﻿ Configuration Interaction

# Configuration Interaction

Configuration Interaction (CI ) is a method of solution of the Schrödinger equation (or their relativistic generalizations ), which is specially used in quantum chemistry. The many-body wave function is thereby developed in a base of Slater determinants, which the Schrödinger equation is reduced to a matrix eigenvalue problem. The ( partial) diagonalization of this matrix then provides the eigenstates of the quantum system.

## Base development, Slater determinants

The Schrödinger equation

Provides an Operator Equation for abstract vectors in a Hilbert space dar. For their solution to choose a certain representation of the wave function. A Einteilchenwellenfunktion provides one example is by developing into a base the size of a Einteilchenhilbertraums,

N- particle wave functions are made, the N -fold Cartesian product of a single-particle Hilbert space. A base of all possible products is given by the Einteilchenbasis so that the wave function can be developed as follows:

Where you, the basis vectors

Hartree products lists.

Due to the Pauli principle, the electronic wave function must be antisymmetric with respect to interchange of two particle coordinates to be, ie alive only in the sub-space of the antisymmetric functions. The Hartree products meet this requirement is not, which is why the wave function does not have to be antisymmetric. To ensure antisymmetrization, one can project the wave function. Far more often, however, one projects previously on the basis vectors, which are obtained from the Hartree- products Slater determinants,

Wherein the sum of all possible permutations is. By Slater determinants obtained a suitable basis for the development of the wave function,

Slater determinants are eigenfunctions of the projected spins, but generally not eigenfunctions of the total spin. In practice, therefore, often also selects ConfigurationState Functions ( CSF ) as base functions. A CSF can be specified as a linear combination of a few Slater determinants. Their advantage lies in the fact that the wave function is automatically an eigenfunction of the spin, and that you need less CSFs as determinants for development. However, it should be noted that the currently most successful CI codes work with Slater determinants.

## Full Configuration Interaction

The Configuration Interaction Method you get now very easy. It requires the development of the wave function into the Schrödinger equation,

And multiply it by. Because of the orthonormality of the Slater determinant ( follows from the orthonormal Einteilchenbasis ), one obtains

And thus a matrix eigenvalue problem,

In quantum chemistry, the Hamiltonian is often given by

That is, as the sum of Einteilchentermen (kinetic potential energy) as well as the two-particle Coulomb interaction. and denote the spin variables.

To determine the eigenvalue problem, matrix elements of the form must

Be calculated. The evaluation of these matrix elements is done with the Slater -Condon rules.

## Properties

The method is, in principle, exactly the only approximation is to select a last large Einteilchenbasis. A major limitation, however, is given by the scaling of the Hamiltonian. For a selected number of particles and number of basis functions, the matrix has the dimension. By taking advantage of symmetries, such as this number can be reduced but this is the exponential scaling remains.

In practice, therefore, used iterative methods for the solution of the eigenvalue problem (eg ARPACK ), or other minimization methods (eg forms of Newton's method ), which is obtained only a few eigenfunctions, typically the ground state.

In many cases, the Hamiltonian matrix, a variant is not explicitly formed, but only calculate their effect on the coefficient vector, which is called " Direct CI ".

Full -CI is size- consistent ( size consistent ), ie the energy of two subsystems is always equal to the energy of the entire system. If a smaller basis set used is the CI method is not size consistent.

## Embedding into quantum chemistry

Related methods are:

• Coupled Cluster (CC),
• Self- consistent field method (SCF ), see Hartree -Fock method,
• Møller -Plesset perturbation theory (MP ) and
• Multiconfiguration self- consistent field algorithms ( MCSCF ).
• Quantum mechanics
• Computational Chemistry
• Theoretical Chemistry
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