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Coupled cluster

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Coupled cluster

Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry. It essentially takes the basic Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation. Some of the most accurate calculations for small to medium-sized molecules use this method.[1][2][3]

The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics phenomena, but became more frequently used when in 1966 Jiři Čížek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation. CC theory is simply the perturbative variant of the Many Electron Theory (MET) of Oktay Sinanoğlu, which is the exact (and variational) solution of the many electron problem, so it was also called "Coupled Pair MET (CPMET)". J. Čížek used the correlation function of MET and used Goldstone type perturbation theory to get the energy expression while original MET was completely variational. Čížek first developed the Linear-CPMET and then generalized it to full CPMET in the same paper in 1966. He then also performed an application of it on benzene molecule with O. Sinanoğlu in the same year. Because MET is somewhat difficult to perform computationally, CC is simpler and thus, in today's computational chemistry, CC is the best variant of MET and gives highly accurate results in comparison to experiments.[4][5][6]

Contents

  • Wavefunction ansatz 1
  • Cluster operator 2
  • Coupled-cluster equations 3
  • Types of coupled-cluster methods 4
  • General description of the theory 5
  • Historical accounts 6
  • Relation to other theories 7
    • Configuration Interaction 7.1
    • Symmetry Adapted Cluster 7.2
  • See also 8
  • References 9
  • External resources 10

Wavefunction ansatz

Coupled-cluster theory provides the exact solution to the time-independent Schrödinger equation

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

where H is the Hamiltonian of the system, \vert{\Psi}\rangle the exact wavefunction, and E the exact energy of the ground state. Coupled-cluster theory can also be used to obtain solutions for excited states using, for example, linear-response,[7] equation-of-motion,[8] state-universal multi-reference coupled cluster,[9] or valence-universal multi-reference coupled cluster[10] approaches.

The wavefunction of the coupled-cluster theory is written as an exponential ansatz:

\vert{\Psi}\rangle = e^{T} \vert{\Phi_0}\rangle ,

where \vert{\Phi_0}\rangle, the reference wave function, which is typically a Slater determinant constructed from Hartree–Fock molecular orbitals, though other wave functions such as Configuration interaction, Multi-configurational self-consistent field, or Brueckner orbitals can also be used. T is the cluster operator which, when acting on \vert{\Phi_0}\rangle, produces a linear combination of excited determinants from the reference wave function (see section below for greater detail).

The choice of the exponential ansatz is opportune because (unlike other ansatzes, for example, configuration interaction) it guarantees the size extensivity of the solution. Size consistency in CC theory, also unlike other theories, does not depend on the size consistency of the reference wave function. This is easily seen, for example, in the single bond breaking of F_2 when using a restricted Hartree-Fock (RHF) reference, which is not size consistent, at the CCSDT level of theory which provides an almost exact, full CI-quality, potential energy surface and does not dissociate the molecule into F^{-} and F^{+} ions, like the RHF wave function, but rather into two neutral F atoms.[11] If one were to use, for example, the CCSD, CCSD[T], or CCSD(T) levels of theory, they would not provide reasonable results for the bond breaking of F_2, with the latter two approaches providing unphysical potential energy surfaces,[12] though this is for reasons other than just size consistency.

A criticism of the method is that the conventional implementation employing the similarity-transformed Hamiltonian (see below) is not variational, though there are bi-variational and quasi-variational approaches that have been developed since the first implementations of the theory. While the above ansatz for the wave function itself has no natural truncation, however, for other properties, such as energy, there is a natural truncation when examining expectation values, which has its basis in the linked- and connected-cluster theorems, and thus does not suffer from issues such as lack of size extensivity, like the variational configuration interaction approach.

Cluster operator

The cluster operator is written in the form,

T=T_1 + T_2 + T_3 + \cdots ,

where T_1 is the operator of all single excitations, T_2 is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are expressed as

T_1=\sum_{i}\sum_{a} t_{a}^{i} \hat{a}^{a}\hat{a}_{i},
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 n-fold cluster operator

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 formulae (\hat{a}^{\dagger}_{a} =) \hat{a}^{a} and \hat{a}_{i} denote the creation and annihilation operators respectively and i, j stand for occupied (hole) and a, b for unoccupied (particle) orbitals (states). The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in the normal order form, with respect to the Fermi vacuum, \vert{\Phi_0}\rangle. Being the one-particle cluster operator and the two-particle cluster operator, T_1 and T_2 convert the reference function \vert{\Phi_0}\rangle into a linear combination of the singly and doubly excited Slater determinants, respectively, if applied without the exponential (such as in CI where a linear excitation operator is applied to the wave function). Applying the exponential cluster operator to the wave function, one can then generate more than doubly excited determinants due to the various powers of T_1 and T_2 that appear in the resulting expressions (see below). Solving for the unknown coefficients t_{a}^{i} and t_{ab}^{ij} is necessary for finding the approximate solution \vert{\Psi}\rangle.

The exponential operator e^{T} may be expanded as a Taylor series and if we consider only the T_1 and T_2 cluster operators of T, we can write:

e^{T} = 1 + T + \frac{1}{2!}T^2 + \cdots = 1 + T_1 + T_2 + \frac{1}{2}T_1^2 + T_1T_2 + \frac{1}{2}T_2^2 + \cdots

Though 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, to the extent that even modern day massively parallel computers are inadequate, except for problems of a dozen or so electrons and very small basis sets, when considering all contributions to the cluster operator and not just T_1 and T_2. Often, as was done above, the cluster operator includes only singles and doubles (see CCSD below) as this offers a computationally affordable method that performs better than MP2 and CISD, but is not very accurate usually. For accurate results some form of triples (approximate or full) are needed, even near the equilibrium geometry (in the Franck-Condon region), and especially when breaking single-bonds or describing diradical species (these latter examples are often what is referred to as multi-reference problems, since more than one determinant has a significant contribution to the resulting wave function). For double bond breaking, and more complicated problems in chemistry, quadruple excitations often become important as well, though usually they are small for most problems, and as such, the contribution of T_5, T_6 etc. to the operator T is typically small. Furthermore, if the highest excitation level in the T operator is n,

T = T_1 + ... + T_n

then Slater determinants for an N-electron system excited more than n (< N) times may still contribute to the coupled cluster wave function \vert{\Psi}\rangle because of the non-linear nature of the exponential ansatz, and therefore, coupled cluster terminated at T_n usually recovers more correlation energy than CI with maximum n excitations.

Coupled-cluster equations

The Schrödinger equation can be written, using the coupled-cluster wave function, as

H \vert{\Psi_{0}}\rangle = H e^{T} \vert{\Phi_0}\rangle = E e^{T} \vert {\Phi_0}\rangle

where there are a total of q coefficients (t-amplitudes) to solve for. To obtain the q equations, first, we multiply the above Schrödinger equation on the left by e^{-T} and then project onto the entire set of up to m-tuply excited determinants, where m is the highest order excitation included in T, that can be constructed from the reference wave function \vert{\Phi_0}\rangle, denoted by \vert{\Phi^{*}}\rangle, and individually, \vert{\Phi_{i}^{a}}\rangle are singly excited determinants where the electron in orbital i has been excited to orbital a; \vert{\Phi_{ij}^{ab}}\rangle are doubly excited determinants where the electron in orbital i has been excited to orbital a and the electron in orbital j has been excited to orbital b, etc. In this way we generate a set of coupled energy-independent non-linear algebraic equations needed to determine the t-amplitudes.

\langle {\Phi_0}\vert e^{-T}He^{T} \vert{\Phi_0}\rangle = E \langle {\Phi_{0}}\vert {\Phi_0}\rangle = E
\langle {\Phi^{*}}\vert e^{-T}He^{T} \vert{\Phi_0}\rangle = E \langle {\Phi^{*}}\vert {\Phi_0}\rangle = 0 ,

(note, we have made use of e^{-T} e^{T}=1, the identity operator, and we are also assuming that we are using orthogonal orbitals, though this does not necessarily have to be true, e.g., valence bond orbitals, and in such cases the last set of equations are not necessarily equal to zero) the latter being the equations to be solved and the former the equation for the evaluation of the energy.

Considering the basic CCSD method:

\langle {\Phi_0}\vert e^{-(T_1+T_2)}He^{(T_1+T_2)} \vert{\Phi_0}\rangle = E ,
\langle {\Phi_{i}^{a}}\vert e^{-(T_1+T_2)}He^{(T_1+T_2)} \vert{\Phi_0}\rangle =0,
\langle {\Phi_{ij}^{ab}}\vert e^{-(T_1+T_2)}He^{(T_1+T_2)} \vert{\Phi_0}\rangle =0,

in which the similarity transformed Hamiltonian, \bar{H}, can be explicitly written down using Hadamard's formula in Lie algebra, also called Hadamard's lemma (see also Baker–Campbell–Hausdorff formula (BCH formula), though note they are different, in that Hadamard's formula is a lemma of the BCH formula):

\bar{H} = e^{-T} H e^{T} = H + [H,T] + \frac{1}{2!}^{a_{1}\ldots a_{n}}\vert(H-E_{0})e^{S}\vert\Phi\rangle = 0,
i_{1}<\cdots, a_{1}<\cdots, n=1,\dots,M_{s},

where \vert\Phi_{i_{1}\ldots i_{n}}^{a_{1}\ldots a_{n}}\rangle are the n-tuply excited determinants relative to \vert\Phi\rangle (usually they are the spin- and symmetry-adapted configuration state functions, in practical implementations), and M_{s} is the highest-order of excitation included in the SAC operator. If all of the nonlinear terms in e^{S} are included then the SAC equations become equivalent to the standard coupled-cluster equations of Jiři Čížek. This is due to the cancellation of the energy-dependent terms with the disconnected terms contributing to the product of He^{S}, resulting in the same set of nonlinear energy-independent equations. Typically, all nonlinear terms, except \frac{1}{2}S_{2}^{2} are dropped, as higher-order nonlinear terms are usually small.[20]

See also

References

  1. ^ Kümmel, H. G. (2002). "A biography of the coupled cluster method". In Bishop, R. F.; Brandes, T.; Gernoth, K. A.; Walet, N. R.; Xian, Y. Recent progress in many-body theories Proceedings of the 11th international conference. Singapore: World Scientific Publishing. pp. 334–348.  
  2. ^ Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 191–232.  
  3. ^ Shavitt, Isaiah; Bartlett, Rodney J. (2009). Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory. Cambridge University Press.  
  4. ^ Čížek, Jiří (1966). "On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods". The Journal of Chemical Physics 45 (11): 4256.  
  5. ^ Sinanoğlu, O.; Brueckner, K. (1971). Three approaches to electron correlation in atoms. Yale Univ. Press.   and references therein
  6. ^ Si̇nanoğlu, Oktay (1962). "Many-Electron Theory of Atoms and Molecules. I. Shells, Electron Pairs vs Many-Electron Correlations". The Journal of Chemical Physics 36 (3): 706.  
  7. ^ Monkhorst, H.J. (1977). "Calculation of properties with the coupled-cluster method". International Journal of Quantum Chemistry. 12, S11: 421.  
  8. ^ Stanton, John F.; Bartlett, Rodney J. (1993). "The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties". The Journal of Chemical Physics 98 (9): 7029.  
  9. ^ Jeziorski, B.; Monkhorst, H. (1981). "Coupled-cluster method for multideterminantal reference states". Physical Review A 24 (4): 1668.  
  10. ^ Lindgren, D.; Mukherjee, Debashis (1987). "On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces". Physics Reports 151 (2): 93.  
  11. ^ Kowalski, K.; Piecuch, P. (2001). "A comparison of the renormalized and active-space coupled-cluster methods: Potential energy curves of BH and F2". Chemical Physics Letters 344: 165. 
  12. ^ Ghose, K.B.; Piecuch, P.; Adamowicz, L. (1995). "Improved computational strategy for the state‐selective coupled‐cluster theory with semi‐internal triexcited clusters: Potential energy surface of the HF molecule". Journal of Physical Chemistry 103: 9331. 
  13. ^ Monkhorst, Hendrik J (1987). "Chemical physics without the Born-Oppenheimer approximation: The molecular coupled-cluster method". Physical Review A 36 (4): 1544–1561.  
  14. ^ Nakai, Hiromi; Sodeyama, Keitaro (2003). "Many-body effects in nonadiabatic molecular theory for simultaneous determination of nuclear and electronic wave functions: Ab initio NOMO/MBPT and CC methods". The Journal of Chemical Physics 118 (3): 1119.  
  15. ^ Paldus, J. (2005). "The beginnings of coupled-cluster theory: an eyewitness account". In Dykstra, C. Theory and Applications of Computational Chemistry: The First Forty Years. Elsivier B.V. p. 115. 
  16. ^ Paldus, J (1981). Diagrammatic Methods for Many-Fermion Systems (Lecture Notes ed.). University of Nijmegen, Njimegen, The Netherlands. 
  17. ^ Bartlett, R.J.; Dykstra, C.E.; Paldus, J. (1984). Dykstra, C.E., ed. Advanced Theories and Computational Approaches to the Electronic Structure of Molecules. p. 127. 
  18. ^ Nakatsuji, H.; Hirao, K. (1977). "Cluster expansion of the wavefunction. Pseduo-orbital theory applied to spin correlation". Chemical Physics Letters 47 (3): 569.  
  19. ^ Nakatsuji, H.; Hirao, K. (1978). "Cluster expansion of the wavefunction. Symmetry‐adapted‐cluster expansion, its variational determination, and extension of open‐shell orbital theory". Journal of Chemical Physics 68 (5): 2053.  
  20. ^ Ohtsuka, Y.; Piecuch, P.; Gour, J.R.; Ehara, M.; Nakatsuji, H. (2007). "Active-space symmetry-adapted-cluster configuration-interaction and equation-of-motion coupled-cluster methods for high accuracy calculations of potential energy surfaces of radicals". Journal of Chemical Physics 126 (16): 164111.  

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