Molecular Dipole Moments within the Incremental Scheme Using the

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Molecular Dipole Moments within the Incremental Scheme using the Domain-Specific Basis-Set Approach Benjamin Fiedler, Sonia Coriani, and Joachim Friedrich J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00076 • Publication Date (Web): 14 Jun 2016 Downloaded from http://pubs.acs.org on June 14, 2016

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Journal of Chemical Theory and Computation

Molecular Dipole Moments within the Incremental Scheme using the Domain-Specific Basis-Set Approach Benjamin Fiedler,a Sonia Coriani,b,c Joachim Friedricha∗ a

Institute for Chemistry, Technische Universit¨at Chemnitz, Straße der Nationen 62, D-09111 Chemnitz, Germany

b

Dipartimento di Scienze Chimiche e Farmaceutiche, Universit`a degli Studi di Trieste, Via L. Giorgieri 1, I-34127 Trieste, Italy

c

Aarhus Institute of Advanced Studies, Aarhus University,

Høegh-Guldbergs Gade 6B, DK-8000 Aarhus C, Denmark

May 24, 2016

∗

Corresponding author. E-mail: [email protected]

ACS Paragon1Plus Environment


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Abstract

We present the first implementation of the fully-automated incremental scheme for CCSD unrelaxed dipole moments using the domain-specific basis set approach. Truncation parameters are varied and the accuracy of the method is statistically analyzed for a test set of 20 molecules. The local approximations introduce small errors at second order and negligible ones at third order. For a third-order incremental CCSD expansion with a CC2 error correction, a cc-pVDZ/SV domain-specific basis set (tmain = 3.5 Bohr) and the truncation parameter f = 30 Bohr, we obtain a mean error of 0.00 mau (-0.20 mau) and a standard deviation of 1.95 mau (2.17 mau) for the total dipole moments (cartesian components of the dipole vectors). By analyzing incremental CCSD energies, we demonstrate that the MP2 and CC2 error correction schemes are an exclusive correction for the domain-specific basis set error. Our implementation of the incremental scheme provides fully automated computations of highly accurate dipole moments at reduced computational cost and is fully parallelized in terms of the calculation of the increments. Therefore one can utilize the incremental scheme, on the same hardware, to extend the basis set in comparison to standard CCSD and thus obtain a better total accuracy.


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1


Introduction

The determination of molecular properties by quantum chemical methods is an important field of research in theoretical and computational chemistry. If experimental measurements of properties shall be replaced or confirmed by computations, quantum chemistry must feature quantitative accuracy to have a predictive character. Coupled cluster (CC) methods like CCSD (coupled-cluster singles and doubles) or CCSD(T) (coupled-cluster singles and doubles with perturbative triple corrections) provide very accurate and systematically improvable results for molecular properties that describe the interaction with an electric (e.g., dipole moments 1–4 and (hyper)polarizabilities 5,6 ) and/or a magnetic field (e.g., magnetizabilities, 7,8 shielding constants 9,10 ) as well as those expressing optical activity (e.g. optical rotations, 11–13 circular dichroisms 14–16 ). For a good overview on the calculation of molecular properties by ab initio methods we recommend the review of Helgaker et al. 17 However, due to the steep scaling of the canonical CC methods with respect to the number of occupied (O) and virtual (V ) molecular orbitals (MOs) (CCSD: O2 V 4 , CCSD(T): O3 V 4 ), 18,19 their applicability is limited to relatively small molecules. Another drawback arises from the slow convergence with respect to the one-particle basis set, which requires relatively large ζ-levels to reach the intrinsic accuracy of coupled cluster. 20 The latter problem can be diminished by complete basis set extrapolations, 21 or by the application of explicitly correlated F12 methods. 22,23 For the solution of the scaling problem a large variety of local correlation methods with a reduced scaling have been developed over the last three decades inspired by the fundamental work of Pulay and Saebø. 24–28 All these methods benefit from the fact that the electron correlation decreases rapidly with increasing electron-electron distances. Therefore, additional approximations can be implemented, if localized MO spaces are used. Significant contribution in this area was done by Werner, Sch¨ utz and coworkers, 29–36 who utilized the



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Pulay-Saebø approach, 24,25 as well as by Neese et al., who realized the local pair-naturalorbital (LPNO) approach. 37–42 Other methods are based on fragmentation schemes, which divide the whole system into parts and examine only the smaller components. Some examples are the fragment-MO approach of Fedorov, Kitaura and coworkers, 43–45 the cluster-in-molecule method (CIM), 46–52 the natural linear scaling coupled cluster, 53–55 the divide-and-conquer scheme, 56–58 the divide-expandconsolidate (DEC) method 59–61 or the electrostatically embedded many-body (EE-MB) expansion of Truhlar et al. 62 All these methods have been frequently utilized for energy calculations, whereas other molecular properties have rarely been treated. Based on the framework of Pulay and Saebø, Werner, Korona and coworkers used local correlation methods at the CCSD level for dipole moments, static polarizabilities and excitation energies, 63,64 whereas Crawford et al. calculated excitation energies, 65 optical rotations 66–69 and frequency-dependent polarizabilities. 6,69 In this work the incremental scheme of Stoll 70–72 is used for the calculation of molecular dipole moments. The fundamental idea behind the incremental scheme was introduced in quantum chemistry by Nesbet 73–75 as a generalized Bethe-Goldstone expansion. The method is based on the partitioning of the system into domains of localized occupied MOs. The total correlation contribution (to whatever energy or property) is calculated in an incremental series from the correlation contribution of one-body, two-body, three-body increments, and so on. The incremental scheme has been successfully used to compute coupled cluster energies for periodic structures, 76–87 metals with multireference-character, 88 closed-shell 89–100 as well as open-shell molecules. 101–104 Additionally, polarizabilities have been calculated by Yang and Dolg, 105 and by Friedrich et al. 106 Dipole and quadrupole moments have been previously obtained within the incremental scheme by Friedrich et al., yet not in a domain-specific basis set. 107


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In this work we test the domain-specific basis-set approach for the computation of dipole moments (and energies). The results are based on a test set of 20 molecules, varying the truncation parameters tmain and f . Furthermore we establish a CC2-based error correction as an alternative for the already proposed MP2-based one. 108,109

2 2.1

Theory Incremental Scheme

In the following, we use the incremental method to calculate correlation energies and correlation contributions to molecular dipole moments. As anticipated in the introduction, the starting point of the incremental scheme is the partitioning of the total system into disjoint sets of occupied localized molecular orbitals (LMOs), the so-called one-site domains. 110–112 Occupied Hartree-Fock molecular orbitals are first localized by a Boys procedure, 113 followed by a graph partitioning with a subsequent refinement 114 in order to obtain a suitable set of one-site domains, denoted as D. The target number of LMOs in a one-site domain, which is required for the partitioning scheme, is given directly by the integer domain size parameter (dsp). 110,114 The correlation energies εi of all one-site domains are calculated and summed up to yield the first-order contribution of the incremental series. Due to the nonadditivity of the total correlation energy, contributions of higher-order domains (e.g. two-body increments,...) must also be included to account for e.g. pair interactions. Thus the one-body (∆εi ) and two-body increments (∆εij ) are given according to ∆εi = εi

∆εij = εij − ∆εi − ∆εj

(1)

The total incremental correlation energy Ecorr of order O is obtained by summing up all



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increments until this order is reached, 70 which is e.g. for second order Ecorr =

X

∆εi +

i

1 X ∆εij 2! ij

(2)

where the prefactor (e.g. 1/2!) prohibits double counting due to index permutation. According to index symmetry (e.g. ∆εij = ∆εji ), we can restrict the indices and eliminate the prefactors in Eq. (3) to get Ecorr =

X

∆εi +

i

X

∆εij

(3)

i

Molecular Dipole Moments within the Incremental Scheme Using the

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