Transcorrelated coupled cluster methods

Transcorrelated coupled cluster and distinguishable cluster methods are presented. The Hamiltonian is similarity transformed with a Jastrow factor in the first quantisation, which results in up to three-body integrals. The coupled cluster with singles and doubles equations on this transformed Hamiltonian are formulated and implemented. It is demonstrated that the resulting methods have a superior basis set convergence and accuracy to the corresponding conventional and explicitly correlated methods. Additionally, approximations for three-body integrals are suggested and tested.


I. INTRODUCTION
An accurate description of the dynamical electron correlation usually requires large one-particle basis sets and high-dimensional tensors to represent the correlated movement of electrons. Various approaches have been developed to accelerate the basis-set and excitation-level convergence. Explicit correlation methods introduce explicit dependence on the electron-electron distance into the wavefunction, which drastically reduces the finite basis-set error.  The coupled cluster hierarchy [30][31][32] is known to converge extremely fast to the full configuration interaction (FCI) results as long as the electron correlation is weak. [33][34][35] However, both methodologies have limitations. Efficient implementations of the explicitly correlated (F12) methods rely on various approximations, and the formalism is difficult to extend to higher than doubles excitation classes. 36 The fixed-amplitude ansatz commonly employed in the F12 methods makes the methods less suitable for pairs containing core orbitals, although the problem can be somewhat mitigated by using different length scales in the correlation factor for valence and core electrons 26 or by partially relaxing the F12 amplitudes 37 . The coupled-cluster methods are sensitive to strong electron correlation and results are often not even qualitatively correct if large amount of static correlation is involved. Besides, at least perturbative triple excitations 32 are needed for accurate results, which makes the calculations expensive. Modified coupled cluster methods exist which demonstrate higher accuracy for a given excitation level. [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] The distinguishable cluster approach is one of these methods, and is known to improve not only the accuracy for weakly correlated systems, but also to yield qualitatively good results for many strongly correlated systems. [51][52][53]56,57 An alternative route to improve the accuracy of electron-correlation treatment is to apply a similarity transformation to the Hamiltonian, which can be used to incorporate some correlation effects into the transformed Hamiltonian and thus simplify the problem for the electron-correlation methods. This idea goes back to the pioneering work of Boys and Handy and is termed transcorrelation, 58 and it has been recently demonstrated that a combination of the transcorrelation with a stochastic full configuration interaction quantum Monte-Carlo (FCIQMC) 59-61 yields very promising results for weakly and strongly correlated systems. [62][63][64][65] Since the Hamiltonian itself is transformed, the transcorrelation approach is not specific for low-order methods and can be used together with any excitation orders. The transcorrelated Hamiltonian has been applied before to linearized coupled cluster with singles and doubles, 66 and for the uniform electron gas for coupled cluster and distinguishable cluster with singles and doubles, 67 however with some approximations to the three-body terms in the transformed Hamiltonian and with a simpler correlation factor. In this communication we explore the quality of full transcorrelated coupled cluster and distinguishable cluster methods as well as their approximated versions.

A. Transcorrelated coupled cluster
The transcorrelated formalism starts by similarity transforming the electronic Hamiltonian with the Jastrow factor, which can be efficiently done before going to the second-quantisation formulation. u(r i , r j ) is a symmetric correlation function, u(r i , r j ) = u(r j , r i ). The specific form of the correlation factor can be adjusted to the problem under consideration. We will be using the Boys-Handy correlation factor, 58 with one (h q p = p|ĥ|q ) and two-(V qs pr = pr|qs ) electron part of the original electronic Hamiltonian, and the additional terms arising from the similarity transformation, P i j andP i jk are permutation operators, e.g., Note that the three-body operator is hermitian, and for real orbitals possesses 48-fold symmetry.
The transformed electronic Schrödinger equation, can approximately be solved using coupled-cluster methods. We restrict ourselves to singles and doubles. However, we would like to stress that the transcorrelated Hamiltonian can be used together with coupled-cluster methods truncated at any excitation level. The coupledcluster amplitude equations including three-body integrals can be derived using the well- We evaluate two types of approximations for the three-body integrals. In the first approximation (denoted in the following as approximation A), only terms involving explicit normalordered dressed three-body integrals are neglected, i.e., the effective 0, 1, and 2-body terms are calculated using dressed three-body integrals and contribute to the amplitude equations. This approximation corresponds to neglecting three-body integrals normal-ordered with respect to "optimized" orbitals. In the second type of approximation (approximation B), only three-body contributions to the 0-2 body terms before dressing are retained, i.e., this approximation corresponds to neglecting three-body terms normal-ordered with respect to the HF determinant. This approach allows for very efficient implementation of transcorrelated methods, since the three-body terms can be self-contracted at the construction time on-the-fly and stored together with effective 0-2 body integrals.

III. TEST CALCULATIONS
The TC-CCSD and TC-DCSD amplitude equations have been implemented using Integrated Tensor Framework (ITF) in Molpro. 68 The integrals are calculated as outlined in Ref. 64 and imported using an FCIDUMP-type interface.
We employ the same Boys-Handy correlation factors as in Ref. 64, First, we compare the absolute atomic energies to TC-FCIQMC and experimental numbers, Table I. Evidently, transcorrelation drastically improves the basis set convergence. The results become very close to experimental values. The explicitly correlated coupled cluster numbers (using F12a approximations) are also given for comparison. It is obvious that the TC methods are much more accurate. The mean absolute errors of the transcorrelated methods compared to experimental numbers are 3-3.5 times smaller than of the explicitly correlated methods, and six times smaller than of the conventional methods (Table II).
Even more interesting is the ability of the transcorrelation to improve the accuracy of the underlying method itself. For all atoms TC-CCSD differs from TC-FCIQMC by less than 1 mE h . This can be compared to the accuracy of the conventional CCSD versus FCI,  The accuracy of relative energies has been evaluated by computing atomic ionisation  Note that in the case of transcorrelated methods the HF orbitals and Jastrow factors are optimized for the neutral atoms, and the same integrals are reused for the cations, i.e., neither orbitals nor Jastrow factors are reoptimized for the cations. Thus, some bias towards the neutral atoms is expected in our results. However, the good agreement with the experimental values and TC-FCIQMC suggests that the partial orbital relaxation coming from the single excitations is sufficient to largely eliminate this problem, and demonstrates the transferability of the approach, even for methods truncated at the singles and doubles level.
As in the case of absolute energies, the IPs from transcorrelated CCSD are closer to the corresponding FCI reference values, Table III, than for the conventional CCSD method,

IV. CONCLUSIONS
The transcorrelated approach combined with the coupled cluster methods shows great promise. Not only does it drastically improve the basis set convergence, but it can also increase the accuracy of the CC method itself. The non-hermiticity of the transcorrelated Hamiltonian does not cause problems. The high degree of orbital invariance of coupled cluster methods allows to use conventional HF orbitals instead of solving bi-orthogonal HF equations.
Our results demonstrate that the expensive three-body terms can be approximated by neglecting their normal-ordered contributions. The self-contraction of the integrals can be done on-the-fly, and the resulting effective 0 -2 body contributions can be stored together with other terms of the Hamiltonian. The quality of the approximation can be improved by using better orbitals, e.g., Brueckner orbitals, as shown by the singles-dressed formulation (approximation A).
The transcorrelated approach can be easily applied to higher order coupled cluster methods, especially if approximated three-body terms are used. Moreover, the use of highly flexible Jastrow functions, which incorporate information on the position of the nuclei as well as electron-electron distances, can be expected to bring further benefits in the treatment of more complex systems.