Configuration interaction
Encyclopedia
Configuration interaction (CI) 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 of Slater determinant
s used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)2(2s)2(2p)1...), interaction means the mixing (interaction) of different electronic configurations (states). Due to the long CPU time and immense hardware required for CI calculations, the method is limited to relatively small systems.
In contrast to the Hartree–Fock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state function
s (CSFs) built from spin orbitals (denoted by the superscript SO),
where Ψ is usually the electronic ground state of the system. If the expansion includes all possible CSF
s of the appropriate symmetry, then this is a full configuration interaction
procedure which exactly solves the electronic Schrödinger equation
within the space spanned by the one-particle basis set. The first term in the above expansion is normally the Hartree–Fock determinant. The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree–Fock determinant. If only one spin orbital differs, we describe this as a single excitation determinant. If two spin orbitals differ it is a double excitation determinant and so on. This is used to limit the number of determinants in the expansion which is called the CI-space.
Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. Single excitations on their own do not mix with the Hartree–Fock determinant. These methods, CID and CISD, are in many standard programs. The Davidson correction
can be used to estimate a correction to the CISD energy to account for higher excitations. An important problem of truncated CI methods is their size-inconsistency
which means the energy of two infinitely separated particles is not double the energy of the single particle.
The CI procedure leads to a general matrix eigenvalue equation:
where c is the coefficient vector, e is the eigenvalue matrix, and the elements of the hamiltonian and overlap matrices are, respectively,
,
.
Slater determinants are constructed from sets of orthonormal spin orbitals, so that
, making 
the identity matrix and simplifying the above matrix equation.
The solution of the CI procedure are some eigenvalues
and their corresponding eigenvectors 
.
The eigenvalues are the energies of the ground and some electronically excited state
s. By this it is possible to calculate energy differences (excitation energies) with CI methods. Excitation energies of truncated CI methods are generally too high, because the excited states are not that well correlated
as the ground state is. For equally (balanced) correlation of ground and excited states (better excitation energies) one can use more than one reference determinant from which all singly, doubly, ... excited determinants are included (multireference configuration interaction
).
MRCI also gives better correlation of the ground state which is important if it has more than one dominant determinant. This can be easily understood because some higher excited determinants are also taken into the CI-space.
For nearly degenerate determinants which build the ground state one should use the multi-configurational self-consistent field
(MCSCF) method because the Hartree–Fock determinant is qualitatively wrong and so are the CI wave functions and energies.
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....
within the Born–Oppenheimer approximation for a quantum chemical
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...
multi-electron system. Mathematically, configuration simply describes the linear combination of 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 used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)2(2s)2(2p)1...), interaction means the mixing (interaction) of different electronic configurations (states). Due to the long CPU time and immense hardware required for CI calculations, the method is limited to relatively small systems.
In contrast to the Hartree–Fock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of 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) built from spin orbitals (denoted by the superscript SO),
where Ψ is usually the electronic ground state of the system. If the expansion includes all possible CSF
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 of the appropriate symmetry, then this is a full configuration interaction
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....
procedure which exactly solves 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....
within the space spanned by the one-particle basis set. The first term in the above expansion is normally the Hartree–Fock determinant. The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree–Fock determinant. If only one spin orbital differs, we describe this as a single excitation determinant. If two spin orbitals differ it is a double excitation determinant and so on. This is used to limit the number of determinants in the expansion which is called the CI-space.
Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. Single excitations on their own do not mix with the Hartree–Fock determinant. These methods, CID and CISD, are in many standard programs. 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...
can be used to estimate a correction to the CISD energy to account for higher excitations. An important problem of truncated CI methods is their size-inconsistency
Size consistency
In quantum chemistry, size consistency is a property that guarantees the consistency of the energy behavior when interaction between the involved molecular system is nullified ....
which means the energy of two infinitely separated particles is not double the energy of the single particle.
The CI procedure leads to a general matrix eigenvalue equation:
where c is the coefficient vector, e is the eigenvalue matrix, and the elements of the hamiltonian and overlap matrices are, respectively,
,.Slater determinants are constructed from sets of orthonormal spin orbitals, so that
, making the identity matrix and simplifying the above matrix equation.The solution of the CI procedure are some eigenvalues
and their corresponding eigenvectors .The eigenvalues are the energies of the ground and some electronically excited state
Excited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
s. By this it is possible to calculate energy differences (excitation energies) with CI methods. Excitation energies of truncated CI methods are generally too high, because the excited states are not that well correlated
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...
as the ground state is. For equally (balanced) correlation of ground and excited states (better excitation energies) one can use more than one reference determinant from which all singly, doubly, ... excited determinants are included (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...
).
MRCI also gives better correlation of the ground state which is important if it has more than one dominant determinant. This can be easily understood because some higher excited determinants are also taken into the CI-space.
For nearly degenerate determinants which build the ground state one should use the 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) method because the Hartree–Fock determinant is qualitatively wrong and so are the CI wave functions and energies.
See also
- 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...
- Electron correlation
- Multireference 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) - 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) - Post-Hartree–Fock
- 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 chemistryQuantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
- 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...