Ab initio quantum chemistry methods
Encyclopedia
Ab initio quantum chemistry methods are computational chemistry
methods based on quantum chemistry
. The term ab initio
was first used in quantum chemistry by Robert Parr
and coworkers, including David Craig
in a semiempirical study on the excited states of benzene.
The background is described by Parr. In its modern meaning ('from first principles of quantum mechanics') the term was used by Chen (when quoting an unpublished 1955 MIT report by Allen and Nesbet), by Roothaan and, in the title of an article, by Allen and Karo, who also clearly define it.
Almost always the basis set
(which is usually built from the LCAO
ansatz
) used to solve the Schrödinger equation is not complete, and does not span the Hilbert space
associated with ionization
and scattering
processes (see continuous spectrum
for more details). In the Hartree–Fock method and the configuration interaction
method, this approximation allows one to treat the Schrödinger equation
as a "simple" eigenvalue equation of the electronic molecular Hamiltonian, with a discrete
set of solutions.
tends toward the limit of a complete set. In this case, configuration interaction
, where all possible configurations are included (called "Full CI
"), tends to the exact non-relativistic solution of the electronic Schrödinger equation
(in the Born-Oppenheimer approximation
). The convergence, however, is usually not monotonic
, and sometimes the smallest calculation gives the best result for some properties.
The downside of ab initio methods is their computational cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales nominally as N4 (N being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete. However in practice it can scale closer to N3 as the program can identify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably: MP2 as N5, MP4 as N6 and coupled cluster as N7. DFT
methods using functionals which include Hartree–Fock exchange scale in a similar manner to Hartree–Fock but with a larger proportionality term and are thus more expensive than an equivalent Hartree–Fock calculation. DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.
s used to describe the interaction between electron pairs are reduced to simpler two- or three-index integrals, by treating the charge densities they contain in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix "df-", for example the density fitting MP2 is df-MP2 (many authors use lower-case to prevent confusion with DFT
). In the local approximation, the molecular orbitals are first localized by a unitary rotation in the orbital space (which leaves the reference wave function invariant, i.e., is not an approximation) and subsequently interactions of distant pairs of localized orbitals are neglected in the correlation calculation. This sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules
. Methods employing this scheme are denoted by the prefix "L", e.g. LMP2. Both schemes can be employed together, as in the df-LMP2 and df-LCCSD(T0) methods. In fact, df-LMP2 calculations are faster than df-Hartree–Fock calculations and thus are feasible in nearly all situations in which also DFT is.
electronic structure calculation is the Hartree–Fock (HF) scheme, in which the instantaneous Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect (mean field) is included in the calculation. This is a variational procedure; therefore, the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree–Fock limit as the size of the basis is increased. Many types of calculations begin with a Hartree–Fock calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation
. Møller–Plesset perturbation theory (MPn) and coupled cluster
theory (CC) are examples of these post-Hartree–Fock methods. In some cases, particularly for bond breaking processes, the Hartree–Fock method is inadequate and this single-determinant reference function is not a good basis for post-Hartree–Fock methods. It is then necessary to start with a wave function that includes more than one determinant such as multi-configurational self-consistent field
(MCSCF) and methods have been developed that use these multi-determinant references for improvements.
Example: Is Si2H2 like acetylene (C2H2)?
A series of ab initio studies of Si2H2 is an example of how ab initio computational chemistry can predict new structures that are subsequently confirmed by experiment. They go back over 20 years, and most of the main conclusions were reached by 1995. The methods used were mostly post-Hartree–Fock, particularly configuration interaction
(CI) and coupled cluster
(CC). Initially the question was whether disilyne
, Si2H2 had the same structure as ethyne (acetylene), C2H2. In early studies, by Binkley and Lischka and Kohler, it became clear that linear Si2H2 was a transition structure between two equivalent trans-bent structures and that the ground state was predicted to be a four-membered ring bent in a 'butterfly' structure with hydrogen atoms bridged between the two silicon atoms. Interest then moved to look at whether structures equivalent to vinylidene (Si=SiH2) existed. This structure is predicted to be a local minimum, i. e. an isomer of Si2H2, lying higher in energy than the ground state but below the energy of the trans-bent isomer. Then a new isomer with an unusual structure was predicted by Brenda Colegrove in Henry F. Schaefer, III
's group. It requires post-Hartree–Fock methods to obtain a local minimum for this structure. It does not exist on the Hartree–Fock energy hypersurface. The new isomer is a planar structure with one bridging hydrogen atom and one terminal hydrogen atom, cis to the bridging atom. Its energy is above the ground state but below that of the other isomers. Similar results were later obtained for Ge2H2.
Al2H2 and Ga2H2 have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules.
The only difference is that the four-membered ring ground state is planar and not bent. The cis-mono-bridged and vinylidene-like isomers are present. Experimental work on these molecules is not easy, but matrix isolation spectroscopy of the products of the reaction of hydrogen atoms and silicon and aluminium surfaces has found the ground state ring structures and the cis-mono-bridged structures for Si2H2 and Al2H2. Theoretical predictions of the vibrational frequencies were crucial in understanding the experimental observations of the spectra of a mixture of compounds. This may appear to be an obscure area of chemistry, but the differences between carbon and silicon chemistry is always a lively question, as are the differences between group 13 and group 14 (mainly the B and C differences). The silicon and germanium compounds were the subject of a Journal of Chemical Education article.
(QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo
integration. Such calculations can be very time-consuming, but they are probably the most accurate methods known today.
Computational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
methods based on quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
. The term ab initio
was first used in quantum chemistry by Robert Parr
Robert Parr
Robert Ghormley Parr is a theoretical chemist. He is a chemistry professor at the University of North Carolina at Chapel Hill.-Career:...
and coworkers, including David Craig
David P. Craig
David Parker Craig is Emeritus Professor at the Australian National University in Canberra, Australia, where he was the Foundation Professor of Physical and Theoretical Chemistry in the Research School of Chemistry....
in a semiempirical study on the excited states of benzene.
The background is described by Parr. In its modern meaning ('from first principles of quantum mechanics') the term was used by Chen (when quoting an unpublished 1955 MIT report by Allen and Nesbet), by Roothaan and, in the title of an article, by Allen and Karo, who also clearly define it.
Almost always the basis set
Basis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...
(which is usually built from the LCAO
Linear combination of atomic orbitals molecular orbital method
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...
ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
) used to solve the Schrödinger equation is not complete, and does not span the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
associated with ionization
Ionization
Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...
and scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
processes (see continuous spectrum
Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers \lambda such that A - \lambda I : H \to H is not...
for more details). In the Hartree–Fock method and the configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
method, this approximation allows one to treat the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
as a "simple" eigenvalue equation of the electronic molecular Hamiltonian, with a discrete
Discrete spectrum
In physics, an elementary explanation of a discrete spectrum is that it is an emission spectrum or absorption spectrum for which there is only an integer number of intensities. Atomic electronic absorption and emission spectrum are discrete, as contrasted with, for example, the emission spectrum...
set of solutions.
Accuracy and scaling
Ab initio electronic structure methods have the advantage that they can be made to converge to the exact solution, when all approximations are sufficiently small in magnitude and when the finite set of basis functionsBasis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...
tends toward the limit of a complete set. In this case, configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
, where all possible configurations are included (called "Full CI
Full configuration interaction
Full configuration interaction is a linear variational approach which provides numerically exact solutions to the electronic time-independent, non-relativistic Schrödinger equation....
"), tends to the exact non-relativistic solution of the electronic Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
(in the Born-Oppenheimer approximation
Born-Oppenheimer approximation
In quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation, named after Max Born and J. Robert Oppenheimer. For instance the benzene molecule consists of 12 nuclei and 42...
). The convergence, however, is usually not monotonic
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
, and sometimes the smallest calculation gives the best result for some properties.
The downside of ab initio methods is their computational cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales nominally as N4 (N being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete. However in practice it can scale closer to N3 as the program can identify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably: MP2 as N5, MP4 as N6 and coupled cluster as N7. DFT
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
methods using functionals which include Hartree–Fock exchange scale in a similar manner to Hartree–Fock but with a larger proportionality term and are thus more expensive than an equivalent Hartree–Fock calculation. DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.
Linear scaling approaches
The problem of computational expense can be alleviated through simplification schemes. In the density fitting scheme, the four-index integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s used to describe the interaction between electron pairs are reduced to simpler two- or three-index integrals, by treating the charge densities they contain in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix "df-", for example the density fitting MP2 is df-MP2 (many authors use lower-case to prevent confusion with DFT
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
). In the local approximation, the molecular orbitals are first localized by a unitary rotation in the orbital space (which leaves the reference wave function invariant, i.e., is not an approximation) and subsequently interactions of distant pairs of localized orbitals are neglected in the correlation calculation. This sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules
Biomolecule
A biomolecule is any molecule that is produced by a living organism, including large polymeric molecules such as proteins, polysaccharides, lipids, and nucleic acids as well as small molecules such as primary metabolites, secondary metabolites, and natural products...
. Methods employing this scheme are denoted by the prefix "L", e.g. LMP2. Both schemes can be employed together, as in the df-LMP2 and df-LCCSD(T0) methods. In fact, df-LMP2 calculations are faster than df-Hartree–Fock calculations and thus are feasible in nearly all situations in which also DFT is.
Hartree–Fock methods
- Hartree–Fock (HF)
- Restricted open-shell Hartree–Fock (ROHF)
- Unrestricted Hartree–Fock (UHF)
Post-Hartree–Fock methods
- Møller–Plesset perturbation theory (MPn)
- Configuration interactionConfiguration interactionConfiguration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
(CI) - Coupled clusterCoupled clusterCoupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry...
(CC) - Quadratic configuration interactionQuadratic configuration interactionQuadratic configuration interaction is an extension of Configuration interaction that corrects for size-consistency errors in the all singles and double excitation CI methods ....
(QCI) - Quantum chemistry composite methodsQuantum chemistry composite methodsQuantum chemistry composite methods are computational chemistry methods that aim for high accuracy by combining the results of several calculations. They combine methods with a high level of theory and a small basis set with methods that employ lower levels of theory with larger basis sets...
Multi-reference methods
- Multi-configurational self-consistent fieldMulti-configurational self-consistent fieldMulti-configurational self-consistent field is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate...
(MCSCF including CASSCF and RASSCF) - Multi-reference configuration interactionMultireference configuration interactionIn quantum chemistry, the multireference configuration interaction method consists in a configuration interaction expansion of the eigenstates of the electronic molecular Hamiltonian in a set of Slater determinants which correspond to excitations of the ground state electronic configuration but...
(MRCI) - n-electron valence state perturbation theoryN-Electron Valence state Perturbation TheoryIn quantum chemistry, n-electron valence state perturbation theory is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known second-order Møller-Plesset perturbation theory to multireference Complete Active Space...
(NEVPT) - Complete active space perturbation theory (CASPTn)
Hartree–Fock and Post-Hartree–Fock methods
The simplest type of ab initioAb initio
ab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...
electronic structure calculation is the Hartree–Fock (HF) scheme, in which the instantaneous Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect (mean field) is included in the calculation. This is a variational procedure; therefore, the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree–Fock limit as the size of the basis is increased. Many types of calculations begin with a Hartree–Fock calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...
. Møller–Plesset perturbation theory (MPn) and coupled cluster
Coupled cluster
Coupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry...
theory (CC) are examples of these post-Hartree–Fock methods. In some cases, particularly for bond breaking processes, the Hartree–Fock method is inadequate and this single-determinant reference function is not a good basis for post-Hartree–Fock methods. It is then necessary to start with a wave function that includes more than one determinant such as multi-configurational self-consistent field
Multi-configurational self-consistent field
Multi-configurational self-consistent field is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate...
(MCSCF) and methods have been developed that use these multi-determinant references for improvements.
Example: Is Si2H2 like acetylene (C2H2)?
A series of ab initio studies of Si2H2 is an example of how ab initio computational chemistry can predict new structures that are subsequently confirmed by experiment. They go back over 20 years, and most of the main conclusions were reached by 1995. The methods used were mostly post-Hartree–Fock, particularly configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
(CI) and coupled cluster
Coupled cluster
Coupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry...
(CC). Initially the question was whether disilyne
Disilyne
A disilyne is a chemical compound that contains a formal silicon - silicon triple bond and as such is formulated R2Si2 and is the silicon analogue of an alkyne. The hypothetical parent hydride, Si2H2 does not actually exist, all attempts to synthesize it results in a diradical disilene, with the...
, Si2H2 had the same structure as ethyne (acetylene), C2H2. In early studies, by Binkley and Lischka and Kohler, it became clear that linear Si2H2 was a transition structure between two equivalent trans-bent structures and that the ground state was predicted to be a four-membered ring bent in a 'butterfly' structure with hydrogen atoms bridged between the two silicon atoms. Interest then moved to look at whether structures equivalent to vinylidene (Si=SiH2) existed. This structure is predicted to be a local minimum, i. e. an isomer of Si2H2, lying higher in energy than the ground state but below the energy of the trans-bent isomer. Then a new isomer with an unusual structure was predicted by Brenda Colegrove in Henry F. Schaefer, III
Henry F. Schaefer, III
Henry "Fritz" Schaefer III is a computational and theoretical chemist. He is the author of a large number of scientific publications, and was the 6th most cited chemist from 1981 to 1997 and the Graham Perdue Professor of Chemistry and Director of the Center for Computational Chemistry at the...
's group. It requires post-Hartree–Fock methods to obtain a local minimum for this structure. It does not exist on the Hartree–Fock energy hypersurface. The new isomer is a planar structure with one bridging hydrogen atom and one terminal hydrogen atom, cis to the bridging atom. Its energy is above the ground state but below that of the other isomers. Similar results were later obtained for Ge2H2.
Al2H2 and Ga2H2 have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules.
The only difference is that the four-membered ring ground state is planar and not bent. The cis-mono-bridged and vinylidene-like isomers are present. Experimental work on these molecules is not easy, but matrix isolation spectroscopy of the products of the reaction of hydrogen atoms and silicon and aluminium surfaces has found the ground state ring structures and the cis-mono-bridged structures for Si2H2 and Al2H2. Theoretical predictions of the vibrational frequencies were crucial in understanding the experimental observations of the spectra of a mixture of compounds. This may appear to be an obscure area of chemistry, but the differences between carbon and silicon chemistry is always a lively question, as are the differences between group 13 and group 14 (mainly the B and C differences). The silicon and germanium compounds were the subject of a Journal of Chemical Education article.
Valence bond methods
Valence bond (VB) methods are generally ab initio although some semi-empirical versions have been proposed. Current VB approaches are:-- Generalized valence bondGeneralized valence bondThe generalized valence bond method is one of the simplest and oldest valence bond method that uses flexible orbitals in the general way used by modern valence bond theory. The method was developed by the group of William A...
(GVB) - Modern valence bond theoryModern valence bond theoryModern valence bond theory is the term used to describe applications of valence bond theory with computer programs that are competitive in accuracy and economy with programs for the Hartree-Fock method and other molecular orbital based methods. The latter methods dominated quantum chemistry from...
(MVBT)
Quantum Monte Carlo methods
A method that avoids making the variational overestimation of HF in the first place is Quantum Monte CarloQuantum Monte Carlo
Quantum Monte Carlo is a large class of computer algorithms that simulate quantum systems with the idea of solving the quantum many-body problem. They use, in one way or another, the Monte Carlo method to handle the many-dimensional integrals that arise...
(QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
integration. Such calculations can be very time-consuming, but they are probably the most accurate methods known today.
See also
- Density functional theoryDensity functional theoryDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
- Quantum chemistry computer programsQuantum chemistry computer programsQuantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock and some post-Hartree–Fock methods. They may also include density functional theory , molecular mechanics or semi-empirical quantum...
- see columns for Hartree–Fock and post-Hartree–Fock methods