Full configuration interaction
Encyclopedia
Full configuration interaction (or full CI) is a linear variational approach
which provides numerically exact solutions (within the infinitely flexible complete basis set
) to the electronic time-independent, non-relativistic Schrödinger equation
.
method in which all Slater determinant
s (or configuration state function
s, CSFs) of the proper symmetry are included in the variational procedure (i.e. all Slater determinants obtained by exciting all possible electrons to all possible virtual orbitals, orbitals which are unoccupied in the electronic ground state configuration). This method is equivalent to computing the eigenvalues of the electronic molecular Hamiltonian within the basis set of the above mentioned configuration state functions.
In a minimal basis set a full CI computation is very easy. But in larger basis sets
this is usually just a limiting case which is not often attained. This is because exact solution of the full CI determinant is NP-complete
, so the existence of a polynomial time algorithm is unlikely. The Davidson correction
is a simple correction which allows one to estimate the value of the full-CI energy from a limited configuration interaction
expansion result.
Because the number of determinants required in the full-CI expansion grows factorially with the number of electrons and orbitals, full CI is only possible for atoms or very small molecules with about a dozen or fewer electrons. Full CI problems including several million up to a few billion determinants are possible using current algorithms. Because full CI results are exact within the space spanned by the orbital basis set, they are invaluable in benchmarking approximate quantum chemical methods. This is particularly important in cases such as bond-breaking reactions, diradicals, and first-row transition metals, where electronic near-degeneracies can invalidate the approximations inherent in many standard methods such as Hartree–Fock theory, multireference configuration interaction
, finite-order Møller–Plesset perturbation theory, and coupled cluster
theory.
Although fewer N-electron functions are required if one employs a basis of spin-adapted functions (Ŝ2 eigenfunctions), the most efficient full CI programs employ a Slater determinant basis because this allows for the very rapid evaluation of coupling coefficients using string-based techniques advanced by Nicholas C. Handy
in 1980. In the 1980s and 1990s, full CI programs were adapted to provide arbitrary-order Møller–Plesset perturbation theory wave functions, and in the 2000s they have been adapted to provide coupled cluster
wave functions to arbitrary orders, greatly simplifying the task of programming these complex methods.
Variational method (quantum mechanics)
In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states...
which provides numerically exact solutions (within the infinitely flexible complete 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...
) to the electronic time-independent, non-relativistic 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....
.
Explanation
It is a special case of the configuration interactionConfiguration 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 in which all Slater determinant
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
s (or configuration state function
Configuration state function
In quantum chemistry, a configuration state function , is a symmetry-adapted linear combination of Slater determinants. A CSF must not be confused with a configuration.-Definition:...
s, CSFs) of the proper symmetry are included in the variational procedure (i.e. all Slater determinants obtained by exciting all possible electrons to all possible virtual orbitals, orbitals which are unoccupied in the electronic ground state configuration). This method is equivalent to computing the eigenvalues of the electronic molecular Hamiltonian within the basis set of the above mentioned configuration state functions.
In a minimal basis set a full CI computation is very easy. But in larger basis sets
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...
this is usually just a limiting case which is not often attained. This is because exact solution of the full CI determinant is NP-complete
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
, so the existence of a polynomial time algorithm is unlikely. The Davidson correction
Davidson correction
The Davidson correction is a simple correction that is often applied in calculations using the method of configuration interaction, which is one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry. It was introduced by Ernest R...
is a simple correction which allows one to estimate the value of the full-CI energy from a limited 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...
expansion result.
Because the number of determinants required in the full-CI expansion grows factorially with the number of electrons and orbitals, full CI is only possible for atoms or very small molecules with about a dozen or fewer electrons. Full CI problems including several million up to a few billion determinants are possible using current algorithms. Because full CI results are exact within the space spanned by the orbital basis set, they are invaluable in benchmarking approximate quantum chemical methods. This is particularly important in cases such as bond-breaking reactions, diradicals, and first-row transition metals, where electronic near-degeneracies can invalidate the approximations inherent in many standard methods such as Hartree–Fock theory, multireference configuration interaction
Multireference configuration interaction
In 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...
, finite-order Møller–Plesset perturbation theory, 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.
Although fewer N-electron functions are required if one employs a basis of spin-adapted functions (Ŝ2 eigenfunctions), the most efficient full CI programs employ a Slater determinant basis because this allows for the very rapid evaluation of coupling coefficients using string-based techniques advanced by Nicholas C. Handy
Nicholas C. Handy
Nicholas Charles Handy is a Professor of quantum chemistry, University of Cambridge, UK.He has written 320 scientific papers published in physical and theoretical chemistry journals. He retired from his Cambridge Professorship in September 2004.-Contributions:He has developed several methods in...
in 1980. In the 1980s and 1990s, full CI programs were adapted to provide arbitrary-order Møller–Plesset perturbation theory wave functions, and in the 2000s they have been adapted to provide 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...
wave functions to arbitrary orders, greatly simplifying the task of programming these complex methods.