Configuration Interaction

Configuration Interaction (CI) describes a method from quantum chemistry . You can see the interaction between correlated particles, e.g. B. describes electrons in a molecule , better than the Hartree-Fock method and belongs to the post-Hartree-Fock methods . To do this, it builds the wave function used from more than one electron configuration , in the form of a linear combination of Slater determinants . Variants of CI relate to the amount and type of additional configurations considered. For example, B. Full-CI all available excited states and is therefore to be calculated for almost all real systems too complex while CISD only single and double ( CI , s ingles, d includes oubles) excited states. With all variants except Full-CI, the energy does not double when the system is doubled - CI is therefore generally not size-consistent.

Base development, Slater determinants

The time-independent Schrödinger equation

{\ displaystyle {\ hat {H}} | \ Psi \ rangle = E | \ Psi \ rangle}

(or its relativistic generalizations), which is used especially in quantum chemistry , represents an operator equation for abstract vectors in a Hilbert space . A certain representation of the wave function is selected for its solution. A single particle wave function can be e.g. B. by expanding into a basis of magnitude on a one-particle Hilbert space : ${\ displaystyle \ {| \ phi _ {k} \ rangle \}}$ ${\ displaystyle N_ {b}}$ ${\ displaystyle {\ mathcal {H}}}$

{\ displaystyle | \ Psi \ rangle = \ sum _ {k} c_ {k} \, | \ phi _ {k} \ rangle}

${\ displaystyle N}$ -Particle wave functions are elements of the tensor product space , which is composed of the respective single-particle Hilbert spaces. A basis of is given by all possible products of the one-particle basis, so that the wave function can be expanded as follows: ${\ displaystyle {\ mathcal {H}} _ {N} = {\ mathcal {H}} \ otimes {\ mathcal {H}} \ otimes \ cdots \ otimes {\ mathcal {H}}}$ ${\ displaystyle {\ mathcal {H}} _ {N}}$

{\ displaystyle | \ Psi \ rangle = \ sum _ {k_ {1} \ cdots k_ {N}} c_ {k_ {1} \ cdots k_ {N}} \, | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle}

The basis vectors

{\ displaystyle | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle \ = \ | \ phi _ {k_ {1}} \ rangle \ cdots | \ phi _ {k_ { N}} \ rangle}

referred to as Hartree products.

Due to the Pauli principle , the electronic wave function must be antisymmetric to the exchange of two particle coordinates, i.e. H. lives only in the subspace of the antisymmetric functions. The Hartree products do not meet this requirement, which is why the wave function does not have to be antisymmetric. In order to ensure the antisymmetrization, the wave function can be projected onto. Much more often, however, the basis vectors are projected beforehand , whereby Slater determinants are obtained from the Hartree products , ${\ displaystyle | \ Psi \ rangle}$ ${\ displaystyle {\ mathcal {H}} _ {N} ^ {-}}$ ${\ displaystyle {\ mathcal {H}} _ {N} ^ {-}}$ ${\ displaystyle {\ mathcal {H}} _ {N} ^ {-}}$ ${\ displaystyle N_ {b} ^ {\, N}}$ ${\ displaystyle {\ binom {2N_ {b}} {N}}}$

{\ displaystyle | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle = {\ frac {1} {\ sqrt {N!}}} \, \ sum _ {\ sigma \ in {\ mathcal {S}} _ {N}} {\ text {sign}} (\ sigma) | \ phi _ {\ sigma (k_ {1})} \ rangle \ cdots | \ phi _ {\ sigma (k_ {N})} \ rangle}

where the sum goes over all possible permutations. The Slater determinants provide a suitable basis for developing the wave function,

{\ displaystyle | \ Psi \ rangle = \ sum _ {k_ {1} <k_ {2} <\ cdots <k_ {N}} c_ {k_ {1} \ cdots k_ {N}} \, | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle}

Slater determinants are eigenfunctions of the projected spin , but generally not eigenfunctions of the total spin . In practice, configuration state functions (CSF) are therefore often chosen as basic functions. A CSF can be expressed as a linear combination of a few Slater determinants. Their advantage is that the wave function is automatically an eigenfunction of the spin and that fewer CSFs are needed as determinants for development. However, it should be mentioned that the most successful CI codes currently work with Slater determinants. ${\ displaystyle {\ hat {S}} _ {z}}$ ${\ displaystyle {\ hat {S}} ^ {2}}$

The orbitals of an optimized Hartree-Fock wave function are usually chosen as the orbital base.

Full configuration interaction

The Configuration Interaction method is now very easy to get. The expansion of the wave function is inserted into the Schrödinger equation,

{\ displaystyle \ sum _ {k_ {1} <\ cdots <k_ {N}} {\ hat {H}} c_ {k_ {1} \ cdots k_ {N}} | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle = E \ sum _ {k_ {1} <\ cdots <k_ {N}} c_ {k_ {1} \ cdots k_ {N}} | \ phi _ { k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle}

and multiply them by . Because of the orthonormality of the Slater determinant (follows from the orthonormal one-particle basis) one obtains ${\ displaystyle \ langle \ phi _ {j_ {1}} \ cdots \ phi _ {j_ {N}} |}$

{\ displaystyle \ sum _ {k_ {1} <\ cdots <k_ {N}} c_ {k_ {1} \ cdots k_ {N}} \ langle \ phi _ {j_ {1}} \ cdots \ phi _ { j_ {N}} | {\ hat {H}} | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle = Ec_ {j_ {1} \ cdots j_ {N}} }

and thus a matrix eigenvalue problem,

{\ displaystyle \ mathbf {H} \ mathbf {c} = E \ mathbf {c}}

The many-body wave function is in this case in a base of Slater determinants developed, whereby the Schrodinger equation to a matrix - eigenvalue problem is reduced. The (partial) diagonalization of this matrix then provides the eigenstates of the quantum mechanical system.

In quantum chemistry, the Hamiltonian is often given by

{\ displaystyle {\ hat {H}} = \ sum _ {pq} h_ {pq} \ sum _ {\ sigma} {\ hat {a}} _ {p \ sigma} ^ {\ dagger} {\ hat { a}} _ {q \ sigma} + {\ frac {1} {2}} \ sum _ {pqrs} g_ {pqrs} \ sum _ {\ sigma \ tau} {\ hat {a}} _ {p \ sigma} ^ {\ dagger} {\ hat {a}} _ {r \ tau} ^ {\ dagger} {\ hat {a}} _ {s \ tau} {\ hat {a}} _ {q \ sigma } \ ,,}

d. H. as the sum of one-particle terms (kinetic + potential energy) and the two -particle Coulomb interaction . and denote the spin variables. ${\ displaystyle \ sigma}$ ${\ displaystyle \ tau}$

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

{\ displaystyle \ langle \ phi _ {j_ {1}} \ cdots \ phi _ {j_ {N}} | {\ hat {H}} | \ phi _ {k_ {1}} \ cdots \ phi _ {k_ {N}} \ rangle}

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

properties

The method is in principle exact, the only approximation consists in the choice of a finitely large single-particle basis. As a result, the wave function is not an eigenfunction of the Hamilton operator. One major limitation is the scaling of the Hamilton matrix. For a selected number of particles and number of basis functions the matrix has the dimension . By taking advantage of symmetries, e.g. B. Although this number can be reduced, the exponential scaling remains. ${\ displaystyle N}$ ${\ displaystyle N_ {b}}$ ${\ displaystyle {\ binom {2N_ {b}} {N}}}$ ${\ displaystyle [{\ hat {H}}, {\ hat {S}} _ {z}] = [{\ hat {H}}, {\ hat {S}} ^ {2}] = \ dots = 0}$

In practice, therefore, iterative methods are used to solve the eigenvalue problem (e.g. Arpack), or other minimization methods (e.g. forms of Newton's method ) with which only a few eigenfunctions are obtained, typically the ground state.

In many cases, the Hamilton matrix is not explicitly formed, but only its effect on the coefficient vector is calculated, a variant called "Direct CI".

Due to the exponential scaling, the CI development is usually terminated at a certain point in practice. The determinants or CSFs are classified according to how many "suggestions" (formal application of ladder operators) they can be generated from the reference determinant. CIS describes a CI development that is terminated after the "singles", i.e. the single stimuli, while CISD also contains the "doubles". CIS often represents a simple approximation for describing the first excited states of molecules, but does not provide an improved description of the ground state (when using converged Hartree-Fock orbitals), since the corresponding matrix elements are equal to zero due to the Brillouin theorem . In the CIS formalism, the excitation energy can be specified as follows:

${\ displaystyle \ omega _ {CIS} = \ sum _ {ia} (c_ {i} ^ {a}) (\ epsilon _ {a} - \ epsilon _ {i}) + \ sum _ {ia, jb} c_ {i} ^ {a} c_ {j} ^ {b} (ia || jb)}$

Here, i , j and occupied a , b unoccupied orbitals. Assuming that a configuration is dominant, the excitation energy results from the sum of the corresponding orbital difference and the two-electron integrals, which (partially) take into account the changed electron-electron interaction (due to the excitation).

Full-CI is size consistent, i.e. H. the energy of two subsystems is always equal to the energy of the entire system. On the other hand, if the CI development is terminated beforehand, the CI method (apart from CIS) is not size-consistent.

Embedding in quantum chemistry

Related methods are:

Coupled Cluster (CC),
Self-Consistent Field Method (SCF),
Møller-Plesset perturbation calculation (MP) and
Multiconfiguration Self-Consistent-Field Algorithms (MCSCF).

Individual evidence

↑ ^a ^b Trygve Helgaker, Jeppe Olsen, Poul Jorgensen: Molecular Electronic Structure Theory . Reprint edition. Wiley-Blackwell, Chichester 2013, ISBN 978-1-118-53147-1 ( amazon.de [accessed December 19, 2018]).
↑ Andreas Dreuw, Martin Head-Gordon: Single-Reference from Initio Methods for the Calculation of Excited States of Large Molecules . In: Chemical Reviews . tape 105 , no. 11 , November 1, 2005, ISSN 0009-2665 , p. 4009-4037 , doi : 10.1021 / cr0505627 .

[:0-1] Trygve Helgaker, Jeppe Olsen, Poul Jorgensen: Molecular Electronic Structure Theory . Reprint edition. Wiley-Blackwell, Chichester 2013, ISBN 978-1-118-53147-1 ( amazon.de [accessed December 19, 2018]).

[2] Andreas Dreuw, Martin Head-Gordon: Single-Reference from Initio Methods for the Calculation of Excited States of Large Molecules . In: Chemical Reviews . tape 105 , no. 11 , November 1, 2005, ISSN 0009-2665 , p. 4009-4037 , doi : 10.1021 / cr0505627 .