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Configuration interaction
 
 
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Configuration interaction (CI) is a post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation
Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
 within 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....
 for a quantum chemical
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
 multi-electron system. Two meanings are connected to the term configuration interaction in this context. Mathematically, configuration simply describes the linear combination of Slater determinant
Slater determinant

In quantum mechanics, a Slater determinant is an expression which describes the wavefunction of a multi-fermionic system that satisfies Skew-symmetric matrix requirements and subsequently the Pauli exclusion principle by changing Plus and minus signs upon exchange of fermions....
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).






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Configuration interaction (CI) is a post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation
Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
 within 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....
 for a quantum chemical
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
 multi-electron system. Two meanings are connected to the term configuration interaction in this context. Mathematically, configuration simply describes the linear combination of Slater determinant
Slater determinant

In quantum mechanics, a Slater determinant is an expression which describes the wavefunction of a multi-fermionic system that satisfies Skew-symmetric matrix requirements and subsequently the Pauli exclusion principle by changing Plus and minus signs upon exchange of fermions....
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
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 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. It is an eigenstate of the square of the angular momentum operator and of the square of spin operator ....
s (CSFs) built from spin orbitals (denoted by the superscript SO),

where ? 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. It is an eigenstate of the square of the angular momentum operator and of the square of spin operator ....
s of the appropriate symmetry, then this is a full configuration interaction
Full configuration interaction

Full configuration interaction is a linear variational method which provides numerically exact solutions to the electronic Schr?dinger equation....
 procedure which exactly solves the electronic Schrödinger equation
Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation 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
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 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....
 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 refers to the interaction between electrons in a quantum mechanics system whose electronic structure is being considered....
 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 also of some excited states....
). MRCI also gives better correlation of the ground state which is important if it has more than one dominant determinant. This can be easily understand 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
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
 determinant is qualitatively wrong and so are the CI wave functions and energies.

See also

  • Electron correlation
  • 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 also of some excited states....
     (MRCI)
  • 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)
  • Post Hartree-Fock
  • Quadratic configuration interaction
    Quadratic configuration interaction

    Quadratic 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
    Quantum chemistry

    Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
  • Quantum chemistry computer programs
    Quantum chemistry computer programs

    Quantum 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....