Free Submitted: 03 February 2022 Accepted: 04 July 2022 Accepted Manuscript Online: 04 July 2022 Published Online: 02 August 2022
J. Chem. Phys. 157, 054101 (2022); https://doi.org/10.1063/5.0087300
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• Benjamin Kincaid: Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Visualization (supporting); Writing – original draft (supporting).
• Haihan Zhou: Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Visualization (supporting); Writing – review & editing (supporting).
• Abdulgani Annaberdiyev: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Writing – original draft (supporting).
• M. Chandler Bennett: Data curation (supporting); Methodology (supporting).
• Jaron T. Krogel: Writing – original draft (supporting); Writing – review & editing (equal).
• Lubos Mitas: Conceptualization (supporting); Funding acquisition (lead); Project administration (equal); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

We introduce new correlation consistent effective core potentials (ccECPs) for the elements I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W with 4d, 5d, 6s, and 6p valence spaces. These ccECPs are given as a sum of spin-orbit averaged relativistic effective potential (AREP) and effective spin–orbit (SO) terms. The construction involves several steps with increasing refinements from more simple to fully correlated methods. The optimizations are carried out with objective functions that include weighted many-body atomic spectra, norm-conservation criteria, and SO splittings. Transferability tests involve molecular binding curves of corresponding hydride and oxide dimers. The constructed ccECPs are systematically better and in a few cases on par with previous effective core potential (ECP) tables on all tested criteria and provide a significant increase in accuracy for valence-only calculations with these elements. Our study confirms the importance of the AREP part in determining the overall quality of the ECP even in the presence of sizable spin–orbit effects. The subsequent quantum Monte Carlo calculations point out the importance of accurate trial wave functions that, in some cases (mid-series transition elements), require treatment well beyond a single-reference.
Most electronic structure calculations aim at valence properties such as bonding, ground and excited states, and related properties. These characteristics are determined by the valence electronic states with the energy scale of eVs and spatial ranges from covalent bonds to delocalized conduction states. On the other hand, the atomic cores of heavier elements are very strongly bonded to nuclei and spatially very localized. Therefore, the cores appear as almost rigid repulsive charge barriers around the nuclei so that, in many quantum chemical and condensed matter electronic structure calculations, the cores are routinely kept static and frozen. Building upon this understanding, it has been realized that the atomic cores can be alternatively represented by properly adjusted effective core potentials (ECPs). The effective potentials mimic the action of the core on valence states and allow for valence-only calculations with resulting gains in efficiency. On a quantitative level, this partitioning is based on significant differences between spatial and energy domains that are occupied by core vs valence states. ECPs and closely related pseudopotentials in the condensed matter context have been developed over several decades and involve a number of complete tables1–141. L. Fernandez Pacios and P. A. Christiansen, J. Chem. Phys. 82, 2664 (1985). https://doi.org/10.1063/1.4482632. M. M. Hurley, L. F. Pacios, P. A. Christiansen, R. B. Ross, and W. C. Ermler, J. Chem. Phys. 84, 6840 (1986). https://doi.org/10.1063/1.4506893. L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler, J. Chem. Phys. 87, 2812 (1987). https://doi.org/10.1063/1.4530694. R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen, J. Chem. Phys. 93, 6654 (1990). https://doi.org/10.1063/1.4589345. W. J. Stevens, M. Krauss, H. Basch, and P. G. Jasien, Can. J. Chem. 70, 612 (1992). https://doi.org/10.1139/v92-0856. N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). https://doi.org/10.1103/PhysRevB.43.19937. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 126, 234105 (2007). https://doi.org/10.1063/1.27415348. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 129, 164115 (2008). https://doi.org/10.1063/1.29878729. A. Bergner, M. Dolg, W. Küchle, H. Stoll, and H. Preuß, Mol. Phys. 80, 1431 (1993). https://doi.org/10.1080/0026897930010312110. M. Dolg, U. Wedig, H. Stoll, and H. Preuss, J. Chem. Phys. 86, 866 (1987). https://doi.org/10.1063/1.45228811. G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982). https://doi.org/10.1103/PhysRevB.26.419912. J. R. Trail and R. J. Needs, J. Chem. Phys. 146, 204107 (2017). https://doi.org/10.1063/1.498404613. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). https://doi.org/10.1103/PhysRevB.41.789214. M. Dolg and X. Cao, Chem. Rev. 112, 403 (2012). https://doi.org/10.1021/cr2001383 as well as computational tools.15,1615. M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119, 67 (1999). https://doi.org/10.1016/s0010-4655(98)00201-x16. See http://opium.sourceforge.net/sci.html for Opium—pseudopotential generation project.
The basic construction of ECPs involves reproducing valence one-particle eigenvalues and closely related one-particle orbital norm conservation, i.e., the amount of valence charge outside an appropriate effective ion radius.1111. G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982). https://doi.org/10.1103/PhysRevB.26.4199 Since the number of core states and their spatial properties vary, each angular momentum symmetry channel requires a different effective potential resulting in semilocal ECPs with corresponding projectors. This simplest construction can be further generalized in several directions.17–2017. L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). https://doi.org/10.1103/PhysRevLett.48.142518. P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). https://doi.org/10.1103/PhysRevB.50.1795319. E. L. Shirley and R. M. Martin, Phys. Rev. B 47, 15413 (1993). https://doi.org/10.1103/PhysRevB.47.1541320. E. L. Shirley, X. Zhu, and S. G. Louie, Phys. Rev. B 56, 6648 (1997). https://doi.org/10.1103/PhysRevB.56.6648 Further improvement has been introduced through the so-called energy consistency that requires reproducing energy differences such as atomic excitations within a given theory, e.g., Hartree–Fock.7,8,11,147. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 126, 234105 (2007). https://doi.org/10.1063/1.27415348. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 129, 164115 (2008). https://doi.org/10.1063/1.298787211. G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982). https://doi.org/10.1103/PhysRevB.26.419914. M. Dolg and X. Cao, Chem. Rev. 112, 403 (2012). https://doi.org/10.1021/cr2001383 Other directions for improvement have been explored within Density Functional Theory (DFT) using many-body perturbation theory.21,2221. E. L. Shirley, D. C. Allan, R. M. Martin, and J. D. Joannopoulos, Phys. Rev. B 40, 3652 (1989). https://doi.org/10.1103/PhysRevB.40.365222. G. Kresse, J. Hafner, and R. J. Needs, J. Phys.: Condens. Matter 4, 7451 (1992). https://doi.org/10.1088/0953-8984/4/36/018
Very recently, we have introduced correlation consistent ECPs (ccECPs)23–2623. M. C. Bennett, C. A. Melton, A. Annaberdiyev, G. Wang, L. Shulenburger, and L. Mitas, J. Chem. Phys. 147, 224106 (2017). https://doi.org/10.1063/1.499564324. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.503813525. A. Annaberdiyev, G. Wang, C. A. Melton, M. C. Bennett, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 134108 (2018). https://doi.org/10.1063/1.504047226. A. Annaberdiyev, C. A. Melton, M. C. Bennett, G. Wang, and L. Mitas, J. Chem. Theory Comput. 16, 1482 (2020). https://doi.org/10.1021/acs.jctc.9b00962 that build upon previous constructions. Our overarching goal was to provide ccECPs that would offer the accuracy needed for many-body and highly accurate calculations. For this purpose, our construction has been based on many-body wave functions. These have enabled us to use nearly exact results to reach higher levels of accuracy and also to ascertain the robustness and transferability of the constructed potentials. The construction of ccECPs for heavier atoms involves several steps. The initial construction involves reproducing atomic excitations across a range of energies and different states including also a number of highly ionized states. We included scalar relativistic effects from the outset, and as explained below, spin–orbit effects were added as they can influence the valence accuracy for heavier elements.
We have taken into account further criteria having in mind the ultimate goal of describing with high accuracy the valence properties in real systems such as molecules or condensed matter systems. Therefore, the transferability of the ECPs has been carefully tested by probing the (effective) ion in bonded environments. In this setting, reproducing molecular bonding curves has become another important criterion both for construction and validation. Therefore, validation tests of the ccECP construction have included hydride and oxide dimers for each element. This provides insights into the simplest bonds with covalent and ionic character. Molecular calculations involved essentially the full binding curve from the stretched bond lengths to the dissociation limit at short interatomic distances so as to test the restructuring of the valence charge in high pressure bonding environments. The typical discrepancies that were observed have been within chemical accuracy (≈0.043 eV). There were a few exceptions where discrepancies were larger due to a small number of valence electrons; especially, at highly compressed bond length, we find the inclusion of norm-conservation is essential to alleviate such discrepancies. Therefore, additional cautions are needed to achieve optimal balance in several properties including spectrum and molecular bindings.
The tests and comparisons with previously generated tables have shown that ccECPs represent a significant step forward in achieving accuracy and fidelity within the ECP effective Hamiltonian model. In addition, we have found that correlated construction also provides welcome and important gains when compared with frozen core, all-electron treatments. In particular, ccECPs capture core-valence correlations that are missed in most frozen core calculations. We have found that using ccECPs provides higher valence accuracy than uncorrelated (self-consistent) cores (UC).
In this work, we extend our generation of ccECPs beyond the 3d and 4s4p elements where explicit spin–orbit interaction is an important ingredient for accurate correlated treatments. The constructed ccECPs are selected heavy elements that include 4d and 5d as well as main group atoms, namely, I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W. This choice has been motivated by several considerations. One is the development of an efficient methodology for 4d and 5d transition elements that require an accurate representation of atomic spin–orbit effects. Another goal was to include elements that are prominent in a number of technologically important 2D materials.27–3027. L. Margulis, G. Salitra, R. Tenne, and M. Talianker, Nature 365, 113 (1993). https://doi.org/10.1038/365113b028. L. L. Handy and N. W. Gregory, J. Am. Chem. Soc. 74, 891 (1952). https://doi.org/10.1021/ja01124a00929. H. Wang, V. Eyert, and U. Schwingenschlögl, J. Phys.: Condens. Matter 23, 116003 (2011). https://doi.org/10.1088/0953-8984/23/11/11600330. X. Li and J. Yang, J. Mater. Chem. C 2, 7071 (2014). https://doi.org/10.1039/c4tc01193g
We mostly employ our previously developed methodology with several updates and adjustments needed due to increasing demands on the correlated treatment of large atomic cores. Intuitively, explicit spin–orbit effects are treated as further refinements and advanced features relying on the accurate spin–orbit averaged relativistic effective potential (AREP) to form the spin–orbit relativistic effective potentials (SOREP). The detailed methods for construction of these heavy element ccECPs are described in Sec. II.
The composition of this paper is as follows: Sec. II describes the form parameterizations of ccECPs. In what follows, we discuss the objective function and optimization procedure for AREP and SOREP. In Sec. III, the results are presented including the atomic properties and selected tests on molecular systems using both correlated methods based on basis set expansions such as coupled cluster with the singles, doubles, and perturbative triples [CCSD(T)] method and fixed-phase spin–orbit diffusion Monte Carlo (FPSODMC) approaches.3131. C. A. Melton, M. Zhu, S. Guo, A. Ambrosetti, F. Pederiva, and L. Mitas, Phys. Rev. A 93, 042502 (2016). https://doi.org/10.1103/PhysRevA.93.042502 Each element is presented in greater detail and this is followed by summaries of ccECP properties. The results are further elaborated in discussion and conclusions in Sec. IV.
A. ECP form and parameterization
The aim of this study is to accurately reproduce the properties of the relativistic all-electron (AE) Hamiltonians with a much smaller valence effective Hamiltonians Hval. Following the Born–Oppenheimer approximation, the valence Hamiltonian Hval in atomic units (a.u.) is expressed as
 $Hval=∑iTikin+ViSOREP+∑i (1)
The full spin–orbit relativistic effective potential (SOREP) ECPs are of the form proposed by Lee,3232. Y. S. Lee, W. C. Ermler, and K. S. Pitzer, J. Chem. Phys. 67, 5861 (1977). https://doi.org/10.1063/1.434793
 $VSOREP=VLJSOREP(r)+∑l=0L−1∑j=|l−1/2|l+1/2∑m=−jj×VljSOREP(r)−VLJSOREP(r)|ljm〉〈ljm|,$ (2)
where r is the electron–ion distance and $VLJSOREP(r)$ is a local potential. This form can be split into the averaged relativistic effective core potential (AREP) and spin–orbit potential (SO) terms,3333. W. C. Ermler, Y. S. Lee, P. A. Christiansen, and K. S. Pitzer, Chem. Phys. Lett. 81, 70 (1981). https://doi.org/10.1016/0009-2614(81)85329-8
 $VSOREP=VAREP+VSO,$ (3)
where $VℓAREP$ is the weighted J-average of VSOREP,
 $VℓAREP(r)=12ℓ+1ℓ⋅Vℓ,ℓ−12SOREP(r)+(ℓ+1)⋅Vℓ,ℓ+12SOREP(r).$ (4)
The set of SO potentials VSO are defined using VSOREP as
 $VSO=s⋅∑ℓ=1L22ℓ+1ΔVℓSOREP(r)⋅∑m=−ℓℓ∑m′=−ℓℓ|ℓm〉〈ℓm|ℓ|ℓm′〉〈ℓm′|,$ (5)
where
 $ΔVℓSOREP(r)=Vℓ,ℓ+12SOREP(r)−Vℓ,ℓ−12SOREP(r).$ (6)
VAREP is defined as follows similar to our previous works:
 $VAREP(r)=VLAREP(r)+∑ℓ=0ℓmax=L−1VℓAREP(r)−VLAREP(r)×∑m|ℓm〉〈ℓm|.$ (7)
The latter part of Eq. (7) involves the non-local |ℓm⟩⟨ℓm| spherical harmonics projectors. $VLAREP(ri)$ is again a local channel that acts on all valence electrons and parameterized as
 $VLAREP(r)=−Zeffr(1−e−αr2)+αZeffre−βr2+∑k=1γke−δkr2,$ (8)
where the Zeff = ZZcore is the effective core charge and α, β, γk, and δk are optimization coefficients. With the given format of local potential, the Coulomb singularity is explicitly canceled out and first derivative at the origin vanishes. The non-local potential is expressed as
 $VℓAREP(r)−VLAREP(r)=∑p=1βℓprnℓp−2e−αℓpr2,$ (9)
where βℓp and αℓp are parameters to be optimized. In most cases, nℓp were set to be nℓp = 2; however, we included nℓp = 4 terms in some cases to achieve the desired accuracy.
The ccECP SO parameters are provided as
 $VℓSO,ccECP=22ℓ+1ΔVℓSOREP,$ (10)
which is also the convention adopted by codes such as dirac, molpro, and nwchem. $VlSO,ccECP$ are parameterized similar to AREP non-local potentials,
 $VℓSO,ccECP(r)=∑p′=1βℓp′rnℓp′−2e−αℓp′r2.$ (11)
Together with parameters described in the AREP part, nℓp, βℓp, and αℓp, are the full sets of variables to be determined and treated by the optimizer in minimizing the chosen objective functions, separately at AREP and SO levels in sequence. The optimization is in such a way that SO parameters are pursued after we have ensured the desirable AREP accuracy. AREP parameters are kept fixed during SO optimizations.
We aimed to keep the parameterization of ECP in a simple and compact form. For the fourth row elements, I, Te, Ag, Pd, and Mo, max = 2 is used as there are 3d electrons being included in the core. For the fifth row elements, Bi, Au, Ir, and W, max = 3 is employed because the 4f electrons are present in the core. For each channel, the chosen number of Gaussian terms varies from 2 to 4 due to different scenarios in the ECP construction of each element. Note that we did not include any core polarization potential (CPP) terms or considered these in our calculations.
B. Objective function and optimization—AREP
The objective functions used to optimize the AREP portion of each ccECP depended on the particular element in question. The general recipe follows our previous studies.23–25,3423. M. C. Bennett, C. A. Melton, A. Annaberdiyev, G. Wang, L. Shulenburger, and L. Mitas, J. Chem. Phys. 147, 224106 (2017). https://doi.org/10.1063/1.499564324. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.503813525. A. Annaberdiyev, G. Wang, C. A. Melton, M. C. Bennett, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 134108 (2018). https://doi.org/10.1063/1.504047234. G. Wang, A. Annaberdiyev, C. A. Melton, M. C. Bennett, L. Shulenburger, and L. Mitas, J. Chem. Phys. 151, 144110 (2019). https://doi.org/10.1063/1.5121006 We include the many-body energy-consistency and single-particle eigenvalues in the definition of the objective function,
 $OAREP2=∑s∈Sws(ΔEAE(s)−ΔEECP(s))2+∑i∈Lwi(ϵAE(i)−ϵECP(i))2,$ (12)
where the $ΔEX(s)$, X ∈ {ECP, AE}, denotes the atomic energy gaps referenced to the ground state for given Hamiltonians. The subset of states is chosen by picking a representative set of states for the pertinent valence space. In particular, the electron affinity (EA), neutral excitations, and various ionization levels (IPn) are included. ws are weights corresponding to spectral states, decreasing as the energy gap increases due to the deep ionizations. $ϵX(i)$, X ∈ {ECP, AE}, labels the one-particle eigenvalue in ground state for ith valence eigenvalue with a weight wi. The ECP parameters were initialized from either MDFSTU3535. H. Stoll, B. Metz, and M. Dolg, J. Comput. Chem. 23, 767 (2002). https://doi.org/10.1002/jcc.10037 or BFD7,87. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 126, 234105 (2007). https://doi.org/10.1063/1.27415348. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 129, 164115 (2008). https://doi.org/10.1063/1.2987872 ECP parameters.
All energies in the AREP case are calculated using the CCSD(T) method with large uncontracted aug-cc-p(wC)VnZ (nT, Q, 5) basis sets while adjustments are made either to add some diffuse primitives or to remove primitives that cause near-linear dependencies, as necessary. Here, AE calculations are fully correlated, which include core–core, core-valence, and valence–valence correlations and use a scalar relativistic 10th-order DKH Hamiltonian.3636. M. Reiher and A. Wolf, J. Chem. Phys. 121, 2037 (2004). https://doi.org/10.1063/1.1768160 All ECP calculations correlate the full valence space as well (including the semi-cores if present). Molpro code3737. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, and M. Schütz, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2, 242 (2012). https://doi.org/10.1002/wcms.82 was used for all AREP calculations.
For the elements Ag, I, Mo, and W, the simple objective function given in Eq. (12) produced accurate ECPs. This objective function includes the single particle eigenvalue that is one term of the norm-conserving objective function used in previous work.2424. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.5038135 Some of the other elements in this series did not yield as accurate ECPs when this method was employed leading to two other ECP optimization schemes being used. The first of these was a modification of the objective function [Eq. (12)] by adding the full norm-conservation criteria. Refer to Ref. 2424. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.5038135 for details. This norm-conserving method was only used for Bi.
Finally, a method similar to the single-electron fit (SEFIT), and multi-electron fit (MEFIT) optimizations developed by the Stuttgart group,1010. M. Dolg, U. Wedig, H. Stoll, and H. Preuss, J. Chem. Phys. 86, 866 (1987). https://doi.org/10.1063/1.452288 has been employed. In this method, the ECP parameters are first optimized using single-particle energies only, and then, the exponents are kept fixed in the MEFIT optimization. This method was used for the elements Au, Ir, and Pd with the objective function given in Eq. (12) being used in the MEFIT step.
C. Objective function and optimization—SOREP
The spin–orbit coupling terms were optimized using the DIRAC3838. R. Bast, A. S. P. Gomes, T. Saue, L. Visscher, and H. J. Aa. Jensen, with contributions from I. A. Aucar, V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, L. Halbert, E. D. Hedegård, T. Helgaker, B. Helmich–Paris, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, J. K. Lærdahl, C. V. Larsen, Y. S. Lee, N. H. List, H. S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, A. Papadopoulos, Y. C. Park, J. K. Pedersen, M. Pernpointner, J. V. Pototschnig, R. di Remigio, M. Repisky, K. Ruud, P. Sałek, B. Schimmelpfennig, B. Senjean, A. Shee, J. Sikkema, A. Sunaga, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, M. L. Vidal, S. Villaume, O. Visser, T. Winther, and S. Yamamoto (2021), “DIRAC, a relativistic ab initio electronic structure program, Release DIRAC21,” Zenodo. http://dx.doi.org/10.5281/zenodo.4836496, see http://www.diracprogram.org. code and the Complete Open-Shell Configuration Interaction (COSCI) method. The reference AE atomic states were calculated using the exact two-component (X2C) Hamiltonian as implemented in DIRAC. The SO parameters were initialized from MDFSTU ECP values. In general, we keep the energy-consistency scheme in spin–orbit terms optimization,
 $OSO2=∑s∈S′ws(ΔEAE(s)−ΔEECP(s))2+∑m∈Mwm(ΔEAE(m)−ΔEECP(m))2,$ (13)
where the $ΔEX(Y)$, X ∈ {ECP, AE} and Y ∈ {s, m} denotes the atomic gaps using COSCI method. Here, Y labels different kinds of states included in the SO optimization. For Y = s, states with different charges referenced to the ground state were included, such as EA, IP, and IP2 similar to the AREP case. For Y = m, states with the same charges were included, particularly the 2S+1LJ multiplet splittings with various S, L, and J values were included by referencing the lowest energy for the given electronic charge. This separation of Y ∈ {s, m} was motivated by the goal of capturing the SO effect better for small gaps. In addition, since COSCI is generally less accurate than CCSD(T), our expectation is that referencing the lowest energy within a given charge will result in a better error cancellation producing accurate multiplet gaps. This will be apparent in Sec. III where COSCI gaps are directly compared to experimental gaps.
We use three different metrics to assess the errors of the pseudoatom spectrum at the AREP level. One is the mean absolute deviation (MAD) of all considered N atomic gaps,
 $MAD=1N∑iNΔEiECP−ΔEiAE.$ (14)
 $LMAD=1n∑inΔEiECP−ΔEiAE.$ (15)
For LMAD states, we chose electron affinity (EA), first ionization potential (IP), and second ionization potential (IP2) states only. Finally, we also consider a weighted-MAD (WMAD) of all considered N gaps as follows:
 $WMAD=1N∑iN100%|ΔEiAE|ΔEiECP−ΔEiAE.$ (16)
The pseudoatom spectrum errors are evaluated also for various other tabulated ECPs so as to assess the quality of our constructions. The ECPs are compared with MWBSTU,3939. T. Leininger, A. Berning, A. Nicklass, H. Stoll, H.-J. Werner, and H.-J. Flad, Chem. Phys. 217, 19 (1997). https://doi.org/10.1016/S0301-0104(97)00043-8 MDFSTU,3535. H. Stoll, B. Metz, and M. Dolg, J. Comput. Chem. 23, 767 (2002). https://doi.org/10.1002/jcc.10037 BFD,7,87. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 126, 234105 (2007). https://doi.org/10.1063/1.27415348. M. Burkatzki, C. Filippi, and M. Dolg, J. Chem. Phys. 129, 164115 (2008). https://doi.org/10.1063/1.2987872 LANL2,4040. T. H. Dunning and P. J. Hay, in Modern Theoretical Chemistry, edited by H. F. Schaefer III (Plenum, New York, 1977), Vol. 3, pp. 1–28. CRENB(S/L),3,43. L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler, J. Chem. Phys. 87, 2812 (1987). https://doi.org/10.1063/1.4530694. R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen, J. Chem. Phys. 93, 6654 (1990). https://doi.org/10.1063/1.458934 and SBKJC55. W. J. Stevens, M. Krauss, H. Basch, and P. G. Jasien, Can. J. Chem. 70, 612 (1992). https://doi.org/10.1139/v92-085 ECPs. In addition, to further demonstrate the improvement of ccECPs, we include uncorrelated-core (UC) calculations that are self-consistent AE calculations but with the only valence electrons correlated, and the cores are frozen with the same size of ECPs.
The summary of defined quantities, MAD, LMAD, and WMAD, for selected elements is provided in Figs. 13, respectively. Clearly, ccECP achieves consistent improvements overall. Detailed discussions for each element will follow in the parts below.
In addition, for each element, we show the transferability tests of all core approximations using molecular binding energies compared to the fully correlated AE case. We use the AE discrepancy of dimer binding energies as one criterion of ECP quality,
 $Δ(r)=D(r)ECP−D(r)AE,$ (17)
where D(r) is the binding energy at given separation r for dimer atoms.
For the SOREP pseudoatom spectrum, we adopt MAD definition defined above in Eq. (14) where the atomic gaps are chosen for the pertinent states from spin–orbit calculations. The MADs from COSCI calculations for MDFSTU and ccECPs are compared with AE values to show the optimization gains. In addition, we provide error analysis by comparing both ECP gaps to experimental values using the FPSODMC method that also provides the spin–orbit splittings.
The SOREP transferability tests are done for several mono-atomic dimers for both MDFSTU and ccECPs using AREP CCSD(T) and SOREP FPSODMC calculations, which are compared to experimental data. The binding curve is fitted to Morse potential,
 $V(r)=Dee−2ar−re−2e−ar−re,$ (18)
where De labels the dissociation energy, re is the equilibrium bond length, and a is a fitting parameter related to vibrations,
 $ωe=2a2Deμ,$ (19)
where μ is the reduced mass of the molecule. Note that we use Morse potential for fitting all molecules both for AREP and SOREP treatments.
All optimized parameters for ccECPs are listed in Tables I and II, respectively, for the fourth row elements and the fifth row elements. Results for each element are discussed separately in Secs. III AIII J.
TABLE I. Parameter values for the fourth row selected heavy-element ccECPs. For all ECPs, the highest value corresponds to the local channel L. Note that the highest non-local angular momentum channel max is related to it as max = L − 1.
AtomZeffHamiltoniannℓkαℓkβℓkAtomZeffHamiltoniannℓkαℓkβℓk
I7AREP023.150 276114.023 762Pd14AREP0212.451 647222.004 870
021.008 0801.974 146029.656 9584.624 354
122.954 190111.466 589026.301 50855.838 586
120.730 8221.710 2401212.376 014166.644 960
222.604 97945.300 844129.850 17928.721 347
220.899 5938.704 290125.706 18742.681 167
3112.001 3377.000 000229.195 34457.976 811
3312.035 88284.009 359227.923 46753.576 592
323.022 296−3.998 559223.358 6924.674 921
320.977 906−3.050 3103115.997 66418.000 000
3315.964 009287.957 945
SO123.086 88255.714 7473216.128 878−176.382 552
122.867 556−54.814 595329.592 539−44.253 623
121.921 809−1.544 859
121.061 6051.770 002SO1217.775 389−4.392 610
222.000 739−8.269 917126.010 0864.167 684
221.969 1228.345 637228.310 412−28.617 555
221.002 536−2.185 466228.113 17828.944 376
220.976 0892.118 147223.313 498−1.901 878
222.152 5120.867 558
Te6AREP022.656 48348.280 562
022.281 974−1.021 525Mo14AREP0210.079 375180.089 303
122.946 98839.656 024026.321 32816.961 392
122.790 00179.544 830024.386 73024.723 630
121.909 579−2.728 455129.027 487123.680 940
121.750 168−2.600 491126.340 37316.933 336
221.107 2336.846 462123.910 53218.784 314
221.084 0599.411 814227.378 96448.289 191
3112.000 0006.000 000226.390 41716.932 154
3312.000 00072.000 000222.774 8307.940 102
3212.000 000−50.505 1263110.993 52714.000 000
3311.015 078153.909 378
SO122.826 025−79.930 5553210.365 343−91.434 311
122.772 90879.814 350326.281 056−16.945 728
122.087 577−1.100 622
121.603 3421.969 873SO129.121 564−82.455 357
221.107 233−5.059 096128.863 22382.452 670
221.084 0594.999 134124.044 948−12.690 183
123.866 65712.458 423
Ag19AREP0212.570 677281.004 418227.535 754−19.308 744
027.075 22840.246 565227.278 97619.318 449
1211.402 841210.992 439222.772 0853.133 446
126.504 91530.829 872222.763 205−3.189 516
2210.792 675101.033 610
224.485 39614.813 640
3111.116 99619.000 000
3311.307 067211.222 924
3210.887 465−111.385 674
326.050 896−10.049 234
SO1214.533 732−7.088 207
127.620 8836.990 650
2211.057 85628.649 549
229.109 402−28.772 257
225.540 539−5.426 715
224.274 6364.873 157
TABLE II. Parameter values for the fifth row selected heavy-element ccECPs. For all ECPs, the highest value corresponds to the local channel L. Note that the highest non-local angular momentum channel max is related to it as max = L − 1.
AtomZeffHamiltoniannℓkαℓkβℓkAtomZeffHamiltoniannℓkαℓkβℓk
Bi5AREP023.398 563137.072 914Au19AREP0213.809 175423.151 508
022.236 39134.391 642026.576 68223.880 430
041.063 682−1.801 931024.315 26612.729 619
121.245 62039.826 9681211.806 942260.255 986
120.366 7620.844 734125.843 81632.797 931
140.823 760−4.391 855124.462 10914.975 886
220.924 628−3.787 918228.633 228135.277 395
220.835 03424.096 314224.081 60513.920 475
241.239 8550.456 375223.946 75312.464 658
321.847 446−0.382 962323.613 93515.143 598
321.069 842−0.546 444323.593 63514.877 722
340.862 193−0.236 708323.467 16312.317 560
418.029 1605.000 000419.985 26219.000 000
437.481 55140.145 802439.536 259189.719 980
421.775 683−6.251 2154210.882 390−140.528 643
420.698 085−0.286 410423.792 290−8.417 696
SO123.624 196−0.056 803SO1215.759 013−22.092 336
122.084 0173.659 507126.041 69422.320 517
141.279 9623.069 035228.904 73520.778 790
221.877 3533.942 473226.570 928−24.832 959
221.118 436−4.025 036223.389 9613.968 015
241.002 374−0.276 729323.387 468−5.901 882
321.013 3771.745 433323.083 2915.872 893
320.959 437−1.645 441
341.004 8950.007 556
Ir17AREP0213.826 287438.873 879W14AREP0211.273 956420.479 803
026.284 09576.599 511028.410 92439.584 943
1211.264 188262.750 135128.517 824320.877 235
125.346 38361.991 944121.608 091−0.472 821
226.815 210123.955 9931410.073 4900.112 929
224.766 74631.123 654226.059 286158.077 959
322.786 35010.329 276221.104 853−0.283 299
322.592 94611.885 698246.262 477−0.459 490
4112.171 07017.000 000322.075 3427.745 405
4312.170 139206.908 194321.845 9476.916 005
4212.320 343−143.208 3794110.188 96014.000 000
425.602 984−23.157 042439.478 240142.645 440
4210.146 880−100.078 727
SO1213.722 145−15.366 434425.491 651−0.941 578
124.201 4279.843 941
2210.062 658−2.838 512SO1211.964 154−0.551 147
225.177 6293.880 360124.897 3251.427 345
322.299 3184.142 597148.302 365194.404 774
322.295 619−4.159 394225.772 0347.045 184
223.175 200−0.115 276
246.155 4843.907 520
322.473 2954.123 819
322.405 627−4.032 609
A. Iodine (I)
1. AREP: I
For iodine, the AREP atomic and molecular results are shared in Figs. 4 and 5. The construction of ccECP for iodine followed some of the adjustments used for 3s, 3p elements that are in the same group of elements. The atomic spectral properties are significantly improved when compared with previous constructions, especially in MAD and WMAD. The low-lying excitations are on par with the best available construction SBKJC while our oxide molecule shows a much improved binding curve (Fig. 5). MWBSTU shows less errors in IO dimers; however, it underbinds in short bond length IH and has larger errors in the atomic spectrum. At low IO distances, we see similar inaccuracies that we observed for phosphorus,2424. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.5038135 which are manifested by overbinding at significantly compressed bond lengths. Within the presented data, our construction represents the most consistent option with clear advantages for a broad range of valence-only calculations.
2. SOREP: I
Table III gives SOREP iodine atomic excitation errors. We see that the spin–orbit splittings are reproduced very well, on par with MDFSTU results (that are very accurate to begin with) or marginally better. However, FPSODMC MAD shows an improvement over MDFSTU.
TABLE III. Iodine atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
5s25p52P3/20.0000.0000.0000.0000.00000.0000
5s25p61S0−2.1850.0220.035−3.0600.05(1)0.06(1)
5s25p43P29.4130.009−0.04610.4510.05(1)0.04(2)
5s25p34S3/227.2620.097−0.05829.5820.33(1)0.08(3)
5s25p52P3/20.0000.0000.0000.0000.00000.0000
2P1/20.9620.0300.0240.9420.01(1)0.00(2)
5s25p43P20.0000.0000.0000.0000.00000.0000
3P00.8630.0460.0470.7990.03(1)0.02(2)
3P10.8690.0190.0180.8780.08(1)0.04(2)
1D21.999−0.029−0.0401.7020.04(1)−0.01(2)
1S04.258−0.104−0.1323.657−0.08(1)−0.13(2)
5s25p34S3/20.0000.0000.0000.0000.00000.0000
2D3/21.959−0.136−0.1591.451−0.24(1)−0.07(2)
2D5/22.348−0.101−0.1191.847−0.14(1)0.08(2)
2P1/23.765−0.183−0.2153.012−0.24(1)−0.12(3)
2P3/24.375−0.116−0.1383.674−0.16(1)−0.06(3)
The overall accuracy is further corroborated in the calculation of the iodine dimer bonding curve that shows significant improvements already at the AREP level (Fig. 6). We see that the calculation reproduces correctly the experimental bond length unlike the previous construction that shows a bias toward smaller values. Furthermore, the two-component spinor calculations show an excellent agreement with experimental atomization energy that clearly demonstrates the quality of ccECP. Note that the inclusion of the explicit spin–orbit effect alleviates the overbinding of about 0.5 eV for both ECPs. The excellent agreement of consistent improvement between explicit spin–orbit FPSODMC and CCSD(T) results is encouraging and suggests comparable quality of the correlation description.
B. Tellurium (Te)
1. AREP: Te
The atomic spectrum and molecular binding discrepancy data for Te are provided in Figs. 7 and 8. In the atomic spectrum, we observe a marginal improvement in ccECP LMAD compared with other ECPs. Regarding MAD and WMAD, ccECP shows remarkable improvement, resulting in a much better description of higher atomic excitations. The molecular data in Fig. 8 demonstrate the transferability test in TeH and TeO molecules. In TeH, we see that ccECP is well within chemical accuracy except very short bond lengths near the dissociation limit. LANL2 and CRENBS appear to be better in TeH; however, they overbind significantly in the oxide dimer TeO. In fact, we see that all other ECPs noticeably overbind in TeO through all geometries, even at the equilibrium bond length. ccECP is very accurate near the equilibrium bond length and minor overbinding results only at the shortest bond lengths. Although BFD behaves slightly better at short bond lengths, it errs significantly at the equilibrium, which is vitally important for molecular and condensed matter properties. Obviously, we achieve a better balance between these two types of bonds. A similar compromise between hydride and oxide molecules has been seen from our previous ccECP constructions in 4p elements.3434. G. Wang, A. Annaberdiyev, C. A. Melton, M. C. Bennett, L. Shulenburger, and L. Mitas, J. Chem. Phys. 151, 144110 (2019). https://doi.org/10.1063/1.5121006
2. SOREP: Te
Tellurium SOREP atomic energy gap errors are provided in Table IV. We note that the MAD of COSCI gaps for ccECP is slightly larger than for MDFSTU, and further improvements without compromising other ccECP aspects proved to be difficult. (Exploiting the greater variational freedom of additional Gaussians could provide some options in the future.) The fixed-phase DMC calculations show similar results for ccECP and MDFSTU, which compare the atomic gaps and spin–orbit splittings as referenced to experimental data. Some of the differences are larger than desired; however, this is to be expected due to small valence space, neglect of core relaxations, and for FPSODMC using a single-reference trial function. The errors are larger for higher excitations, and at this point, it is not clear how much would that affect accuracy in bonded settings. Further research might be necessary if tests in bonded systems of interest would indicate that higher ccECP accuracy might be required.
TABLE IV. Tellurium atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
5s25p43P20.00000.00000.00000.0000.000.00
5s25p52P3/2−0.9927−0.02440.0254−1.9700.01(1)−0.02(1)
5s25p34S3/27.8285−0.06800.00219.0100.21(1)0.17(1)
5s25p23P225.47020.05050.210327.6100.55(1)0.38(1)
5s25p43P20.00000.00000.00000.0000.000.00
3P10.5773−0.06510.01330.5890.13(1)0.11(1)
3P00.6304−0.04540.02010.5840.01(1)0.04(1)
1D21.5787−0.00450.07251.3090.11(1)0.09(1)
1S03.45140.04410.17142.876−0.04(1)−0.09(1)
5s25p52P3/20.00000.00000.00000.0000.000.00
2P1/20.6346−0.06600.0139
5s25p34S3/20.00000.00000.00000.0000.000.00
2D3/21.75730.11260.11501.267−0.18(1)−0.18(1)
2D5/22.01170.07290.12111.540−0.10(1)−0.13(1)
2P1/23.26030.13720.19862.547−0.21(1)−0.25(1)
2P3/23.65110.07290.20972.980−0.12(1)−0.19(1)
5s25p23P00.00000.00000.00000.0000.000.00
3P10.5445−0.06490.01910.5890.09(1)0.11(1)
3P21.0371−0.08120.04691.0120.07(1)0.12(1)
1D22.4298−0.05270.13522.1520.12(1)0.10(1)
1S04.47710.06560.2803
C. Bismuth (Bi)
1. AREP: Bi
Figure 9 shows the Bi atomic spectral errors of all considered core approximations models considered. Our ccECP displays the smallest MAD and WMAD of all the approximations, while LMAD is also within chemical accuracy. Similarly, molecular errors are shown in Fig. 10 for varying bond lengths. In BiH, ccECP results in the smallest errors that are mostly within the chemical accuracy with a slight underbinding existing near the dissociation limit. For BiH, SBKJC and LANL2 ECPs show competitive errors; however, they significantly overbind in BiO molecule with up to 2 eV errors. On the other hand, ccECP errors in BiO are mostly within the chemical accuracy throughout the whole binding energy curve. Note that UC severely underbinds in both molecules and shows larger errors in the atom compared to ccECP. Overall, the data suggest that it might be possible to achieve better accuracy with proper form and optimizations even compared to AE systems with the same active space.
Another point of interest is that, in this case, the core/valence partitioning is not as clear-cut as for 5d elements that include the semi-core 5s, 5p subshells into the valence space. Specifically, this requires partitioning of n = 5 principal quantum number, where 5d10 is in the core while 5f14 could be considered semi-core or valence space. Note that this type of partitioning (i.e., the lowest one-particle eigenvalue does not correspond to = 0 channel) could result in significant errors in transition metals.1414. M. Dolg and X. Cao, Chem. Rev. 112, 403 (2012). https://doi.org/10.1021/cr2001383 However, the errors seen in this case are similar to what was observed in isovalent elements, such as N, P, and As,23,24,3423. M. C. Bennett, C. A. Melton, A. Annaberdiyev, G. Wang, L. Shulenburger, and L. Mitas, J. Chem. Phys. 147, 224106 (2017). https://doi.org/10.1063/1.499564324. M. C. Bennett, G. Wang, A. Annaberdiyev, C. A. Melton, L. Shulenburger, and L. Mitas, J. Chem. Phys. 149, 104108 (2018). https://doi.org/10.1063/1.503813534. G. Wang, A. Annaberdiyev, C. A. Melton, M. C. Bennett, L. Shulenburger, and L. Mitas, J. Chem. Phys. 151, 144110 (2019). https://doi.org/10.1063/1.5121006 with analogous core/valence definitions.
2. SOREP: Bi
The SOREP spectral data for Bi are given in Table V. Our chosen subset of valence states is composed of EA, IP, and single d/f excitations. The inclusion of single d/f excitation states is essential to constrain the spin–orbit splitting bias for corresponding channels. Overall, we see a marginal reduction in MAD and a more noticeable improvement in the ground state and in the first IP multiplet splitting.
TABLE V. Bismuth atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
6s26p34S3/20.0000.0000.0000.0000.00000.0000
6s26p43P20.034−0.0620.0080.942−0.04(1)0.08(1)
6s26p23P06.663−0.079−0.1137.2850.03(1)0.10(1)
6s26p12P1/222.621−0.052−0.16623.9880.33(1)0.26(1)
6s26p34S3/20.0000.0000.0000.0000.00000.0000
2D3/21.5490.007−0.0361.4150.07(1)0.03(2)
2D5/22.1400.011−0.0471.9140.07(1)0.04(2)
2P1/23.1100.002−0.0932.6850.08(1)−0.02(2)
2P3/24.4880.060−0.0504.1110.11(1)0.06(2)
6s26p23P00.0000.0000.0000.0000.00000.0000
3P11.5330.0640.0501.6520.14(1)0.01(2)
3P22.1460.0590.0192.1110.10(1)−0.03(2)
1D24.3070.1260.0414.2070.21(1)0.01(2)
1S05.9440.095−0.0675.4760.18(1)0.05(2)
6s26p12P1/20.0000.0000.0000.0000.00000.0000
2P3/22.5970.1030.0572.5770.12(1)0.10(1)
6s25d12D3/20.0000.0000.0000.0000.00000.0000
2D5/20.1780.020−0.0000.7800.61(1)0.53(2)
6s25f12F7/20.0000.0000.0000.0000.00000.0000
2F5/20.0110.004−0.0020.0120.03(1)−0.03(1)
Figure 11 shows the bismuth dimer binding curve. The molecular calculations include AREP CCSD(T) and SOREP FPSODMC using PBE0 trial wave functions for both MDFSTU and ccECP. For both ECPs, the two-component spinor FPSODMC calculations show significant alleviation of overbinding that is present in AREP UCCSD(T). Our ccECP outperforms MDFSTU in both AREP and SOREP calculations being very close to experiments.
D. Silver (Ag)
1. AREP: Ag
For silver, the averaged relativistic atomic and molecular results are shown in Figs. 12 and 13. All ECPs show quantitatively good accuracy for LMAD due to the simple closed d shell. If we consider broader states in the spectrum, MAD and WMAD reveal significant improvement achieved comparing to other ECPs. In Fig. 13, AgH is quite well described by most ECPs, and ccECP is among the best ones. Improvement is more noticeable in AgO; although most core approximations maintain the accuracy for all geometries, ccECP and MWBSTU show the best performance with almost perfect agreement with AE binding energies. Note that, in both molecules, ccECP is the closest to AE results while keeping the discrepancies flat throughout; therefore, it provides the highest accuracy of all core approximations.
2. SOREP: Ag
The SOREP atomic and molecular results are given in Tables VI and VII. The COSCI agreement of MDFSTU and ccECP with AE gaps is remarkably good, with MAD less than 0.01 eV. The closed-shell electronic configurations lead to Ag being a special case resulting in a single determinant COSCI wave function even in spin–orbit relativistic REL-CCSD(T) calculations (Au is a similar case, too). This enables us to perform explicit spin–orbit relativistic REL-CCSD(T) calculations for the lowest charged states with results collected in Table VII. We observe the gaps errors smaller than chemical accuracy when compared to experimental data for both ECPs.
TABLE VI. Silver atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
4d105s12S1/20.0000.0000.0000.0000.0000.000
4d105s21S0−0.1170.0000.001−1.304−0.35(1)−0.27(3)
4d105p12P1/23.013−0.0010.0023.6640.06(1)0.12(3)
4d101S06.340−0.002−0.0027.5760.19(1)0.28(3)
4d92D5/225.999−0.0060.02029.061−0.28(1)0.34(3)
4d105p12P1/20.0000.0000.0000.0000.0000.000
2P3/20.0740.0010.0020.1140.01(1)−0.01(3)
4d92D5/20.0000.0000.0000.0000.0000.000
2D3/20.5710.0300.0110.5710.01(1)0.03(3)
TABLE VII. Silver atomic excitation errors for MDFSTU vs ccECP in SOREP forms. The set of errors are calculated using REL-CCSD(T) from the DIRAC code and compared to experiments. All values are in eV.
REP-CCSD(T)
StateTermExpt.STUccECP
4d105s12S1/20.0000.0000.000
4d105s21S0−1.3040.0050.002
4d105p12P1/23.664−0.029−0.024
4d101S07.576−0.034−0.027
4d92D5/229.0610.0070.031
For the same states, we carry out FPSODMC calculations with single-reference trial functions in order to probe for the corresponding fixed-phase biases (Table VI). Here, we see discrepancies from 0.1 to 0.34 eV for the highest state. These types of errors are not unexpected for single-reference due to increased mixing of higher excitations, which results from lowering the symmetry from LS-coupling to J-coupling. We verified this argument in the part devoted to tungsten where we constructed trial functions based on medium size Configuration Interaction (CI) expansions and we observed corresponding diminishing of fixed-phase biases. On the other hand, the FPSODMC method shows more favorable results of meV bias in multiplet splittings whereas the REL-CCSD(T) method proved to be problematic in the DIRAC code. This demonstrates the quality level in dealing with spin–orbit splittings with the developed fixed-phase method.3131. C. A. Melton, M. Zhu, S. Guo, A. Ambrosetti, F. Pederiva, and L. Mitas, Phys. Rev. A 93, 042502 (2016). https://doi.org/10.1103/PhysRevA.93.042502 Therefore, we employed J-MAD that includes only bias from spin–orbit splitting states for FPSODMC accuracy assessment from Expt. values. Obviously, the J-MAD does not represent the ultimate accuracy but gives a reasonable approximation for the accuracy of SO terms.
E. Gold (Au)
1. AREP: Au
Figure 14 shows the spectral errors for each Au ECP investigated in this work. Our ccECP far outperforms the others in MAD and WMAD, with the LMAD remaining well within the chemical accuracy. For the molecular binding energy curves in Fig. 15, our ccECP remained well within chemical accuracy over the range of geometries tested. Specifically, for AuH, our ccECP has the lowest discrepancy from the equilibrium bond length to the most compressed geometry we tested, whereas most other ECPs show pronounced underbinding. For AuO, the ccECP performs consistently with the discrepancy remaining very small at all bond lengths. In both molecules and in the atomic spectrum, the UC approximation is significantly outperformed by our ccECP.
2. SOREP: Au
The atomic data for Au MDFSTU and ccECP are listed in Tables VIII and IX. In Table VIII, we observe that, for multiplet splittings, both ECPs show close performance using the FPSODMC approach. For the atomic calculations, we provide additional REP-CCSD(T) charged states for both ECPs (Table IX). Although FPSODMC results for some charged states show biases of ≈0.1–0.3 eV due to limits of single-reference trial functions, REL-CCSD(T) calculations exhibit uniform consistency almost fully within the chemical accuracy bounds. The remarkable agreement of REL-CCSD(T) calculations with experimental excitations demonstrates the quality of both ECPs. The scattered FPSODMC biases for charged states that result from varying mixing of higher excitations of the same symmetry clearly require a more thorough study with trial functions that include sufficiently large active spaces. Overall, ccECP and MDFSTU show similar excellent accuracy for both charged excitations and multiplet splittings.
TABLE VIII. Gold atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
5d106s12S1/20.0000.0000.0000.0000.0000.000
5d106s21S0−0.6480.0160.022−2.310−0.35(4)−0.22(3)
5d96s22D5/21.284−0.040−0.0381.140−0.25(4)−0.14(3)
5d106p12P1/24.0240.0080.0034.630−0.10(3)−0.10(3)
5d101S07.7040.0260.0059.2300.26(4)0.25(3)
5d92D5/226.2660.026−0.03329.4300.15(3)0.16(3)
5d96s22D5/20.0000.0000.0000.0000.0000.000
2D3/21.4800.061−0.0201.5200.02(3)−0.06(3)
5d106p12P1/20.0000.0000.0000.0000.0000.000
2P3/20.3490.011−0.0050.4700.02(3)0.00(3)
5d92D5/20.0000.0000.0000.0000.0000.000
2D3/21.5580.071−0.010
5d105f12F7/20.0000.0000.0000.0000.0000.000
2F5/20.1030.0180.0080.000−0.09(4)−0.05(4)
TABLE IX. Gold atomic excitation errors for MDFSTU vs ccECP in SOREP forms. The set of errors are calculated using REL-CCSD(T) from the DIRAC code and compared to experiments. All values are in eV.
REP-CCSD(T)
StateTermExpt.STUccECP
5d106s12S1/20.0000.0000.000
5d106s21S0−2.3100.0860.025
5d96s22D5/21.1400.0100.035
5d106p12P1/24.630−0.044−0.046
5d101S09.230−0.004−0.025
5d92D5/229.430−0.010−0.052
Further tests were carried out in Au2 dimer (Fig. 16). Here, we find a similar and satisfying agreement with the experiment for the equilibrium bond length and also binding energy using the AREP-CCSD(T) method. However, for ccECP, the ultimate accuracy manifests in the full SOREP setting. Note that STU underbinds the dimer by about 0.25 eV while ccECP improves slightly the bond length equilibrium and the binding energy with very negligible constant overbinding.
F. Tungsten (W)
1. AREP: W
Figures 17 and 18 present the bias of W atomic spectra and discrepancies for W molecular dimers, respectively. This is again a case where we show the accuracy of our constructed ccECP to be higher consistently for atomic and molecular properties than other core approximations included. Figure 17 shows that the MAD and WMAD are significantly reduced for ccECP. Also, as the metrics for evaluating the low-lying state errors, ccECP LMAD is contained within the chemical accuracy. The other ECPs do not maintain the errors within the chemical accuracy. We have also achieved substantial improvement in molecular properties. In Fig. 18, there is a clear tendency for underbinding of the hydride dimer and overbinding the oxide dimer for all ECPs. In the WH molecule, our constructed ccECP retains the bias inside the chemical accuracy band for the entire binding curve. For WO, ccECP also stands out when compared with previously tabulated ECPs and overall provides the best balance of accuracy in all tested systems.
2. SOREP: W
Using the SOREP Hamiltonian, the tungsten atomic energy gaps are shared in Table X for MDFSTU vs ccECP. Our optimization of ccECP reduces MAD to approximately one-third of MDFSTU when referenced to COSCI/AE gaps. Not surprisingly, fixed-phase calculations show the encouraging agreement of multiplet splitting gaps with experimental values but significant errors appear for charged states, which were scrutinized also in constructions for gold and silver. To further investigate the origin of these errors, we extend our calculations to CI expansions and to related DMC calculations of the ground state (5d46s2, 5D0) and excited state (5d56s1, 7S3) for W ccECP (Table XI). Clearly, we show that, in AREP with substantial multi-reference wave functions, the gap approaches the AREP estimated experimental value. Similarly, SOREP calculations reveal that applying CI expansions boost the accuracy further toward the experimental value. As it has been observed also previously, both FPSODMC and CI with restricted COSCI trial function are inadequate and exhibit the incorrect ground state occupancy 5d56s1. This shows that both the explicit treatment of spin–orbit and accurate correlation are crucial for the atomic spectrum calculations, and this, in particular, is true for 5d mid-series elements. It is reassuring that both MR-CISD and subsequent FPSODMC/MR-CISD calculations correctly predict the order of these two states. Therefore, our estimation of FPSODMC gap errors excludes the charged configuration gaps and only keeps the J-splitting gaps in J-MAD.
TABLE X. Tungsten atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
5s25p65d46s25D00.0000.0000.0000.0000.000.00
5s25p65d56s26S5/20.0160.0780.021−0.8150.14(3)0.01(2)
5s25p65d56s17S3−0.7330.1130.0570.3660.64(4)0.48(2)
5s25p65d46s16D1/25.8500.031−0.0057.8640.52(3)0.44(2)
5s25p65d45D020.6850.020−0.05324.2(2)1.1(2)1.0(2)
5s25p65d46s25D00.0000.0000.0000.0000.000.00
5D10.1230.026−0.0030.2070.14(3)0.04(2)
5D20.2960.052−0.0080.4120.11(3)0.03(2)
5D30.4860.072−0.0140.5990.14(3)−0.00(2)
5D40.6830.086−0.0210.7710.16(3)−0.02(2)
5s25p65d46s16D1/20.0000.0000.0000.0000.000.00
6D3/20.1160.020−0.0030.1880.02(3)0.05(2)
6D5/20.2730.041−0.0090.3930.12(3)0.08(2)
6D7/20.4540.061−0.0150.5850.14(3)0.02(2)
6D9/20.6480.076−0.0220.7620.13(3)0.00(2)
5s25p65d46s15D00.0000.0000.0000.0000.000.00
5D10.1480.031−0.0040.2800.18(3)0.10(1)
5D20.3480.060−0.0100.5530.22(3)0.09(2)
5D30.5610.081−0.0190.7780.22(3)0.10(2)
5D40.7770.094−0.0270.9530.19(3)0.05(2)
5s25p66s26p12P1/20.0000.0000.000
2P3/21.6970.023−0.054
5s25p66s26f12F5/20.0000.0000.000
2F7/20.0080.003−0.028
5s25p52P3/20.0000.0000.000
2P1/211.0590.1440.020
TABLE XI. Atomic excitations of the W ground state 5d46s2 and excited state 5d56s1 of ccECP pseudoatom using various methods. Note the improvement of gaps as more accurate methods are used. “AREP Est. Expt.” represents the estimated experimental value for the AREP gap by weighted J-averaging.
MethodGap (eV)
AREP
FNDMC/HF−0.65(2)
FNDMC/PBE0−0.54(2)
FNDMC/CASCIa−0.42(3)
CCSD(T)−0.31
AREP Est. Expt.−0.18
SOREP
COSCI−0.790
FPSODMC/COSCI−0.11(3)
FPSODMC/CISDb0.169
Expt.0.366
aActive space = (30o, 6e) $≈10$k dets.
bFrozen-core, truncated CISD $≈10$k dets.
Figure 19 shows the W dimers in AREP CCSD(T) and two-component FPSODMC calculations. To the best of our knowledge, experimental data are lacking here since the most accurate experimental value was estimated to be 5(1) eV.4141. M. D. Morse, Chem. Rev. 86, 1049 (1986). https://doi.org/10.1021/cr00076a005 Interestingly, we find MDFSTU and ccECP cross-validate each other in both AREP and REP calculations with very similar binding curves. All of the binding energies for both ECPs are inside the estimations from the previous work. Both ECPs predict the equilibrium bond length near 1.95 Å.
1. AREP: Pd
Figure 20 shows atomic excitation errors for Pd ccECP. The Pd ccECP outperforms most of the other ECPs in all metrics with a couple of notable exceptions. The CRENBL ECP had a slight advantage with the raw MAD from all of the states chosen, but the ccECP performed better at replicating the energies at low-lying states and this is demonstrated by the WMAD statistic that weights the errors of smaller gaps more heavily.
For the molecular binding, Pd was one of the elements where we used the SEFIT/MEFIT method. Compared to our initial optimizations using our more conventional spectral fitting method, the SEFIT/MEFIT method led to much better molecular binding curve discrepancies with comparable or unchanged atomic spectrum performance and provided energy curves with greater accuracy than most contending ECPs.
Figure 21 provides the PdH and PdO molecular binding curve plots. Here, most ECPs remain within the bounds of chemical accuracy over the entire range of geometries tested with the exception of SBKJC and LANL2. Overall, CRENBL and ccECP show the smallest errors for these molecules.
2. SOREP: Pd
Table XII provides the SOREP atomic gap errors using COSCI and FPSODMC. We again see only minor improvements from MDFSTU in the COSCI method while FPSODMC errors are comparable. Overall, both SOREP ECPs show high quality in terms of multiplet splittings. We expect to see even smaller errors for all gaps as the trial wave function quality is increased.
TABLE XII. Palladium atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors is calculated using FPSODMC with reference to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
4s24p64d101S30.0000.0000.0000.0000.0000.000
4s24p64d105s12S1/20.2360.0090.008−0.562−0.33(3)−0.36(2)
4s24p64d95s12[5/2]3−0.067−0.012−0.0360.814−0.22(2)−0.23(2)
4s24p64d92S5/26.273−0.009−0.0448.3370.05(2)0.03(2)
4s24p64d83F423.6620.034−0.07827.7700.26(2)0.12(2)
4s24p64d95s12[5/2]30.0000.0000.0000.0000.0000.000
2[5/2]20.1750.010−0.0080.148−0.02(2)0.01(2)
4s24p64d92S5/20.0000.0000.0000.0000.0000.000
2S3/20.4360.042−0.0380.439−0.02(3)−0.03(2)
4s24p64d83F40.0000.0000.0000.0000.0000.000
3F30.3940.041−0.0150.4000.10(2)0.05(2)
3F20.6040.050−0.0240.5810.02(2)−0.06(2)
4s24p12P1/20.0000.0000.000
2P3/26.6400.0930.000
H. Iridium (Ir)
1. AREP: Ir
Figure 22 shows the Ir spectral errors of various ECPs tested within this work. The Ir ccECP exceeded the accuracy of all the other ECPs for this element in all metrics. The LMAD was comfortably within the chemical accuracy, and the MAD was much smaller than most other contenders apart from the MDFSTU ECP, which achieved a similar MAD. Figure 23 shows the molecular binding energy discrepancy for both IrH and IrO. Half of the ECPs tested were outside of the chemical accuracy for IrH over the entire range of geometries. The Ir ccECP does not have the smallest discrepancy at all points for either molecule, but when considering both systems, it has the most balanced biases that are always remaining within the chemical accuracy.
2. SOREP: Ir
We provide Ir SOREP atomic excitation errors in Table XIII. The optimization of ccECP provides an accurate spectrum for COSCI atomic gaps for the entire set of states with the MAD of 0.018 eV, which is less than one-third of MDFSTU MAD. However, considering the higher accuracy of FPSODMC calculations, we see that ccECP and STU give comparable MADs for spin–orbit splitting states referenced to experimental data, about 0.2 eV. Although we see an improvement in MAD of J-splitting in the first IP state, the large discrepancies in ground state multiplet gaps overshadow this improvement. Further inspection shows that the ground state multiplet splittings change order when going from COSCI to experimental values. This is another indication that electron correlations must be accurately accounted to properly describe these low-lying states. Clearly, a more extensive study of correlation for these states is required and we plan to address it in the future. A similar case was observed in the W atom where the incorrect ordering of states was obtained unless higher order CI expansions were used (Table XI).
TABLE XIII. Iridium atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors are calculated using FPSODMC and compared to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
5s25p65d76s24F9/20.0000.0000.0000.0000.000.00
5s25p65d86s26S5/20.1200.0000.039−1.565−0.33(3)−0.28(3)
5s25p65d86s14F9/20.284−0.0010.0450.3510.08(3)0.11(3)
5s25p65d76s15F56.9970.0120.0048.9670.34(4)0.35(3)
5s25p65d74F9/223.2900.020−0.01026.0(3)−0.1(3)−0.1(3)
5s25p65d76s24F9/20.0000.0000.0000.0000.000.00
4F7/20.7820.0880.0090.7840.01(3)−0.05(3)
4F5/21.0750.080−0.0040.717−0.35(3)−0.36(3)
4F3/21.2020.052−0.0210.506−0.66(3)−0.71(3)
5s25p65d86s14F9/20.0000.0000.0000.0000.000.00
4F7/20.5120.0380.0020.5300.07(3)0.05(3)
4F5/20.9400.068−0.0030.873−0.06(3)−0.02(3)
4F3/21.0840.056−0.0101.1150.06(3)0.09(3)
5s25p65d76s15F50.0000.0000.0000.0000.000.00
5F40.5960.0550.0030.5940.05(4)−0.07(3)
5F31.0150.0920.0031.0150.09(4)−0.03(3)
5F21.2410.096−0.0031.4020.26(4)0.14(3)
5F11.3760.099−0.0061.4830.21(4)0.05(3)
5s25p65d74F9/20.0000.0000.000
4F7/20.8480.0940.008
4F5/21.1590.082−0.007
4F3/21.2890.048−0.027
5s25p66s26p12P1/20.0000.0000.000
2P3/23.6010.0160.031
5s25p66s26f12F5/20.0000.0000.000
2F7/20.1060.040−0.001
5s25p52P3/20.0000.0000.000
2P1/216.0840.169−0.114
I. Molybdenum (Mo)
1. AREP: Mo
Figures 24 and 25 show the Mo atomic spectral errors and molecular binding discrepancies for various ECPs, respectively. Figure 24 clearly shows that the developed Mo ccECP outperforms all the other ECPs in MAD and WMAD. Note that the MAD of our ccECP is refined to chemical accuracy, which indicates that the high accuracy is achieved for the full span of deep ionizations. Moreover, the LMAD, representing the precision of the low-lying state energies, is within the chemical accuracy at a notably low level. MDFSTU ECP has a slightly better LMAD, yet the MAD and WMAD are not comparable with ccECP, suggesting that ccECP is a more comprehensive solution as a robust effective core potential. Figure 25 shows that the ccECP binding energy discrepancies are within chemical accuracy for all geometries in both MoH and MoO molecules. In hydride, MWBSTU shows flatter and much smaller errors in shorter bond lengths. However, in oxide, ccECP is accurate for all bond lengths, while all other core approximations (including MWBSTU) deviate outside of chemical accuracy in some parts of the curve.
2. SOREP: Mo
Table XIV provides the Mo atomic excitations errors for MDFSTU and ccECP. Obviously, the optimization of SO ccECP results in a more accurate spectrum. The MAD for COSCI calculations of ccECP is decreased by almost a magnitude when compared with MDFSTU results. Although the accuracy is not fully achieved in fixed-phase calculations with restricted quality trial function, the MAD of the low-lying spin–orbit splitting spectrum referenced in experiments is about 0.08 eV, which is significantly smaller than for MDFSTU.
TABLE XIV. Molybdenum atomic excitation errors for MDFSTU vs ccECP in SOREP forms. One set of errors are shown for full-relativistic X2C AE gaps using COSCI. Another set of errors is calculated using FPSODMC with reference to experiments. All values are in eV.
COSCIFPSODMC
StateTermAESTUccECPExpt.STUccECP
4s24p64d55s17S30.0000.0000.0000.0000.0000.000
4s24p64d55s26S5/20.693−0.007−0.007−0.747−1.21(3)−0.67(5)
4s24p64d45s25D02.189−0.068−0.0391.360−1.18(3)−0.74(4)
4s24p64d56S5/28.2740.629−0.0787.092−0.62(5)−0.16(5)
4s24p64d45D021.200−0.057−0.02423.252−0.46(4)0.25(5)
4s24p64d55s17S30.0000.0000.0000.0000.0000.00
5S21.870−0.009−0.0091.335−0.4(1)−0.24(4)
5G22.725−0.066−0.0632.063−0.66(6)−0.34(4)
4s24p64d45s25D00.0000.0000.0000.0000.0000.00
5D10.0240.0050.0050.022−0.14(3)0.03(5)
5D20.0700.0150.0130.061−0.05(4)0.09(4)
5D30.1330.0270.0240.111−0.28(4)0.16(4)
5D40.2090.0400.0360.171−0.08(3)0.07(3)
4s24p64d56S5/20.0000.0000.0000.0000.0000.000
4G5/20.063−0.5070.0091.884−0.26(3)0.00(4)
4G7/20.195−0.9660.0301.901−0.20(4)−0.05(6)
4G9/20.431−0.7730.0821.913−0.26(4)0.00(4)
4G11/20.726−0.570−0.0201.915−0.34(2)−0.16(4)
4s24p64d45D00.0000.0000.0000.0000.0000.000
5D10.0260.0060.0050.030−0.04(3)−0.016(4)
5D20.0760.0160.0140.0830.04(6)0.01(4)
5D30.1430.0280.0250.1520.14(5)0.01(3)
5D40.2260.0420.0370.2320.11(5)−0.07(4)
4s24p65s26p12P1/20.0000.0000.000
2P3/20.5640.097−0.006
4s24p52P3/20.0000.0000.000
2P1/22.9480.050−0.013
J. Summary of results
1. Averages: AREP
Figure 26 and Table XV give the summary of AREP atomic spectrum errors and AREP molecular property discrepancies for all elements considered in this work. In general, we have achieved substantial improvements in both atomic and molecular results. For the atomic spectrum, our ccECP shows significant improvement in all metrics, LMAD, MAD, and WMAD. The LMAD metric includes only low-lying states composed of EA, IP, and IP2, and it is well within chemical accuracy and slightly better than the rest of the core approximations. The MAD and WMAD show a similar picture to previous elements with the lowest errors for ccECP, with an overall dramatic improvement in a wide range of excitation energies. Table XV provides the collected results of various binding parameter errors for all elements as obtained by fits in Eq. (18). Clearly, our ccECPs give the highest accuracy with the lowest errors for all parameters characterizing the molecular bonding. In addition, considering both atomic and molecular errors, ccECPs show higher accuracy than the existing ECPs, but also than UC results as discussed earlier.
TABLE XV. Mean absolute deviations of binding parameters for various core approximations with respect to AE correlated data for I, Te, Ag, Pd, Mo, Bi, Au, Ir, and W related molecules. All parameters were obtained using the Morse potential fit. The parameters shown are dissociation energy De, equilibrium bond length re, vibrational frequency ωe, and binding energy discrepancy at dissociation bond length Ddiss.
De (eV)re (Å)ωe (cm−1)Ddiss (eV)
BFD0.078(6)0.018(1)22(3)0.41(5)
CRENBL(S)0.115(5)0.0183(9)26(3)0.40(4)
LANL20.118(5)0.0122(9)16(3)0.36(4)
MDFSTU0.096(5)0.0094(9)10(3)0.24(4)
MWBSTU0.050(5)0.0056(9)10(3)0.10(4)
SBKJC0.089(4)0.0120(8)19(2)0.30(4)
UC0.040(5)0.0104(9)13(3)0.19(4)
ccECP0.018(5)0.0022(9)6(3)0.07(4)
2. Averages: SOREP
We provide the summary of SOREP atomic excitations MADs for MDFSTU and ccECP in Fig. 27. The atomic COSCI gaps are used for the optimization of spin–orbit terms and they are plotted in Fig. 27(a). As commented above, ccECPs show mildly higher or on par MADs for main group elements and consistent significant improvements for transition elements. Not surprisingly, the slightly higher MADs in main group elements are due to the fixed AREP part in spin–orbit terms optimization since that restricts the total variational freedom; however, it helps the optimization efficiency. Although further minor refinements might be possible, the overall expected gains are deemed as marginal due to the dominant source of bias from large cores. In transition metals, the larger valence space alleviates this deficiency and provides the COSCI MADs with chemical accuracy for all cases. Especially, for the Mo atom, we see a dramatic improvement of MADs from previously observed ≈0.2 eV, which is reduced by almost an order of magnitude. Figure 27(b) presents the assessments of MDFSTU and ccECPs in FPSODMC calculations referenced to experimental data. Note that, for the transition metals, we include only spin–orbit splitting states since we find the accuracy of FPSODMC as being somewhat limited for the charged states in the single-reference trial setting. Clearly, this calls for a more elaborated QMC study with multi-reference trial functions. We see that ccECP shows overall reduced MAD compared to MDFSTU except for Te with similar large MADs and Ag with MADs maintained within desirable chemical accuracy. Interestingly, though Fig. 27(a) shows a slightly larger COSCI MAD for the main group elements, the FPSODMC MAD is slightly lower. We anticipate that overall our AREP ccECPs will boost the accuracy of spin–orbit calculations, especially for the charged states where both charge relaxation and correlation play significant roles. For transition metals, noticeable gains in accuracy and consistency are obtained for Au, Ag, W, and Mo while Ir shows modest improvements. This has been discussed in the Ir section where the ground state spin–orbit splitting appears to be significantly different from other cases, and perhaps, it might be further refined in the future. Generally, ccECP shows consistent improvements in accuracy compared to MDFSTU.
We show the SOREP transferability tests for I, Bi, Au, and W in Figs. 6, 11, 16, and 19, which are monoatomic dimer binding curves, respectively. For I2 and Bi2, we see consistent improvements from AREP CCSD(T) to SOREP FPSODMC calculations as well as going from MDFSTU to ccECP. Especially, in I2, the constructed ccECP in FPSODMC shows near exact equilibrium bond length and binding energy compared to Expt. In certain aspects, the full d shell in Au leads to a simpler binding picture for Au2, and we, indeed, see excellent accuracy of both MDFSTU and ccECP in the CCSD(T)/AREP binding parameters agreeing with experimental data. When we further consider the explicit spin–orbit effect, the exceptional accuracy is not maintained for MDFSTU while ccECP shows very consistent performance with desired properties in both AREP and SOREP levels. For W2, we have provided our calculations though experimental data are lacking to our best knowledge. The estimated binding energy from the previous work is given 5(1) eV, in which all calculations are inside the range. In this case, MDFSTU and ccECP behave almost the same and cross-validate each other. We believe that the calculations provide a new reliable reference for further studies of this system.
Basis sets and K–B formats. The derived ccECPs are accompanied by basis sets up to 6Z level for main group elements and 5Z level for transition metal elements.4242. See https://pseudopotentiallibrary.org for Pseudopotential Library: A community website for pseudopotentials/effective core potentials developed for high accuracy correlated many-body methods such as quantum Monte Carlo and quantum chemistry; accessed 01 May 2021. The cited library includes also Kleinmann–Bylander transformed forms and corresponding files for use with plane wave codes. In general, very good convergence is achieved for cut-offs below ≈200 Ry for main group elements and ≈400 Ry for transition metal elements, which enable routine calculations of solids and 2D materials. Further details can be found in the supplementary material. All ccECP and corresponding basis sets in various code formats can be found at https://pseudopotentiallibrary.org.
In this work, we present newly constructed correlation consistent effective core potentials for heavy elements I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W. Following the same convention of our previous constructions for the first three row elements, the valence spaces are the most generally used for main groups elements, I, Bi, and Te, including only nth s and p electrons, where n = {5, 6} is the largest main quantum number. For the 4d and 5d transition metal elements, we chose a larger valence space by incorporating the semi-core s and p electrons with the outer-layer s and d electrons.
Our primary goal was to generate highly accurate ccECPs for the mentioned elements incorporating many-body theories and explicit spin–orbit effect. Intuitively, the construction is partitioned into the AREP (spin-averaged) part and subsequent SO (spin–orbit) part. Such methodology relies on the quality of AREP since the SO part plays the role as extensive refinements. To obtain highly accurate AREP ccECPs, we follow the previous scheme of the many-body construction method that involves an iterative process in corresponding calculations of all-electron atoms using coupled cluster methods and optimizations of objective functions that include weighted atomic spectra, norm-conservations, and extensive quality/transferability tests in molecular binding curves for hydride and oxide dimers. The spin–orbit optimization applies the iso-spectrality of low-lying states and corresponding spin–orbit splittings. Further assessments of the constructed ccECPs from spin–orbit splittings and several molecular dimer binding calculations are carried out for the two-component spinor FPSODMC calculations.
We find that the main source of biases is the AREP part. This appears to be similar to the observation for 3d transition metal elements where the Hartree–Fock levels produced the dominant source of inaccuracies.
The ccECPs enable finding spin–orbit splittings with errors of 0.05–0.1 eV when compared with experiments in most cases and show remarkable accuracy in the dimer molecular binding for the cases we tested.
The comparisons with previously constructed sets (CRENBL, STU, and SBKJC) show that ccECPs are overall much more accurate and consistent in minimizing the biases. The main group elements show minor errors beyond chemical accuracy in dimers at very small bond lengths, which have been observed already for 3s3p main group elements although, for those cases, the errors are generally larger.
We also test the ECPs on FPSODMC calculations for particular homonuclear dimers and with single-reference trial functions. The agreement with CCSD(T) is reasonably good although biases of a few tenths of eV occur in particular cases. Similar to some of the atomic calculations, these discrepancies are assigned to the limited accuracy of the trial fixed-phase generated by single-reference as the tests on selected tungsten systems with CI trial states illustrate. More systematic studies of the fixed-phase biases with more accurate CI trial functions are left for future work due to the already significant length of this paper. We also mention that considering the lack of accurate experimental data, our calculations of the W2 dimer provide an independent prediction of the binding parameters.
We believe that our study paves the way for accurate many-body valence space calculations with heavy atoms by providing a new generation of effective core potentials for several elements that are present in technologically important materials. The spin–orbit terms are included in a two-component spinor formalism. The new ccECPs are systematically more accurate and show better consistency with all-electron settings for atoms as well as molecular oxides and hydrides with additional benchmarks and periodic system calculations left for future work.
Additional information about ccECPs can be found in the supplementary material. Therein, calculated AE spectra are given for each element and also corresponding discrepancies of various core approximations. AE, UC, and various ECP molecular fit parameters for hydrides, oxides, or dimers are provided. The ccECPs in semi-local and Kleinman–Bylander projected forms as well as optimized Gaussian valence basis sets in various input formats (molpro, gamess, and nwchem) can be found in Ref. 4242. See https://pseudopotentiallibrary.org for Pseudopotential Library: A community website for pseudopotentials/effective core potentials developed for high accuracy correlated many-body methods such as quantum Monte Carlo and quantum chemistry; accessed 01 May 2021..
The authors thank Cody A. Melton for kind help with spin–orbit QMC calculations. We are also grateful to Paul R. C. Kent for reading the manuscript and for helpful suggestions.
This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, as part of the Computational Materials Sciences Program and Center for Predictive Simulation of Functional Materials.
This research used resources from the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. This research used resources from the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC02-06CH11357. This research also used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract No. DE-AC05-00OR22725.
This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
The authors have no conflicts to disclose.
The input/output files and supporting data generated in this work are published in Materials Data Facility43,4443. B. Blaiszik, K. Chard, J. Pruyne, R. Ananthakrishnan, S. Tuecke, and I. Foster, JOM 68, 2045 (2016). https://doi.org/10.1007/s11837-016-2001-344. B. Blaiszik, L. Ward, M. Schwarting, J. Gaff, R. Chard, D. Pike, K. Chard, and I. Foster, MRS Commun. 9, 1125 (2019). https://doi.org/10.1557/mrc.2019.118 and can be found in Ref. 4545. G. Wang, B. Kincaid, H. Zhou, A. Annaberdiyev, M. C. Bennett, J. T. Krogel, and L. Mitas (2022), “A new generation of effective core potentials from correlated and spin-orbit calculations: Selected heavy elements,” Dataset. https://doi.org/10.18126/IMHE-8CSM.
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