#### ABSTRACT

We introduce new correlation consistent effective core potentials (ccECPs) for the elements I, Te, Bi, Ag, Au, Pd, Ir, Mo, and W with 4

*d*, 5*d*, 6*s*, and 6*p*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 tables

^{1–14}as well as computational tools.^{15,16}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.

^{11}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.^{17–20}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.^{7,8,11,14}Other directions for improvement have been explored within Density Functional Theory (DFT) using many-body perturbation theory.^{21,22}Very recently, we have introduced correlation consistent ECPs (ccECPs)

^{23–26}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 3

*d*and 4*s*4*p*elements where explicit spin–orbit interaction is an important ingredient for accurate correlated treatments. The constructed ccECPs are selected heavy elements that include 4*d*and 5*d*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 4*d*and 5*d*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.^{27–30}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.

^{31}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

The full spin–orbit relativistic effective potential (SOREP) ECPs are of the form proposed by Lee,

where

where ${V}_{\ell}^{\mathit{AREP}}$ is the weighted

The set of SO potentials

where

*H*_{val}. Following the Born–Oppenheimer approximation, the valence Hamiltonian*H*_{val}in atomic units (a.u.) is expressed as$${H}_{\mathit{val}}=\sum _{i}\left[{T}_{i}^{\text{kin}}+{V}_{i}^{\mathrm{S}\mathrm{O}\mathrm{R}\mathrm{E}\mathrm{P}}\right]+\sum _{i<j}1/{r}_{ij}.$$ | (1) |

^{32}$$\begin{array}{ll}\hfill {V}^{\mathit{SOREP}}& ={V}_{LJ}^{\mathit{SOREP}}(r)+\sum _{l=0}^{L-1}\sum _{j=|l-1/2|}^{l+1/2}\sum _{m=-j}^{j}\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}\times \left[{V}_{lj}^{\mathit{SOREP}}(r)-{V}_{LJ}^{\mathit{SOREP}}(r)\right]|ljm\u3009\u3008ljm|,\hfill \end{array}$$ | (2) |

*r*is the electron–ion distance and ${V}_{LJ}^{\mathit{SOREP}}(r)$ is a local potential. This form can be split into the averaged relativistic effective core potential (AREP) and spin–orbit potential (SO) terms,^{33}$${V}^{\mathit{SOREP}}={V}^{\mathit{AREP}}+{V}^{SO},$$ | (3) |

*J*-average of*V*^{SOREP},$${V}_{\ell}^{\mathit{AREP}}(r)=\frac{1}{2\ell +1}\left[\ell \cdot {V}_{\ell ,\ell -\frac{1}{2}}^{\mathit{SOREP}}(r)+(\ell +1)\cdot {V}_{\ell ,\ell +\frac{1}{2}}^{\mathit{SOREP}}(r)\right].$$ | (4) |

*V*^{SO}are defined using*V*^{SOREP}as$${V}^{SO}=s\cdot \sum _{\ell =1}^{L}\frac{2}{2\ell +1}\mathrm{\Delta}{V}_{\ell}^{\mathit{SOREP}}(r)\cdot \sum _{m=-\ell}^{\ell}\sum _{{m}^{\prime}=-\ell}^{\ell}|\ell m\u3009\u3008\ell m|\ell |\ell {m}^{\prime}\u3009\u3008\ell {m}^{\prime}|,$$ | (5) |

$$\mathrm{\Delta}{V}_{\ell}^{\mathit{SOREP}}(r)={V}_{\ell ,\ell +\frac{1}{2}}^{\mathit{SOREP}}(r)-{V}_{\ell ,\ell -\frac{1}{2}}^{\mathit{SOREP}}(r).$$ | (6) |

*V*^{AREP}is defined as follows similar to our previous works:$$\begin{array}{ll}\hfill {V}^{\mathit{AREP}}(r)& ={V}_{L}^{\mathit{AREP}}(r)+\sum _{\ell =0}^{{\ell}_{\mathrm{max}}=L-1}\left[{V}_{\ell}^{\mathit{AREP}}(r)-{V}_{L}^{\mathit{AREP}}(r)\right]\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}\times \sum _{m}|\ell m\u3009\u3008\ell m|.\hfill \end{array}$$ | (7) |

The latter part of Eq. (7) involves the non-local |

where the

where

*ℓm*⟩⟨*ℓm*| spherical harmonics projectors. ${V}_{L}^{\mathit{AREP}}({r}_{i})$ is again a local channel that acts on all valence electrons and parameterized as$${V}_{L}^{\mathit{AREP}}(r)=-\frac{{Z}_{\text{eff}}}{r}(1-{e}^{-\alpha {r}^{2}})+\alpha {Z}_{\text{eff}}r{e}^{-\beta {r}^{2}}+\sum _{k=1}{\gamma}_{k}{e}^{-{\delta}_{k}{r}^{2}},$$ | (8) |

*Z*_{eff}=*Z*−*Z*_{core}is the effective core charge and*α*,*β*,*γ*_{k}, and*δ*_{k}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$${V}_{\ell}^{\mathit{AREP}}(r)-{V}_{L}^{\mathit{AREP}}(r)=\sum _{p=1}{\beta}_{\ell p}{r}^{{n}^{\ell p}-2}{e}^{-{\alpha}_{\ell p}{r}^{2}},$$ | (9) |

*β*_{ℓp}and*α*_{ℓp}are parameters to be optimized. In most cases,*n*_{ℓp}were set to be*n*_{ℓp}= 2; however, we included*n*_{ℓp}= 4 terms in some cases to achieve the desired accuracy.The ccECP SO parameters are provided as

which is also the convention adopted by codes such as dirac, molpro, and nwchem. ${V}_{l}^{SO,ccECP}$ are parameterized similar to AREP non-local potentials,

Together with parameters described in the AREP part,

$${V}_{\ell}^{SO,ccECP}=\frac{2}{2\ell +1}\mathrm{\Delta}{V}_{\ell}^{\mathit{SOREP}},$$ | (10) |

$${V}_{\ell}^{SO,ccECP}(r)=\sum _{{p}^{\prime}=1}{\beta}_{\ell {p}^{\prime}}{r}^{{n}^{\ell {p}^{\prime}}-2}{e}^{-{\alpha}_{\ell {p}^{\prime}}{r}^{2}}.$$ | (11) |

*n*_{ℓp′},*β*_{ℓp′}, and*α*_{ℓp′}, 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,

*ℓ*_{max}= 2 is used as there are 3*d*electrons being included in the core. For the fifth row elements, Bi, Au, Ir, and W,*ℓ*_{max}= 3 is employed because the 4*f*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.

where the $\mathrm{\Delta}{E}_{X}^{(s)}$,

^{23–25,34}We include the many-body energy-consistency and single-particle eigenvalues in the definition of the objective function,$${\mathcal{O}}_{\mathit{AREP}}^{2}=\sum _{s\in S}{w}_{s}{(\mathrm{\Delta}{E}_{AE}^{(s)}-\mathrm{\Delta}{E}_{\mathit{ECP}}^{(s)})}^{2}+\sum _{i\in L}{w}_{i}{({\u03f5}_{AE}^{(i)}-{\u03f5}_{\mathit{ECP}}^{(i)})}^{2},$$ | (12) |

*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 (IP*n*) are included.*w*_{s}are weights corresponding to spectral states, decreasing as the energy gap increases due to the deep ionizations. ${\u03f5}_{X}^{(i)}$,*X*∈ {*ECP*,*AE*}, labels the one-particle eigenvalue in ground state for*i*th valence eigenvalue with a weight*w*_{i}. The ECP parameters were initialized from either MDFSTU^{35}or BFD^{7,8}ECP parameters.All energies in the AREP case are calculated using the CCSD(T) method with large uncontracted aug-cc-p(wC)V

*n*Z (*n*∈*T*,*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.^{36}All ECP calculations correlate the full valence space as well (including the semi-cores if present). Molpro code^{37}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.

^{24}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. 24 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,

^{10}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 DIRAC

where the $\mathrm{\Delta}{E}_{X}^{(Y)}$,

^{38}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,$${\mathcal{O}}_{SO}^{2}=\sum _{s\in {S}^{\prime}}{w}_{s}{(\mathrm{\Delta}{E}_{AE}^{(s)}-\mathrm{\Delta}{E}_{\mathit{ECP}}^{(s)})}^{2}+\sum _{m\in M}{w}_{m}{(\mathrm{\Delta}{E}_{AE}^{(m)}-\mathrm{\Delta}{E}_{\mathit{ECP}}^{(m)})}^{2},$$ | (13) |

*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+1}*L*_{J}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

Another metric is the MAD of selected low-lying

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*atomic gaps,$$\mathrm{M}\mathrm{A}\mathrm{D}=\frac{1}{N}\sum _{i}^{N}\left|\mathrm{\Delta}{E}_{i}^{\text{ECP}}-\mathrm{\Delta}{E}_{i}^{\text{AE}}\right|.$$ | (14) |

*n*gaps (LMAD),$$\mathrm{L}\mathrm{M}\mathrm{A}\mathrm{D}=\frac{1}{n}\sum _{i}^{n}\left|\mathrm{\Delta}{E}_{i}^{\text{ECP}}-\mathrm{\Delta}{E}_{i}^{\text{AE}}\right|.$$ | (15) |

*N*gaps as follows:$$\mathrm{W}\mathrm{M}\mathrm{A}\mathrm{D}=\frac{1}{N}\sum _{i}^{N}\frac{100\%}{\sqrt{|\mathrm{\Delta}{E}_{i}^{\text{AE}}|}}\left|\mathrm{\Delta}{E}_{i}^{\text{ECP}}-\mathrm{\Delta}{E}_{i}^{\text{AE}}\right|.$$ | (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,

^{39}MDFSTU,^{35}BFD,^{7,8}LANL2,^{40}CRENB(S/L),^{3,4}and SBKJC^{5}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. 1–3, 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,

where

$$\mathrm{\Delta}(r)=D{(r)}^{\mathit{ECP}}-D{(r)}^{AE},$$ | (17) |

*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,

where

where

$$V(r)={D}_{e}\left({e}^{-2a\left(r-{r}_{e}\right)}-2{e}^{-a\left(r-{r}_{e}\right)}\right),$$ | (18) |

*D*_{e}labels the dissociation energy,*r*_{e}is the equilibrium bond length, and*a*is a fitting parameter related to vibrations,$${\omega}_{e}=\sqrt{\frac{2{a}^{2}{D}_{e}}{\mu}},$$ | (19) |

*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 A–III J.

Atom | Z_{eff} | Hamiltonian | ℓ | n_{ℓk} | α_{ℓk} | β_{ℓk} | Atom | Z_{eff} | Hamiltonian | ℓ | n_{ℓk} | α_{ℓk} | β_{ℓk} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I | 7 | AREP | 0 | 2 | 3.150 276 | 114.023 762 | Pd | 14 | AREP | 0 | 2 | 12.451 647 | 222.004 870 |

0 | 2 | 1.008 080 | 1.974 146 | 0 | 2 | 9.656 958 | 4.624 354 | ||||||

1 | 2 | 2.954 190 | 111.466 589 | 0 | 2 | 6.301 508 | 55.838 586 | ||||||

1 | 2 | 0.730 822 | 1.710 240 | 1 | 2 | 12.376 014 | 166.644 960 | ||||||

2 | 2 | 2.604 979 | 45.300 844 | 1 | 2 | 9.850 179 | 28.721 347 | ||||||

2 | 2 | 0.899 593 | 8.704 290 | 1 | 2 | 5.706 187 | 42.681 167 | ||||||

3 | 1 | 12.001 337 | 7.000 000 | 2 | 2 | 9.195 344 | 57.976 811 | ||||||

3 | 3 | 12.035 882 | 84.009 359 | 2 | 2 | 7.923 467 | 53.576 592 | ||||||

3 | 2 | 3.022 296 | −3.998 559 | 2 | 2 | 3.358 692 | 4.674 921 | ||||||

3 | 2 | 0.977 906 | −3.050 310 | 3 | 1 | 15.997 664 | 18.000 000 | ||||||

3 | 3 | 15.964 009 | 287.957 945 | ||||||||||

SO | 1 | 2 | 3.086 882 | 55.714 747 | 3 | 2 | 16.128 878 | −176.382 552 | |||||

1 | 2 | 2.867 556 | −54.814 595 | 3 | 2 | 9.592 539 | −44.253 623 | ||||||

1 | 2 | 1.921 809 | −1.544 859 | ||||||||||

1 | 2 | 1.061 605 | 1.770 002 | SO | 1 | 2 | 17.775 389 | −4.392 610 | |||||

2 | 2 | 2.000 739 | −8.269 917 | 1 | 2 | 6.010 086 | 4.167 684 | ||||||

2 | 2 | 1.969 122 | 8.345 637 | 2 | 2 | 8.310 412 | −28.617 555 | ||||||

2 | 2 | 1.002 536 | −2.185 466 | 2 | 2 | 8.113 178 | 28.944 376 | ||||||

2 | 2 | 0.976 089 | 2.118 147 | 2 | 2 | 3.313 498 | −1.901 878 | ||||||

2 | 2 | 2.152 512 | 0.867 558 | ||||||||||

Te | 6 | AREP | 0 | 2 | 2.656 483 | 48.280 562 | |||||||

0 | 2 | 2.281 974 | −1.021 525 | Mo | 14 | AREP | 0 | 2 | 10.079 375 | 180.089 303 | |||

1 | 2 | 2.946 988 | 39.656 024 | 0 | 2 | 6.321 328 | 16.961 392 | ||||||

1 | 2 | 2.790 001 | 79.544 830 | 0 | 2 | 4.386 730 | 24.723 630 | ||||||

1 | 2 | 1.909 579 | −2.728 455 | 1 | 2 | 9.027 487 | 123.680 940 | ||||||

1 | 2 | 1.750 168 | −2.600 491 | 1 | 2 | 6.340 373 | 16.933 336 | ||||||

2 | 2 | 1.107 233 | 6.846 462 | 1 | 2 | 3.910 532 | 18.784 314 | ||||||

2 | 2 | 1.084 059 | 9.411 814 | 2 | 2 | 7.378 964 | 48.289 191 | ||||||

3 | 1 | 12.000 000 | 6.000 000 | 2 | 2 | 6.390 417 | 16.932 154 | ||||||

3 | 3 | 12.000 000 | 72.000 000 | 2 | 2 | 2.774 830 | 7.940 102 | ||||||

3 | 2 | 12.000 000 | −50.505 126 | 3 | 1 | 10.993 527 | 14.000 000 | ||||||

3 | 3 | 11.015 078 | 153.909 378 | ||||||||||

SO | 1 | 2 | 2.826 025 | −79.930 555 | 3 | 2 | 10.365 343 | −91.434 311 | |||||

1 | 2 | 2.772 908 | 79.814 350 | 3 | 2 | 6.281 056 | −16.945 728 | ||||||

1 | 2 | 2.087 577 | −1.100 622 | ||||||||||

1 | 2 | 1.603 342 | 1.969 873 | SO | 1 | 2 | 9.121 564 | −82.455 357 | |||||

2 | 2 | 1.107 233 | −5.059 096 | 1 | 2 | 8.863 223 | 82.452 670 | ||||||

2 | 2 | 1.084 059 | 4.999 134 | 1 | 2 | 4.044 948 | −12.690 183 | ||||||

1 | 2 | 3.866 657 | 12.458 423 | ||||||||||

Ag | 19 | AREP | 0 | 2 | 12.570 677 | 281.004 418 | 2 | 2 | 7.535 754 | −19.308 744 | |||

0 | 2 | 7.075 228 | 40.246 565 | 2 | 2 | 7.278 976 | 19.318 449 | ||||||

1 | 2 | 11.402 841 | 210.992 439 | 2 | 2 | 2.772 085 | 3.133 446 | ||||||

1 | 2 | 6.504 915 | 30.829 872 | 2 | 2 | 2.763 205 | −3.189 516 | ||||||

2 | 2 | 10.792 675 | 101.033 610 | ||||||||||

2 | 2 | 4.485 396 | 14.813 640 | ||||||||||

3 | 1 | 11.116 996 | 19.000 000 | ||||||||||

3 | 3 | 11.307 067 | 211.222 924 | ||||||||||

3 | 2 | 10.887 465 | −111.385 674 | ||||||||||

3 | 2 | 6.050 896 | −10.049 234 | ||||||||||

SO | 1 | 2 | 14.533 732 | −7.088 207 | |||||||||

1 | 2 | 7.620 883 | 6.990 650 | ||||||||||

2 | 2 | 11.057 856 | 28.649 549 | ||||||||||

2 | 2 | 9.109 402 | −28.772 257 | ||||||||||

2 | 2 | 5.540 539 | −5.426 715 | ||||||||||

2 | 2 | 4.274 636 | 4.873 157 |

Atom | Z_{eff} | Hamiltonian | ℓ | n_{ℓk} | α_{ℓk} | β_{ℓk} | Atom | Z_{eff} | Hamiltonian | ℓ | n_{ℓk} | α_{ℓk} | β_{ℓk} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Bi | 5 | AREP | 0 | 2 | 3.398 563 | 137.072 914 | Au | 19 | AREP | 0 | 2 | 13.809 175 | 423.151 508 |

0 | 2 | 2.236 391 | 34.391 642 | 0 | 2 | 6.576 682 | 23.880 430 | ||||||

0 | 4 | 1.063 682 | −1.801 931 | 0 | 2 | 4.315 266 | 12.729 619 | ||||||

1 | 2 | 1.245 620 | 39.826 968 | 1 | 2 | 11.806 942 | 260.255 986 | ||||||

1 | 2 | 0.366 762 | 0.844 734 | 1 | 2 | 5.843 816 | 32.797 931 | ||||||

1 | 4 | 0.823 760 | −4.391 855 | 1 | 2 | 4.462 109 | 14.975 886 | ||||||

2 | 2 | 0.924 628 | −3.787 918 | 2 | 2 | 8.633 228 | 135.277 395 | ||||||

2 | 2 | 0.835 034 | 24.096 314 | 2 | 2 | 4.081 605 | 13.920 475 | ||||||

2 | 4 | 1.239 855 | 0.456 375 | 2 | 2 | 3.946 753 | 12.464 658 | ||||||

3 | 2 | 1.847 446 | −0.382 962 | 3 | 2 | 3.613 935 | 15.143 598 | ||||||

3 | 2 | 1.069 842 | −0.546 444 | 3 | 2 | 3.593 635 | 14.877 722 | ||||||

3 | 4 | 0.862 193 | −0.236 708 | 3 | 2 | 3.467 163 | 12.317 560 | ||||||

4 | 1 | 8.029 160 | 5.000 000 | 4 | 1 | 9.985 262 | 19.000 000 | ||||||

4 | 3 | 7.481 551 | 40.145 802 | 4 | 3 | 9.536 259 | 189.719 980 | ||||||

4 | 2 | 1.775 683 | −6.251 215 | 4 | 2 | 10.882 390 | −140.528 643 | ||||||

4 | 2 | 0.698 085 | −0.286 410 | 4 | 2 | 3.792 290 | −8.417 696 | ||||||

SO | 1 | 2 | 3.624 196 | −0.056 803 | SO | 1 | 2 | 15.759 013 | −22.092 336 | ||||

1 | 2 | 2.084 017 | 3.659 507 | 1 | 2 | 6.041 694 | 22.320 517 | ||||||

1 | 4 | 1.279 962 | 3.069 035 | 2 | 2 | 8.904 735 | 20.778 790 | ||||||

2 | 2 | 1.877 353 | 3.942 473 | 2 | 2 | 6.570 928 | −24.832 959 | ||||||

2 | 2 | 1.118 436 | −4.025 036 | 2 | 2 | 3.389 961 | 3.968 015 | ||||||

2 | 4 | 1.002 374 | −0.276 729 | 3 | 2 | 3.387 468 | −5.901 882 | ||||||

3 | 2 | 1.013 377 | 1.745 433 | 3 | 2 | 3.083 291 | 5.872 893 | ||||||

3 | 2 | 0.959 437 | −1.645 441 | ||||||||||

3 | 4 | 1.004 895 | 0.007 556 | ||||||||||

Ir | 17 | AREP | 0 | 2 | 13.826 287 | 438.873 879 | W | 14 | AREP | 0 | 2 | 11.273 956 | 420.479 803 |

0 | 2 | 6.284 095 | 76.599 511 | 0 | 2 | 8.410 924 | 39.584 943 | ||||||

1 | 2 | 11.264 188 | 262.750 135 | 1 | 2 | 8.517 824 | 320.877 235 | ||||||

1 | 2 | 5.346 383 | 61.991 944 | 1 | 2 | 1.608 091 | −0.472 821 | ||||||

2 | 2 | 6.815 210 | 123.955 993 | 1 | 4 | 10.073 490 | 0.112 929 | ||||||

2 | 2 | 4.766 746 | 31.123 654 | 2 | 2 | 6.059 286 | 158.077 959 | ||||||

3 | 2 | 2.786 350 | 10.329 276 | 2 | 2 | 1.104 853 | −0.283 299 | ||||||

3 | 2 | 2.592 946 | 11.885 698 | 2 | 4 | 6.262 477 | −0.459 490 | ||||||

4 | 1 | 12.171 070 | 17.000 000 | 3 | 2 | 2.075 342 | 7.745 405 | ||||||

4 | 3 | 12.170 139 | 206.908 194 | 3 | 2 | 1.845 947 | 6.916 005 | ||||||

4 | 2 | 12.320 343 | −143.208 379 | 4 | 1 | 10.188 960 | 14.000 000 | ||||||

4 | 2 | 5.602 984 | −23.157 042 | 4 | 3 | 9.478 240 | 142.645 440 | ||||||

4 | 2 | 10.146 880 | −100.078 727 | ||||||||||

SO | 1 | 2 | 13.722 145 | −15.366 434 | 4 | 2 | 5.491 651 | −0.941 578 | |||||

1 | 2 | 4.201 427 | 9.843 941 | ||||||||||

2 | 2 | 10.062 658 | −2.838 512 | SO | 1 | 2 | 11.964 154 | −0.551 147 | |||||

2 | 2 | 5.177 629 | 3.880 360 | 1 | 2 | 4.897 325 | 1.427 345 | ||||||

3 | 2 | 2.299 318 | 4.142 597 | 1 | 4 | 8.302 365 | 194.404 774 | ||||||

3 | 2 | 2.295 619 | −4.159 394 | 2 | 2 | 5.772 034 | 7.045 184 | ||||||

2 | 2 | 3.175 200 | −0.115 276 | ||||||||||

2 | 4 | 6.155 484 | 3.907 520 | ||||||||||

3 | 2 | 2.473 295 | 4.123 819 | ||||||||||

3 | 2 | 2.405 627 | −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 3

*s*, 3*p*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,^{24}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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

5s^{2}5p^{5} | ^{2}P_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

5s^{2}5p^{6} | ^{1}S_{0} | −2.185 | 0.022 | 0.035 | −3.060 | 0.05(1) | 0.06(1) |

5s^{2}5p^{4} | ^{3}P_{2} | 9.413 | 0.009 | −0.046 | 10.451 | 0.05(1) | 0.04(2) |

5s^{2}5p^{3} | ^{4}S_{3/2} | 27.262 | 0.097 | −0.058 | 29.582 | 0.33(1) | 0.08(3) |

5s^{2}5p^{5} | ^{2}P_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}P_{1/2} | 0.962 | 0.030 | 0.024 | 0.942 | 0.01(1) | 0.00(2) | |

5s^{2}5p^{4} | ^{3}P_{2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{3}P_{0} | 0.863 | 0.046 | 0.047 | 0.799 | 0.03(1) | 0.02(2) | |

^{3}P_{1} | 0.869 | 0.019 | 0.018 | 0.878 | 0.08(1) | 0.04(2) | |

^{1}D_{2} | 1.999 | −0.029 | −0.040 | 1.702 | 0.04(1) | −0.01(2) | |

^{1}S_{0} | 4.258 | −0.104 | −0.132 | 3.657 | −0.08(1) | −0.13(2) | |

5s^{2}5p^{3} | ^{4}S_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}D_{3/2} | 1.959 | −0.136 | −0.159 | 1.451 | −0.24(1) | −0.07(2) | |

^{2}D_{5/2} | 2.348 | −0.101 | −0.119 | 1.847 | −0.14(1) | 0.08(2) | |

^{2}P_{1/2} | 3.765 | −0.183 | −0.215 | 3.012 | −0.24(1) | −0.12(3) | |

^{2}P_{3/2} | 4.375 | −0.116 | −0.138 | 3.674 | −0.16(1) | −0.06(3) | |

MAD | 0.074 | 0.085 | 0.12(1) | 0.06(2) |

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.

^{34}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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

5s^{2}5p^{4} | ^{3}P_{2} | 0.0000 | 0.0000 | 0.0000 | 0.000 | 0.00 | 0.00 |

5s^{2}5p^{5} | ^{2}P_{3/2} | −0.9927 | −0.0244 | 0.0254 | −1.970 | 0.01(1) | −0.02(1) |

5s^{2}5p^{3} | ^{4}S_{3/2} | 7.8285 | −0.0680 | 0.0021 | 9.010 | 0.21(1) | 0.17(1) |

5s^{2}5p^{2} | ^{3}P_{2} | 25.4702 | 0.0505 | 0.2103 | 27.610 | 0.55(1) | 0.38(1) |

5s^{2}5p^{4} | ^{3}P_{2} | 0.0000 | 0.0000 | 0.0000 | 0.000 | 0.00 | 0.00 |

^{3}P_{1} | 0.5773 | −0.0651 | 0.0133 | 0.589 | 0.13(1) | 0.11(1) | |

^{3}P_{0} | 0.6304 | −0.0454 | 0.0201 | 0.584 | 0.01(1) | 0.04(1) | |

^{1}D_{2} | 1.5787 | −0.0045 | 0.0725 | 1.309 | 0.11(1) | 0.09(1) | |

^{1}S_{0} | 3.4514 | 0.0441 | 0.1714 | 2.876 | −0.04(1) | −0.09(1) | |

5s^{2}5p^{5} | ^{2}P_{3/2} | 0.0000 | 0.0000 | 0.0000 | 0.000 | 0.00 | 0.00 |

^{2}P_{1/2} | 0.6346 | −0.0660 | 0.0139 | ||||

5s^{2}5p^{3} | ^{4}S_{3/2} | 0.0000 | 0.0000 | 0.0000 | 0.000 | 0.00 | 0.00 |

^{2}D_{3/2} | 1.7573 | 0.1126 | 0.1150 | 1.267 | −0.18(1) | −0.18(1) | |

^{2}D_{5/2} | 2.0117 | 0.0729 | 0.1211 | 1.540 | −0.10(1) | −0.13(1) | |

^{2}P_{1/2} | 3.2603 | 0.1372 | 0.1986 | 2.547 | −0.21(1) | −0.25(1) | |

^{2}P_{3/2} | 3.6511 | 0.0729 | 0.2097 | 2.980 | −0.12(1) | −0.19(1) | |

5s^{2}5p^{2} | ^{3}P_{0} | 0.0000 | 0.0000 | 0.0000 | 0.000 | 0.00 | 0.00 |

^{3}P_{1} | 0.5445 | −0.0649 | 0.0191 | 0.589 | 0.09(1) | 0.11(1) | |

^{3}P_{2} | 1.0371 | −0.0812 | 0.0469 | 1.012 | 0.07(1) | 0.12(1) | |

^{1}D_{2} | 2.4298 | −0.0527 | 0.1352 | 2.152 | 0.12(1) | 0.10(1) | |

^{1}S_{0} | 4.4771 | 0.0656 | 0.2803 | ||||

MAD | 0.049 | 0.079 | 0.14(1) | 0.14(3) |

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 5

*d*elements that include the semi-core 5*s*, 5*p*subshells into the valence space. Specifically, this requires partitioning of*n*= 5 principal quantum number, where 5*d*^{10}is in the core while 5*f*^{14}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*ℓ*= 0 channel) could result in significant errors in transition metals.^{14}However, the errors seen in this case are similar to what was observed in isovalent elements, such as N, P, and As,^{23,24,34}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.COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

6s^{2}6p^{3} | ^{4}S_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

6s^{2}6p^{4} | ^{3}P_{2} | 0.034 | −0.062 | 0.008 | 0.942 | −0.04(1) | 0.08(1) |

6s^{2}6p^{2} | ^{3}P_{0} | 6.663 | −0.079 | −0.113 | 7.285 | 0.03(1) | 0.10(1) |

6s^{2}6p^{1} | ^{2}P_{1/2} | 22.621 | −0.052 | −0.166 | 23.988 | 0.33(1) | 0.26(1) |

6s^{2}6p^{3} | ^{4}S_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}D_{3/2} | 1.549 | 0.007 | −0.036 | 1.415 | 0.07(1) | 0.03(2) | |

^{2}D_{5/2} | 2.140 | 0.011 | −0.047 | 1.914 | 0.07(1) | 0.04(2) | |

^{2}P_{1/2} | 3.110 | 0.002 | −0.093 | 2.685 | 0.08(1) | −0.02(2) | |

^{2}P_{3/2} | 4.488 | 0.060 | −0.050 | 4.111 | 0.11(1) | 0.06(2) | |

6s^{2}6p^{2} | ^{3}P_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{3}P_{1} | 1.533 | 0.064 | 0.050 | 1.652 | 0.14(1) | 0.01(2) | |

^{3}P_{2} | 2.146 | 0.059 | 0.019 | 2.111 | 0.10(1) | −0.03(2) | |

^{1}D_{2} | 4.307 | 0.126 | 0.041 | 4.207 | 0.21(1) | 0.01(2) | |

^{1}S_{0} | 5.944 | 0.095 | −0.067 | 5.476 | 0.18(1) | 0.05(2) | |

6s^{2}6p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}P_{3/2} | 2.597 | 0.103 | 0.057 | 2.577 | 0.12(1) | 0.10(1) | |

6s^{2}5d^{1} | ^{2}D_{3/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}D_{5/2} | 0.178 | 0.020 | −0.000 | 0.780 | 0.61(1) | 0.53(2) | |

6s^{2}5f^{1} | ^{2}F_{7/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.0000 | 0.0000 |

^{2}F_{5/2} | 0.011 | 0.004 | −0.002 | 0.012 | 0.03(1) | −0.03(1) | |

MAD | 0.053 | 0.053 | 0.15(1) | 0.09(2) |

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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

4d^{10}5s^{1} | ^{2}S_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

4d^{10}5s^{2} | ^{1}S_{0} | −0.117 | 0.000 | 0.001 | −1.304 | −0.35(1) | −0.27(3) |

4d^{10}5p^{1} | ^{2}P_{1/2} | 3.013 | −0.001 | 0.002 | 3.664 | 0.06(1) | 0.12(3) |

4d^{10} | ^{1}S_{0} | 6.340 | −0.002 | −0.002 | 7.576 | 0.19(1) | 0.28(3) |

4d^{9} | ^{2}D_{5/2} | 25.999 | −0.006 | 0.020 | 29.061 | −0.28(1) | 0.34(3) |

4d^{10}5p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}P_{3/2} | 0.074 | 0.001 | 0.002 | 0.114 | 0.01(1) | −0.01(3) | |

4d^{9} | ^{2}D_{5/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}D_{3/2} | 0.571 | 0.030 | 0.011 | 0.571 | 0.01(1) | 0.03(3) | |

J-MAD | 0.01(1) | 0.02(3) | |||||

MAD | 0.007 | 0.006 |

REP-CCSD(T) | ||||
---|---|---|---|---|

State | Term | Expt. | STU | ccECP |

4d^{10}5s^{1} | ^{2}S_{1/2} | 0.000 | 0.000 | 0.000 |

4d^{10}5s^{2} | ^{1}S_{0} | −1.304 | 0.005 | 0.002 |

4d^{10}5p^{1} | ^{2}P_{1/2} | 3.664 | −0.029 | −0.024 |

4d^{10} | ^{1}S_{0} | 7.576 | −0.034 | −0.027 |

4d^{9} | ^{2}D_{5/2} | 29.061 | 0.007 | 0.031 |

CC-MAD | 0.015 | 0.017 |

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.^{31}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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

5d^{10}6s^{1} | ^{2}S_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

5d^{10}6s^{2} | ^{1}S_{0} | −0.648 | 0.016 | 0.022 | −2.310 | −0.35(4) | −0.22(3) |

5d^{9}6s^{2} | ^{2}D_{5/2} | 1.284 | −0.040 | −0.038 | 1.140 | −0.25(4) | −0.14(3) |

5d^{10}6p^{1} | ^{2}P_{1/2} | 4.024 | 0.008 | 0.003 | 4.630 | −0.10(3) | −0.10(3) |

5d^{10} | ^{1}S_{0} | 7.704 | 0.026 | 0.005 | 9.230 | 0.26(4) | 0.25(3) |

5d^{9} | ^{2}D_{5/2} | 26.266 | 0.026 | −0.033 | 29.430 | 0.15(3) | 0.16(3) |

5d^{9}6s^{2} | ^{2}D_{5/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}D_{3/2} | 1.480 | 0.061 | −0.020 | 1.520 | 0.02(3) | −0.06(3) | |

5d^{10}6p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}P_{3/2} | 0.349 | 0.011 | −0.005 | 0.470 | 0.02(3) | 0.00(3) | |

5d^{9} | ^{2}D_{5/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}D_{3/2} | 1.558 | 0.071 | −0.010 | ||||

5d^{10}5f^{1} | ^{2}F_{7/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}F_{5/2} | 0.103 | 0.018 | 0.008 | 0.000 | −0.09(4) | −0.05(4) | |

J-MAD | 0.05(3) | 0.03(3) | |||||

MAD | 0.031 | 0.016 |

REP-CCSD(T) | ||||
---|---|---|---|---|

State | Term | Expt. | STU | ccECP |

5d^{10}6s^{1} | ^{2}S_{1/2} | 0.000 | 0.000 | 0.000 |

5d^{10}6s^{2} | ^{1}S_{0} | −2.310 | 0.086 | 0.025 |

5d^{9}6s^{2} | ^{2}D_{5/2} | 1.140 | 0.010 | 0.035 |

5d^{10}6p^{1} | ^{2}P_{1/2} | 4.630 | −0.044 | −0.046 |

5d^{10} | ^{1}S_{0} | 9.230 | −0.004 | −0.025 |

5d^{9} | ^{2}D_{5/2} | 29.430 | −0.010 | −0.052 |

CC-MAD | 0.026 | 0.031 |

Further tests were carried out in Au

_{2}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 (5

*d*^{4}6*s*^{2},^{5}*D*_{0}) and excited state (5*d*^{5}6*s*^{1},^{7}*S*_{3}) 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 5*d*^{5}6*s*^{1}. 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 5*d*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.COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

5s^{2}5p^{6}5d^{4}6s^{2} | ^{5}D_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

5s^{2}5p^{6}5d^{5}6s^{2} | ^{6}S_{5/2} | 0.016 | 0.078 | 0.021 | −0.815 | 0.14(3) | 0.01(2) |

5s^{2}5p^{6}5d^{5}6s^{1} | ^{7}S_{3} | −0.733 | 0.113 | 0.057 | 0.366 | 0.64(4) | 0.48(2) |

5s^{2}5p^{6}5d^{4}6s^{1} | ^{6}D_{1/2} | 5.850 | 0.031 | −0.005 | 7.864 | 0.52(3) | 0.44(2) |

5s^{2}5p^{6}5d^{4} | ^{5}D_{0} | 20.685 | 0.020 | −0.053 | 24.2(2) | 1.1(2) | 1.0(2) |

5s^{2}5p^{6}5d^{4}6s^{2} | ^{5}D_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{5}D_{1} | 0.123 | 0.026 | −0.003 | 0.207 | 0.14(3) | 0.04(2) | |

^{5}D_{2} | 0.296 | 0.052 | −0.008 | 0.412 | 0.11(3) | 0.03(2) | |

^{5}D_{3} | 0.486 | 0.072 | −0.014 | 0.599 | 0.14(3) | −0.00(2) | |

^{5}D_{4} | 0.683 | 0.086 | −0.021 | 0.771 | 0.16(3) | −0.02(2) | |

5s^{2}5p^{6}5d^{4}6s^{1} | ^{6}D_{1/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{6}D_{3/2} | 0.116 | 0.020 | −0.003 | 0.188 | 0.02(3) | 0.05(2) | |

^{6}D_{5/2} | 0.273 | 0.041 | −0.009 | 0.393 | 0.12(3) | 0.08(2) | |

^{6}D_{7/2} | 0.454 | 0.061 | −0.015 | 0.585 | 0.14(3) | 0.02(2) | |

^{6}D_{9/2} | 0.648 | 0.076 | −0.022 | 0.762 | 0.13(3) | 0.00(2) | |

5s^{2}5p^{6}5d^{4}6s^{1} | ^{5}D_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{5}D_{1} | 0.148 | 0.031 | −0.004 | 0.280 | 0.18(3) | 0.10(1) | |

^{5}D_{2} | 0.348 | 0.060 | −0.010 | 0.553 | 0.22(3) | 0.09(2) | |

^{5}D_{3} | 0.561 | 0.081 | −0.019 | 0.778 | 0.22(3) | 0.10(2) | |

^{5}D_{4} | 0.777 | 0.094 | −0.027 | 0.953 | 0.19(3) | 0.05(2) | |

5s^{2}5p^{6}6s^{2}6p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{3/2} | 1.697 | 0.023 | −0.054 | ||||

5s^{2}5p^{6}6s^{2}6f^{1} | ^{2}F_{5/2} | 0.000 | 0.000 | 0.000 | |||

^{2}F_{7/2} | 0.008 | 0.003 | −0.028 | ||||

5s^{2}5p^{5} | ^{2}P_{3/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{1/2} | 11.059 | 0.144 | 0.020 | ||||

J-MAD | 0.15(3) | 0.05(2) | |||||

MAD | 0.059 | 0.021 |

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.

^{41}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.G. Palladium (Pd)

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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

4s^{2}4p^{6}4d^{10} | ^{1}S_{3} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

4s^{2}4p^{6}4d^{10}5s^{1} | ^{2}S_{1/2} | 0.236 | 0.009 | 0.008 | −0.562 | −0.33(3) | −0.36(2) |

4s^{2}4p^{6}4d^{9}5s^{1} | ^{2}[5/2]_{3} | −0.067 | −0.012 | −0.036 | 0.814 | −0.22(2) | −0.23(2) |

4s^{2}4p^{6}4d^{9} | ^{2}S_{5/2} | 6.273 | −0.009 | −0.044 | 8.337 | 0.05(2) | 0.03(2) |

4s^{2}4p^{6}4d^{8} | ^{3}F_{4} | 23.662 | 0.034 | −0.078 | 27.770 | 0.26(2) | 0.12(2) |

4s^{2}4p^{6}4d^{9}5s^{1} | ^{2}[5/2]_{3} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}[5/2]_{2} | 0.175 | 0.010 | −0.008 | 0.148 | −0.02(2) | 0.01(2) | |

4s^{2}4p^{6}4d^{9} | ^{2}S_{5/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{2}S_{3/2} | 0.436 | 0.042 | −0.038 | 0.439 | −0.02(3) | −0.03(2) | |

4s^{2}4p^{6}4d^{8} | ^{3}F_{4} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{3}F_{3} | 0.394 | 0.041 | −0.015 | 0.400 | 0.10(2) | 0.05(2) | |

^{3}F_{2} | 0.604 | 0.050 | −0.024 | 0.581 | 0.02(2) | −0.06(2) | |

4s^{2}4p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{3/2} | 6.640 | 0.093 | 0.000 | ||||

J-MAD | 0.05(2) | 0.05(2) | |||||

MAD | 0.033 | 0.028 |

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).COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

5s^{2}5p^{6}5d^{7}6s^{2} | ^{4}F_{9/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

5s^{2}5p^{6}5d^{8}6s^{2} | ^{6}S_{5/2} | 0.120 | 0.000 | 0.039 | −1.565 | −0.33(3) | −0.28(3) |

5s^{2}5p^{6}5d^{8}6s^{1} | ^{4}F_{9/2} | 0.284 | −0.001 | 0.045 | 0.351 | 0.08(3) | 0.11(3) |

5s^{2}5p^{6}5d^{7}6s^{1} | ^{5}F_{5} | 6.997 | 0.012 | 0.004 | 8.967 | 0.34(4) | 0.35(3) |

5s^{2}5p^{6}5d^{7} | ^{4}F_{9/2} | 23.290 | 0.020 | −0.010 | 26.0(3) | −0.1(3) | −0.1(3) |

5s^{2}5p^{6}5d^{7}6s^{2} | ^{4}F_{9/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{4}F_{7/2} | 0.782 | 0.088 | 0.009 | 0.784 | 0.01(3) | −0.05(3) | |

^{4}F_{5/2} | 1.075 | 0.080 | −0.004 | 0.717 | −0.35(3) | −0.36(3) | |

^{4}F_{3/2} | 1.202 | 0.052 | −0.021 | 0.506 | −0.66(3) | −0.71(3) | |

5s^{2}5p^{6}5d^{8}6s^{1} | ^{4}F_{9/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{4}F_{7/2} | 0.512 | 0.038 | 0.002 | 0.530 | 0.07(3) | 0.05(3) | |

^{4}F_{5/2} | 0.940 | 0.068 | −0.003 | 0.873 | −0.06(3) | −0.02(3) | |

^{4}F_{3/2} | 1.084 | 0.056 | −0.010 | 1.115 | 0.06(3) | 0.09(3) | |

5s^{2}5p^{6}5d^{7}6s^{1} | ^{5}F_{5} | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 | 0.00 |

^{5}F_{4} | 0.596 | 0.055 | 0.003 | 0.594 | 0.05(4) | −0.07(3) | |

^{5}F_{3} | 1.015 | 0.092 | 0.003 | 1.015 | 0.09(4) | −0.03(3) | |

^{5}F_{2} | 1.241 | 0.096 | −0.003 | 1.402 | 0.26(4) | 0.14(3) | |

^{5}F_{1} | 1.376 | 0.099 | −0.006 | 1.483 | 0.21(4) | 0.05(3) | |

5s^{2}5p^{6}5d^{7} | ^{4}F_{9/2} | 0.000 | 0.000 | 0.000 | |||

^{4}F_{7/2} | 0.848 | 0.094 | 0.008 | ||||

^{4}F_{5/2} | 1.159 | 0.082 | −0.007 | ||||

^{4}F_{3/2} | 1.289 | 0.048 | −0.027 | ||||

5s^{2}5p^{6}6s^{2}6p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{3/2} | 3.601 | 0.016 | 0.031 | ||||

5s^{2}5p^{6}6s^{2}6f^{1} | ^{2}F_{5/2} | 0.000 | 0.000 | 0.000 | |||

^{2}F_{7/2} | 0.106 | 0.040 | −0.001 | ||||

5s^{2}5p^{5} | ^{2}P_{3/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{1/2} | 16.084 | 0.169 | −0.114 | ||||

J-MAD | 0.20(4) | 0.17(3) | |||||

MAD | 0.060 | 0.018 |

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.

COSCI | FPSODMC | ||||||
---|---|---|---|---|---|---|---|

State | Term | AE | STU | ccECP | Expt. | STU | ccECP |

4s^{2}4p^{6}4d^{5}5s^{1} | ^{7}S_{3} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

4s^{2}4p^{6}4d^{5}5s^{2} | ^{6}S_{5/2} | 0.693 | −0.007 | −0.007 | −0.747 | −1.21(3) | −0.67(5) |

4s^{2}4p^{6}4d^{4}5s^{2} | ^{5}D_{0} | 2.189 | −0.068 | −0.039 | 1.360 | −1.18(3) | −0.74(4) |

4s^{2}4p^{6}4d^{5} | ^{6}S_{5/2} | 8.274 | 0.629 | −0.078 | 7.092 | −0.62(5) | −0.16(5) |

4s^{2}4p^{6}4d^{4} | ^{5}D_{0} | 21.200 | −0.057 | −0.024 | 23.252 | −0.46(4) | 0.25(5) |

4s^{2}4p^{6}4d^{5}5s^{1} | ^{7}S_{3} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 |

^{5}S_{2} | 1.870 | −0.009 | −0.009 | 1.335 | −0.4(1) | −0.24(4) | |

^{5}G_{2} | 2.725 | −0.066 | −0.063 | 2.063 | −0.66(6) | −0.34(4) | |

4s^{2}4p^{6}4d^{4}5s^{2} | ^{5}D_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.00 |

^{5}D_{1} | 0.024 | 0.005 | 0.005 | 0.022 | −0.14(3) | 0.03(5) | |

^{5}D_{2} | 0.070 | 0.015 | 0.013 | 0.061 | −0.05(4) | 0.09(4) | |

^{5}D_{3} | 0.133 | 0.027 | 0.024 | 0.111 | −0.28(4) | 0.16(4) | |

^{5}D_{4} | 0.209 | 0.040 | 0.036 | 0.171 | −0.08(3) | 0.07(3) | |

4s^{2}4p^{6}4d^{5} | ^{6}S_{5/2} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{4}G_{5/2} | 0.063 | −0.507 | 0.009 | 1.884 | −0.26(3) | 0.00(4) | |

^{4}G_{7/2} | 0.195 | −0.966 | 0.030 | 1.901 | −0.20(4) | −0.05(6) | |

^{4}G_{9/2} | 0.431 | −0.773 | 0.082 | 1.913 | −0.26(4) | 0.00(4) | |

^{4}G_{11/2} | 0.726 | −0.570 | −0.020 | 1.915 | −0.34(2) | −0.16(4) | |

4s^{2}4p^{6}4d^{4} | ^{5}D_{0} | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

^{5}D_{1} | 0.026 | 0.006 | 0.005 | 0.030 | −0.04(3) | −0.016(4) | |

^{5}D_{2} | 0.076 | 0.016 | 0.014 | 0.083 | 0.04(6) | 0.01(4) | |

^{5}D_{3} | 0.143 | 0.028 | 0.025 | 0.152 | 0.14(5) | 0.01(3) | |

^{5}D_{4} | 0.226 | 0.042 | 0.037 | 0.232 | 0.11(5) | −0.07(4) | |

4s^{2}4p^{6}5s^{2}6p^{1} | ^{2}P_{1/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{3/2} | 0.564 | 0.097 | −0.006 | ||||

4s^{2}4p^{5} | ^{2}P_{3/2} | 0.000 | 0.000 | 0.000 | |||

^{2}P_{1/2} | 2.948 | 0.050 | −0.013 | ||||

J-MAD | 0.200 | 0.080 | |||||

MAD | 0.199 | 0.027 |

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.

*D*

_{e}, equilibrium bond length

*r*

_{e}, vibrational frequency

*ω*

_{e}, and binding energy discrepancy at dissociation bond length

*D*

_{diss}.

D_{e} (eV) | r_{e} (Å) | ω_{e} (cm^{−1}) | D_{diss} (eV) | |
---|---|---|---|---|

BFD | 0.078(6) | 0.018(1) | 22(3) | 0.41(5) |

CRENBL(S) | 0.115(5) | 0.0183(9) | 26(3) | 0.40(4) |

LANL2 | 0.118(5) | 0.0122(9) | 16(3) | 0.36(4) |

MDFSTU | 0.096(5) | 0.0094(9) | 10(3) | 0.24(4) |

MWBSTU | 0.050(5) | 0.0056(9) | 10(3) | 0.10(4) |

SBKJC | 0.089(4) | 0.0120(8) | 19(2) | 0.30(4) |

UC | 0.040(5) | 0.0104(9) | 13(3) | 0.19(4) |

ccECP | 0.018(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 I

_{2}and Bi_{2}, we see consistent improvements from AREP CCSD(T) to SOREP FPSODMC calculations as well as going from MDFSTU to ccECP. Especially, in I_{2}, 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 Au_{2}, 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 W_{2}, 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.

^{42}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

*n*th*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 3

*d*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 3

*s*3*p*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 W

_{2}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. 42.

#### ACKNOWLEDGMENTS

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).

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