CoupledCluster

Table of Contents

1. Brief description

This algorithm solves the CCSD amplitude equations using the Coulomb integrals and Hartree–Fock eigenenergies.

2. Algorithm call

A typical input file snippet to call the CoupledCluster algorithm is given below.

- name: CoupledCluster
  in:
    method: Ccsd
    integralsSliceSize: 100
    slicedEigenEnergies: EigenEnergies
    coulombIntegrals: CoulombIntegrals
    slicedCoulombVertex: CoulombVertex
    maxIterations: 20
    energyConvergence: 1.0E-4
    amplitudesConvergence: 1.0E-4
    mixer:
      type: DiisMixer
      maxResidua: 5
    #or for instance a linear mixer
    #mixer:
    #  type: LinearMixer
    #  ratio: 1.0
  out:
    amplitudes: Amplitudes

3. Algorithm input

Table 1: Input keywords
Keyword Value
method Type of coupled cluster algorithm
linearized Use linearized or fully non-linear coupled cluster
integralsSliceSize Integral slice size
slicedEigenEnergies Sliced one-electron energies
coulombIntegrals Coulomb Integrals
slicedCoulombVertex Sliced Coulomb vertex
maxIterations Maximum number of iterations
energyConvergence Energy convergence threshold
amplitudesConvergence Amplitude convergence threshold
mixer Mixer type and parameters
initialAmplitudes initial Amplitudes to begin the iterations

3.1. method

method specifies which approximation of the coupled-cluster Amplitudes should be used. See the Amplitudes for the specification of each approximation. The currently available approximations are:

Ccsd
Coupled Cluster singles and doubles. See (Bartlett and Musiał 2007) and references therin.
CcsdReference
Canonical and unoptimized version of the Ccsd algorithm. This algorithm has the same inputs as Ccsd with the exception that the arguments integralSliceSize and slicedCoulombVertex and are ignored.
Drccd
Unoptimized implementation of Direct-Ring Coupled Cluster singles and doubles giving the Random Phase Approximation (RPA) with the Second Order Screened Exchange (SOSEX) correction (Freeman 1977; Grüneis et al. 2009).

3.2. linearzied

linearized A non-zero value indicates that the linearized coupled-cluster amplitude equations should be solved, rather than the fully non-linear ones. The default is linearized: 0. Note: currently only method: Drccd supports linearized amplitude equations.

3.3. integralsSliceSize

integralsSliceSize controls the slice size of the \(V_{cd}^{ab}\) integrals, which are computed on-the-fly to reduce the memory footprint. The integer value specified for integralsSliceSize refers to the dimension size used for the \(a\) and \(b\) index. We recommend setting integralsSliceSize: 100 to balance computational efficency with memory usage. Smaller/larger values reduce/increase the memory footprint.

3.4. maxIterations

maxIterations controls the maximum number of iterations allowed to solve the \(t_{ij}^{ab}\) and \(t_i^a\) amplitude equations. If convergence of the energy and residual vectors within the specified thresholds is achieved using fewer iterations than maxIterations, the algorithm was successfull and will stop. If maxIterations is reached without achieving energyConvergence and amplitudesConvergence , the algorithm was not successful and will stop. We recommend to set maxIterations: 20, which is ususally enough to achieve reasonable convergence thresholds.

3.5. energyConvergence

energyConvergence specifies the convergence threshold for the correlation energy. If energyConvergence and amplitudesConvergence is achieved, the iterative solution was successful and the algorithm will stop.

3.6. amplitudesConvergence

amplitudesConvergence specifies the convergence threshold for the residual vector of the singles and doubles amplitude equations. If energyConvergence and amplitudesConvergence is achieved, the iterative solution was successful and the algorithm will stop.

3.7. initialAmplitudes

initialAmplitudes specifies amplitudes from a precedent calculations. This allows restarting a self consistent calculation from a former checkpoint.

3.8. mixer

mixer specifies mixer-specific parameters used to solve the amplitude equations iteratively. The direct inversion iterative subspace (Diis) algorithm is the default algorithm used to mix amplitude guesses from previous iterations to improve the guess for the next iteration. mixer: type can currently be set to *DiisMixer or *LinearMixer.

3.8.1. DiisMixer

For mixer: type: DissMixer, it is possible to specify mixer: maxResidua, which controls the number of residual vectors used by the Diis mixer. We recommend to set the maximum number of residual vectors used in the DiisMixer to 5. More residual vectors result in a larger memory footprint.

Example
mixer:
  type: DiisMixer
  maxResidua: 5

3.8.2. LinearMixer

For mixer: type: LinearMixer it is possible to specify mixer: ratio, which controls the mixing ratio used to update the new guess of the amplitudes based on estimates from the previous and current iteration. The LinearMixer exhibits the smallest possible memory footprint from all currently available mixers, keeping two sets of amplitudes in memory at once only. We recommend to set mixer: ratio: 1.0. Smaller ratios result in a slower but perhaps more stable convergence.

Example
mixer:
  type: LinearMixer
  ratio: 1.0

4. Algorithm output

Table 2: Output keywords for CoupledCluster
Keyword Value
amplitudes Amplitudes

The output of the CoupledCluster algorithm includes energy and amplitudes . The amplitudes output contains the converged singles and doubles amplitude tensors. The amplitudes can be used as input for algorithms that estimate the finite simulation cell size error (FiniteSizeCorrection) and the basis set incompleteness errors (BasisSetCorrection).

4.1. Sample stdout

Below an example standard output stream is shown for a successful CoupledCluster algorithm run.

step: 6, CoupledCluster
Using method Ccsd. integralsSliceSize: 100
Using mixer DiisMixer. maxResidua: 5
Maximum number of iterations: 30
Unless reaching energy convergence dE: 1e-05
and amplitudes convergence dR: 1e-05
Iter         Energy         dE           dR         time   GF/s/core
   1  -2.43605043e+01  -2.4361e+01   4.3924e-01      0.5    1.0
   2  -2.47577534e+01  -3.9725e-01   7.4733e-02      0.8    4.8
   3  -2.53776918e+01  -6.1994e-01   1.8674e-02      0.7    5.4
   4  -2.54455925e+01  -6.7901e-02   6.4132e-03      0.7    5.5
   5  -2.54445080e+01   1.0845e-03   2.2120e-03      0.7    5.4
   6  -2.54458312e+01  -1.3232e-03   1.0304e-03      0.7    5.4
   7  -2.54448941e+01   9.3705e-04   5.0727e-04      0.7    5.4
   8  -2.54452894e+01  -3.9521e-04   1.9694e-04      0.7    5.4
   9  -2.54454262e+01  -1.3682e-04   7.7180e-05      0.7    5.4
  10  -2.54455328e+01  -1.0663e-04   3.0247e-05      0.7    5.4
  11  -2.54455929e+01  -6.0110e-05   1.1758e-05      0.7    5.4
  12  -2.54456151e+01  -2.2218e-05   5.0053e-06      0.7    5.4
  13  -2.54456249e+01  -9.7454e-06   2.2689e-06      0.7    5.4

Ccsd correlation energy:          -25.4456248862
2nd-order correlation energy:     -24.3605043096
realtime 9.189542891 s
--

5. Sample yaml output

Below an example yaml output stream is shown for a successful CoupledCluster algorithm run.

name: CoupledCluster
out:
  amplitudes: 0x26e4758
  convergenceReached: 1
  energy:
    correlation: -25.445624886202758
    direct: -38.822491455744313
    exchange: 13.376866569541555
    secondOrder: -24.360504309639897
    unit: 0.036749322175638782
realtime: 9.189542891

6. Computational complexity

This section explains computational or memory footprints for the various methods implemented in CoupledCluster (see method).

Ccsd

The computational bottle neck of a typical Ccsd calculation originates from the following contraction, which is part of the doubles amplitude equations: \(V_{cd}^{ab} t_{ij}^{cd}\). The computational cost for evaluating this expression scales as \(\mathcal{O}(N_{\rm o}^2 N_{\rm v}^4)\). To avoid a memory footprint of \(\mathcal{O}(N_{\rm v}^4)\) in storing \(V_{cd}^{ab}\), slices \(V_{cd}^{xy}\) are computed on-the-fly and used in the contraction, reducing the corresponding memory footprint to \(\mathcal{O}(N_{\rm v}^2 N_{\rm s}^2)\), where \(N_{\rm s}\) is controlled using the integralsSliceSize keyword.

We note that required storage of a set of doubles amplitudes adds substantially to the memory footprint in Ccsd calculations. The Diis algorithm requires the storage of both doubles residua and amplitudes maxResidua times. We recommend to choose the type of mixer and its parameters carefully to reduce the memory footprint if necessary.

Drccd
The computational complexity is \(\mathcal O(N_\mathrm{o}^3N_\mathrm{v}^3)\). The implementation is not optimized for large systems. The memory requirement scales as \(\mathcal O(N_\mathrm{o}^2 N_\mathrm{v}^2)\)

7. Theory

Coupled-cluster employs the exponential Ansatz for the correlated wave function

\begin{equation} | \Psi \rangle = e^{\hat T} | \Phi \rangle, \end{equation}

where \(|\Phi\rangle\) denotes the single Hartree–Fock slater determinant. The cluster operator \(\hat T = \hat T_1 + \hat T_2 + \ldots\) is expanded in increasing number of excitation levels. The single and double exciation parts of the cluster operator are given by

\begin{eqnarray} \hat T_1 = \sum_{ai} t^a_i \hat\tau^a_i, \\ \hat T_2 = \sum_{abij} t^{ab}_{ij} \hat\tau^{ab}_{ij}, \end{eqnarray}

where \(\hat \tau^{a\ldots}_{i\ldots} = \hat c^\dagger_a\ldots \hat c_i\ldots\) denotes the exciation operator. The coefficients \(t^a_i\), \(t^{ab}_{ij}\), \(\ldots\) are called coupled-cluster Amplitudes. They are found by projecting the stationary Schrödinger equation for the coupled-cluster Ansatz \(E|\Psi\rangle = \hat H|\Psi\rangle\) on \(\langle \Phi|e^{-\hat T}\). The resulting over-determined set of equations is truncated at the same level as the cluster-operator \(\hat T\), giving as many equations as there are coefficients in \(\hat T\). The Amplitudes are generated by the CoupledCluster algorithm by solving the amplitude equation of the employed coupled-cluster method, described below:

7.1. Coupled-Cluster Singles Doubles (Ccsd)

The cluster-operator is truncated containing only single and double excitations. The projections on the singly and doubly excited slater-determinants \(\langle \Phi^a_i|\) and \(\langle \Phi^{ab}_{ij}|\) give

\begin{align} \big\langle \Phi^{a}_{i} \big| e^{-\hat T} \hat H e^{\hat T} \big| \Phi \big\rangle &= 0 \quad \forall ai, \\ \label{eqn:t2} \big\langle \Phi^{ab}_{ij} \big| e^{-\hat T} \hat H e^{\hat T} \big| \Phi \big\rangle &= 0 \quad \forall abij. \end{align}

The above equations expand to a finite set of a few dozens of, in general, non-linear algebraic contractions of \(t^a_i\) and \(t^{ab}_{ij}\) with \(\varepsilon_p\) and \(V^{pq}_{sr}\). See (Shavitt and Bartlett 2009) for a text-book introduction of the original works of (Coester and Kümmel 1960) and (Čížek 1969).

7.2. Direct-Ring Couple-Cluster Doubles (Drccd)

This method uses only the double excitation part of the cluster-operator. From the full doubles amplitude equations in Eq. (\ref{eqn:t2}) only those contractions are considered where pairs of particle and hole contractions originate from a common vertex and terminate in another common vertex, forming direct-rings in the diagrammatic notation. Only five terms remain in a canonical calculation and they read

\begin{equation} \Delta^{ab}_{ij} t^{ab}_{ij} + V^{ab}_{ij} + V^{kb}_{cj} t^{ac}_{ik} + V^{al}_{id} t^{db}_{lj} + V^{kl}_{cd} t^{ac}_{ik} t^{db}_{lj} = 0 \quad \forall abij, \end{equation}

with \(\Delta^{ab}_{ij} = \varepsilon_a+\varepsilon_b-\varepsilon_i-\varepsilon_j\) and where a sum over all indices that occurr only on the left-hand-side is implied. These terms are the dominant terms of coupled-cluster singles doubles in the hight-density limit (Gell-Mann and Brueckner 1957). The direct correlation contribution of the Drccd method \(\sum_{abij} 2t^{ab}_{ij}V^{ij}_{ab}\) gives the Random Phase Approximation (RPA), the remaining exchange contribution is often termed Second Order Screened Exchange (SOSEX) correction (Freeman 1977; Grüneis et al. 2009). See (Furche 2008; Chen, Agee, and Furche 2018) for a review on the RPA and its corrections, as well as (Macke 1950; Pines and Bohm 1952) for the original work on the RPA.

We recommend the following review article and references therein to get started with coupled-cluster theory (Bartlett and Musiał 2007) .

8. Literature

Bartlett, R.J., and Monika Musiał. 2007. “Coupled-cluster theory in quantum chemistry.” Rev. Mod. Phys. 79 (1): 291–352. doi:10.1103/RevModPhys.79.291.
Chen, Guo P., Matthew M. Agee, and Filipp Furche. 2018. “Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies.” J. Chem. Theory Comput. 14: 5701–14. doi:10.1021/acs.jctc.8b00777.
Čížek, Jirí. 1969. “On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules.” In Advances in Chemical Physics, edited by R. LeFebvre and C. Moser, 35–89. John Wiley & Sons, Inc.
Coester, F., and H. Kümmel. 1960. “Short-Range Correlations in Nuclear Wave Functions.” Nucl. Phys. 17 (June): 477–85. doi:10.1016/0029-5582(60)90140-1.
Freeman, David. 1977. “Coupled-Cluster Expansion Applied to the Electron Gas: Inclusion of Ring and Exchange Effects.” Physical Review B 15 (12): 5512–21. doi:10.1103/PhysRevB.15.5512.
Furche, Filipp. 2008. “Developing the Random Phase Approximation into a Practical Post-KohnSham Correlation Model.” J. Chem. Phys. 129: 114105. doi:10.1063/1.2977789.
Gell-Mann, Murray, and Keith A. Brueckner. 1957. “Correlation Energy of an Electron Gas at High Density.” Phys. Rev. 106 (2): 364–68. doi:10.1103/PhysRev.106.364.
Grüneis, Andreas, Martijn Marsman, Judith Harl, Laurids Schimka, and Georg Kresse. 2009. “Making the Random Phase Approximation to Electronic Correlation Accurate.” J. Chem. Phys. 131 (15): 154115. doi:10.1063/1.3250347.
Macke, W. 1950. “Über Die Wechselwirkungen Im Fermi-Gas, Polarisationserscheinungen, Correlationsenergie, Elektronenkondensation.” Z. Naturforsch. 5a (8): 192–208. http://zfn.mpdl.mpg.de/data/Reihe_A/5/ZNA-1950-5a-0192.pdf.
Pines, David, and David Bohm. 1952. “A Collective Description of Electron Interactions: II. Collective Vs Individual Particle Aspects of the Interactions.” Phys. Rev. 85 (2): 338–53. doi:10.1103/PhysRev.85.338.
Shavitt, Isaiah, and Rodney J Bartlett. 2009. Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory. Cambridge; New York: Cambridge University Press.

Created: 2022-09-19 Mon 15:00