Ground and low-lying electronic states of the diatomic molecule TiV from quantitative multireference ab initio calculations

Abstract

The lowest electronic states of the transition intermetallic TiV molecule have been studied by first principles employing the multireference configuration interaction technique and large correlation consistent basis sets. The ground state was found to be of X⁴Σ⁻ symmetry with a binding energy of D⁰₀ = 41.1 kcal mol⁻¹ and r_e = 1.910 Å. Full potential energy curves were constructed for a total of 45 low-lying Λ–S states of TiV, extracting spectroscopic constants, as well. In addition, an effort was made to rationalize the nature of the chemical bond in the different states of the system.

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1. Introduction

Titanium and vanadium, two of the early first-row transition elements, are known for forming very useful alloys with significant technological applications in fields ranging from aerospace engineering and tools to bioimplants. They are, also, highly involved in organometallic chemistry and heterogeneous catalysis. However, the “simple” intermetallic TiV diatomic species has been poorly investigated so far. Certainly, technical difficulties, both experimental and theoretical, make it rather hard to study this system in depth. TiV was first observed in cryogenic argon matrices at 4 K, in 1984 by Van Zee and Weltner Jr.¹ These workers studied the ESR spectrum of ⁴⁷Ti⁵¹V and established that its ground state is of ⁴Σ symmetry. Based on the observed hyperfine parameters, they proposed a sσ²dσ¹dπ⁴dδ² electron configuration with multiple bonding involving sσ and dσ bonds. At that time an estimate of 43.7 kcal mol⁻¹ for the Ti−V binding energy had been published by Miedema.² In 1985 Walsch and Bauschlicher³ using the MCSCF method presented small portions of potential energy curves for three electronic states of ⁴Σ⁻, ²Δ, and ⁴Π symmetries. The energetically lowest state was found to be ⁴Σ⁻ consistent with the experimental observation of Van Zee and Weltner Jr.¹ For this state they calculated a TiV bondlength of 1.86 Å and a binding energy D_e = 0.80 eV with respect to the excited Ti(⁵F;3d³4s¹) + V(⁶D;3d⁴4s¹) separated atoms. Seven years later, in 1992, Spain and Morse⁴ used resonant two-photon ionization (R2PI) spectroscopy and determined an experimental value D⁰₀ = 2.068 eV. In the same year Mattar and Hamilton⁵ performed LSD-LCAO calculations on the ⁴Σ⁻ state of TiV and reported r_e = 1.77 Å and a binding energy D⁰_e = 5.88 eV relative to the ground state Ti(³F) + V(⁴F) asymptote. Gutzev et al.⁶ in a 2004 paper titled “Periodic Table of 3d-metal dimers” presented DFT/BPW91 results on the ground state of TiV. They found for the X⁴Σ⁻ state: r_e = 1.78 Å, ω_e = 562 cm⁻¹, and D⁰₀ = 2.78 eV. They also claimed that the lowest excited state is of ²Γ symmetry lying 0.29 eV higher than the ground-state. Finally, in 2023, Jiang and Liu⁷ using DFT with eight different exchange–correlation functionals and several basis sets, found for the TiV binding energy very different values ranging from 0.70 to 1.90 eV depending on the functional/basis they used. To the best of our knowledge, these are all the experimental or theoretical data in the literature on the TiV dimer to date.

We undertook the present work with the aim of producing a comprehensive mapping and characterization of numerous low-lying bound electronic states of the TiV species. This is the only reliable way to assign the ground state and predict the correct ordering of the excited states of the system. It is well known that the interaction between two transition metals can give rise to a veritable zoo of hundreds of molecular states with different spatial and spin symmetries that are very closely spaced. This is a consequence of the existence of many asymptotic atomic channels very close to each other within a narrow energy interval of a few eV.

The Ti(³F) and V(⁴F) atomic ground states both have a 3dⁿ⁻²4s² valence electron distribution, where n is the number of valence electrons. The much larger spatial extent of the doubly occupied 4s orbitals, as compared to the 3d orbitals, prevents the shielded 3d electrons from interacting effectively in covalent bonding. As a result, the interaction of the two ground state atoms is only expected to be of the van der Waals (vdW) type. Bonding should therefore involve excited states of one or both atoms with half-filled or empty 4s subshells. In a recent paper on the ScV,⁸ it was shown that the majority of its lowest bound states stem from the Sc(²D;3d¹4s²) + V(⁶D;3d⁴4s¹) asymptote, where the vanadium atom is in its first excited state only 0.245 eV above its ground state (M_J-averaged value).⁹ In that case the ground X⁷Σ⁺ state is formed mainly through a 2 center-3 electron (2c-3e) interaction leading to a relatively weakly bound system. Of course, the combination Sc(⁴F;3d²4s¹) + V(⁶D;3d⁴4s¹), with all electrons uncoupled, should be more favorable for stronger covalent bonding. Unfortunately, the Sc(⁴F) state is located 1.427 eV higher than its ground state. Therefore the Sc(⁴F) + V(⁶D) atomic limit, which lies 1.672 eV above the ground state atoms, cannot form molecular states that are sufficiently bound to become the ground state. Now, returning to the TiV system we find that Ti has its first excited state, ⁵F, significantly lower, at 0.833 eV,⁹ which gives an energy gap of 1.051 eV for the Ti(⁵F;3d³4s¹) + V(⁶D;3d⁴4s¹) asymptote relative to the ground-state separated atomic limit. It is, therefore, expected to interfere significantly in the formation of low-lying electronic states of TiV. Indeed, as we will see below, many of them, as well as the ground state of TiV, originate from that atomic limit. To make things clearer and also to show the complexity arising from the Ti + V interaction, all Λ–Σ states of TiV resulting from the low-lying atomic asymptotes are given below:

Ti(³F) + V(⁴F) → ^2,4,6[Σ⁺(4), Σ⁻(3), Π(6), Δ(5), Φ(4), Γ(3), H(2), I] (84 states)

Ti(³F) + V(⁶D) → ^4,6,8[Σ⁺(2), Σ⁻(3), Π(5), Δ(4), Φ(3), Γ(2), H] (0.245 eV, 60 states)

Ti(⁵F) + V(⁴F) → ^2,4,6,8[Σ⁺(4), Σ⁻(3), Π(6), Δ(5), Φ(4), Γ(3), H(2),I] (0.806 eV, 112 states)

Ti(¹D) + V(⁴F) → ⁴[Σ⁺(2), Σ⁻(3), Π(5), Δ(4), Φ(3), Γ(2), H] (0.871 eV, 20 states)

Ti(³F) + V(⁴D) → ^2,4,6[Σ⁺(2), Σ⁻(3), Π(5), Δ(4), Φ(3), Γ(2), H] (1.026 eV, 60 states)

Ti(⁵F) + V(⁶D) → ^2,4,6,8,10[Σ⁺(2), Σ⁻(3), Π(5), Δ(4), Φ(3), Γ(2), H] (1.051 eV, 100 states)

i.e. a total of 436 ^2S+1Λ electronic states within an asymptotic energy window of ∼1 eV. This number increases dramatically if we take into account the spin–orbit coupling (SOC) which, however, is not so important for such light atoms and we believe that considering the Λ–S coupling picture is a very good first-order approach. See also some SOC results discussed at the end of the Results and Discussion section.

From the above it appears that only a robust multireference strategy is appropriate to reliably address this situation. Thus, in what follows, the MRCISD methodology in conjunction with large basis sets was employed in order to calculate potential energy curves (PEC), spectroscopic constants, dipole moments, and binding energies for a total of 45 ^2S+1Λ electronic states of the TiV molecular system. Moreover, a discussion is held with the aim of elucidating the way bonds are formed in the different states of the molecule. Finally, an effort was made to determine the binding energy of the ground and first excited states of TiV as accurately and systematically as possible.

2. Computational outline

For all calculated states and corresponding potential energy curves (PEC) the correlation consistent basis sets of quadruple cardinality, cc-pVQZ, were used for both atoms generally contracted to [8s, 7p, 5d, 3f, 2g]¹⁰ comprising 186 contracted spherical Gaussians. The choice of the cc-pVQZ basis set for the construction of potential energy curves was made to achieve an acceptable compromise between the quality of results and computational cost (computing time and memory usage).

For the ground and two excited states the triple cc-pVTZ and the quintuple cc-pV5Z basis sets¹⁰ contracted as [7s, 6p, 4d, 2f] and [9s, 8p, 6d, 4f, 3g, 2h], respectively, were also employed at equilibrium geometries. For the scalar relativistic Douglas–Kroll–Hess (DKH)^11,12 calculations we used the same basis sets, cc-pVnZ-DK with n = T, Q, 5, appropriately contracted.

Our computational approach was based on the internally contracted¹³ multireference configuration interaction method, CASSCF + single + double replacements ≡ MRCI. The reference complete active space (CAS) was constructed by distributing the nine valence electrons (3d²4s² + 3d³4s²) of TiV to 15 orbitals correlating with the [4s(1) + 3d(5)] × 2 valence space of the Ti and V fragments supplemented with three orbitals of A₁, B₁, and B₂ symmetries (under C_2v restrictions) in order to provide greater flexibility to the active space. The size of the resulting CAS spaces under C_2v symmetry restrictions were, roughly, 2.3 × 10⁴ configuration state functions (CSF) for octets, 14.5 × 10⁴ CSFs (sextets), 4 × 10⁵ CSFs (quartets), and 4.6 × 10⁵ CSFs (doublets). During the CASSCF optimization a number of states from all four irreducible representations (A₁, A₂, B₁, B₂) were state averaged in order to assure a smooth evolution of the PECs along the internuclear distance and also correct Λ values. More specifically, states from A₁ and A₂ irreps were averaged for the description of Σ, Δ, Γ molecular C_∞v symmetries whilst B₁ and B₂ were used for the Π, Φ, and H states.

In the present calculations we did not consider the known double d-shell effect, (see for instance ref. 14) because it doesn’t seem important for the Ti and V atoms. For example the [V(⁴F) ← V(⁶D)] energy gap is very accurately predicted with the present computational scheme (vide infra).

The subsequent MRCI expansions contain 0.24–2.54 × 10⁹ uncontracted CSFs internally contracted¹³ to 9–55 × 10⁶ CSFs.

At all stages of our calculations, MCSCF and MRCI, the convergence thresholds were 10⁻⁷ for energies and 10⁻⁶ for the coefficients gradient or the density.

Size non-extensivity was taken into account by applying the multireference Davidson correction¹⁵ for unlinked quadruples, MRCI + Q.

Spectroscopic constants were extracted by numerically solving the nuclear Schrödinger equation using a Numerov procedure with a code developed in our laboratory and employing the masses of the ⁴⁸Ti and ⁵¹V isotopes.

All calculations were carried out with the MOLPRO2025.1-4 program.^16,17

3. Results and discussion

Table 1 collects the numerical data obtained for 45 electronic states of TiV while Fig. 1, 3, 4 and 5 display the corresponding PECs for spin multiplicities 4, 2, 6, and 8, respectively. The different states are numbered in ascending order of their absolute electronic energies as obtained at the MRCI level. In some cases of near-degeneracy the ordering based on the MRCI energies is reversed at the MRCI + Q level.

Table 1 Energy E (E_h), bond length r_e (Å), binding energies D_e, D₀ (kcal mol⁻¹), Harmonic frequency ω_e (cm⁻¹), anharmonicity constant ω_ex_e (cm⁻¹), centrifugal distortion constant D̄e (cm⁻¹), rotation-vibration coupling constant a_e (cm⁻¹), net Mulliken charge q_Ti (e) on Ti, dipole moment μ (Debye) and energy separation T_e (cm⁻¹) for 45 low-lying electronic states of TiV at the MRCI(+Q)/cc-pVQZ level of theory

Method	−E	r e	D e	D 0	ω e	ω e x e	D̄ e (×10^−7)	a e (×10^−3)	q Ti	μ	T e
a Binding energies with respect to Ti(^3F) + V(^6D) for all octet states and with respect to Ti(^3F) + V(^4F) for all other states.
Xs ^4 Σ ^−
MRCI	1791.46830	1.897	29.3	28.8	391	1.576	1.78	1.37	−0.08	0.37	0
MRCI + Q (secondary minimum)	1791.4796	1.900	31.0	30.5	377	0.846	1.80	1.38			0
MRCI	1791.43625	2.729	9.2	9.0	173	3.620	1.10	0.29	+0.04	0.13	7034
MRCI + Q	1791.4483	2.708	11.4	11.2	178	3.300	1.20	0.61			6869
1 ^2 Δ
MRCI	1791.46407	1.920	27.4	26.8	476	8.887	1.12	1.52	−0.12	0.23	928
MRCI + Q	1791.4739	1.936	28.3	27.7	496	7.269	1.03	2.21			1251
2 ^2 Γ
MRCI	1791.45268	1.903	20.2	19.8	271	1.721	3.19	3.15	−0.09	0.69	3428
MRCI + Q	1791.4642	1.903	22.2	21.8	307	0.832	2.73	1.95			2941
3 ^2 Σ ^+
MRCI	1791.45207	1.903	19.6	19.2	239	2.654	4.70	9.61	−0.13	0.73	3562
MRCI + Q	1791.4636	1.903	21.5	21.0	288	7.102	3.05	−1.08			3512
4 ^4 Π
MRCI	1791.44998	1.993	17.9	17.3	360	11.990	1.56	2.44	−0.08	0.21	4021
MRCI + Q (secondary minimum)	1791.4641	2.028	21.4	20.1	337	8.860	1.61	1.57			3402
MRCI	1791.43776	2.686	10.2	10.0	164	3.630	1.35	0.61	0.02	0.03	6703
MRCI + Q	1791.4491	2.695	12.0	11.7	163	8.010	0.43	−1.27			6694
5 ^2 Σ ^−
MRCI	1791.44691	1.827	16.4	15.8	429	9.781	2.32	−0.82	−0.05	0.24	4695
MRCI + Q	1791.4580	1.827	17.9	17.4	387	2.425	2.69	−0.33			4741
6 ^8 Δ
MRCI	1791.44595	2.644	20.9	20.6	219	8.992	0.50	0.13	−0.05	0.34	4905
MRCI + Q	1791.4583	2.630	25.6	25.3	231	7.309	0.37	−0.14			4675
7 ^4 Δ
MRCI	1791.44440	2.163	14.5	14.2	185	1.312	3.48	−2.55	−0.07	0.23	5245
MRCI + Q	1791.4545	2.171	15.4	15.1	180	1.189	3.85	−0.03			5509
8 ^8 H
MRCI	1791.44124	2.801	17.9	17.6	197	0.798	0.63	0.37	−0.01	0.34	5939
MRCI + Q	1791.4526	2.788	22.0	21.7	203	0.346	0.69	0.47			5926
9 ^6 Π
MRCI	1791.44082	2.641	12.3	12.1	126	4.884	1.48	−2.92	−0.01	0.08	6031
MRCI + Q	1791.4519	2.641	14.0	13.8	116	3.953	1.95	−5.05			6079
10 ^6 H
MRCI	1791.44080	2.707	12.3	12.0	171	0.483	1.26	0.23	0.00	0.14	6036
MRCI + Q	1791.4518	2.715	13.9	13.7	176	1.925	0.63	0.03			6101
11 ^8 Π
MRCI	1791.43999	2.812	17.1	16.8	186	0.951	0.53	0.28	0.00	0.36	6213
MRCI + Q	1791.4515	2.791	21.3	21.0	191	0.711	1.22	32.5			6167
12 ^2 Π
MRCI	1791.43942	2.049	11.2	10.8	327	0.926	1.59	0.36	−0.14	0.83	6338
MRCI + Q	1791.4500	2.069	12.5	12.1	275	5.973	2.10	2.20			6496
13 ^2 H
MRCI	1791.43894	2.066	10.9	10.5	288	10.977	1.76	3.67	−0.15	0.83	6444
MRCI + Q	1791.4494	2.089	12.1	11.7	269	4.729	2.29	1.13			6628
14 ^6 Π
MRCI	1791.43835	2.666	10.8	10.6	160	−0.304	1.60	0.23	−0.02	0.01	6573
MRCI + Q	1791.4497	2.682	12.5	12.3	163	−0.496	1.31	0.22			6562
15 ^6 Σ ^+
MRCI	1791.43828	2.677	10.5	10.3	127	1.058	2.36	−0.20	0.01	0.11	6589
MRCI + Q	1791.4505	2.654	12.8	12.6	124	2.445	2.38	−0.07			6387
16 ^8 Σ ^+
MRCI	1791.43775	2.800	15.7	15.5	189	7.059	0.01	−0.28	0.00	0.60	6705
MRCI + Q	1791.4490	2.750	19.8	19.5	189	3.019	0.85	−0.76			6716
17 ^6 Φ
MRCI	1791.43771	2.690	10.4	10.2	163	0.921	1.26	0.16	0.00	0.15	6714
MRCI + Q	1791.4491	2.682	12.2	11.9	165	0.769	1.24	0.12			6694
18 ^4 H
MRCI	1791.43757	2.692	10.2	9.9	163	0.119	1.14	0.38	0.02	0.03	6744
MRCI + Q	1791.4489	2.703	11.9	11.6	164	0.649	1.20	0.09			6738
19 ^6 Σ ^−
MRCI	1791.43751	2.717	10.0	9.8	154	7.066	1.32	1.94	0.01	0.17	6758
MRCI + Q	1791.4495	2.709	12.2	11.9	171	1.437	1.09	0.36			6606
20 ^2 Δ
MRCI	1791.43736	2.205	10.7	10.4	237	4.180	1.80	0.92	−0.08	0.18	6791
MRCI + Q	1791.4496	2.226	12.6	12.3	202	8.050	2.38	2.92			6584
21 ^6 Γ
MRCI	1791.43704	2.717	9.7	9.5	177	6.741	0.84	0.32	0.02	0.16	6861
MRCI + Q	1791.4489	2.709	11.8	11.5	180	6.913	0.98	0.84			6738
22 ^8 Π
MRCI	1791.43698	2.820	15.2	15.0	184	0.984	0.41	0.70	0.00	0.40	6874
MRCI + Q	1791.4485	2.800	19.4	19.1	197	2.750	0.57	0.73			6826
23 ^8 Γ
MRCI	1791.43690	2.822	15.2	15.0	195	2.249	0.61	1.06	0.00	0.58	6892
MRCI + Q	1791.4480	2.800	19.2	18.9	202	2.496	0.57	0.84			6935
24 ^4 Φ
MRCI	1791.43677	2.628	9.9	9.7	139	0.748	2.01	−0.96	0.00	0.04	6920
MRCI + Q	1791.4487	2.620	11.8	11.6	155	8.126	1.07	−1.01			6782
25 ^8 Σ ^−
MRCI	1791.43676	2.761	15.1	14.8	196	0.756	0.66	0.26	0.00	0.52	6922
MRCI + Q	1791.4480	2.753	19.1	18.7	203	0.471	0.63	0.11			6935
26 ^4 Π
MRCI	1791.43658	2.653	9.8	9.6	224	8.991	0.61	−1.03	0.02	0.03	6962
MRCI + Q	1791.4485	2.645	11.7	11.5	220	8.548	0.73	−0.67			6826
27 ^4 Γ
MRCI	1791.43615	2.748	9.2	9.0	164	−0.127	1.08	1.50	−0.14	0.87	7056
MRCI + Q	1791.4483	2.726	11.4	11.2	188	5.774	0.78	−1.10			6870
28 ^8 Φ
MRCI	1791.43606	2.851	14.7	14.4	174	0.314	0.76	0.44	0.00	0.44	7076
MRCI + Q	1791.4474	2.835	18.7	18.4	182	0.821	0.74	0.75			7067
29 ^4 Σ ^+
MRCI	1791.43603	2.734	9.1	8.8	170	2.713	0.93	0.88	−0.16	0.87	7082
MRCI + Q	1791.4481	2.717	11.2	11.0	174	3.647	0.75	1.69			6913
30 ^2 Φ
MRCI	1791.43590	2.057	9.0	8.6	293	7.000	1.92	1.89	−0.11	0.83	7111
MRCI + Q	1791.4467	2.092	10.4	10.0	283	8.049	1.93	1.00			7221
31 ^8 Δ
MRCI	1791.43452	2.881	13.7	13.5	181	0.602	0.60	0.24	0.00	0.66	7414
MRCI + Q	1791.4460	2.857	17.9	17.6	186	0.614	0.68	0.74			7374
32 ^6 Δ
MRCI	1791.43433	2.748	8.0	7.8	171	4.329	0.91	0.80	0.02	0.16	7456
MRCI + Q	1791.4466	2.741	10.4	10.1	177	3.338	0.96	0.06			7243
33 ^2 Π
MRCI	1791.43419	2.631	8.0	7.6	226	7.594	0.75	2.16	0.03	0.04	7486
MRCI + Q	1791.4455	2.632	9.6	9.3	216	5.825	0.82	0.79			7484
34 ^4 Π
MRCI	1791.43257	2.419	7.0	6.8	141	1.420	2.06	−0.68	−0.10	0.07	7842
MRCI + Q	1791.4433	2.419	8.6	8.3	83	−9.138	4.48	−2.34			7967
35 ^4 Σ ^−
MRCI	1791.43214	2.699	6.7	6.4	179	—	—	—	0.03	0.07	7936
MRCI + Q	1791.4444	2.725	9.0	8.8	125	—	—	—			7726
36 ^4 H
MRCI	1791.43187	2.655	4.6	4.3	178	0.767	1.15	0.51	0.01	0.11	7995
MRCI + Q	1791.4430	2.652	4.5	4.2	160	−1.105	1.08	0.93			8033
37 ^4 Γ
MRCI	1791.43183	2.702	6.5	6.2	180	2.084	0.53	0.29	0.03	0.05	8004
MRCI + Q	1791.4457	2.705	9.8	9.6	186	2.741	1.02	0.17			7440
38 ^6 Δ
MRCI	1791.43176	2.733	6.4	6.2	130	1.224	1.90	0.92	−0.03	0.17	8020
MRCI + Q	1791.4443	2.725	8.9	8.7	157	7.379	1.34	0.72			7747
39 ^2 Σ ^−
MRCI	1791.43001	2.063	8.3	7.7	422	9.835	0.89	0.44	−0.10	0.73	8404
MRCI + Q	1791.4428	2.053	9.6	9.1	411	4.829	1.21	1.41			8077
40 ^4 Π
MRCI	1791.43001	2.633	4.5	4.2	164	0.714	1.43	0.91	−0.01	0.10	8404
MRCI + Q	1791.4418	2.654	4.0	3.8	164	0.426	1.28	0.26			8296
41 ^8 Π
MRCI	1791.42962	2.938	10.7	10.4	160	3.980	0.80	0.41	−0.01	0.48	8489
MRCI + Q	1791.4420	2.875	15.4	15.1	193	3.264	0.59	0.40			8252
42 ^8 Σ ^+
MRCI	1791.42754	2.985	9.4	9.2	152	7.556	0.63	2.15	−0.01	0.71	8946
MRCI + Q	1791.4389	2.929	13.4	13.2	184	7.119	0.38	−1.62			8933
43 ^8 Γ
MRCI	1791.42732	3.081	9.3	9.0	149	−1.920	1.04	−0.53	−0.02	0.67	8994
MRCI + Q	1791.4382	2.993	13.0	12.8	159	−0.881	0.65	0.36			9086
44 ^8 Δ
MRCI	1791.42650	2.993	8.7	8.5	149	3.433	0.78	0.82	0.00	0.41	9174
MRCI + Q	1791.4378	2.937	12.8	12.5	182	1.843	0.59	0.31			9174

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Fig. 1 Potential energy curves of the thirteen lowest quartet states of TiV at the MRCI/cc-pVQZ level of theory: (a) X⁴Σ⁻, 7⁴Δ, 27⁴Γ, 29⁴Σ⁺, 35⁴Σ⁻, 37⁴Γ and (b) 4⁴Π, 18⁴H, 24⁴Φ, 26⁴Π, 34⁴Π, 36⁴H, 40⁴Π.

From Table 1, it is clear that the ground state of TiV is of X⁴Σ⁻ symmetry with a ²Δ state lying slightly higher. Thus, we discuss first quartet states, then doublets, and finally sextets and octets. We conclude this section with a discussion on the binding energies of the X⁴Σ⁻, 1²Δ, and 4⁴Π electronic states.

3.1 Quartets

We first present all quartet states studied in this work, as the TiV ground state was found to be a ⁴Σ⁻, in complete agreement with the observation of Van Zee and Weltner Jr.¹

In Fig. 1a, we can see its PEC, which adiabatically correlates with the ground atomic states Ti(³F) + V(⁴F). This curve has a rather unusual shape, indicative of successive avoided crossings. Therefore, it will be enlightening to explore the nature of its morphology as a function of the interatomic Ti−V separation. As stated in the Introduction section, the ground atomic states Ti(³F;3d²4s²) + V(⁴F;3d³4s²) can only lead to weak vdW interactions. Indeed, at r(Ti–V) = ∼3.40 Å, we observe a very shallow minimum of ∼3 kcal mol⁻¹. Then, at this point, an abrupt change occurs, resulting in the formation of a relatively shallow potential well with a local minimum at r(Ti–V) = 2.729 (2.708) Å and a binding energy D⁰₀ = 9.0 (11.2) kcal mol⁻¹ at the MRCI(+Q)/cc-pVQZ level, Table 1. This is apparently the result of an avoided crossing with an incoming curve of same symmetry that stems from a higher atomic asymptote. This is clearly illustrated in Fig. 1a, where it is shown that this local minimum non-adiabatically traces its origin to the first atomic asymptote Ti(³F) + V(⁶D), lying 0.245 eV above the ground-state atomic limit. This quasi-diabatic curve (grey line) was produced by a CASSCF reference accidentally locked into this minimum's electronic configuration. This means that, at the crossing point, the MCSCF reference, instead of following the lowest adiabatic path, continues onto the higher asymptotic limit, which corresponds to the equilibrium electronic configuration. Furthermore, inspection of the electronic wavefunction shows an in situ electronic distribution corresponding to Ti(3d²4s²)V(3d⁴4s¹). The bonding occurs, mainly, through a 2 center-3 electron (2c-3e) interaction of the Ti(4s²)−(4s¹)V electrons, with minor contributions from the 3d electrons, resulting in a weak bond. We will discuss this binding mode further in the discussion of sextet and octet states.

Now, moving further left on the same adiabatic curve, we come across a second sharp change at r(Ti–V) = 2.40 Å, obviously a second avoided crossing, leading to a much deeper potential energy well. It forms the global minimum, hence the ground X⁴Σ⁻ state of TiV. Thus, we are going to investigate it in detail.

The equilibrium TiV(X⁴Σ⁻) MRCI wavefunction is dominated by the configuration

|(core)1σ²2σ¹1π⁴1δ¹₊1δ¹₋〉

(where only valence orbitals are counted) with a coefficient of 0.83. This electronic configuration is identical with the one proposed in ref. 1 on the basis of ESR observations, see Inroduction. The corresponding atomic Mulliken electronic distributions at equilibrium are:

These populations coupled to the active orbitals participating in the leading configuration (Fig. 2) suggest a bonding situation which can be depicted by Scheme 1.

Fig. 2 Illustration of the 15 natural orbitals participating in the CASSCF and MRCI active space for the TiV(X⁴Σ⁻) state at equilibrium. Their shapes are practically the same for all other states studied.

Scheme 1

In this Scheme, the 3d orbitals are represented by overextended sticks in order to clearly show the formal interactions between them. It becomes apparent from the overall electronic arrangement that the in situ atomic states correspond to the Ti(⁵F) + V(⁶D) excited asymptote located 1.051 eV (see Inroduction) above the ground separated atomic limit. A significant stabilization takes place through a formally 3-bond + 3-half-bond, formation, making this configuration the ground state of TiV after crossing all states of the same symmetry that arise from intermediate atomic channels shown in the Introduction section.

The equilibrium bond length was found r_e = 1.900 Å. This is close to the MCSCF result, r_e = 1.86 Å, of ref. 8 but larger by ∼0.1 Å than the DFT findings of ref. 5 and 6. The binding energy with respect to the ground state adiabatic limit was computed D⁰₀ = 30.5 kcal mol⁻¹ = 1.323 eV at the MRCI + Q/cc-pVQZ level of theory, Table 1. This is much lower than the most recent experimental value, D⁰₀ = 2.068 eV, of Spain and Morse.⁴ As of the DFT values of 5.88 eV⁵ and 2.78 eV,⁶ they are deemed rather excessive. We will return to this binding energy problem with a dedicated section at the end of this discussion.

On Table 1, are also reported spectroscopic constants and the dipole moment of TiV(X⁴Σ⁻), as they were computed at the MRCI(+Q)/cc-pVQZ level. We note that our harmonic frequency, ω_e = 377 cm⁻¹, is much lower than the DFT value, ω_e = 562 cm⁻¹, of Gutzev et al.⁶ The very low value of the dipole moment, μ = 0.37 D, reflects a very small overall charge transfer, in agreement with Scheme 1 and the corresponding Mulliken populations.

We can see from Fig. 1 that the states 4⁴Π, 7⁴Δ, 27⁴Γ, and 29⁴Σ⁺, also have PECs exhibiting the same double-well aspect as the ground state, with a first minimum at ∼2.7–2.8 Å and a second one at around 1.9–2.0 Å. Just like for the ground state, we find out that the first minima originate from the first excited atomic channel Ti(³F) + V(⁶D), while the second ones come from the Ti(⁵F) + V(⁶D) asymptote. All states with a single minimum at ∼2.70 Å correspond to Ti(³F) + V(⁶D).

We are going, now, to deal with the 4⁴Π state, Fig. 1b, which is suitable (dipole allowed) for a spectroscopic study of TiV. Its PEC has same shape as the one of the ground state. The global minimum is at r_e = 1.993 (2.028) Å with binding energy D⁰₀ = 17.3 (20.1) kcal mol⁻¹ at the MRCI(+Q)/cc-pVQZ level. Its equilibrium wavefunction, although of a somewhat multireference nature, is mainly characterized by the configuration

|(core)1σ²2σ¹1π³1δ¹₊1δ¹₋〉

with a coefficient of 0.75. Taking into account and the atomic Mulliken populations (B₁ component):

the binding can be described by Scheme 2, from which it is clear again, that the two atoms are in their excited Ti(⁵F) + V(⁶D) states.

Scheme 2

We have calculated for this state an excitation energy T_e = 4021 (3402) cm⁻¹ at the MRCI(+Q)/cc-pVQZ level (Table 1); thus it could be accessed spectroscopically in the infrared.

3.2 Doublets

Fig. 3 shows potential energy curves for 10 doublet states. We can see from this figure and also from Table 1 that the first three excited states of TiV are the 1²Δ, 2²Γ, and 3²Σ⁺ doublets, with the 1²Δ state being located energetically very close to the ground state X⁴Σ⁻. We have computed for this state T_e = 928 (1251) cm⁻¹ at the MRCI(+Q)/cc-pVQZ level and a bondlength r_e = 1.920 (1.936) Å (Table 1). The proximity of this state to the ground state raises some doubt as to what the true ground state of the molecule is and whether it might change at a higher computational level. However, as we will see below, this energy difference persists at higher levels of theory. Its equilibrium MRCI wavefunction has one leading configuration, i.e.

|(²Δ〉∼0.80|(core)1σ²2σ¹1π⁴1δ¹〉

which combined with the following Mulliken atomic populations (A₁ component):

lead to a binding mode illustrated in Scheme 3. From this scheme it is clear that the in situ atomic states are Ti(⁵F; M_L = 0) + V(⁶D; M_L = ±2). The 1²Δ state resembles the ground state (Scheme 1) with the only difference being that one δ electron has moved to the σ frame to form a 2-electron bond. Its potential energy curve suffers an avoided crossing with the 20²Δ state (Fig. 3a) at r(Ti−V) = ∼2.40 Å, so it forms a shelf in the region 2.40–3.00 Å before crossing the curve coming from the ground state asymptote.

Fig. 3 Potential energy curves of the ten lowest doublet states of TiV at the MRCI/cc-pVQZ level of theory: (a) 1²Δ, 2²Γ, 3²Σ⁺, 5²Σ⁻, 20²Δ, 39²Σ⁻ and (b) 12²Π, 13²H, 30²Φ, 33²Π.

Scheme 3

Other doublet curves have the same double-well profile as described for the quartets (vide supra). The rationalization of this feature is the same as for the latter, and is omitted.

In concluding this section, we wish to remark that the first excited state of TiV is 1²Δ, in contrast to ref. 6, which claims that it is a ²Γ. The latter lies higher in energy and is nearly degenerate with a ²Σ⁺ state (Table 1).

3.3 Sextets and octets

We discuss sextet and octet states together in this section because, as we will see, they all originate from the same asymptote, namely Ti(³F) + V(⁶D). The corresponding potential energy curves are shown in Fig. 4 and 5 for sextets and octets, respectively. We observe that, unlike the quartets and doublets, they all exhibit single minima in the vicinity of 2.70–2.80 Å. All octet PECs adiabatically correlate with the Ti(³F) + V(⁶D) asymptotic channel since the ground atomic states cannot yield such a spin multiplicity. However, the sextet states adiabatically go to the ground-state atomic limit after suffering avoided crossings at about r(Ti−V) = 3.50 Å (Fig. 4). All states have relatively weak binding energies ranging from ∼15–25 kcal mol⁻¹ (Table 1). As mentioned before, this happens because bonding occurs primarily through a 2c-3e interaction between the more spatially extended 4s orbitals, namely Ti(4s²)−(4s¹)V, supplemented by additional weak 3d–3d interactions. As an illustrative example, we will consider the 6⁸Δ state. At equilibrium, its MRCI wavefunction, although multireference in character, is dominated by the following configuration

|(core)1σ²2σ¹3σ¹1π¹₊1π¹₋1δ¹₊1δ¹₋2δ¹〉

with a coefficient 0.65. The corresponding atomic Mulliken populations (A₁ component) are:

Fig. 4 Potential energy curves of the nine lowest sextet states of TiV at the MRCI/cc-pVQZ level of theory: (a) 15⁶Σ⁺, 19⁶Σ⁻, 21⁶Γ, 32⁶Δ, 38⁶Δ and (b) 9⁶Π, 10⁶H, 14⁶Π, 17⁶Π.

Fig. 5 Potential energy curves of the thirteen lowest octet states of TiV at the MRCI/cc-pVQZ level of theory: (a) 6⁸Δ, 16⁸Σ⁺, 23⁸Γ, 25⁸Σ⁻, 31⁸Δ, 42⁸Σ⁺, 43⁸Γ, 44⁸Δ and (b) 8⁸H, 11⁸Π, 22⁸Π, 28⁸Φ, 41⁸Π.

From these findings we infer the binding mode of Scheme 4. In this scheme the 2c−3e interaction is represented as a two-electron σ-bond with the third electron promoted to a non-bonding orbital (3σ¹) (symbolized by an up arrow). This type of bonding is supplemented by additional 3d-3d half-bond interactions (dashed lines) finally yielding a binding energy D_e = 25.6 kcal mol⁻¹ relative to the Ti(³F) + V(⁶D) asymptote (Table 1). All other octet states depicted in Fig. 5, arise from different arrangements of the 3d electrons while the overall bonding motif remains unchanged. It is interesting to note here that the ground X⁷Σ⁺ state of ScV is formed through a similar mechanism and has a comparable binding energy D_e = 28 kcal mol⁻¹.⁸

Scheme 4

Now, turning to the sextet states, we see from Fig. 4 that they all present minima at about r(Ti−V) = 2.70 Å with energies very close to those of the octets. Although they adiabatically correlate with the ground state atomic asymptote, their origin can be traced to the first excited channel, Ti(³F) + V(⁶D), through avoided crossings at r(Ti−V) = 3.20-3.50 Å. Their equilibrium MRCI wavefunctions and Mulliken populations support a binding mode consistent with Scheme 4 but with the unpaired electrons coupled into a sextet.

Finally, it should be noted that all secondary minima in the quartet and doublet PECs mentioned before, exhibit exactly the same characteristics; therefore they correspond to the first excited atomic asymptote, as well.

3.4 Binding energies of X⁴Σ⁻, 1²Δ, and 4⁴Π

In the previous discussion of the TiV ground state it was pointed out that the calculated binding energy at the MRCI(+Q)/cc-pVQZ level (Table 1) was significantly lower than the existing experimental value, D⁰₀ = 2.068 eV = 47.7 kcal mol⁻¹.⁴ Now, in this section we will examine this discrepancy in detail. To this end, complete basis set (CBS) extrapolation calculations were performed using the cc-pVnZ bases, with n = T, Q, 5 (CBS-TQ5), at the MRCI and MRCI + Q level. Scalar relativistic effects were, also, included via the third order Douglas–Kroll–Hess (DKH) correction, MRCI + DKH and MRCI + DKH + Q (Tables 2 and 3).

Table 2 Equilibrium bond lengths r_e (Å) and energies E (E_h) of TiV(X⁴Σ⁻), asymptotic energies E_∞ and (E_h) of the atomic channels Ti(³F) + V(⁴F) and Ti(⁵F) + V(⁶D), respectively, and corresponding energy separations ΔE_∞ (eV), adiabatic and non-adiabatic binding energies D_e and (eV) of TiV(X⁴Σ⁻) at different levels of theory and with cc-pVnZ basis sets, with n = T (Tζ), Q (Qζ), 5(5ζ). Extrapolation to the complete basis set (CBS) limit are also given

Basis set	r e	−E	−E ∞		ΔE∞	D e
MRCI
Tζ	1.921	1791.45399	1791.41832	1791.36829	1.361	0.971	2.332
Qζ	1.899	1791.47043	1791.42060	1791.37935	1.123	1.356	2.479
5ζ	1.894	1791.47585	1791.42527	1791.37996	1.233	1.377	2.610
CBS		1791.47852	1791.42790	1791.37999	1.304	1.378	2.681
MRCI + Q
Tζ	1.928	1791.4618	1791.4211	1791.3732	1.301	1.110	2.411
Qζ	1.907	1791.4793	1791.4292	1791.3849	1.205	1.365	2.569
5ζ	1.902	1791.4851	1791.4320	1791.3858	1.258	1.444	2.701
CBS		1791.4879	1791.4335	1791.3859	1.297	1.478	2.775
MRCI + DKH
Tζ	1.929	1801.06546	1801.02110	1800.97344	1.297	1.152	2.449
Qζ	1.908	1801.08044	1801.03325	1800.98326	1.360	1.284	2.644
5ζ	1.904	1801.08632	1801.03424	1800.98618	1.308	1.415	2.725
CBS		1801.08944	1801.03424	1800.98742	1.274	1.502	2.776
MRCI + DKH + Q
Tζ	1.936	1801.0714	1801.0230	1800.9789	1.200	1.319	2.519
Qζ	1.916	1801.0894	1801.0356	1800.9894	1.256	1.466	2.722
5ζ	1.912	1801.0957	1801.0364	1800.9926	1.193	1.613	2.806
CBS		1801.0990	1801.0365	1800.9939	1.159	1.701	2.860

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Table 3 Equilibrium bond lengths r_e (Å) and energies E (E_h), adiabatic and non-adiabatic binding energies D_e and (eV) (with respect to Ti(³F) + V(⁴F) and Ti(⁵F) + V(⁶D), respectively) and energy separations T_e (cm⁻¹) of TiV(1²Δ and 4⁴Π) at different levels of theory and with cc-pVnZ basis sets, with n = T (Tζ), Q (Qζ), 5 (5ζ). Extrapolation to the complete basis set (CBS) limit are also given

1^2Δ						4^4Π
Basis set	r e	−E	D e		T e	r e	−E	D e		T e
MRCI
Tζ	1.942	1791.44837	0.818	2.179	1233	2.029	1791.43790	0.533	1.894	3531
Qζ	1.925	1791.46440	1.192	2.314	1324	2.009	1791.45280	0.876	1.999	3869
5ζ	1.923	1791.47022	1.223	2.456	1237	2.005	1791.45799	0.890	2.123	3921
CBS		1791.47353	1.242	2.545	1095		1791.46076	0.894	2.198	3902
MRCI + Q
Tζ	1.946	1791.4563	0.978	2.259	1207	2.042	1791.4466	0.695	1.996	3336
Qζ	1.931	1791.4732	1.199	2.404	1339	2.023	1791.4625	0.906	2.111	3687
5ζ	1.927	1791.4794	1.290	2.547	1251	2.018	1791.4680	0.980	2.237	3753
CBS		1791.4803	1.273	2.570	1668		1791.4710	1.019	2.316	3709
MRCI + DKH
Tζ	1.943	1801.05872	1.024	2.321	1478	2.030	1801.04791	0.729	2.649	3851
Qζ	1.929	1801.07565	1.154	2.514	1051	2.011	1801.06356	0.825	2.185	3705
5ζ	1.927	1801.08175	1.293	2.601	1003	2.007	1801.06908	0.948	2.256	3784
CBS		1801.08519	1.386	2.661	933		1801.07209	1.030	2.304	3808
MRCI + DKH + Q
Tζ	1.949	1801.0667	1.190	2.390	1032	2.043	1801.0568	0.920	2.119	3204
Qζ	1.935	1801.0846	1.334	2.590	1054	2.024	1801.0734	1.030	2.286	3512
5ζ	1.932	1801.0910	1.487	2.680	1032	2.020	1801.0793	1.166	2.360	3599
CBS		1801.0947	1.584	2.742	944		1801.0825	1.252	2.411	3621

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For the extrapolation, a simple exponential formula was employed, namely

E(n) = E(CBS) + A exp(−Bn)

where n is the basis set cardinal number and A, B fitting parameters.

All our results for TiV(X⁴Σ⁻) are summarized in Table 2. This Table reports absolute energies at equilibrium, as well as those of the asymptotic Ti(³F) + V(⁴F) and Ti(⁵F) + V(⁶D) channels. The corresponding energy separations are also given, along with the adiabatic and non-adiabatic binding energies of TiV(X⁴Σ⁻). We note here that the computation of the Ti(⁵F) + V(⁶D) asymptotic energy was technically feasible using a spin multiplicity of 10 at infinite separation.

In Table 2, we see a gradual increase in binding energy with increasing quality of the basis set and also a small additional improvement from the introduction of scalar relativistic corrections. At the MRCI + DKH + Q/CBS-TQ5 level, the non-adiabatic (intrinsic) binding energy of TiV(X⁴Σ⁻) is and taking into account the energy difference between the two asymptotic channels, ΔE_∞ = 1.159 eV (Table 2), we obtain D⁰_e = 1.701 eV, with respect to the ground state atomic limit Ti(³F) + V(⁴F). This yields D⁰₀ = 1.678 eV using the spectroscopic constants of Table 1. A further correction can result from the small error in the calculation of the energy gap between the Ti(³F) + V(⁴F) and Ti(⁵F) + V(⁶D) asymptotes: 1.159 eV (Table 2) compared to the experimental (M_J averaged) 1.051 value.⁹ This difference of 0.108 eV should be added to D⁰₀ to arrive at a final value of D⁰₀ = 1.786 eV = 41.1 kcal mol⁻¹. This is our best value for the binding energy of TiV(X⁴Σ⁻) but it is still lower by 0.282 eV than the experimental one reported by Spain and Morse⁴ using the R2PI technique. Certainly, other subtle factors could influence our results, e.g. core correlation effects which, however, do not have such a significant impact on the binding energies. In the present work, it was beyond our computational possibilities to correlate 16 additional core electrons even at the CISD level.

A final remark concerns the fact that our small difference, of 0.282 eV, from the experimental value, closely matches the first excitation energy of V, ΔE(⁴F ← ⁶D) = 0.245 eV.⁹ If, in the R2PI experiment, the outgoing V atom were produced in its first excited state, ⁶D, the experimentally determined D⁰₀ value would be larger by this amount, thereby accounting for our computed value. However, this interpretation remains speculative.

In Table 3, we present a similar analysis for the 1²Δ and 4⁴Π states. Our primary objective was to assess whether the 1²Δ state remains above the ground state at higher levels of theory and, also, to determine an accurate ΔE(X⁴Σ⁻ ← 4⁴Π) excitation energy. From Table 3, at the MRCI + DKH + Q/CBS-TQ5 level, we have T_e values: 944 cm⁻¹ (1²Δ) and 3621 cm⁻¹ (4⁴Π). These results are in close agreement with the corresponding results of Table 1.

The binding energies, D_e, of both states increase by, roughly, the same amount as that of the ground state at the MRCI + DKH + Q/CBS-TQ5 level. We recall here that both states 1²Δ and 4⁴Π originate non-adiabatically from the excited Ti(⁵F) + V(⁶D) asymptote, like the ground TiV(X⁴Σ⁻) state. As a general rule, we could say that these observations apply to all states intrinsically correlating with the same dissociation channel.

Concluding this section, we found it interesting to examine the effect of spin–orbit coupling (SOC) on some of the lowest-lying states of TiV. To this end, the five lowest Λ–S states were used to diagonalize the Breit–Pauli operator in the region around their equilibrium distances at the MRCI/cc-pVQZ level. Our results are summarized in Fig. 6. As shown, SOC splittings are generally very small, and cannot in any way quantitatively alter the above results. For example, the ²Δ_3/2–²Δ_5/2 splitting is approximately 250 cm⁻¹, while for the ground state, which is a Σ state, the splitting is only 10 cm⁻¹. The overall picture remains largely unchanged, except for a few avoided crossings between some higher states with Ω = 1/2 near r(Ti–V) = 2.00 Å. These findings indicate that inclusion of SOC does not significantly alter our potential energy curve profiles. The TiV molecule is primarily described by Hund's case (a); however, SOC can induce couplings between energetically close-lying states, thereby subtly modifying the qualitative picture and accounting for some weak features in the TiV spectrum. A treatment of all 45 states, including SOC effects, would constitute a separate future study.

Fig. 6 Potential energy curves of the five lowest-lying Λ–S states of TiV around their equilibrium distances (left) and corresponding Ω states (right) as they were calculated by including spin–orbit coupling at the MRCI/cc-pVQZ level of theory.

4. Summary and conclusions

We have employed the multireference complete active space MRCISD methodology in conjunction with high quality correlation consistent basis sets to study 45 low-lying Λ–S electronic states of the TiV diatomic system.

For the first time full potential energy curves were constructed for all states considered, and corresponding spectroscopic constants were extracted.

The ground state was unambiguously found to be of X⁴Σ⁻ symmetry, in agreement with the experimental observations.¹ Its potential energy curve adiabatically correlates with the ground state atomic asymptote, Ti(³F) + V(⁴F). However, its morphology is modulated by successive avoided crossings with curves originating from higher atomic channels, namely Ti(³F) + V(⁶D) and Ti(⁵F) + V(⁶D). As a result it exhibits a double-well shape with an additional shallow vdW minimum at long distance. The global minimum corresponding to TiV(X⁴Σ⁻) is located at r_e = 1.910 Å, while our best value for its adiabatic binding energy was found D⁰₀ = 1.786 eV = 41.1 kcal mol⁻¹, relative to the ground state Ti(³F) + V(⁴F) atomic limit, in reasonable agreement with the corresponding experimental value.⁴ We note that, since X⁴Σ⁻ stems from the Ti(⁵F) + V(⁶D) asymptote, its intrinsic bond strength can be evaluated as D₀ = 1.786 eV + 1.051 eV = 2.837 eV = 65.4 kcal mol⁻¹; a rather large value.

Several other states, doublets and quartets, exhibit the same double-well feature. A similar rationale applies in these cases.

All sextet and octet states considered in the present work, trace their lineage to the first excited atomic asymptote Ti(³F) + V(⁶D) and have relatively small binding energies.

The first excited state of TiV is 1²Δ, lying only 944 cm⁻¹ above the X⁴Σ⁻ ground state, with r_e = 1.930 Å, and exhibiting a similar electronic structure, reminiscent of its origin in the Ti(⁵F) + V(⁶D) atomic configuration.

Finally, we mention the 4⁴Π state which is located approximately 3500–4000 cm⁻¹ higher than X⁴Σ⁻, with r_e = 2.020 Å, and could be of interest in a spectroscopic study of TiV. It has a double-well potential energy profile, similar to that of the ground state.

We conclude this paper with the hope that our results will be helpful in a future investigation of the experimentally poorly explored TiV system.

Author contributions

Magdalene Liosi: writing – original draft, visualization, investigation, formal analysis, data curation. Aristotle Papakondylis: writing – review & editing, writing – original draft, supervision, conceptualization.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The coordinates of all potential energy curves presented, are provided in the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d6cp00590j.

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