different types of atomic orbitals can be used: Gaussian type orbitals , Slater type orbitals or numerical atomic orbitals. Of these three, the Gaussian Type Orbitals are most frequently used, as they allow efficient implementations of Post-Hartree-Fock methods .

introduction

In modern computational chemistry , quantum chemical calculations are carried out with a finite set of basic functions . If this finite set is expanded in the direction of an (infinite) complete set of functions, this is called an approximation to the complete basis set (CBS, German: entire basis set) limit. In this article, basis function and atomic orbitals are used interchangeably, although the basis functions are usually not true atomic orbitals, as many basis functions are used to describe polarization effects in molecules.

Within a basic set, the wave function is represented as a vector ; the components of this correspond to the coefficients of the basic function in the linear extension. In such a basis, one-electron operators correspond to matrices (also called tensor of level 2), whereas two-electron operators are tensors of level 4.

When performing molecular calculations, it is common to use a base of atomic orbitals centered on the nuclei within the molecule (LCAO method). The physically best basis sets are Slater Type Orbitals (STOs), which are solutions of the Schrödinger equation of hydrogen-like atoms and which decrease exponentially with the distance from the nucleus. It can be shown that the molecular orbitals of Hartree-Fock and density functional theory also decrease exponentially. In addition, S-type STOs satisfy Kato's theorem at the nucleus, which means that they are able to accurately describe the electron density near the nucleus. Nevertheless, hydrogen-like atoms lack many multi-electron interactions because the orbitals do not accurately describe the electronic correlation.

Unfortunately, the computation of integrals using STOs is technically difficult and Frank Boys was later able to show that linear combinations of Gaussian Type Orbitals can serve as approximations instead. Since the product of two GTOs can be written as a linear combination of GTOs, integrals with a Gaussian basis function can be written in closed form, which leads to a great computational simplification (see John Pople ).

Dozens of Gaussian Type Orbital basis sets are known in the literature. Basic sets are typically published in different sizes, which enables systematically more accurate results to be obtained, but at higher costs.

The smallest basic sets are called the minimum basic sets. The minimum basic set is understood to be a basic set in which a single basic function for each orbital in a Hartree-Fock calculation on the free atom is used for each atom in the molecule. Since, for example, lithium has a 1s2p bond state, p-like basic functions are added for such atoms, which interact with the 1s and 2s orbitals of the free atom. Each atom of the second period of the periodic table (lithium to neon) would therefore have a basis set of five functions (two s and three p functions).

A d polarization function is added to a p orbital

Minimum basis sets are approximately exact for atoms in the gas phase. At the next higher level, additional functions are introduced to describe the polarization of the electron density of the atom in the molecule. These functions are called polarization functions. For hydrogen, for example, the minimal basis set includes the 1s atomic orbital, a simple polarized basis set typically has two s and one p functions, which in turn consists of three basis functions: px, py and pz. This makes the basic set more flexible, since molecular orbitals that contain hydrogen can behave more asymmetrically around the hydrogen nucleus. This behavior is particularly important for representing chemical bonds, as these are often polarized. Similarly, d-like functions can be added to a basis set with p-valence orbitals and f-functions can be added to a basis set with d-like orbitals, and so on.

It is also common to add diffuse functions . These are Gaussian basis functions with a small exponent that make the end of the atomic orbitals, which is far from the nucleus, more flexible. Diffuse basis functions are important for describing dipole moments, but they also play a decisive role in the accurate formation of intra- and intermolecular bonds.

Minimal basic rates

The most widespread minimal basis sets are the STO-nG basis sets ( Slater Type Orbital) , where n is an integer here. This n value represents the number of primitive Gaussian functions ( G ) that make up a single basis function. In these basic theorems, the same number of primitive Gaussian functions encompasses both core and valence orbitals. Minimal basic rates typically lead to approximate results that are insufficient for a qualitative publication, but they are significantly cheaper than their larger counterparts. Frequently used minimal basic sentences of this type are:

STO-3G
STO-4G
STO-6G
STO-3G * - Polarized version of STO-3G

However, a large number of other minimal basic sets such as MidiX are also used.

Split-valence basic rates

In most molecular bonds, the valence electrons take on the majority of the bond. For this reason it is common to have the valence orbitals mapped by more than one basic function ( split-valence ), each of which in turn can be composed of a fixed linear combination of primitive Gaussian functions. Basic sets in which several basic functions belong to a valence orbital are called valence double-zeta, triple-zeta, quadruple-zeta and so on basic sets. Zeta, ζ, was widely used to represent the exponent of an STO basis function. Since the different orbitals of the partition have a different spatial extent, the combination of the electron density allows the spatial extent to be adapted so that it fits the respective molecular environment. In contrast, the minimal basis sets lack this flexibility.

Pople basic sets

The nomenclature of these split-valence basic clauses comes from the research group of John Pople and is typically X-YZg . X stands for the number of primitive Gaussian functions from which each basic function of the nuclear atomic orbitals is composed. Y and Z indicate that the valence orbitals result from two base functions each. One is in turn made up of a linear combination of Y , the other a linear combination of Z primitive Gaussian functions. In this case, the number after the hyphen implies that it is a split-valence double-zeta basic set. Split-valence triple- and quadruple-zeta basic sentences are also used and described in the form X-YZWg , X-YZWVg . Commonly used split-valence basic sentences of this kind are:

3-21G
3-21G * - polarization functions on heavy atoms
3-21G ** - polarization functions on heavy atoms and hydrogen
3-21 + G - Diffuse functions on heavy atoms
3-21 ++ G - Diffuse functions on heavy atoms and hydrogen
3-21 + G * - polarization and diffuse functions on heavy atoms
3-21 + G ** - polarization functions on heavy atoms and hydrogen and diffuse functions only 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 6-31G * basic set is defined for the atoms hydrogen to zinc and a polarized split-valence double-zeta basic set, which, compared to the 6-31G basic set, has six more d -like Cartesian Gaussian polarization functions for each of the atoms lithium to calcium and ten has further f -like Cartesian Gaussian polarization functions for each of the atoms scandium to zinc.

Nowadays, the Pople basic sets are considered obsolete, since correlation-consistent or polarization-consistent basic sets typically lead to better results with the aid of similar resources. It should also be noted that some of Pople's basic sets have serious shortcomings that can lead to incorrect results.

Correlation-consistent basic sets

Some of the most widely used basis sets are those developed by Thom Dunning and his co-workers, as they were designed to systematically converge post-Hartree-Fock calculations to the basis set limit using empirical extrapolation techniques.

For the atoms of the first and second periods they are called cc-pVNZ, where N = D, T, Q, 5.6 and so on, which stands for double, triple etc. according to the general nomenclature. The "cc-p" is the abbreviation for correlation-consistent polarized (German: correlation-consistent polarized) and the V indicates that it is valence-only basic sentences. They contain successive larger shells based on polarization (dependent) functions ( d , f , g ...). Recently, these correlation-consistent polarized basic sets have become widespread and are considered state of the art for correlation or post-Hartee-Fock calculations. Examples of this class of basic sentences are:

cc-pVDZ - Double-zeta
cc-pVTZ - Triple-zeta
cc-pVQZ - quadruple zeta
cc-pV5Z - quintuple zeta
aug-cc-pVDZ, etc. - Extended versions of the previous basic sets with additional diffuse functions
cc-pCVDZ - Double-zeta with core correlation

For the atoms of the third period (aluminum to argon), it has been found that additional functions are necessary, which is why the cc-pV (N + d) Z basis sets were developed. Even larger atoms require the use of pseudopotential-based basis sets, cc-pVNZ-PP, or the relativistically contracted Douglas-Kroll basis sets cc-pVNZ-DK. Further modifications are available that include the core electrons in the form of core-valence basic sets, cc-pCVXZ or their weighted counterparts cc-pwCVXZ. Furthermore, there are many possibilities for adding diffuse functions to enable a better description of anions and long-range interactions (e.g. van der Waals forces).

Caution is required when extrapolating energy differences, as the individual components that contribute to the total energy converge differently. The Hartree-Fock energy converges exponentially, while the correlation energy converges polynomially.

Polarization-consistent basic sets

A lot of density functional theory is used in computational chemistry. However, the correlation-consistent basis sets described above are suboptimal for this, as they were designed for post-Hartee-Fock calculations and density functional theory shows a significantly faster basis set convergence than wave function-based methods.

Using a methodology similar to the correlation-consistent basic sets, Frank Jensen's working group presented the polarization-consistent (in German: polarization-consistent ) basic sets. These form a way that density functional theory computations can quickly converge to the entire basis set limit. Analogous to the Dunning basic sets, the pc-n basic sets can be combined with extrapolation techniques in order to obtain CBS values.

In addition, the pc-n basic sets can be expanded with diffuse functions (English: augmented), whereby aug-pc-n basic sets are obtained.

Karlsruhe basic rates

Some of the many valence adjustments of the Karlsruhe basic sets are:

def2-SV (P) - Split valence with polarization function on heavy atoms (not on hydrogen)
def2-SVP - split valence polarization
def2-SVPD - Split valence polarization with diffuse functions
def2-TZVP - triple-zeta valence polarization
def2-TZVPD - Triple-zeta valence polarization with diffuse functions
def2-TZVPP - Triple-zeta valence with two sets of polarization functions
def2-TZVPPD - Triple-zeta valence with two sets of polarization functions and one set of diffuse functions
def2-QZVP - Quadruple-zeta valence polarization
def2-QZVPD - Quadruple-zeta valence polarization with diffuse functions
def2-QZVPP - Quadruple zeta with two sets of polarization functions
def2-QZVPPD - Quadruple-zeta valence with two sets of polarization and one set of diffuse functions

Completeness-optimized basic sets

Gaussian-type orbitals are typically optimized to reproduce the lowest possible energies for the systems used to train the basis set. Nevertheless, a convergence of energy does not necessarily imply the convergence of other properties, such as the magnetic shielding of the core, the dipole moment or the density of the electron pulse, which are related to different aspects of the electronic wave function.

Manninen and Vaara have therefore proposed the completeness-optimized basic sets in which the exponents are obtained by maximizing the one-electron completeness profile instead of minimizing the energy. Complenetess-optimized basic sets are a possibility to easily reach the entire basic set limit for any property to any precision. The procedure is also easy to automate.

Completeness-optimized basic sets are tailored to a specific property. In this way, the flexibility of the basic set can be focused on the computational requirements, which typically leads to a significantly faster convergence to the full basic set limit than is possible with energy-optimized basic sets.

Plane-wave basic sets

In addition to the localized basic sets, plane-wave basic sets can also be used in quantum chemical calculations. Typically, the choice of the plane-wave basis set depends on a limit energy. The plane waves in the simulation environment that match the energy criterion are then used in the calculation. These basis theorems are particularly popular in calculations that have three-dimensional periodic boundary conditions .

The great advantage of a plane-wave basis lies in the guarantee of convergence to the target wave function in a smooth, consistent manner. In contrast, constant convergence to the basis set limit for localized basis sets can be difficult due to problems with over-completeness , since in a large basis set the functions for different atoms begin to look the same and many eigenvalues of the overlap matrix approach zero .

In addition, with plane-wave basic functions, certain integrations and processes are much easier to program and execute compared to their localized counterparts. For example, the kinetic energy operator in reciprocal space is diagonal. Integrations via operators of real space can be carried out efficiently using fast Fourier transforms . The properties of the Fourier transformation make it possible to calculate a vector that represents the gradient of the total energy, including the plane-wave coefficients, with a computational effort that scales according to NPW * In (NPW). NPW is the number of plane-waves (English: N umber of P Lane- W aves). If this property is combined with separable pseudopotentials of the Kleinman-Bylander type and prefabricated gradient, conjugate solution techniques, the dynamic simulation of periodic problems affecting hundreds of atoms becomes possible.

In practice, plane-wave basic sentences are often used in combination with an effective core potential or pseudopotential , so that the plane-waves are only used to describe the valence charge density. This has to do with the fact that the nuclear electrons are very highly concentrated near the atomic nucleus, which leads to large wave functions and density gradients near the nucleus, which is not easy to describe using plane-wave basic theorems, unless it becomes a very high energy criterion and thus a small wavelength used. This combined method of a plane-wave basic theorem with a nuclear pseudopotential is often abbreviated as the PSPW calculation.

Since all functions in the basis are still jointly orthogonal and not associated with a specific atom, plane-wave basis sets do not have the basis set superposition error. Nevertheless, the plane-wave basis set depends on the size of the simulated environment, which complicates the optimization of the environment size.

Due to the assumption of periodic boundary conditions, plane-wave basis sets are less suitable for gas phase calculations than localized basis sets. In order to avoid interactions with the molecule and its periodic copies, large regions of vacuum must therefore be added on all sides of the gas phase molecule. Since the plane waves use a similar accuracy to the regions in which the molecule is located to describe the vacuum regions, it can be computationally expensive to reach the true limit of non-interaction.

Real-space basic rates

Analogous to the plane-wave basic sets, in which the basic function is an eigenfunction of the momentum operator, there are basic sets whose functions are eigenfunctions of the position operator, which means points on a uniform network in real space. Real implementations can however use finite differences , finite elements or Lagrangian sinc functions , or wavelets .

Sinc functions form an orthonormal, analytic and complete basis set. The convergence towards the full base rate limit is systematic and relatively simple. Similar to plane-wave basic sets, the accuracy of Sinc basic sets is given by an energy limit.

In the case of wavelets and finite elements, it is possible to make the network adaptable so that more points are used near the core. Wavelets rely on the use of localized functions that enable the development of nonlinear scaling methods.

Individual evidence

^ Frank Jensen: Atomic orbital basis sets . In: WIREs Comput. Mol. Sci. . 3, No. 3, 2013, pp. 273-295. doi : 10.1002 / wcms.1123 .
^ Ernest R. Davidson, David Feller: Basis set selection for molecular calculations . In: Chemical Reviews . tape 86 , no. 4 , August 1986, pp. 681-696 , doi : 10.1021 / cr00074a002 .
↑ Ernest Davidson , David Feller: Basis set selection for molecular calculations . In: Chem Rev.. . 86, No. 4, 1986, pp. 681-696. doi : 10.1021 / cr00074a002 .
↑ R. Ditchfield, WJ Hehre, JA Pople: Self ‐ Consistent Molecular ‐ Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules . In: The Journal of Chemical Physics . tape 54 , no. 2 , January 15, 1971, p. 724-728 , doi : 10.1063 / 1.1674902 .
↑ Damian Moran, Andrew C. Simmonett, Franklin E. III Leach, Wesley D. Allen, Paul v. R. Schleyer, Henry F. Schaefer: Popular theoretical methods predict benzene and arenes to be nonplanar . In: J. Am. Chem. Soc. . 128, No. 29, 2006, pp. 9342-9343. doi : 10.1021 / ja0630285 .
↑ Thomas H. Dunning: Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen . In: J. Chem. Phys. . 90, No. 2, 1989, pp. 1007-1023. bibcode : 1989JChPh..90.1007D . doi : 10.1063 / 1.456153 .
^ Frank Jensen: Polarization consistent basis sets: Principles . In: J. Chem. Phys. . 115, No. 20, 2001, pp. 9113-9125. bibcode : 2001JChPh.115.9113J . doi : 10.1063 / 1.1413524 .
↑ Pekka Manninen, Juha Vaara: Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties . In: J. Comput. Chem. . 27, No. 4, 2006, pp. 434-445. doi : 10.1002 / jcc.20358 . PMID 16419020 .
↑ Delano P. Chong: Completeness profiles of one-electron basis sets . In: Can. J. Chem. . 73, No. 1, 1995, pp. 79-83. doi : 10.1139 / v95-011 .
↑ Susi Lehtola: Automatic algorithms for completeness optimization of Gaussian basis sets . In: J. Comput. Chem. . 36, No. 5, 2015, pp. 335–347. doi : 10.1002 / jcc.23802 .

The numerous basic sentences discussed here and others are discussed in the sources below with reference to the original articles:

Ira N. Levine: Quantum Chemistry . Prentice Hall, Englewood Cliffs, New Jersey 1991, ISBN 978-0-205-12770-2 , pp. 461-466.
Christopher J. Cramer: Essentials of Computational Chemistry . John Wiley & Sons, Ltd., Chichester 2002, ISBN 978-0-471-48552-0 , pp. 154-168.
Frank Jensen: Introduction to Computational Chemistry . John Wiley and Sons, 1999, ISBN 978-0471980858 , pp. 150-176.
Andrew R. Leach: Molecular Modeling: Principles and Applications . Longman, Singapore 1996, ISBN 978-0-582-23933-3 , pp. 68-77.
Warren J. Hehre: A Guide to Molecular Mechanics and Quantum Chemical Calculations . Wavefunction, Inc., Irvine, California 2003, ISBN 978-1-890661-18-2 , pp. 40-47.
https://web.archive.org/web/20070830043639/http://www.chem.swin.edu.au/modules/mod8/basis1.html
Damian Moran, Andrew C. Simmonett, Franklin E. Leach, Wesley D. Allen, Paul v. R. Schleyer, Henry F. Schaefer: Popular Theoretical Methods Predict Benzene and Arenes To Be Nonplanar . In: Journal of the American Chemical Society . 128, No. 29, 2006, pp. 9342-3. doi : 10.1021 / ja0630285 . PMID 16848464 .
Sunghwan Choi, Hong Kwangwoo, Kim Jaewook, Kim Woo Youn: Accuracy of Lagrange-sinc functions as a basis set for electronic structure calculations of atoms and molecules . In: The Journal of Chemical Physics . 142, No. 9, 2015, p. 094116. bibcode : 2015JChPh.142i4116C . doi : 10.1063 / 1.4913569 . PMID 25747070 .

Web links

[1] Frank Jensen: Atomic orbital basis sets . In: WIREs Comput. Mol. Sci. . 3, No. 3, 2013, pp. 273-295. doi : 10.1002 / wcms.1123 .

[2] Ernest R. Davidson, David Feller: Basis set selection for molecular calculations . In: Chemical Reviews . tape 86 , no. 4 , August 1986, pp. 681-696 , doi : 10.1021 / cr00074a002 .

[3] Ernest Davidson , David Feller: Basis set selection for molecular calculations . In: Chem Rev.. . 86, No. 4, 1986, pp. 681-696. doi : 10.1021 / cr00074a002 .

[4] R. Ditchfield, WJ Hehre, JA Pople: Self ‐ Consistent Molecular ‐ Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules . In: The Journal of Chemical Physics . tape 54 , no. 2 , January 15, 1971, p. 724-728 , doi : 10.1063 / 1.1674902 .

[5] Damian Moran, Andrew C. Simmonett, Franklin E. III Leach, Wesley D. Allen, Paul v. R. Schleyer, Henry F. Schaefer: Popular theoretical methods predict benzene and arenes to be nonplanar . In: J. Am. Chem. Soc. . 128, No. 29, 2006, pp. 9342-9343. doi : 10.1021 / ja0630285 .

[6] Thomas H. Dunning: Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen . In: J. Chem. Phys. . 90, No. 2, 1989, pp. 1007-1023. bibcode : 1989JChPh..90.1007D . doi : 10.1063 / 1.456153 .

[7] Frank Jensen: Polarization consistent basis sets: Principles . In: J. Chem. Phys. . 115, No. 20, 2001, pp. 9113-9125. bibcode : 2001JChPh.115.9113J . doi : 10.1063 / 1.1413524 .

[8] Pekka Manninen, Juha Vaara: Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties . In: J. Comput. Chem. . 27, No. 4, 2006, pp. 434-445. doi : 10.1002 / jcc.20358 . PMID 16419020 .

[9] Delano P. Chong: Completeness profiles of one-electron basis sets . In: Can. J. Chem. . 73, No. 1, 1995, pp. 79-83. doi : 10.1139 / v95-011 .

[10] Susi Lehtola: Automatic algorithms for completeness optimization of Gaussian basis sets . In: J. Comput. Chem. . 36, No. 5, 2015, pp. 335–347. doi : 10.1002 / jcc.23802 .