Coupled Cluster

Coupled Cluster (CC) describes a method for solving the Schrödinger equation , which is used especially in quantum chemistry as one of many post-Hartee-Fock ab initio methods. It is a procedure similar to Configuration Interaction . 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.

Since the accuracy z. B. the Hartree-Fock (HF) solutions is usually not sufficient must be followed by z. B. an HF calculation a correlated calculation can be carried out, whereby the previously calculated unoccupied orbitals are used.

Basics

The CC theory is the perturbative variant of the many-electron theory (MET) by Oktay Sinanoğlu . Because the MET is difficult to calculate, CC is used in today's computational chemistry.

The CC theory promises an exact solution to the time-independent Schrödinger equation:

${\displaystyle H\vert {\Psi }\rangle =E\vert {\Psi }\rangle }$

The wave function is described in the CC theory as an exponential approach:

${\displaystyle \vert {\Psi }\rangle =e^{T}\vert {\Phi _{0}}\rangle }$,

Where , is the reference wave function (or the reference orbital), which is normally obtained as a Slater determinant from HF molecular orbitals (but other wave functions can also be used, e.g. from MCSCF calculations). is the cluster operator which, when applied to a linear combination, generates excited determinants. ${\displaystyle \vert {\Phi _{0}}\rangle }$ ${\displaystyle T}$ ${\displaystyle \vert {\Phi _{0}}\rangle }$

In contrast to other approaches such as B. the configuration interaction (CI), the exponential approach is size-consistent and -extensive.

The cluster operator is written in the following form:

${\displaystyle T=T_{1}+T_{2}+T_{3}+\cdots }$,

where the operator of all single excitations (singles), the operator of all double excitations (doubles), ... is. ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$

These excitation operators are expressed as:

${\displaystyle T_{1}=\sum _{i}\sum _{a}t_{a}^{i}{\hat {a}}^{a}{\hat {a}}_{i},}$

${\displaystyle T_{2}={\frac {1}{4}}\sum _{i,j}\sum _{a,b}t_{ab}^{ij}{\hat {a}}^{a}{\hat {a}}^{b}{\hat {a}}_{j}{\hat {a}}_{i},}$

and for the general -fold cluster operator: ${\displaystyle n}$

${\displaystyle T_{n}={\frac {1}{(n!)^{2}}}\sum _{i_{1},i_{2},\ldots ,i_{n}}\sum _{a_{1},a_{2},\ldots ,a_{n}}t_{a_{1},a_{2},\ldots ,a_{n}}^{i_{1},i_{2},\ldots ,i_{n}}{\hat {a}}^{a_{1}}{\hat {a}}^{a_{2}}\ldots {\hat {a}}^{a_{n}}{\hat {a}}_{i_{n}}\ldots {\hat {a}}_{i_{2}}{\hat {a}}_{i_{1}}.}$

In the above expression, and denote the creation and annihilation operators. And stand for occupied and as well as unoccupied (particle) orbitals. ${\displaystyle ({\hat {a}}_{a}^{\dagger }=){\hat {a}}^{a}}$ ${\displaystyle {\hat {a}}_{i}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle T_{1}}$(One-Particle Cluster Operator) and (Two-Particle Cluster Operator) convert the reference function into a linear combination of the singly and doubly excited Slater determinants when applied without an exponential function. Applying the exponential cluster operator to the wave function, one can produce determinants more than doubly excited due to the different powers of and appearing in the resulting expressions. Determining the unknown coefficients and is necessary to find the approximate solution . ${\displaystyle T_{2}}$ ${\displaystyle \vert {\Phi _{0}}\rangle }$ ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$ ${\displaystyle t_{a}^{i}}$ ${\displaystyle t_{ab}^{ij}}$ ${\displaystyle \vert {\Psi }\rangle }$

The exponential operator can be expressed in a Taylor series (only and are considered here): ${\displaystyle e^{T}}$ ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$

${\displaystyle e^{T}=1+T+{\frac {1}{2!}}T^{2}+\cdots =1+T_{1}+T_{2}+{\frac {1}{2}}T_{1}^{2}+T_{1}T_{2}+{\frac {1}{2}}T_{2}^{2}+\cdots }$

Although this series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations, it is still very large, so that even modern high-performance computers have difficulties with the calculation. The calculation is therefore terminated after a certain number of higher types of excitation. So z. B. with CCSD higher excitation types approximated by products of the singles and doubles coefficients. The development runs to infinity in singles and doubles combinations, but has no "pure" triples or quadruples etc.

nomenclature

The classification of traditional CC methods is based on the highest number of suggestions allowed in the definition of . The abbreviations for Coupled Cluster methods usually begin with the letters "CC", followed by: ${\displaystyle T}$

1. S - for a single suggestion (abbreviated to singles in the terminology of the CC method )
2. D - for double excitation ( doubles )
3. T - for triple excitation ( triples )
4. Q - for fourfold excitation ( quadruples )

So the - operator in CCSDT has the form: ${\displaystyle T}$

${\displaystyle T=T_{1}+T_{2}+T_{3}.}$

Letters in parentheses indicate that these terms are calculated based on perturbation theory. For example, CCSD (T) means:

1. CC with a full treatment of singles and doubles.
2. An estimate of the associated triple contribution is computed non-iteratively using multibody perturbation theory.

application

A CCSD calculation is often sufficient for sufficiently accurate results. However, better results will e.g. B. achieved with CCSDT or CCSDTQ, but their disadvantage is the sharp rise in costs.

Individual evidence

1. ^ ^{A b} Young, David C .: Computational chemistry: a practical guide for applying techniques to real world problems . Wiley, New York 2001, ISBN 0-471-33368-9 , pp. 25 .
2. ↑ Jiří Čížek: On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods . In: The Journal of Chemical Physics . 45, No. 11, 1966, p. 4256. bibcode : 1966JChPh..45.4256C . doi : 10.1063 / 1.1727484 .
3. ^ O. Sinanoğlu, K. Brueckner: Three approaches to electron correlation in atoms . Yale Univ. Press, 1971, ISBN 0-300-01147-4 . and references therein
4. ↑ Oktay Sinanoglu: Many-Electron Theory of Atoms and Molecules. I. Shells, Electron Pairs vs. Many-Electron Correlations . In: The Journal of Chemical Physics . 36, No. 3, 1962, p. 706. bibcode : 1962JChPh..36..706S . doi : 10.1063 / 1.1732596 .
5. ↑ ^{a b c d e f g} Jensen, Frank: Introduction to computational chemistry . Third ed. Chichester West Sussex, UK, ISBN 978-1-118-82599-0 , pp. 169-174 .
6. ↑ Trygve Helgaker, Jeppe Olsen, Poul Jorgensen: Molecular Electronic Structure Theory . Reprint edition. Wiley-Blackwell, Chichester 2013, ISBN 978-1-118-53147-1 .