Electronic Structure and Chemical Bonding of the First-, Second-, and Third-Row-Transition-Metal Monoborides: The Formation of Quadruple Bonds in RhB, RuB, and TcB

Abstract

Boron presents an important role in chemistry, biology, and materials science. Diatomic transition-metal borides (MBs) are the building blocks of many complexes and materials, and they present unique electronic structures with interesting and peculiar properties and a variety of bonding schemes which are analyzed here. In the first part of this paper, we present a review on the available experimental and theoretical studies on the first-row-transition-metal borides, i.e., ScB, TiB, VB, CrB, MnB, FeB, CoB, NiB, CuB, and ZnB; the second-row-transition-metal borides, i.e., YB, ZrB, NbB, MoB, TcB, RuB, RhB, PdB, AgB, and CdB; and the third-row-transition-metal borides, i.e., LaB, HfB, TaB, WB, ReB, OsB, IrB, PtB, AuB, and HgB. Consequently, in the second part, the second- and third-row MBs are studied via DFT calculations using the B3LYP, TPSSh, and MN15 functionals and, in some cases, via multi-reference methods, MRCISD+Q, in conjunction with the aug-cc-pVQZ-PP_M/aug-cc-pVQZ_B basis sets. Specifically, bond distances, dissociation energies, frequencies, dipole moments, and natural NPA charges are reported. Comparisons between MB molecules along the three rows are presented, and their differences and similarities are analyzed. The bonding of the diatomic borides is also described; it is found that, apart from RhB(X¹Σ⁺), which was just recently found to form quadruple bonds, RuB(X²Δ) and TcB(X³Σ⁻) also form quadruple σ²σ²π²π² bonds in their X states. Moreover, to fill the gap existing in the current literature, here, we calculate the TcB molecule.

1. Introduction

Boron has an important role in chemistry, biology, and materials science [1]. It is well known that it forms single, double, and triple bonds, but it was only recently found that it can form quadruple bonds in specific diatomic molecules [2,3,4,5]. Additionally, its chemistry is quite interesting to preparative chemists, theoreticians, industrial chemists, and technologists. It is noteworthy that it is the only non-metal in group 13 of the periodic table, and it presents many similarities to its neighbor, carbon, and its diagonal relative, silicon. Hence, like C and Si, it showcases a marked propensity to form covalent, molecular compounds, but it differs greatly from them in having one less valence electron, a situation sometimes referred to as “electron deficiency”. This deficiency plays a key role in its chemistry [1].

Transition-metal borides have received considerable attention since they present common catalytic properties for the hydrogenation of alkenes and alkynes, the reduction of nitrogenous functional groups, and deoxygenation reactions [6]. They are important building blocks in many complexes and materials. Moreover, they possess remarkable physical properties, such as very high conductivity (TiB₂) [7]—even superconductivity (MgB₂) [8]—as well as super hardness (ReB₂) [9]. In solid state, many computational and experimental studies have been carried out; see, for instance, [10,11,12,13,14,15,16,17,18]. Computationally, the DFT methodology is applied to determine the bond lengths, frequencies, and vibrational properties of solids [10,13]; density of state; bond population; charge density maps [14]; relative stability; mechanical, electronic, and magnetic properties [15]; elastic behavior; and elastic anisotropy [16]. Furthermore, ab initio molecular dynamic (AIMD) simulations at finite temperature have also been employed in order to investigate the structural stability of materials, for instance, those of Ʋ₂B (Ʋ = Ti, Cr, Nb, Mo, Ta, and W) [18].

It has been reported that the electronic structure and the chemical bonding of diatomic and triatomic molecules are strongly related to their structure, the variety of their morphologies, and the properties of their 2D materials and solid state [19,20]. Therefore, an investigation of the electronic structure and the bonding of the diatomic transition-metal borides, which constitute the simplest building blocks of the compounds or materials in question, would lay the foundation for understanding the very complex solid-metal borides and even their bulk properties. Finally, it should be noted that the diatomic transition-metal borides showcase unique electronic structures presenting interesting and peculiar properties and a variety of bonding schemes.

The present work has two aims. In the first part, we present a review of the experimental and theoretical studies on the first-row-transition-metal borides, i.e., ScB, TiB, VB, CrB, MnB, FeB, CoB, NiB, CuB, and ZnB; the second-row-transition-metal borides, i.e., YB, ZrB, NbB, MoB, TcB, RuB, RhB, PdB, AgB, and CdB; and the third-row-transition-metal borides, i.e., LaB, HfB, TaB, WB, ReB, OsB, IrB, PtB, AuB, and HgB. In the second part, the second- and third-row-transition-metal borides, MBs, are studied via DFT calculations using the B3LYP, TPSSh, and MN15 functionals in conjunction with the aug-cc-pVQZ-PP_M/aug-cc-pVQZ_B basis sets. Additionally, MRCISD(+Q) calculations were carried out to clarify the ground states of the MB molecules when their identity was not known. Bond distances, dissociation energies, frequencies, dipole moments, and Mulliken and natural NPA charges are presented. Comparisons between the MB molecules of all three rows are presented, and their differences and similarities are analyzed. Finally, transition-metal borides forming quadruple bonds are described and analyzed here, and for the first time, we report on the RuB and TcB molecules.

2. Previous Studies on Transition-Metal Monoborides, MBs

2.1. First-Row-Transition-Metal MBs

All previous theoretical and experimental data on the ground states of the first-row-transition-metal borides are summarized in

Table 1. Previous theoretical and experimental data on the ground states of the 1st-row-transition-metal boride molecules, MBs (M = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn): bond lengths r_e (Å), dissociation energies D_e (eV) and/or D₀ (eV) with respect to the adiabatic products, vibrational frequencies ω_e (cm⁻¹), anharmonic corrections ω_ex_e (cm⁻¹), and dipole moments μ (μ_FF = δE/δε) (Debye).

MB	Methodology	Ref.	State	re	De a	D0	ωe	ωexe	μ (μFF)
ScB	icMRCISD/cc-pV5Z	[21]	X5Σ–	2.128	3.257	3.220	584.5	3.8	4.02(4.16)
	icMRCI+Q/cc-pV5Z	[21]		2.132	3.287	3.251	579	3.8	(4.23)
	icMRCISD+DKH+Q/cc-pV5Z	[21]		2.094	3.309	3.271	603	2.6	(4.13)
	B3LYP/6-311++G(3df)	[22]	X5Σ–	2.084	1.90 a		612		3.95
	MRCI+Q	[23]				1.787 a
	Pauling method	[24]				2.8 a
	R2PI spectroscopy	[25]				1.72(6) a
TiB	icMRCISD/cc-pV5Z	[21]	X6Δ	2.077	2.723	2.687	587.6	3.5	3.13(3.42)
	icMRCI+Q/cc-pV5Z	[21]		2.080	2.797	2.761	583	3.5	(3.51)
	B3LYP/6-311++G(3df)	[22]	X6Δ	2.039	2.40	2.362	621		3.26
	Pauling method	[24]				<3.1
	R2PI spectroscopy	[25]				1.956(16)
VB	icMRCISD/cc-pV5Z	[21]	X7Σ+	2.039	2.268	2.231	589.9	3.75	2.73(3.20)
	icMRCI+Q/cc-pV5Z	[21]		2.043	2.381	2.344	585	3.8	(3.30)
	B3LYP/6-311++G(3df)	[22]	X7Σ+	2.011	2.17	2.132 a	615		2.82
	R2PI spectroscopy	[25]				2.150(16) a
CrB	icMRCISD/cc-pV5Z	[21]	X6Σ+	2.166	1.141		420	18.5	2.05
	icMRCI+Q/cc-pV5Z	[21]	X6Σ+	2.183	1.353		405	20	1.43
	B3LYP/6-311++G(3df)	[22]	X6Σ+	2.187			415		2.41
MnB	icMRCISD/cc-pV5Z	[21]	X5Π	2.190	0.846	0.821	391.9	3.79	2.21(2.45)
	icMRCI+Q/cc-pV5Z	[21]		2.183	0.854	0.831	383	3.9	(2.39)
	B3LYP/6-311++G(3df)	[22]		1.828	1.27	1.232	621		2.47
FeB	R2PI spectroscopy	[26]	GS			2.43(2)
	B3LYP/aug-6-311++G(3df)	[22]	Χ4Σ−	1.695	2.22		743		2.27
	icMRCISD/cc-pV5Z	[21]	Χ4Σ−	1.743	2.303		642.7	12.50	1.67(2.13)
	icMRCI+Q/cc-pV5Z	[21]	Χ4Σ−	1.747	2.346		645	13.20	(2.20)
CoB	R2PI spectroscopy	[26]	GS			2.954(3)
	LIF spectroscopy	[27]	Χ3Δ3	1.705
	B3LYP/aug-6-311++G(3df)	[22]	Χ3Δ	1.676	2.62		783		1.89
	icMRCISD/cc-pV5Z	[21]	Χ3Δ	1.696	2.736		756.9	6.14	1.03(1.83)
	icMRCI+Q/cc-pV5Z	[21]	Χ3Δ	1.700	2.849		757	6.10	(1.98)
NiB	R2PI spectroscopy	[26]	GS			3.431(4)
	LIF spectroscopy	[28]	Χ2Σ+	1.698			778	4.90
	B3LYP/aug-6-311++G(3df)	[22]	Χ2Σ+	1.676	2.83		793		1.66
	icMRCISD/cc-pV5Z	[21]	Χ2Σ+	1.676	3.239		805.1	3.93	0.80(2.16)
	icMRCI+Q/cc-pV5Z	[21]	Χ2Σ+	1.681	3.434		803	3.98	(2.41)
CuB	icMRCISD/cc-pV5Z	[21]	Χ1Σ+	1.934	1.839		496.9	5.54	1.21(1.71)
	icMRCI+Q/cc-pV5Z	[21]	Χ1Σ+	1.922	2.129		553	4.80	(1.62)
	Nonrelativistic CASPT2/PolMe	[29]	Χ1Σ+	1.910	2.806		518	4.25
	No-pair DK CASPT2/NpPolMe	[29]	Χ1Σ+	1.865	2.354		563	4.34
	No-pair DK CCSD(T)-20/NpPolMe	[29]	Χ1Σ+	1.909	1.522		555.0	4.33
	R2PI spectroscopy	[30]	Χ1Σ+			2.26(15)
	B3LYP/aug-6-311++G(3df)	[22]	Χ1Σ+	1.952	2.12		513		1.61
ZnB	B3LYP/aug-6-311++G(3df)	[22]	Χ2Π	2.274	0.370		286		1.70
	B3LYP/aug-cc-pVQZ(-PP)	b	Χ2Π	2.258	0.373	0.329	282.5		1.65
	TPSSh/aug-cc-pVQZ(-PP)	b	Χ2Π	2.217	0.573	0.529	322.7		1.84
	MN15/aug-cc-pVQZ(-PP)	b	Χ2Π	2.330	0.374	0.317	234.1		1.32

^a Dissociation energy with respect to the atomic ground state products. ^b This work.

[21,22,23,24,25,26,27,28,29,30]. The first study of each first-row-transition-metal boride (MB) was carried out by Wu in 2005, who studied the MB molecules via DFT methodology, i.e., B3LYP/6-311++G(3df) [22]. In 2008, Tzeli and Mavridis, using multireference methods (MRCI), systematically studied the electronic structure and bonding of the ground and some low-lying states, up to twenty-four excited states, of all first-row-transition-metal borides (MBs). They used multireference methods employing correlation-consistent basis sets of quintuple cardinality (cc-pV5Z) [21]. Full potential-energy curves were constructed at the MRCI/cc-pV5Z level for the lowest up to five states, while about twenty states for every MB species were examined at the MRCI/cc-pVQZ_BANO-4Z_M level of theory. At the MRCI/cc-pV5Z level, total energies, dissociation energies, dipole moments, and common spectroscopic parameters of the nine diatomic borides, MBs, M = Sc–Cu, were reported. Ground states of the MBs along with “recommended” bond distances, dissociation energies, and dipole moments were calculated, and boron atoms’ exceptional ability to participate in a variety of bonding schemes was stressed; the bonding in these MB series varied from three half bonds to full triple bonds. Furthermore, the core correlation using the cc-pwCV5Z basis set was calculated for specific molecules as well as the Scalar relativistic effects through the second-order Douglas–Kroll–Hess approximation, i.e., at C-MRCI+DKH2/cc-pwCV5Z-DK. This study [21], using very accurate methodology, calculated bond distances, dissociation energies, dipole moments, and spectroscopic parameters in excellent agreement with experimental studies which were subsequently conducted the following decade; see Table 1 and discussion below.

ScB: In 2008, the electronic structure and bonding of the ground and some low-lying states of ScB were calculated employing MRCI methodology, including the scalar relativistic effects and the correlation of the core electrons [21]. The ground state, X⁵Σ^–, was calculated at the C-MRCISD+DKH+Q/cc-pV5Z level of theory, and a value of 2.094 Å was found for the bond length, 3.309 eV was found for the dissociation energy with respect to the adiabatic atomic products, and 603 cm⁻¹ was found for the vibrational frequency. In 2021, Merriles et al. performed resonant two-photon ionization spectroscopy (R2PI) experiments to measure the predissociation threshold in ScB and found a value of D₀ (ScB) = 1.72(6) eV with respect to the ground state products [25], while the theoretical dissociation energy with respect to the atomic ground state was 1.74 eV [21], in excellent agreement with the experimental value of 1.72(6) eV [25]. The bonding in the ground state consists of three half bonds [21].

TiB: The ground state of TiB is the X⁶Δ state. Its bond distance was calculated to be 2.080 Å with a dissociation energy of 2.797 eV and a vibrational frequency of 583 cm⁻¹ using icMRCISD/cc-pV5Z [21]. Recently, Merriles et al. measured the predissociation threshold in TiB to be D₀ (TiB) = 1.956(16) eV via R2PI spectroscopy [25]. Finally, in the ground state, three half bonds are formed.

VB: The its ground state of VB, X⁷Σ⁺, presents similar bonding behavior to the ground state of the ScB and TiB molecules, i.e., three half bonds are formed [21]. Its bond distance is 2.043 Å, with a dissociation energy, D_e, of 2.381 eV with respect to the adiabatic products, and 2.143 eV with respect to the atomic ground state products, while the vibrational frequency, ω_e, is 589.9 cm⁻¹ [21]. The R2PI predissociation threshold in VB, D₀, was measured to be 2.150(16) eV [25], in excellent agreement with the theoretical value.

CrB: The bonding of the CrB ground state, X⁶Σ⁺, is a σ² bond and two half π bonds, i.e., σ²π¹π¹ [21]. Using the icMRCISD+Q/cc-pV5Z methodology, the Cr-B distance was calculated to be 2.183 Å with a binding energy of 1.353 eV, a rather small dissociation energy, while the ω_e value was 405 cm⁻¹. The B3LYP bond distance, 2.187 Å [22], is in very good agreement with the corresponding value obtained using the icMRCISD+Q/cc-pV5Z methodology [21].

MnB: The relative ordering of twenty-six states of MnB has been calculated [21]. The ground state is the X⁵Π state, with a value of 2.183 Å for the bond length, 0.854 eV for the dissociation energy, and 391.9 cm⁻¹ for the vibrational frequency. The bonding is σ²π¹π¹; however, the dissociation energy is very small. On the contrary, the first excited state, A⁵Σ⁻, which lies 0.09 eV above the X state, has three bonds, σ²π²π², a short bond length of 1.832 Å, and a significant larger dissociation energy of 3.131 eV than the X state obtained using the icMRCISD+Q/cc-pV5Z methodology [21]. Note that in the A⁵Σ⁻ state, Mn is excited at its ⁵D atomic state.

FeB: In 2005, the ground state, ⁴Σ⁻, of the FeB molecule was calculated at the DFT(B3LYP)/6-311++G(3df) level [22]. Then, in 2008, nineteen electronic states were calculated using MRCI, while the electronic structure and chemical bonding of the ground and the first excited states were examined [21]. At the icMRCISD+Q/cc-pV5Z level of theory, the X⁴Σ⁻ ground state values are r_e = 1.743 Å and D_e = 2.346 eV. In 2019, Merriles et al. measured the predissociation threshold in FeB via R2PI spectroscopy and found a value of D₀ = 2.43(2) eV [26]. This work included the first experimental measurement of the BDE of FeB.

CoB: The ground state of CoB, X³Δ, was calculated in 2005 via the DFT methodology [26], and in 2008, it was calculated using icMRCISD+Q/cc-pV5Z [21]. Eighteen states were calculated. In the ground state, a triple bond is formed with a bond distance of 1.700 Å and a binding energy of 2.849 eV. In 2011, Ng et al. observed and analyzed the electronic transition spectrum of CoB in the visible region between 495 and 560 nm using laser-induced fluorescence spectroscopy [27]. The ground state of CoB was identified to be X³Δ₃ with r_e = 1.705 Å. The ground state’s identity was reconfirmed to be X³Δ₃ by Dore et al. [31]. In 2019, Merriles et al. performed resonant two-photon ionization spectroscopy experiments to measure the predissociation threshold of CoB, obtaining a value of D₀ = 2.954(3) eV [26].

NiB: The X²Σ⁺ state of NiB is the ground state, presenting a triple bond. It was calculated via DFT methodology in 2005 [22] and the MRCI method in 2008 [21]. Via MRCI, twenty states were studied. At the icMRCISD+Q/cc-pV5Z level, its bond length was calculated to be 1.681 Å with a binding energy of 3.434 eV and ω_e = 803 cm⁻¹. Also in 2008, Balfour et al. characterized NiB spectroscopically using laser-induced fluorescence spectroscopy [28]. The ground state of NiB was identified to be X²Σ⁺ with an electronic configuration of 1σ²2σ²1π⁴1δ⁴3σ¹, r_e = 1.698 Å, ω_e = 778.0 cm⁻¹, and ω_ex_e = 4.90 cm⁻¹ [28]. In 2010, Zhen et al. investigated NiB using LIF spectroscopy to resolve the rotational structure of a band belonging to a newly discovered band system with a ²Π_3/2 upper state [32]. In 2015, Goudreau et al. [33] investigated the 0-0, 2-0, and 3-0 bands of NiB belonging to the ²Π^3/2 ← X²Σ⁺ band system assigned by Zhen et al. [32] at high resolution. The fine structure splitting in each state was determined for the first time, confirming the assignment of the ground state as ²Σ⁺ with an electronic configuration of 1σ²2σ²1π⁴1δ⁴3σ¹. Finally, in 2019, Merriles et al., via R2PI spectroscopy, measured the predissociation threshold in NiB to be D_{0 =} 3.431(4) eV [26].

CuB: In 1997, Barysz and Urban investigated the spectroscopic constants and dipole moment curves of the ground states, Χ¹Σ⁺, of the coinage metal diatomic molecules with boron, i.e., BCu, using high-level-correlated methods combined with quasi-relativistic Douglas–Kroll (no-pair) spin-averaged approximation [29]. At the CCSD(T)/[9s7p3d2f/_Cu5s3p2d/_B] computational level, the values r_e = 1.909 Å, D_e = 1.522 eV, and ω_e = 555.0 cm⁻¹ were found. In 2005, Wu also studied the X state via DFT [22], while in 2008, Tzeli and Mavridis investigated nineteen states using the MRCI+Q methodology [21]. At the icMRCISD+Q/cc-pV5Z level of theory, they found values of r_e = 1.922 Å, D_e = 2.129 eV, and ω_e = 553 cm⁻¹. The D_e value at the icMRCISD/cc-pV5Z level was significantly larger than the corresponding values at the CCSD(T)/[9s7p3d2f/_Cu5s3p2d/_B] level due to its significantly larger basis set. In 2023, Merriles and Morse studied CuB experimentally for the first time via R2PI spectroscopy and obtained the first BDE measurement for this molecule. They found that CuB remains bound at energies that far surpass its bond dissociation energy (BDE), and bonds break only when excited at or above an excited sharp predissociation threshold (SAL). Nevertheless, a BDE value of D₀ = 2.26(15) eV was derived [30], which was in very good agreement with the calculated value of 2.129 eV using icMRCISD+Q/cc-pV5Z [21].

ZnB: Only one theoretical study was found for ZnB. Its ground state, Χ²Π, was calculated via the DFT methodology, B3LYP/6-311++G(3df) [22], obtaining a value of r_e = 2.274 Å with a very small binding energy of 0.370 eV. Here, we calculated the X state at the B3LYP, TPSSh, and MN15/aug-cc-pVQZ(-PP) levels of theory. Both B3LYP and TPSSh provided the same D_e values, i.e., 0.373 eV, while TPSSh overestimated it. Here, we found that the B3LYP/aug-cc-pVQZ(-PP) methodology is in very good agreement with the available experimental data on MBs compared with the other functionals. Thus, we consider it to be our best DFT methodology. The bond distance was found to be 2.258 Å and the dipole moment was found at 1.65 D.

2.2. Second-Row-Transition-Metal MBs

All previous theoretical and experimental data on the ground states of the diatomic metal borides including the second-row transition metals are summarized in

Table 2. Previous theoretical and experimental data on the ground states of the 2nd-row-transition-metal boride molecules, MBs (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd): bond lengths r_e (Å), dissociation energies D_e (eV) and/or D₀ (eV), vibrational frequencies ω_e (cm⁻¹), anharmonic corrections ω_ex_e (cm⁻¹), and dipole moments μ (μ_FF = δE/δε) (Debye).

MB	Methodology	Ref.	State	re	De a	D0	ωe	ωexe	μ (μFF)
YB	R2PI spectroscopy	[25]				2.057(3)
	DFT: B97-1/AVTZ-PPY/VTZB	[25]	X5Σ−	2.306	(1.99)	1.96	517
	DFT: B3LYP/LANL2DZ	[34]	S = 2	2.254	2.17		582.4		4.65
	Pauling method	[24]				2.99
ZrB	R2PI spectroscopy	[25]				2.573(5)
	DFT: B97-1/AVTZ-PPZr/VTZB	[25]	X6Δ	2.159	(2.65)	2.61
	DFT: B3LYP/LANL2DZ	[34]	S = 2.5	2.189	3.92		610.2		3.48
NbB	R2PI spectroscopy	[25]				2.989(12)
	DFT: B97-1/AVTZ-PPNb/VTZB	[25]	5Π/5Φ	1.988	(3.11)	3.07	698
	DFT: B3LYP/LANL2DZ	[34]	S = 1	1.996	3.40		662.7		3.84
	MRCISD+Q/aug-cc-pVQZ(-PP)	b	X5Π	2.018	2.901		708.9		3.10
	MRCISD+Q/aug-cc-pVQZ(-PP)	b	A5Φ	2.019	2.808		710.3		2.92
MoB	CASPT2/CASSCF/ANO-RCC-4ζ	[35]	X6Π	1.968	2.18		664		2.7
	DFT: B3LYP/LANL2DZ	[34]	S = 0.5	1.817	6.40		826		4.05
TcB	DFT: B97-1/(A)TcVTZB-(PP)Tc	[25]	X5Σ−			3.31
RuB	R2PI spectroscopy	[26]				4.815(3)
	Knudsen effusion	[36]	X2Σ	1.75		4.59(22)
	LIF spectroscopy	[37]	2Δ5/2	1.7099
	DFT: B3LYP/LANL2DZ	[34]	S = 0.5	1.761	6.48		910.8		3.49
RhB	R2PI spectroscopy	[26]				5.252(3)
	Knudsen effusion	[36]	X1Σ	1.75		4.89(22)	915
	LIF spectroscopy	[38]	X1Σ+	1.691(2)
	MS-CASPT2/ANO-RCC-4ζ	[39]	X1Σ+	1.698					4.42
	LIF spectroscopy	[40]	X1Σ+	1.691(2)
	MS-CASPT2/ANO-RCC-4ζ	[41]	X1Σ+	1.694	(5.7)	5.6	924		4.54
	MRCISD+Q/AV5Z-PPRh AV5ZB	[4]	X1Σ+	1.6873	5.473	5.414	938.3	4.32	(3.160)
	RCCSD(T)/AV5Z-PPRh AV5ZB	[4]	X1Σ+	1.6872	5.490	5.431	942.1	3.78	(2.865)
	ADF/PBE/TZ2P	[2]	X1Σ+		5.27
	DFT: TPSSh/AVQZ-PPRh AVQZB	[2]	X1Σ+	1.685
	CCSD(T)/AVQZ-PPRh AVQZB	[2]	X1Σ+	1.689
	MRCI/AVQZ-PPRh AVQZB	[2]	X1Σ+	1.687
	MCSCF/Sapporo-(DKH)Rh-TZP	[42]	X1Σ+	1.701	5.165
	DFT: B3LYP/LANL2DZ	[34]	S = 0	1.745	4.96		932.7		2.84
PdB	ESR spectroscopy	[43]	X2Σ
	UHF/STO-3G*	[43]		1.608
	LIF spectroscopy	[44]	X2Σ+	1.7278			650
	Knudsen effusion	[36]	X2Σ	2.00		3.37(22)
	DFT: B3LYP/LANL2DZ	[34]	S = 0.5	1.856	3.33		725.6		1.44
AgB	Nonrelativistic CASPT2/PolMe	[29]	X1Σ+	2.258	1.248	1.23	341	2.32	?
	No-pair DK CASPT2/NpPolMe	[29]	X1Σ+	2.098	1.684	1.66	425	3.41	?
	No-pair DK CCSD(T)-20/NpPolMe	[29]	X1Σ+	2.115	0.910	0.883	440	3.26	?
	DFT: B3LYP/LANL2DZ	[34]	S = 0	2.187	1.60		415.6		1.41
CdB	DFT: B3LYP/LANL2DZ	[34]	S = 0.5	2.668	0.22		198.3		1.67

^a Dissociation energy with respect to the atomic ground state products. ^b This work.

. Below, they are discussed in detail. There is no experimental or theoretical study on TcB except a D₀ value for the ⁵Σ⁻, obtained via DFT(B97-1). Note that Tc is a synthetic element, and all its isotopes are radioactive. In this paper, we fill this gap by studying the TcB molecule.

YB: The first report on YB was in 1969 by K. A. Gingerich [24], who estimated the dissociation energy of the molecule to be 2.99 eV using the Pauling method of electronegativities [24,25]. In 2009, Kharat et al. calculated the ground spin states of the second-row (4d) transition metals (except for Tc) and their cationic and anionic counterparts at the DFT(B3LYP)/LANL2DZ level. They calculated the bond distances, r_e; binding energies, D_e; electron affinities (EA); ionization potentials (IP); vibrational frequencies, ω_e; and dipole moments, μ [34]. For the diatomic YB, a quintet (S = 2) ground spin state was established. The report lacks details about its electronic configuration, and as such, its ground spin state electronic symmetry is not included. The associated bond distance was found to be 2.254 Å, the binding energy was 2.17 eV, the EA was 0.69 eV, the IP was 6.16 eV, ω_e = 582.4 cm⁻¹, and μ = 4.65 D. The most recent investigation into the electronic structure of the YB dimer was made in 2021 by Merriles et al. [25]. They measured the predissociation thresholds of several early-transition-metal boride diatomics using resonant two-photon ionization (R2PI) spectroscopy. For the YB molecule, a D₀ value of 2.057(3) eV was obtained. Additionally, they provided an insight into the chemical bonding and electronic structures of those same species by performing quantum chemical calculations using the DFT (B97-1) methodology. Computational results showed excellent agreement with the measurements for YB, and its ground state was computed to be the wavefunction X⁵Σ⁻ (with a dominating 1σ²2σ¹3σ¹1π² configuration determinant), resulting in a bond distance r_e of 2.306 Å, a dissociation energy D₀ of 1.96 eV, and ω_e = 517 cm⁻¹.

ZrB: This molecule has only been studied together with other similar species, once in 2009 by Kharat et al. [34] and in 2021 by Merriles et al. [25]. In the first study, a sextet (S = 5/2) ground spin state was determined, with r_e = 2.189 Å, D_e =3.92 eV, EA = 0.48 eV, IP = 7.03 eV, ω_e = 582.4 cm⁻¹, and μ = 3.48 D. In the latter, the predissociation threshold revealed a D₀ value of 2.573(5) eV, and the ground electronic spin state corresponded to the X⁶Δ symmetry wavefunction (with a dominating 1σ²2σ¹3σ¹1π²1δ¹ configuration determinant), resulting in a 2.159 Å bond distance, a 2.61 eV dissociation energy, D₀, and an ω_e value of 610 cm⁻¹.

NbB: Similarly to the previous molecule, there are two studies on NbB [25,34]. In the first study [34], a triplet (S = 1) ground spin state was found, with a bond distance of 1.996 Å, a binding energy of 3.40 eV, EA = 1.05 eV, IP = 7.03 eV, ω_e = 662.7 cm⁻¹, and μ = 3.84 D. In the latter [25], the predissociation threshold revealed a D₀ value of 2.989(12) eV, and the computed ground spin state corresponded to the superposition X⁵Π/⁵Φ (with a 1σ²2σ¹3σ¹1π³1δ¹ configuration determinant), due to the calculations using real forms of the 1π and 1δ orbitals, providing a bond distance of 1.988 Å, a dissociation energy of 3.07 eV, and a vibrational frequency of 698 cm⁻¹. Here, we carried out MRCISD(+Q)/aug-cc-PVQZ(-PP) calculations, and we clarified that the X state is the X⁵Π state, while the A⁵Φ is located 0.084(0.093) eV above the X state; see discussion below.

MoB: In the first report of MoB [34], it was claimed that the ground state is doublet (S = 1/2), with a 1.817 Å bond distance, a 6.40 eV binding energy, EA = 0.21 eV, IP = 8.68 eV, ω_e = 826 cm⁻¹, and μ = 4.05 D. In 2011, Borin and Gobbo [35] investigated the electronic structure of the Χ state and the low-lying electronic states of MoB and its cationic MoB⁺ counterpart by employing quantum computational CASSCF protocols on the CASPT2+DKH/4ζ-8s7p5d3f2g-ANO-RCC_Mo/4ζ-5s4p3d2f_B level. MoB’s ground spin state was computed to be the wavefunction X⁶Π (with a dominant 1σ²2σ¹3σ¹1δ²1π³ configuration determinant) with a bond distance, r_e, of 1.968 Å, a vibrational frequency, ω_e, of 664 cm⁻¹, and a dipole moment, μ, of 2.7 D. The binding energy was also determined to be 2.18 eV.

TcB: TcB is the least studied molecule. Its only study resulted in a D₀ value calculated to be 3.31 eV [25] via the B97-1/(aug)-cc-pVTZ-(PP) methodology for the X⁵Σ⁻ state. Tc is the lightest element, and all its isotopes are radioactive. In this paper, we fill this research gap, and we study three states of the TcB molecule. The main spectroscopic data are provided; see discussion below and tables of Section 3 below. Here, we found that the ground state is an X³Σ⁻ state which presents a quadruple bond; see discussion below.

RuB: This molecule was studied experimentally via mass spectrometry for the first time by Auwera-Mahieu et al. in 1970 [36]. Via the Knudsen effusion method, they determined its dissociation energy, D₀, to be 4.59(22) eV. The ω_e value of 915 cm⁻¹ was estimated from the values of the corresponding carbides using the D₀(A)/D₀(B) = μ_Aω_A²/μ_Bω_B² equation, where μ is the reduced mass. The internuclear distance was estimated from the values of the carbides using the formula r_MB = r_MC + ½ (r_B₂ − r_C₂), resulting in a value of 1.75 Å. Finally, they proposed that the X state is a ²Σ state. It took nearly four decades for this molecule to be inspected again, and in the 2009 work of Kharat et al. [34], a doublet (S = ½) ground spin state was reported for the diatomic RuB, with a bond distance of 1.761 Å, a binding energy of 6.48 eV, an EA of 0.35 eV, an IP of 9.06 eV, an ω_e of 910.8 cm⁻¹, and a dipole moment of 3.49 D. In 2012, Wang et al. [37] studied the laser-induced fluorescence spectrum of RuB in the visible region between 500 nm and 575 nm. They determined that the ground state symmetry is X²Δ, consistent with an electronic configuration obtained using molecular orbital energy level diagrams, while the bond length, r₀, is 1.7099 Å. In 2019, Merriles et al. [26] used R2PI spectroscopy and accurately assigned the bond dissociation energies of the diatomic late-transition-metal monoborides from the measurement of a predissociation threshold. The measured predissociation threshold resulted in D₀ = 4.815(3) eV. Continuing their work, in 2022, Merriles et al. [45] investigated the ionization energies and the cationic dissociation energies of the diatomic second- and third-row-late-transition-metal borides they had previously examined. Resonant two-photon ionization spectroscopy was employed, and the ionization threshold of RuB was measured to be 7.879(9) eV. Regarding the ground state, it was characterized as X²Σ via Knudsen effusion [36] and ²Δ_5/2 via LIF spectroscopy. Here, we clarify that the X²Δ state is the ground state, and that it presents a quadruple bonding; see discussion in the next section.

RhB: In 1970, Auwera-Mahieu et al. [36], via the Knusden effusion method, yielded a value of 4.89(22) eV for the D₀, dissociation energy, of RhB. The vibrational frequency, the bond distance, and the ground state were proposed to be 915 cm⁻¹, 1.75 Å, and ¹Σ, respectively. In 2006, Chowdhury and Balfour [38] measured the gas phase electronic spectrum of RhB in the visible region; it was elucidated that the ground electronic state is of X¹Σ⁺ symmetry, with an internuclear distance of 1.691(2) Å. The following year, in 2007, Gobbo and Borin [39] studied the low-lying ¹Σ⁺ states of RhB at the CASSCF/MS-CASPT2/4ζ-ANO-RCC_Rh/14s9p4d3f_B level to answer some questions raised by the previous experiments. In agreement with the experiment, their results indicated that the ground electronic state corresponded to the X¹Σ⁺ wavefunction (with a dominant 1σ²2σ²1π⁴1δ⁴ configuration determinant) with an internuclear distance, r₀, of 1.698 Å. In the same year, another study by Chowdhury and Balfour [40] resumed their previous spectroscopic studies, with a clear focus on the intrinsic details of the emerging bands. In 2008, A.C. Borin and J.P. Gobbo [41], in order to gain further insight into the structural and spectroscopic properties of RhB, investigated its first two atomic dissociation channels. The first regarded the adiabatic coupling of the two atoms in their ground atomic states, B(2s²2p;²P) and Rh(4d⁸(³F)5s;⁴F), while in the second, the rhodium atom participated in its first excited atomic electronic state, Rh(4d⁹;²D), at the CASSCF/MS-CASPT2/4ζ-ANO-RCC_Rh/14s9p4d3f_B level of theory. Results showcased that X¹Σ⁺ correlates with the second atomic dissociation limit. The researchers predicted a 5.6 eV dissociation energy, D₀; a 924 cm⁻¹ vibrational frequency, ω_e; and a dipole moment of 4.54 D. The Mulliken population analysis yielded a charge of +0.35e on Rh. In the 2009 work of Kharat et al. [34], a singlet (S = 0) ground state was reported for RhB, with a 1.745 Å bond distance, a 6.48 eV binding energy, EA = 0.85 eV, IP = 8.19 eV, and μ = 2.84 D. Later, in the 2019 work by Merriles et al. [26], a predissociation threshold of D₀ = 5.252(3) eV was measured with the use of R2PI spectroscopy. In 2020, two works concerning the bonding structure of RhB were published, where the formation of a quadruple bond was reported. The first study conducted by Cheung et al. [2] explored the bonding nature of RhB(BO)⁻ and RhB species. With the use of PES, an electronic fingerprint was obtained, and the electron affinity of the dimer was measured experimentally to be 0.961 eV. In all computational levels, ADF, DFT, CCSD(T), it was found that the electronic ground state corresponds to an X¹Σ⁺ wavefunction (with a dominant 1σ²1π⁴2σ²1δ⁴ configuration determinant), resulting in a quadruple bond consisting of two π bonds formed between the Rh 4d_xz/4d_yz and B 2p_x/2p_y orbitals and two σ bonds between the Rh 4d_z² and B 2s/2p_z orbitals, followed by internuclear distances ranging from 1.685 Å to 1.689 Å. At the ADF/PBE/TZ2P level, it was also possible to obtain a 5.27 eV value for the dissociation energy. The second study, carried out by Tzeli and Karapetsas [4], investigated the bond occurring inside isoelectronic species between transition metals and the main group elements TcN, RuC, RhB and PdBe. For the RhB molecule, at various levels of theory, i.e., MRCISD, MRCISD+Q, and RCCSD(T)/aug-cc-pV5Z-PP_Rh aug-cc-pV5Z_B, the common spectroscopic constants were computed, presenting great agreement among themselves, as well as with the experimental values. Specifically, values of r_e =1.6872 Å, D_e = 5.490 eV, and ω_e = 942.1 cm⁻¹ were found, along with anharmonic corrections of ω_ex_e =3.78 cm⁻¹ and μ = 2.965 D [4]. It was deduced that the ground electronic state (X¹Σ⁺) has a dominant 1σ²2σ²1π⁴1δ⁴ configuration determinant and correlates adiabatically to the atomic electronic spin states B(X²P;2s²2p) + Rh(a²D;4d⁹), resulting in a four-fold quadruple bond. Tzeli and Karapetsas [4] found that, except for the ground state of RhB, its two lowest excited states, i.e., a³Δ and A¹Δ, also present quadruple bonds [4]. Additionally, in the ground and the first excited states of the RhB⁻ anion, Χ²Σ⁺ and A²Δ quadruple bonds are also formed [5]. In 2021, Schoendorff et al. [42] also studied the bonding in RhB both qualitatively and quantitively at a MCSCF-IOTC/DKH-TZP-2012_Rh/Sapporo-TZP-2012_B level of theory and reached the same results as those in previous works. They concluded that the ground electronic spin state corresponded to the X¹Σ⁺ symmetry wavefunction (with a dominant 1σ²2σ²1π⁴1δ⁴ configuration determinant), with an equilibrium bond length of 1.701 Å and a binding energy of 5.165 eV. Finally, in 2022, Merriles and Morse [45] measured the ionization potential and obtained a value of 8.234(10) eV.

PdB: Via the Knusden effusion method, the dissociation energy, D₀, of PdB was measured to be 4.89(22) eV, with ω = 650 cm⁻¹ and a bond distance of 2.00 Å, while the proposed ground state was ²Σ [36]. In 1992, Knight Jr. et al. [43], via electron spin resonance (ESR) spectroscopy, revealed that the ground electronic state of the dimer is X²Σ. Unrestricted Hartree–Fock (UHF) calculations were also carried out and a 1.608 Å internuclear distance was deduced. In 2009, a DFT study [34] predicted a doublet (S = ½) ground state with a bond distance of 1.856 Å and a binding energy of 3.33 eV. The most recent study on PdB was published in 2012 by Ng et al. [44], marking its first electronic spectroscopic investigation using laser-induced fluorescence spectroscopy in the visible region between 465 and 520 nm. An X²Σ⁺ ground state was revealed, with r₀ = 1.7278 Å. Moreover, a molecular-orbital-energy-level diagram was designed to understand the observed ground state, and the proposed configuration was determined to be 1σ²2σ²1π⁴1δ⁴3σ¹.

AgB: In 1997, Barysz and Urban [29] investigated the spectroscopic constants of AgB at many levels of theory and the plotted dipole moment curves of AgB using high-level-correlated methods combined with quasi-relativistic Douglas–Kroll (no-pair) spin-averaged approximation. The obtained ground state was X¹Σ⁺ in all of them. At the DK-CCSD(T)-20/NpPolMe level of theory, a bond length of 2.115 Å and a dissociation energy of 0.910 eV were obtained, while the corresponding values at the DK-CASPT2/NpPolMe level were 2.098 Å and 1.684 eV. It was highly advised that both relativistic as well as correlation effects be carefully considered to obtain accurate results. Note that due to the relativistic shrinkage of s valence shell electrons, stronger bonds are formed which would, otherwise, not be described successfully. Finally, Kharat et al. [34] predicted via DFT a singlet (S = 0) ground state with a 2.187 Å bond distance and a 1.60 eV binding energy.

CdB: There is only one DFT study in the literature. Kharat et al. [34] reported a doublet (S = ½) ground state with a 2.668 Å bond distance, a 0.22 eV binding energy, an EA of 0.09 eV, an IP of 7.05 eV, ω_e = 198.3 cm⁻¹, and μ = 1.67 D.

2.3. Third-Row-Transition-Metal MBs

Table 3 summarizes the previous experimental and theoretical data obtained for LaB, HfB, TaB, WB, ReB, OsB, IrB, PtB, AuB, and HgB. The calculations are mainly DFT apart from those of HfB [46], PtB [47], and AuB [29,47], for which CCSD(T), CASPT2, and MRCI calculations were carried out.

LaB: In 1969, K. A. Gingerich [24] estimated the binding energy at 3.51 eV. In 2010, Kalamse et al. [48] studied the 5d-metal mononitrides and monoborides using DFT methodology with the B3LYP functional set and both LANL2DZ and SDD basis sets. The lowest electronic spin states at these two DFT levels of theory, as well as EA, IP, binding energies, and electronic configurations of the MBs, were discussed, while the orbitals involved in bond formation were identified. At the B3LYP/LANL2DZ level, it was deduced that the most stable state for LaB is X⁵Σ^–, with a bond length of 2.435 Å, a binding energy of 2.49 eV, μ = 4.29 D, EA = 1.01 eV, and IP = 5.61 eV. On the other hand, for the B3LYP/SDD level, the most stable state was found to be X³Π, with r_e = 2.336 Å and D_e = 2.49 eV. In 2018, Elkahwagy et al. [49] studied LaB and its anionic LaB⁻ counterpart with the diffusion Monte Carlo method in combination with three different trial functions to calculate the potential energy curves for the lowest electronic states of those species, along with some spectroscopic constants of neutral LaB. Irrespectively of the functional used, it was found that the quintet state of LaB is the ground state, while the triplet state is higher in energy, elucidating the mystery that the previous work arose. The 2021 study by Merriles et al. [25] also suggested that the ground spin state corresponded to an X⁵Σ⁻ symmetry wavefunction (with a dominant 2σ¹3σ¹1π² configuration determinant) with a dissociation energy, D₀, of 2.54 eV; a bond length, r_e, of 2.372 Å; and a vibrational frequency, ω_e, of 521 cm⁻¹. The experimental part of their study yielded a 2.086(18) eV value for the dissociation energy from the predissociation threshold. Here, both the lowest triplet and quintet states were calculated, and we found that the ground one is the X⁵Σ⁻ state.

Table 3. Previous theoretical and experimental data on the ground states of the 3rd-row-transition-metal boride molecules, MBs (M = La, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg): bond lengths r_e (Å), dissociation energies D_e (eV) and/or D₀ (eV), electron affinities EA (eV), ionization potentials IP (eV), vibrational frequencies ω_e (cm⁻¹), anharmonic corrections ω_ex_e (cm⁻¹), and dipole moments μ (μ_FF = δE/δε) (Debye).

MB	Methodology	Ref.	State	re	De a	D0	ωe	ωexe	μ (μFF)
LaB	DFT: B3LYP/LANL2DZ	[48]	X5Σ−	2.435	2.49		511		4.29
	DFT/B3LYP/SDD	[48]	X3Π	2.336	2.10		496		5.02
	DMC(B3LYP)/CRENBS-ECPLaBurkatzki-PPB	[49]	S = 2	2.150	3.37				4.22
	DMC(B3PW91)/CRENBS-ECPLa Burk.-PPB	[49]	S = 2	2.145	3.81				4.19
	R2PI spectroscopy	[25]				2.086(18)
	DFT: UB97-1/AVTZ-PPLa VTZB	[25]	X5Σ−	2.372	2.57	2.54	521
	Pauling method	[24]				3.51
HfB	DFT: B3LYP/LANL2DZ	[48]	X4Σ–	2.157	2.70		613		2.44
	DFT/B3LYP/SDD	[48]	X4Σ–	2.195	2.64		580		2.68
	R2PI spectroscopy	[25]				2.593(3)
	DFT: UB97-1/AVTZ-PPLa VTZB	[25]	X4Σ–	2.128	2.64	2.60	584
	c-CCSD(T)/wCV5Z-PPHf AV5ZB	[46]	X4Σ−	2.144	2.841	2.840	607	2.8
	MRCI/c-CCSD(T)/wCVQZ-PPHf AVQZB	[46]	X4Σ−	2.174			610	3.0
	BP86/def2-QZVP	[46]			3.182
	BLYP/def2-QZVP	[46]			2.803
	BPE/def2-QZVP	[46]			3.311
	MN15-L/def2-QZVP	[46]			3.071
	LRC-ωPBEh/def2-QZVP	[46]			2.174
	DSD-PBEB95-D3BJ/def2-QZVP	[46]			1.923
TaB	DFT: B3LYP/LANL2DZ	[48]	X3Σ+	2.001	2.49		721		2.68
	DFT/B3LYP/SDD	[48]	X5Δ	2.184	2.48		555		1.44
	R2PI spectroscopy	[25]				2.700(3)
	DFT: UB97-1/AVTZ-PPTa VTZB	[25]	X5Δ	2.085	3.00	2.95	754
WB	DFT: B3LYP/LANL2DZ	[48]	X6Σ−	2.161	2.74		526		2.61
	DFT/B3LYP/SDD	[48]	X6Σ−	1.990	2.88		725		2.67
	R2PI spectroscopy	[25]				2.730(4)
	DFT: UB97-1/AVTZ-PPW VTZB	[25]	X6Σ+	1.891	2.94	2.89	730
ReB	DFT: B3LYP/LANL2DZ	[48]	X3Σ−	1.842	2.77		867		2.99
	DFT: B3LYP/SDD	[48]	X5Σ−	1.875	4.18		853		2.29
OsB	R2PI spectroscopy	[26]	GS			4.378(3)
	B3LYP/aug-cc-pVQZ-PP	[45]	X4Σ−	1.770			938		2.24
	B3LYP/LANL2DZ	[48]	Χ4Δ	1.813	3.99		955		2.17
IrB	LIF spectroscopy	[50]	Χ3Δ3	1.7675
	B2PLYP/AVQZ(-PP)M	[47]	Χ3Δ	1.763
	CCSD(T)/AVQZ(-PP)	[47]	Χ3Δ			5.085
	R2PI spectroscopy	[26]	GS			4.928(10)
	Mass spectrometry	[36]	GS			5.27(18)
	B3LYP/LANL2DZ	[48]	Χ3Δ	1.806	4.86		936		1.72
PtB	R2PI spectroscopy	[26]	GS			5.235(3)
	Mass spectrometry	[51]	GS			4.91(17)
	LIF spectroscopy	[52]	Χ2Σ+	1.741			903.6
	B3LYP/LANL2DZ	[48]	Χ2Σ+	1.809	5.43		906		1.18
	B2PLYP/AVQZ(-PP)	[47]	Χ2Σ+	1.755
	CCSD(T)/AVQZ(-PP)	[47]	Χ2Σ+			5.668
AuB	Mass spectrometry	[36]	GS			3.50(16)
	Knudsen cell experiment	[53]	GS			3.773
	B3LYP/LANL2DZ	[48]	Χ1Σ+	1.997	2.96		559		0.68
	B2PLYP/AVQZ(-PP)	[47]	Χ1Σ+	1.906			710
	CCSD(T)/AVQZ(-PP)	[47]	Χ1Σ+			3.734
	Nonrelativistic CASPT2/PolMe	[29]	Χ1Σ+	2.256	1.261		362	3.06
	No-pair DK CASPT2/NpPolMe	[29]	Χ1Σ+	1.931	3.519		676	4.76
	No-pair DK CCSD(T)-20/NpPolMe	[29]	Χ1Σ+	1.960	2.709		663	4.01
	R2PI spectroscopy	[30]	Χ1Σ+			3.724(3)
HgB	B3LYP/LANL2DZ	[48]	Χ2Σ+	4.381	0.002		19		0.27

^a Dissociation energy with respect to the atomic ground state products.

Table 3. Previous theoretical and experimental data on the ground states of the 3rd-row-transition-metal boride molecules, MBs (M = La, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg): bond lengths r_e (Å), dissociation energies D_e (eV) and/or D₀ (eV), electron affinities EA (eV), ionization potentials IP (eV), vibrational frequencies ω_e (cm⁻¹), anharmonic corrections ω_ex_e (cm⁻¹), and dipole moments μ (μ_FF = δE/δε) (Debye).

MB	Methodology	Ref.	State	re	De a	D0	ωe	ωexe	μ (μFF)
LaB	DFT: B3LYP/LANL2DZ	[48]	X5Σ−	2.435	2.49		511		4.29
	DFT/B3LYP/SDD	[48]	X3Π	2.336	2.10		496		5.02
	DMC(B3LYP)/CRENBS-ECPLaBurkatzki-PPB	[49]	S = 2	2.150	3.37				4.22
	DMC(B3PW91)/CRENBS-ECPLa Burk.-PPB	[49]	S = 2	2.145	3.81				4.19
	R2PI spectroscopy	[25]				2.086(18)
	DFT: UB97-1/AVTZ-PPLa VTZB	[25]	X5Σ−	2.372	2.57	2.54	521
	Pauling method	[24]				3.51
HfB	DFT: B3LYP/LANL2DZ	[48]	X4Σ–	2.157	2.70		613		2.44
	DFT/B3LYP/SDD	[48]	X4Σ–	2.195	2.64		580		2.68
	R2PI spectroscopy	[25]				2.593(3)
	DFT: UB97-1/AVTZ-PPLa VTZB	[25]	X4Σ–	2.128	2.64	2.60	584
	c-CCSD(T)/wCV5Z-PPHf AV5ZB	[46]	X4Σ−	2.144	2.841	2.840	607	2.8
	MRCI/c-CCSD(T)/wCVQZ-PPHf AVQZB	[46]	X4Σ−	2.174			610	3.0
	BP86/def2-QZVP	[46]			3.182
	BLYP/def2-QZVP	[46]			2.803
	BPE/def2-QZVP	[46]			3.311
	MN15-L/def2-QZVP	[46]			3.071
	LRC-ωPBEh/def2-QZVP	[46]			2.174
	DSD-PBEB95-D3BJ/def2-QZVP	[46]			1.923
TaB	DFT: B3LYP/LANL2DZ	[48]	X3Σ+	2.001	2.49		721		2.68
	DFT/B3LYP/SDD	[48]	X5Δ	2.184	2.48		555		1.44
	R2PI spectroscopy	[25]				2.700(3)
	DFT: UB97-1/AVTZ-PPTa VTZB	[25]	X5Δ	2.085	3.00	2.95	754
WB	DFT: B3LYP/LANL2DZ	[48]	X6Σ−	2.161	2.74		526		2.61
	DFT/B3LYP/SDD	[48]	X6Σ−	1.990	2.88		725		2.67
	R2PI spectroscopy	[25]				2.730(4)
	DFT: UB97-1/AVTZ-PPW VTZB	[25]	X6Σ+	1.891	2.94	2.89	730
ReB	DFT: B3LYP/LANL2DZ	[48]	X3Σ−	1.842	2.77		867		2.99
	DFT: B3LYP/SDD	[48]	X5Σ−	1.875	4.18		853		2.29
OsB	R2PI spectroscopy	[26]	GS			4.378(3)
	B3LYP/aug-cc-pVQZ-PP	[45]	X4Σ−	1.770			938		2.24
	B3LYP/LANL2DZ	[48]	Χ4Δ	1.813	3.99		955		2.17
IrB	LIF spectroscopy	[50]	Χ3Δ3	1.7675
	B2PLYP/AVQZ(-PP)M	[47]	Χ3Δ	1.763
	CCSD(T)/AVQZ(-PP)	[47]	Χ3Δ			5.085
	R2PI spectroscopy	[26]	GS			4.928(10)
	Mass spectrometry	[36]	GS			5.27(18)
	B3LYP/LANL2DZ	[48]	Χ3Δ	1.806	4.86		936		1.72
PtB	R2PI spectroscopy	[26]	GS			5.235(3)
	Mass spectrometry	[51]	GS			4.91(17)
	LIF spectroscopy	[52]	Χ2Σ+	1.741			903.6
	B3LYP/LANL2DZ	[48]	Χ2Σ+	1.809	5.43		906		1.18
	B2PLYP/AVQZ(-PP)	[47]	Χ2Σ+	1.755
	CCSD(T)/AVQZ(-PP)	[47]	Χ2Σ+			5.668
AuB	Mass spectrometry	[36]	GS			3.50(16)
	Knudsen cell experiment	[53]	GS			3.773
	B3LYP/LANL2DZ	[48]	Χ1Σ+	1.997	2.96		559		0.68
	B2PLYP/AVQZ(-PP)	[47]	Χ1Σ+	1.906			710
	CCSD(T)/AVQZ(-PP)	[47]	Χ1Σ+			3.734
	Nonrelativistic CASPT2/PolMe	[29]	Χ1Σ+	2.256	1.261		362	3.06
	No-pair DK CASPT2/NpPolMe	[29]	Χ1Σ+	1.931	3.519		676	4.76
	No-pair DK CCSD(T)-20/NpPolMe	[29]	Χ1Σ+	1.960	2.709		663	4.01
	R2PI spectroscopy	[30]	Χ1Σ+			3.724(3)
HgB	B3LYP/LANL2DZ	[48]	Χ2Σ+	4.381	0.002		19		0.27

^a Dissociation energy with respect to the atomic ground state products.

HfB: The B3LYP/SDD methodology [48] was used to predict X⁴Σ^– as the ground state of HfB with a bond length of 2.195 Å, μ = 2.60 D, EA = 1.05 eV, IP = 5.01 eV, and a binding energy of 2.64 eV. The researchers found that, in contrast to the rest of the 5d-metal monoborides, it is the 5d orbital of the metal that loses an electron in the case of HfB. In the 2021 study by Merriles et al. [25], the authors measured the predissociation threshold, i.e., D₀(HfB) = 2.593(3) eV, and the calculated ground spin state corresponded to the X⁴Σ⁻ symmetry wavefunction (with a dominant 2σ¹3σ²1π² configuration determinant) with a bond distance of 2.128 Å, a dissociation energy of 2.60 eV, and a vibrational frequency of 584 cm⁻¹ at UB97-1/AVTZ-PP_LaVTZ_B. In 2022, Ariyarathna et al. [46] conducted a comparative study on the outcomes of several computational methods on the basis of high-level, multi-reference configuration interaction theory and coupled cluster quantum chemical calculations, with large quadruple-ζ and quintuple-ζ quality-correlation-consistent basis sets, as well as DFT, with numerous exchange-correlation functionals that span multiple rungs of “Jacob’s ladder”, through the inspection of HfO and HfB, to investigate their performance on species containing third-row transition metals. Ab initio studies of HfB showed, unanimously, that X⁴Σ^– is the ground electronic state, owing to a 1σ²2σ¹3σ²1π² configuration dominant determinant with a bond length of 2.144 Å, dissociation energy at 2.841 eV, and ω_e = 607 cm⁻¹ at the c-CCSD(T)/wCV5Z-PPHf AV5ZB level. DFT calculations were performed on the closed-shell single-reference ground state to evaluate the errors in the predictions from distinct density functionals. The resulting trends were discussed, plotted, and compared.

TaB: DFT calculations predict different states to be the ground state of TaB depending on the methodology. B3LYP/LANL2DZ predicted X³Σ⁺ to be the ground state, with a 2π⁴5σ¹6σ¹ electronic configuration, r_e = 2.001 Å, and D_e = 2.49 eV. However, at the B3LYP/SDD level, the calculated ground state was X⁵Δ state, with r_e = 2.184 Å, μ = 1.44 D, ΕA = 1.43 eV, ΙP = 7.35 eV, and D_e = 2.48 eV [48]. In 2021, Merriles et al. [25] determined the D₀ value to be 2.700(3) eV, and they found that at the UB97-1/AVTZ-PP_TaVTZ_B level, the ground state corresponded to X⁵Δ (with a dominant 2σ¹3σ²1π²1δ¹ configuration determinant) with r_e = 2.085 Å, D_e = 2.95 eV, and ω_e = 754 cm⁻¹.

WB: In WB, as in TaB, DFT calculations predict different states to be the ground state, depending on the methodology. Kalamse et al. [48], at both the B3LYP/LANL2DZ and SDD levels, found that the ground state is X⁶Σ⁻, where r_e = 1.990 Å, μ = 2.67 D, and D_e = 2.88 eV at the B3LYP/SDD level [50]. In 2021, Merriles et al. [25] measured D₀ = 2.730(4) eV and calculated the ground state to be X⁶Σ⁺, with a dominant 2σ¹3σ²1π²1δ² configuration determinant, r_e = 1.891 Å, D_e = 2.89 eV, and ω_e = 730 cm⁻¹, using the UB97-1/AVTZ-PP_WVTZ_B methodology [25]. Here, we found that the ground state is the X⁶Π state, while the X⁶Σ⁺ state lies 0.137 eV above the X state; see tables of the Section 3 and discussion below.

ReB: There is only a single theoretical study on the ReB molecule at a DFT level; however, different basis sets predict different states to be the ground state. B3LYP/LANL2DZ predicted the ground state to be X³Σ⁻, with r_e = 1.842 Å and D_e = 2.77 eV, while B3LYP/SDD predicted X⁵Σ⁻ to be the ground state, with r_e = 1.875 Å and D_e = 4.18 eV [48]. Here, we found that the ground state is X⁵Σ⁻, while the a³Σ⁻ state lies 0.099 eV above the X state; see tables of Section 3 and discussion below.

OsB: Kalamse et al., via B3LYP/LANL2DZ and B3LYP/SDD, predicted that the ground state of OsB is the Χ⁴Δ state. It corresponds with a π⁴δ³5σ¹6σ¹ electronic configuration, with r_e = 1.813 Å, ω_e = 955 cm⁻¹, μ = 2.17 D, and D_e = 3.99 eV at the B3LYP/LANL2DZ level [48]. In 2019, Merriles et al. performed R2PI spectroscopy experiments to measure the predissociation threshold, and it was found to be D₀ = 4.378(3) eV [26]. Furthermore, they investigated the electronic ground state at the B3LYP/Def2TZVPP [26] and B3LYP/aug-cc-pVQZ-PP levels [45], and they found that it is a triply bonded 1σ²2σ²1π⁴1δ²3σ¹, at the X⁴Σ⁻ state, which contradicts with the prediction of Kalamse et al. [48], with a bond distance of 1.770 Å. In 2022, Merriles et al. measured, for the first time, the ionization energy of OsB using resonant two-photon ionization spectroscopy, obtaining a value of IE (OsB) = 7.955(9) eV [45].

IrB: In 1969, Auwera-Mahieu et al. determined the dissociation energy of IrB to be 5.27(18) eV via a mass spectrometric study at high temperatures [36]. In 2005, Ye et al. investigated IrB using LIF spectroscopy [50]. In this study, the ground state of IrB was found to be X³Δ₃, with an electronic configuration of 1σ²1π⁴1δ³2σ¹ and r_e = 1.7675 Å. In 2010, using the B3LYP/LANL2DZ methodology, it was also found that the most stable state for IrB is Χ³Δ raised from a π⁴δ³5σ²6σ¹ electronic configuration, with r_e = 1.806 Å, μ = 1.72 D, ΕA = 5.06 eV, ΙP = 4.66 eV, and D_e = 4.86 eV [48]. In 2011, Pang et al. investigated the electronic transition spectrum of IrB in the spectral region between 400 and 545 nm using LIF spectroscopy [54]. Its ground state was also identified to be X³Δ₃ with an electronic configuration of 1σ²2σ²π⁴δ³3σ¹. In 2019, Merriles et al., using R2PI spectroscopy, measured the predissociation threshold of IrB, obtaining a value of D₀ = 4.928(10) eV [26]. In 2020, Wang et al. investigated the nature of the chemical bonding in IrB, employing high-resolution photoelectron imaging and theoretical B2PLYP and CCSD(T)/aug-cc-pVQZ(-PP) calculations. They calculated a bond distance of 1.763 Å using B2PLYP and a dissociation energy of 5.085 eV at the CCSD(T) level of theory, in good agreement with the experimental values [47]. In 2022, Merriles and Morse measured, for the first time, the ionization energy of IrB using R2PI spectroscopy, obtaining IE = 8.301(15) eV [45].

PtB: In 1968, via Knudsen effusion mass spectrometry, McIntyre et al. measured the dissociation energy of gaseous PtB to be 4.91(17) eV [51]. Via the B3LYP/LANL2DZ (B3LYP/SDD) levels of theory, the ground state was assigned to be Χ²Σ⁺, derived from a π⁴5σ²δ⁴6σ¹ electronic configuration, with r_e = 1.809(1.815) Å and D_e = 5.43(4.86) eV [48]. In 2012, Ng et al. investigated the optical spectrum of PtB in the visible region between 455 and 520 nm using LIF spectroscopy [52]. The ground state of PtB was identified to be Χ²Σ⁺, determined using an electronic configuration of 1σ²2σ²1π⁴1δ⁴3σ¹, with r_e = 1.741 Å and ω_e = 903.60 cm⁻¹. In 2019, Merriles et al. performed R2PI spectroscopy measurements, and they found D₀ = 5.235(3) eV [26]. In 2020, Wang et al. investigated the nature of the chemical bonding in PtB, employing high-resolution photoelectron imaging and theoretical calculations using the B2PLYP and CCSD(T)/aug-cc-pVQZ(-PP) methodologies [47]. They calculated a bond distance of 1.755 Å using B2PLYP and a dissociation energy of 5.668 eV at the CCSD(T) level of theory [47]. In 2022, Merriles and Morse measured, for the first time, the ionization energy of PtB using R2PI spectroscopy, obtaining a value of IE = 8.524(10) eV [45].

AuB: In 1969, via mass spectrometry at high temperatures, Auwera-Mahieu et al. measured the dissociation energy of AuB to be 3.50(16) eV [36]. In 1971, Gingerich investigated AuB using a combination of Knudsen effusion and mass spectroscopic techniques. The reaction enthalpies determined by the second and third law method yielded D₀ = 3.77 eV [53]. In 1997, Barysz and Urban investigated the spectroscopic constants and dipole moment curves of the AuB ground state, Χ¹Σ⁺, using high-level-correlated methods combined with quasi-relativistic Douglas–Kroll (no-pair) spin-averaged approximation [29]. At the CCSD(T)/[13s11p7d4f/_Au5s3p2d/_B] computational level, they found values of r_e = 1.960 Å, D_e = 2.709 eV, ω_e = 663.0 cm⁻¹, and ω_ex_e = 4.01 cm⁻¹. In 2010, via B3LYP/LANL2DZ and SSD, values of D_e = 2.96 eV and 3.04 eV, respectively, were calculated [48]. In 2020, Wang et al. investigated the nature of the chemical bonding in AuB, employing high-resolution photoelectron imaging and theoretical calculations [47]. They calculated r_e = 1.906 Å using B2PLYP)/aug-cc-pVQZ(-PP) and a dissociation energy of 3.734 eV at the CCSD(T) level of theory [47]. This dissociation energy was in excellent agreement with the experimental value of 3.724(3) eV measured by Merriles and Morse in 2023 [30], who examined the AuB molecule experimentally using R2PI spectroscopy. They found that it remains bound at energies that far surpass its bond dissociation energy (BDE), and bonds break only when excited at or above an excited sharp predissociation threshold (SAL) [30].

HgB: There is only one theoretical study on HgB. The ground state was assigned to be the Χ²Σ⁺ via the B3LYP/LANL2DZ and B3LYP/SDD methodologies. Values of r_e = 4.381 (2.535) Å, ω_e = 19 (177) cm⁻¹, μ = 0.27 (1.20) D, EA = 0.63 (0.90) eV, IP = 7.10 (7.40) eV, and D_e = 0.002(0.17) eV were calculated at the B3LYP/LANL2DZ(B3LYP/SDD) levels of theory [48]. The differences between the two methods indicate that more accurate calculations should be performed using larger basis sets than the double zeta quality of the LANL2DZ and SDD basis sets.

3. Results and Discussion

DFT calculations were carried out for the ground states of the ZnB and the second- and third-row-transition-metal monoborides, MBs, and for some low-lying excited states of the NbB, TcB, LaB, TaB, ReB, and HgB molecules. Three functionals were used, i.e., B3LYP [55], MN15 [56], and TPSSh [57], in conjunction with the aug-cc-pVQZ basis set for B and the aug-cc-pVQZ-PP basis set [58,59,60,61] for all M except La; for La, the def2-QZVPPD basis set was used [62]. Bond distances, dissociation energies with respect to the adiabatic products and ground state products, frequencies, dipole moments, and natural NPA charges were calculated; Table 4 and Table 5. In the case of the NbB molecule, additional MRCISD(+Q)/aug-cc-pVQZ(-PP) calculations were carried out to clarify its ground state. Finally, the chemical bonding was analyzed, and it is presented in Table 6.

3.1. Second-Row-Transition-Metal Borides

The least studied molecule is the TcB species, where only a calculated D₀ value of 3.31 eV via the B97-1/(aug)-cc-pVTZ-(PP) methodology has been reported [25]. Here, we calculated three electronic states, X³Σ⁻, ⁵Σ⁻, and ⁷Σ⁻, and all the main spectroscopic data are provided; see Table 4. The ground state is a triplet state with a bond distance of 1.746 Å and a dissociation energy of 4.838 eV with respect to the adiabatic products Tc(⁴D)[4d⁶(⁵D)5s] + B(³P). At equilibrium, the Tc atom is excited at the ⁴F[4d⁷] state and it forms a quadruple bond, σ²σ²π_x²π_y², with the boron atom; see Table 6. The dissociation energy of the ground state with the atomic ground state atoms is 3.894 eV; see discussion below. The lowest quintet and septet states are of Σ⁻ symmetry, i.e., ⁵Σ⁻, and ⁷Σ⁻, and they lie 0.312 eV and 2.512 eV above the ground state. They are adiabatically correlated with the Tc(⁶S[4d⁵5s²]) + B(³P) dissociation channel, but at equilibrium, the Tc atom is excited at the Tc(⁶D[4d⁶(⁵D)5s]) + B(³P) channel. Their dissociation energies with respect to the atomic ground state products are 3.582 eV and 1.382 eV, respectively. The corresponding bond distances are 1.829 Å and 2.080 Å and the formed bonds are σ¹π_x²π_y² and σ¹π¹, respectively. The decrease in the formed bonds from the triplet to the quintet and septet can be perceived in the corresponding decrease in the dissociation energy and the corresponding increase in the bond distance. The dissociation energy per bond is about 1.2 eV in all three states, i.e., 1.21 eV (X³Σ⁻), 1.02 eV (⁵Σ⁻), and 1.38 eV (⁷Σ⁻).

Table 4. Bond lengths r_e (Å), dissociation energies D_e (eV) with respect to the adiabatic atomic products (with respect to the ground state atomic products, in parenthesis), vibrational frequencies ω_e (cm⁻¹), dipole moments μ (Debye), and charge on metal q_M via natural population analysis of ground and some low-lying states of the 2nd-row-transition-metal boride molecules, MBs (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd) at the B3LYP, TPSSh, and MN15/aug-cc-pVQZ(-PP) levels of theory and for the NbB molecule using the MRCISD and MRCISD+Q/aug-cc-pVQZ(-PP) methodologies.

MB	State	re	De	ωe	μ	qM
			B3LYP
YB	X5Σ−	2.231	3.261 (2.094)	571.9	4.894	+0.59
ZrB	X6Δ	2.162	3.035 (2.787)	604.2	3.550	+0.30
NbB	X5Π a	1.988	2.862	692.5	3.025	+0.15
	a3Σ+	1.870	2.755 (2.460)	805.8	2.984	+0.18
MoB	X6Π	1.973	2.247	654.9	2.221	+0.00
TcB	X3Σ−	1.746	4.838 (3.894)	848.3	3.767	+0.00
	a5Σ−	1.829	3.582	796.3	1.785	−0.13
	7Σ−	2.080	1.382	516.0	2.240	+0.23
RuB	X2Δ	1.700	5.210 (4.463)	935.6	3.480	−0.09
RhB	X1Σ+	1.679	5.491 (4.969)	960.6	2.877	−0.18
PdB	X2Σ+	1.777	3.781	759.7	1.281	−0.26
AgB	X1Σ+	2.070	1.900	457.5	1.296	−0.11
CdB	X2Π	2.466	0.305	237.1	1.684	+0.24
			TPSSh
YB	X5Σ−	2.235	3.418 (2.475)	574.1	4.893	+0.60
ZrB	X6Δ	2.166	3.247 (3.301)	608.4	3.691	+0.31
NbB	X5Π a	1.992	3.010	690.6	3.229	+0.17
MoB	X6Π	1.982	2.338	646.0	2.416	+0.02
TcB	X3Σ−	1.765	5.210 (4.222)	835.7	3.766	+0.01
RuB	X2Δ	1.711	5.518 (4.443)	915.0	3.604	−0.07
RhB	X1Σ+	1.685	5.412 (4.991)	958.4	3.079	−0.16
PdB	X2Σ+	1.781	4.020	768.8	1.543	−0.24
AgB	X1Σ+	2.056	1.980	474.9	1.555	−0.10
CdB	X2Π	2.413	0.483	274.7	1.953	+0.26
			MN15
YB	X5Σ−	2.210	3.649 (1.969)	590.9	5.095	+0.59
ZrB	X6Δ	2.137	3.347 (2.651)	624.7	3.670	+0.29
NbB	X5Π a	1.968	3.056	718.3	3.006	+0.12
MoB	X6Π	1.942	2.465	709.0	1.991	−0.06
TcB	X3Σ−	1.724	5.390 (4.899)	915.1	3.589	−0.06
RuB	X2Δ	1.684	5.221 (5.169)	986.9	3.375	−0.14
RhB	X1Σ+	1.666	6.064 (5.895)	1002.6	2.850	−0.22
PdB	X2Σ+	1.762	4.056	783.4	1.084	−0.30
AgB	X1Σ+	2.053	2.151	470.5	1.155	−0.15
CdB	X2Π	2.462	0.366	218.5	1.238	+0.20
			MRCISD
NbB	X5Π	2.017	2.799 b	709.0	3.096	+0.16
	A5Φ	2.018	2.715 b	710.4	2.921	+0.16
			MRCISD+Q
	X5Π	2.018	2.901 b	708.9
	A5Φ	2.019	2.808 b	710.3

^a The X⁵Π and ⁵Φ are almost energetically degenerate. A⁵Φ lies 0.084(0.093) eV above the X⁵Π state at the MRCISD(MRCISD+Q) level of theory. ^b Dissociation energy obtained via potential energy cure at infinite r_e values.

Table 4. Bond lengths r_e (Å), dissociation energies D_e (eV) with respect to the adiabatic atomic products (with respect to the ground state atomic products, in parenthesis), vibrational frequencies ω_e (cm⁻¹), dipole moments μ (Debye), and charge on metal q_M via natural population analysis of ground and some low-lying states of the 2nd-row-transition-metal boride molecules, MBs (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd) at the B3LYP, TPSSh, and MN15/aug-cc-pVQZ(-PP) levels of theory and for the NbB molecule using the MRCISD and MRCISD+Q/aug-cc-pVQZ(-PP) methodologies.

MB	State	re	De	ωe	μ	qM
			B3LYP
YB	X5Σ−	2.231	3.261 (2.094)	571.9	4.894	+0.59
ZrB	X6Δ	2.162	3.035 (2.787)	604.2	3.550	+0.30
NbB	X5Π a	1.988	2.862	692.5	3.025	+0.15
	a3Σ+	1.870	2.755 (2.460)	805.8	2.984	+0.18
MoB	X6Π	1.973	2.247	654.9	2.221	+0.00
TcB	X3Σ−	1.746	4.838 (3.894)	848.3	3.767	+0.00
	a5Σ−	1.829	3.582	796.3	1.785	−0.13
	7Σ−	2.080	1.382	516.0	2.240	+0.23
RuB	X2Δ	1.700	5.210 (4.463)	935.6	3.480	−0.09
RhB	X1Σ+	1.679	5.491 (4.969)	960.6	2.877	−0.18
PdB	X2Σ+	1.777	3.781	759.7	1.281	−0.26
AgB	X1Σ+	2.070	1.900	457.5	1.296	−0.11
CdB	X2Π	2.466	0.305	237.1	1.684	+0.24
			TPSSh
YB	X5Σ−	2.235	3.418 (2.475)	574.1	4.893	+0.60
ZrB	X6Δ	2.166	3.247 (3.301)	608.4	3.691	+0.31
NbB	X5Π a	1.992	3.010	690.6	3.229	+0.17
MoB	X6Π	1.982	2.338	646.0	2.416	+0.02
TcB	X3Σ−	1.765	5.210 (4.222)	835.7	3.766	+0.01
RuB	X2Δ	1.711	5.518 (4.443)	915.0	3.604	−0.07
RhB	X1Σ+	1.685	5.412 (4.991)	958.4	3.079	−0.16
PdB	X2Σ+	1.781	4.020	768.8	1.543	−0.24
AgB	X1Σ+	2.056	1.980	474.9	1.555	−0.10
CdB	X2Π	2.413	0.483	274.7	1.953	+0.26
			MN15
YB	X5Σ−	2.210	3.649 (1.969)	590.9	5.095	+0.59
ZrB	X6Δ	2.137	3.347 (2.651)	624.7	3.670	+0.29
NbB	X5Π a	1.968	3.056	718.3	3.006	+0.12
MoB	X6Π	1.942	2.465	709.0	1.991	−0.06
TcB	X3Σ−	1.724	5.390 (4.899)	915.1	3.589	−0.06
RuB	X2Δ	1.684	5.221 (5.169)	986.9	3.375	−0.14
RhB	X1Σ+	1.666	6.064 (5.895)	1002.6	2.850	−0.22
PdB	X2Σ+	1.762	4.056	783.4	1.084	−0.30
AgB	X1Σ+	2.053	2.151	470.5	1.155	−0.15
CdB	X2Π	2.462	0.366	218.5	1.238	+0.20
			MRCISD
NbB	X5Π	2.017	2.799 b	709.0	3.096	+0.16
	A5Φ	2.018	2.715 b	710.4	2.921	+0.16
			MRCISD+Q
	X5Π	2.018	2.901 b	708.9
	A5Φ	2.019	2.808 b	710.3

^a The X⁵Π and ⁵Φ are almost energetically degenerate. A⁵Φ lies 0.084(0.093) eV above the X⁵Π state at the MRCISD(MRCISD+Q) level of theory. ^b Dissociation energy obtained via potential energy cure at infinite r_e values.

Regarding the ground state of NbB, the X state correlates to the atomic ground state products Nb(⁶D[4d⁴5s¹]) + B(²P) and it maintains this character in the minimum. DFT cannot predict if the state is a ⁵Π(1) or ⁵Φ(3), i.e., m_ℓ(Nb) = ±2, +m_ℓ(B) = ±1, and thus multireference techniques are necessary to clarify the electronic state. MRCISD(+Q)/aug-cc-pVQZ(-PP) calculations were carried out and it was found that the global minimum is an X⁵Π state and the A⁵Φ lies 0.084(0.093) eV above the X state. Their r_e values were calculated to be 2.017(2.018) Å for the X⁵Π state and 2.018(2.019) Å for the A⁵Φ, respectively. The lowest triplet state is the a³Σ⁺, which lies 0.402 eV above the X state. The bonding in the X state is $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.(\left|{\sigma }^1{{\pi }_x}^2{{\pi }_y}^1⟩\right.\pm \left|{\sigma }^1{{\pi }_x}^1{{\pi }_y}^2⟩\right.)$, while in the a³Σ⁺ it is σ¹π_x²π_y²; see Table 6. The multiple bonding of 2 in the X state in contradiction to 2.5 in the ³Σ⁺ affects the corresponding bond distances, i.e., 1.988 vs 1.870, where a decrease of 0.1 Å is observed.

The bond distances of the ground states of the second-row-transition-metal borides, MBs, with respect to the different M atoms, are plotted in Figure 1a. Note that all three functionals present the same geometry; see Table 4. YB has a bond distance of 2.231 Å, and as the M changes along the row of the periodic table, the bond distance decreases up to RhB (1.679 Å) and then increases up to CdB at 2.466 Å.

The dissociation energy with respect to the adiabatic products (D_e) as the M atom changes from left to right is plotted in Figure 1d. The D_e value is decreased from YB up to MoB; then, the TcB, RuB, and RhB molecules have similar D_e values at ~5eV, i.e., 4.838 eV, 5.210 eV, and 5.491 eV, respectively; finally, the D_e values decrease constantly up to CdB at 0.305 eV. The dissociation energy with respect to the atomic ground state products at the zero-vibrational level, D_0,GS, with respect to the change in the M atom from left to right, is plotted in Figure 1e, where we observe an increase in the D_0,GS value up to NbB, then a small decease for the MoB, and then an increase up to Rh. Furthermore, the available experimental D₀ values determined via R2PI and Knudsen spectrometry are plotted in Figure 1e. The calculated D_0,GS values are in very good agreement with the experimental values; the % differences range from 0.00 to 0.09%.

The dipole moment, μ, follows a decreasing pattern from 5.10 Debye (YB) to 1.99 D (MoB), followed by an immediate jump at 3.59 D (TcB), and then by a small decrease at 3.38 D (RuB) and at 2.85 D (RhB). Note that a quadruple bond is formed in these three molecules. Finally, the last three MBs have similar dipole moments, i.e., 1.08 D (PdB), 1.16 D (AgB), and 1.24 D CdB; see Figure 1b.

The NPA charge on metal is positive in YB, +0.6e, i.e., about half of the electron has been moved to the B atom, and it decreases to 0 for MoB and TcB. In these two molecules, there is a charge transfer from M to B and back. Then, from Ru to Ag, the metal has a total negative charge, i.e., the charge is transferred from B to M, while in CdB, again the M is charged positively, which is expected since the M has an s²d¹⁰ atomic configuration (Figure 1c). The opposite trend is observed in harmonic vibrational frequencies, where the ω_e values increase from Y (571.9 cm⁻¹) to Rh (960.6 cm⁻¹) and then decreased to Cd (237.1 cm⁻¹); see Figure 1f.

The bonding of the ground and excited states of the MBs is reported in Table 6. In the X states of the first two MB molecules, YB(X⁵Σ⁻) and ZrB(X⁶Δ), three half bonds, σ¹π¹π¹, are formed. The additional electron in ZrB is added to the non-bonding δ(d_x²−y² or d_xy) orbital. As a result, they have r_e values that differ only by about 0.07 Å, and their D_e values differ only by about 0.2 eV. The next NbB(X⁵Π) and MoB(X⁶Π) form two half bonds and one whole bond, σ¹π²π¹. The additional e⁻ from Zr to Nb is placed in the half-occupied π orbital, while the additional electron of MoB is added to the non-bonding δ(d_x²−y² or d_xy) orbital. As a result, the X states of NbB and MoB have r_e values that differ only by about 0.01 Å and D_e values that differ by about 0.6 eV.

Then, in TcB(X³Σ⁻), RuB(X²Δ), and RhB(X¹Σ⁺), there is a significant change in the bonding compared to the first four MBs of the second row. The bonding is formed from an atomic state which has an empty 5s orbital from, for example, Tc(⁴F[4d⁷]), Ru(b³F[4d⁸]), and Rh(a²D[4d⁹]). As a result, the 5s of the metal is empty and thus it can accept electrons, forming a dative bond. The bonding in X states is 1σ²2σ²1π_x²1π_y²; see Figure 2. Specifically, the bonding is 1σ^{2 =} (M4d_z²)²→(B2p_z)⁰, which is a dative bond from the M to the B atom; 2σ^{2 =} (M5sM4d_z²)⁰←(B2s)², also a dative bond from the B to the M; and 1π_x² = (M4d_xz)¹ − (B2p_x)¹ and 1π_y² = (M4d_yz)¹ − (B2p_y)¹, which are both π covalent bonds. The similar bonding leads to similar short r_e values, smaller up to 0.3 Å with respect to MoB, and dissociation energies of about 5 eV (4.84 up to 5.49 eV). Note that there is a 5s5d_z² hybridization in M and a 2s2p_z hybridization in B. The added electrons from Tc to Rh are added to non-bonding δ orbitals, i.e., δ¹δ¹ in TcB, δ²δ¹ in RuB, and δ²δ² in RhB.

The PdB(X²Σ⁺) also has an X state that is formed from an atomic state with an empty 5s orbital in Pd(¹S[4d¹⁰]). However, there is a triple bond which consists of three dative bonds, i.e., 1σ² = (Μ5s)⁰←(Β2s)², 1π_x² = (Μ4d_xz)²→(Β2p_x)⁰, and 1π_y² = (Μ4d_yz)²→(Β2p_y)⁰, while there is a non-bonding σ¹ which corresponds to the (Β2p_z)¹ of B. The AgB(X¹Σ⁺) molecule also has a triple bond with one covalent and two dative bonds, i.e., 1σ^{2 =} (M5s)¹-(B2p_z)¹, 1π_x² = (M4d_xz)²→(B2p_x)⁰ and 1π_y² = (M4d_yz)²→(B2p_y)⁰. Finally, CdB(X²Π) has two dative bonds, i.e., 1π_x² = (M4d_xz)²→(B2p_x)⁰ and 1π_y² = (M4d_yz)²→(B2p_y)⁰, resulting in a positive charge of +0.24e on Cd.

3.2. Third-Row-Transition-Metal Borides

The ground states of the third-row MB molecules were calculated, while for the LaB, TaB, ReB, and HgB molecules, an additional electronic state was also included; see Table 5 and Table 6. The ground state of the LaB is the Χ⁵Σ⁻ state, which correlates to La(⁴F[5d²6s]) + B(²P) and maintains this character in the minimum. Its dissociation energy, D₀, with respect to the ground state products is 2.086 eV, in excellent agreement with the R2PI experimental value of 2.086 eV [25]. On the contrary, the ³Π state, which is 0.448 eV above the X state, correlates to an excited La(²D[5d²6s]), but the in situ La atom is b⁴F[5d³]. The X state has a σ¹π_x¹π_y¹ bonding scheme, while the ³Π state presents a σ¹π_x²π_y¹ bonding scheme i.e., one less bonding electron, and, as a result, a shorter bond distance by 0.1 Å. The X state of the HfB is a Χ⁴Σ⁻ state and it has three half bonds with a dissociation energy of 2.810 eV, in good agreement with experimental value of 2.593 eV [25]. Two states have been calculated for the TaB; the X⁵Δ state, which has a whole bond and two half bonds, and the ³Σ⁺ state, which lies 0.260 eV above the ground and presents two and a half bonds. Again, the addition of one electron in the bonding results in a decrease in the bond distance by 0.1 Å. The dissociation energy of X⁵Δ with respect to the adiabatic products is 3.495 eV. The X state of WB is the Χ⁶Π, where a σ¹π²π¹ bond scheme is formed. It presents a dissociation energy of 2.907 eV and r_e = 1.955 Å. The ⁶Σ⁺ lies 0.137 eV above the X state; it forms a σ²π_x¹π_y¹ bond with a dissociation energy of 2.770 eV and r_e = 2.125 Å. The ground state of ReB is the Χ⁵Σ⁻, state which correlates to Re(⁶S[5d⁵6s²]), but the in situ atom is the Re(⁶D[5d⁶6s¹]), which forms two and a half bonds. The dissociation energy is 3.106 eV and the bond distance at 1.834 Å. The a³Σ⁻ state lies only 0.099 eV above the ground state, and it has a stronger bonding, i.e., three bonds (σ²π_x²π_y²). Its bond distance is 1.809 Å, and its dissociation energy is 4.967 eV with respect to the correlated products, while with respect to the atomic ground state products, it is 3.008 eV.

Table 5. Bond lengths, r_e (Å); dissociation energies, D_e (eV) with respect to the adiabatic atomic products (with respect to the ground state atomic products, in parenthesis); vibrational frequencies, ω_e (cm⁻¹); dipole moments, μ (Debye); and charge on metal q_M via natural population analysis of ground and some low-lying states of the 3rd-row-transition-metal boride molecules, MBs (M = La, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg): at the B3LYP, TPSSh, and MN15/aug-cc-pVQZ_B(-PP)_M and def2-QZVPPD_La levels of theory.

Molecule	State	re (Å)	De	ωe	μ	qM
			B3LYP
LaB	Χ5Σ−	2.384	2.874(2.528)	512.7	4.03	+0.55
	a3Π	2.263	2.080	478.5	6.08	+0.74
HfB	Χ4Σ−	2.151	2.810	594.8	2.55	+0.35
TaB	X5Δ	1.974	3.495	739.6	3.58	+0.22
	a3Σ+	1.870	3.236	838.2	3.78	+0.28
WB	Χ6 Π	1.955	2.907 a	735.0	2.63	+0.04
	6Σ+	2.125	2.770 a	548.2	2.86	+0.10
ReB	Χ5Σ−	1.834	3.106	865.5	2.28	−0.07
	a3Σ−	1.809	4.967(3.008)	850.7	2.85	−0.05
OsB	Χ4Σ−	1.772	4.482	935.1	2.18	−0.13
IrB	Χ3Δ	1.762	5.338	926.9	1.62	−0.24
PtB	Χ2Σ+	1.759	5.377	905.2	1.08	−0.32
AuB	Χ1Σ+	1.925	3.451	642.2	0.87	−0.26
HgB	X2Π	2.397	0.237	223.4	1.38	+0.17
HgB	A2Σ+	4.029	0.003	31.7	0.35	0.00
			TPSSh
LaB	Χ5Σ−	2.384	3.032(2.786)	518.6	4.40	+0.56
HfB	Χ4Σ−	2.149	3.004	606.1	2.55	+0.36
TaB	X5Δ	1.979	3.585	733.5	3.65	+0.23
WB	Χ6 Π	1.963	2.938 a	720.5	2.74	+0.06
	6Σ+	2.126	2.731 a	549.5	2.99	+0.12
ReB	Χ5Σ−	1.838	3.519	863.3	2.41	−0.05
OsB	Χ4Σ−	1.779	4.646	909.9	2.59	−0.07
IrB	Χ3Δ	1.765	5.300	928.1	1.81	−0.22
PtB	Χ2Σ+	1.759	5.547	917.4	1.19	−0.31
AuB	Χ1Σ+	1.921	3.604	657.6	1.04	−0.25
HgB	X2Π	2.341	0.395	266.5	1.62	+0.19
HgB	A2Σ+	3.658	0.016	42.9	0.50	−0.009
			MN15
LaB	Χ5Σ−	2.365	3.237(2.402)	526.1	4.64	+0.55
HfB	Χ4Σ−	2.133	3.123	603.5	2.46	+0.33
TaB	X5Δ	1.957	3.560	767.0	3.42	+0.19
WB	Χ6 Π	1.932	3.031 a	778.1	2.47	0.00
ReB	Χ5Σ−	1.814	3.346	904.4	2.02	−0.11
OsB	Χ4Σ−	1.758	4.943	958.7	1.98	−0.17
IrB	Χ3Δ	1.750	5.633	944.7	1.46	−0.27
PtB	Χ2Σ+	1.744	5.744	934.8	0.87	−0.35
AuB	Χ1Σ+	1.902	3.933	673.9	0.69	−0.30
HgB	X2Π	2.337	0.364	234.4	1.12	+0.14
HgB	A2Σ+	3.388	0.085	66.9	0.68	−0.02

^a With respect to the W(⁷S) state, which is the lowest atomic state with respect to the average J term.

Table 5. Bond lengths, r_e (Å); dissociation energies, D_e (eV) with respect to the adiabatic atomic products (with respect to the ground state atomic products, in parenthesis); vibrational frequencies, ω_e (cm⁻¹); dipole moments, μ (Debye); and charge on metal q_M via natural population analysis of ground and some low-lying states of the 3rd-row-transition-metal boride molecules, MBs (M = La, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg): at the B3LYP, TPSSh, and MN15/aug-cc-pVQZ_B(-PP)_M and def2-QZVPPD_La levels of theory.

Molecule	State	re (Å)	De	ωe	μ	qM
			B3LYP
LaB	Χ5Σ−	2.384	2.874(2.528)	512.7	4.03	+0.55
	a3Π	2.263	2.080	478.5	6.08	+0.74
HfB	Χ4Σ−	2.151	2.810	594.8	2.55	+0.35
TaB	X5Δ	1.974	3.495	739.6	3.58	+0.22
	a3Σ+	1.870	3.236	838.2	3.78	+0.28
WB	Χ6 Π	1.955	2.907 a	735.0	2.63	+0.04
	6Σ+	2.125	2.770 a	548.2	2.86	+0.10
ReB	Χ5Σ−	1.834	3.106	865.5	2.28	−0.07
	a3Σ−	1.809	4.967(3.008)	850.7	2.85	−0.05
OsB	Χ4Σ−	1.772	4.482	935.1	2.18	−0.13
IrB	Χ3Δ	1.762	5.338	926.9	1.62	−0.24
PtB	Χ2Σ+	1.759	5.377	905.2	1.08	−0.32
AuB	Χ1Σ+	1.925	3.451	642.2	0.87	−0.26
HgB	X2Π	2.397	0.237	223.4	1.38	+0.17
HgB	A2Σ+	4.029	0.003	31.7	0.35	0.00
			TPSSh
LaB	Χ5Σ−	2.384	3.032(2.786)	518.6	4.40	+0.56
HfB	Χ4Σ−	2.149	3.004	606.1	2.55	+0.36
TaB	X5Δ	1.979	3.585	733.5	3.65	+0.23
WB	Χ6 Π	1.963	2.938 a	720.5	2.74	+0.06
	6Σ+	2.126	2.731 a	549.5	2.99	+0.12
ReB	Χ5Σ−	1.838	3.519	863.3	2.41	−0.05
OsB	Χ4Σ−	1.779	4.646	909.9	2.59	−0.07
IrB	Χ3Δ	1.765	5.300	928.1	1.81	−0.22
PtB	Χ2Σ+	1.759	5.547	917.4	1.19	−0.31
AuB	Χ1Σ+	1.921	3.604	657.6	1.04	−0.25
HgB	X2Π	2.341	0.395	266.5	1.62	+0.19
HgB	A2Σ+	3.658	0.016	42.9	0.50	−0.009
			MN15
LaB	Χ5Σ−	2.365	3.237(2.402)	526.1	4.64	+0.55
HfB	Χ4Σ−	2.133	3.123	603.5	2.46	+0.33
TaB	X5Δ	1.957	3.560	767.0	3.42	+0.19
WB	Χ6 Π	1.932	3.031 a	778.1	2.47	0.00
ReB	Χ5Σ−	1.814	3.346	904.4	2.02	−0.11
OsB	Χ4Σ−	1.758	4.943	958.7	1.98	−0.17
IrB	Χ3Δ	1.750	5.633	944.7	1.46	−0.27
PtB	Χ2Σ+	1.744	5.744	934.8	0.87	−0.35
AuB	Χ1Σ+	1.902	3.933	673.9	0.69	−0.30
HgB	X2Π	2.337	0.364	234.4	1.12	+0.14
HgB	A2Σ+	3.388	0.085	66.9	0.68	−0.02

^a With respect to the W(⁷S) state, which is the lowest atomic state with respect to the average J term.

The OsB(Χ⁴Σ⁻), IrB(Χ³Δ), and PtB(Χ²Σ⁺) molecules present triple bonds in their ground states, with similar bond distances of 1.772 Å, 1.762 Å, and 1.759 Å, which is expected due to the same bonding. Note that the additional electron moving from Os to Pt is added to the non-bonding d₊₂ or d₋₂ atomic orbital. The dissociation energies with respect to the adiabatic products are 4.482 eV for OsB, 5.338 eV for IrB, and 5.377 eV for PtB. The IrB and OsB molecules correlate to their in situ atoms, while for OsB, the state of the M atom in the minimum differs from the correlated one. Thus, the dissociation energy of OsB with respect to its in situ atoms is 5.12 eV, only 0.2 eV less than the D_e value of IrB and PtB. On the contrary, whilst a triple bond is also formed in the X state of AuB (Χ¹Σ⁺), the two π² bonds which are dative, i.e., Au(5d_xz)²→B(2p_x)² and Au(5d_yz)²→B(2p_y)⁰, are weak with a small charge transfer. Finally, two states were calculated for the HgB molecule, X²Π and A²Σ⁺. The X state has a weak double bond with an elongated bond distance of 2.397 Å and a dissociation energy of 0.237 eV, while the A²Σ⁺ is a van der Waals state, where very weak interactions are formed between atoms. Comparing the D₀ dissociation energies (with respect to the ground state products) with the corresponding experimental values, there is a very good agreement, i.e., calculated[expt] values of 4.424[4.378] eV for OsB, 5.281[4.928] eV for IrB, 5.321[5.235] eV for PtB, and 3.411[3.724] eV for AuB. Thus, differences in D₀ that range from 0.05 eV to 0.35 eV are observed.

The bond distances of the ground states of the third-row-transition-metal MB molecules, with respect to the different M atoms, are plotted in Figure 3a. Note that all three functionals present the same geometry (Table 5). The first molecule of the row, LaB, has a bond distance of 2.384 Å, and, as the M changes along the row of the periodic table, the bond distance decreases up to PtB (1.759 Å) and then increases up to HgB at 2.397 Å. It should be noted that OsB(Χ⁴Σ⁻), IrB(Χ³Δ), and PtB(Χ²Σ⁺) present similar bond distances, i.e., 1.772 Å, 1.762 Å, and 1.759 Å, respectively, and similar triple bonding, σ²π²π².

The dipole moment, μ, follows a decreasing pattern from 4.03 Debye (LaB) to 0.87 D (AuB), except for the increase from Hf to Ta. Finally, there is an increase from AuB to HgB at 1.38 D; see Figure 3b. The NPA charge on the metal is positive in LaB, +0.55e, i.e., about half of the electrons are moved to the B atom, and the charge decreases to WB (+0.04 e); see Figure 3c. Then, from Re to Au, the metal has a total negative charge, i.e., the charge is transferred from B to M, while in HgB, again, the M is charged positively (+0.17e), which is expected since the M has an s²d² atomic configuration. In PtB, the Pt demonstrates the most negative charge of all the metals. Finally, the harmonic vibrational frequencies generally increase from La to OsB. OsB, IrB, and PtB have similar ω_e values at about ~920 cm⁻¹; then, they decrease to HgB (223.4 cm⁻¹); see Figure 3f.

The dissociation energy with respect to the adiabatic products, D_e, as the M atom changes from left to right is plotted in Figure 3d. The first five MBs have similar D_e values, with some discrepancies. Then, in OsB, IrB, and PtB, the D_e value increases; their values are 4.482 eV, 5.338 eV, and 5.377 eV, respectively, and finally, the D_e values decrease constantly up to HgB at 0.237 eV. The dissociation energy with respect to the atomic ground state products at the zero-vibrational level, D_0,GS, with respect to the change in the M atom is plotted in Figure 3e. The available experimental D₀ values are also plotted in Figure 3e via R2PI and Knudsen spectrometry. The calculated D_0,GS values for HfB, OsB, WB, and PtB are in excellent agreement, and for LaB and IrB, they are in good agreement, while the largest difference is observed for TaB. Finally, it should be noted that the B3LYP-obtained values are in better agreement with the experimental ones than the values obtained via the MN15 and TPSSh functional methods.

The bonding of the ground and excited states of the MBs is reported in Table 6. In the ground states of the first two MB molecules, i.e., LaB and HfB, three half bonds, σ¹π_x¹π_y¹, are formed. As a result, they have the same dissociation energies, i.e., 2.874 eV and 2.810 eV. It should be noted that there is a strong 5d_z²6s hybridization in the M atom. The TaB and WB form two half bonds and one whole bond, i.e., σ¹π_x²π_y¹. The following ReB has one half and two bonds, i.e., σ¹π_x¹π_y², while the next four molecules, OsB, IrB, PtB, and AuB, form a triple bond, i.e., σ²π_x²π_y². Note that, contrary to the second row, where the atomic states of M atoms with empty 5s orbitals are involved in the bond, the corresponding atomic states of the M atoms occupy the 6s atomic orbital. Finally, the last MB of this row, i.e., HgB, has two dative bonds, i.e., 1π_x² = M(5d_xz)²→B(2p_x)⁰ and 1π_y² = M(5d_yz)²→B(2p_y)⁰, similar to the CdB of the second row, thus leading to similar bond lengths, 2.466 Å (CdB) and 2.397 Å (HgB), and dissociation energies, 0.305 eV and 0.237 eV, respectively.

3.3. Comparison of MBs of All Three Rows and Bonding Analysis

Quantum chemical computations provide details on the chemical bonds and electronic structures of these species. In general, the bond lengths of the transition-metal borides increase in the periodic table from left to right, or as one goes down a group of elements. Of course, anomalies occur, but the data are the result of the variety of bonding schemes that are formed in the MB molecules. These bonding schemes depend on the bonding and the filling of σ, π, and δ orbitals. MBs can form one-and-a-half, double, triple, and even quadruple bonds, with the latter being recently discovered in RhB and RhB⁻ [2,4,5], while here, we found that RuB and TcB also form quadruple bonds.

The bond distances of the ground states of the MBs of all three rows, with respect to the different M atoms, are plotted in Figure 4a. Moving from up (first row) to down (third row) the bond distances, in the cases where there is the same bonding, the r_e values increase by about 1 Å. Also, moving from left to right, the r_e value decreases up to the seventh or eighth MB, and then the r_e value increases sharply. All differences are a result of the type of bonding. In all rows, there are three MBs that present similar strong bonding. Thus, along the first row, from the sixth to the eight MB, i.e., FeB, CoB, and NiB, a triple bond and similar r_e values, around 1.7 Å, are observed. Similarly, along the third row, from the sixth to the eighth MB, OsB, IrB, and PtB, a triple bond and similar r_e values, around 1.76 Å, are found. On the contrary, in the second row, from the fifth to the seventh MB, i.e., TcB, RuB, and RhB, a quadruple bond is formed with similar r_e values, around 1.7 Å.

In Figure 4d,e, the dissociation energies with respect to the adiabatic products (D_e) and the atomic ground state products at the zero-vibrational level (D_0,GS) are plotted with respect to the change in the M atom from left to right for the three rows. The general shape is the same as that of the lowest dissociation energies being observed for the M, with atomic configurations of (ds)⁶ or (ds)⁷ and d¹⁰s², while the highest values are observed for the M with atomic configurations of (ds)^8–10 for the first and third rows and (ds)^7–9 for the second row, and the largest values are found for the second and third rows. Finally, it should be noted that the first two and the last two MBs present similar dissociation energy regardless of the row.

The dipole moment, μ, with respect to the M follows roughly the same trend as the M changes from left to right in each row, i.e., a general reduction in the μ value. Any differences are related to the differences in the bonding; see Figure 4b. The smallest differences among the three rows are observed in d¹s², (ds)⁵, and d¹⁰s², while the first and third rows present similar μ values for the d⁵s² metals, where the second-row MB presents a μ value larger by 1.5 D than the corresponding MnB and ReB values due to the quadruple bond of TcB.

Table 6. Atomic states of transition metals forming the bonding (in situ M) of the ground MB state (X) and the atomic state of M in R_M-B infinity; and of the atomic ground state X_M and the energy difference between the ground atomic state and the atomic state forming the bonding T_e (eV) (M_J-averaged experimental).

MB	X	Configuration	Bond	In Situ M	In RM-B Infinity	XM	Τe b
		1st row		MB		M
ScB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[3d24s1] a	4F[3d24s1]	2D[3d14s2]	1.428(1.427)
TiB	X6Δ	1σ22σ13σ11π11π11δ1	σ1πx1πy1	5F[3d3(4F)4s1] a	5F[3d3(4F)4s1]	a3F[3d24s2 ]	0.813(0.806)
VB	X7Σ+	1σ22σ13σ11π11π11δ11δ1	σ1πx1πy1	6D[3d4(5D)4s1] a	6D[3d4(5D)4s1]	a4F[3d34s2]	0.262(0.245)
CrB	X6Σ+	1σ22σ23σ11π11π11δ11δ1	σ2πx1πy1	7S[3d5(6S)4s1] a	7S[3d5(6S)4s1]	7S[3d5(6S)4s1]	0
MnB	X5Π	1σ22σ23σ11π21π11δ11δ1	σ2πx2πy1	6S[3d54s2] a	6S[3d54s2]	a6S[3d54s2]	0
FeB	Χ4Σ−	1σ22σ23σ11π21π21δ11δ1	σ2πx2πy2	a5F[3d7(4F)4s1] a	a5D[3d64s2]	a5D[3d64s2]	0.859(0.875)
CoB	Χ3Δ	1σ22σ23σ11π21π21δ21δ1	σ2πx2πy2	b4F[3d8(3F)4s1] a	a4F[3d74s2]	a4F[3d74s2]	0.432(0.417)
NiB	Χ2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	a3D[3d9(2D)4s1] a	a3F[3d8(3F)4s2]	a3F[3d8(3F)4s2]	0.025(-0.030)
CuB	Χ1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[3d10(1S)4s1] a	2S[3d10(1S)4s1]	2S[3d10(1S)4s1]	0
		2nd row
ZnB	Χ2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[3d104s2]	1S[3d104s2]	1S[3d104s2]	0
YB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[4d25s1]	a4F[4d2(3F)5s]	a2D[4d5s2]	1.356(1.359)
ZrB	X6Δ	1σ22σ13σ11π11π11δ1	σ1πx1πy1	5F[4d3(4F)5s1]	a5F[4d3(4F)5s1]	a3F[4d25s2]	0.604(0.588)
NbB	5Π/5Φ	1σ22σ13σ11π21π11δ1	σ1πx2πy1	a6D[4d4(5D)5s1]	a6D[4d4(5D)5s1]	a6D[4d4(5D)5s1]	0
	3Σ+	1σ22σ13σ11π21π2	σ1πx2πy2	a4D[4d45s1]	a4F[4d35s3]	a6D[4d4(5D)5s1]	1.043(1.049)
MoB	X6Π	1σ22σ13σ11π21π11δ11δ1	σ1πx2πy1	a7S[4d5(6S)5s]	a7S[4d5(6S)5s]	a7S[4d5(6S)5s]	0
TcB	X3Σ−	1σ22σ21π21π21δ11δ1	σ2σ2πx2πy2	4F[4d7]	4D[4d6(5D)5s]	6S[4d55s2]	1.827(2.332)
	5Σ−	1σ22σ13σ11π21π21δ11δ1	σ1πx2πy2	6D[4d65s1]	6S[4d55s2]	6S[4d55s2]	0.319(0.406)
	7Σ−	1σ22σ13σ11π21π12π11δ11δ1	σ1π1	6D[4d65s1]	6S[4d55s2]	6S[4d55s2]	0.319(0.406)
RuB	X2Δ	1σ22σ21π21π21δ21δ1	σ2σ2πx2πy2	b3F[4d8]	a3F[4d7(a4F)5s]	a5F[4d7(a4F)5s]	1.131(1.092)
RhB	X1Σ+	1σ22σ21π21π21δ21δ2	σ2σ2πx2πy2	a2D[4d9]	a2D[4d9]	a4F[4d8(3F)5s]	0.410(0.342)
PdB	X2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	1S[4d10]	1S[4d10]	1S[4d10]	0
AgB	X1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[4d105s]	2S[4d105s]	2S[4d105s]	0
CdB	X2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[4d105s2]	1S[4d105s2]	1S[4d105s2]	0
		3rd row
LaB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[5d2(3F)6s]	4F[5d2(3F)6s]	2D[5d6s2]	0.331(0.355)
	3Π	1σ22σ11π21π1	σ1πx2πy1	b4F[5d3]	2D[5d6s2]	2D[5d6s2]	1.541(1.560)
HfB	X4Σ−	1σ22σ23σ11π11π1	σ1πx1πy1	a3F[5d26s2]	a3F[5d26s2]	a3F[5d26s2]	0
TaB	X5Δ	1σ22σ13σ11π21π11δ1	σ1πx2πy1	a6D[5d46s1]	a4F[5d36s2]	a4F[5d36s2]	1.210(1.038)
	3Σ+	1σ22σ13σ11π21π2	σ1πx2πy2		a4F[5d36s2]	a4F[5d36s2]
WB	X6Π	1σ22σ13σ11π21π11δ11δ1	σ1πx2πy1	7S[5d56s1]	5D[5d46s2]	5D[5d46s2]	0.366(−0.187)
	6Σ+	1σ22σ23σ11π11π11δ11δ1	σ2πx1πy1	7S[5d56s1]	5D[5d46s2]	5D[5d46s2]	0.366(−0.187)
ReB	X5Σ−	1σ22σ13σ11π21π21δ11δ1	σ1πx2πy2	a6D[5d66s1]	a6S[5d56s2]	a6S[5d56s2]	1.457(1.759)
	3Σ−	1σ22σ21π21π21δ11δ1	σ2πx2πy2	a4P[5d56s2]	a4P[5d56s2]	a6S[5d56s2]	1.436(1.603)
OsB	Χ4Σ−	1σ22σ23σ11π21π21δ11δ1	σ2πx2πy2	a5F[5d7(4F)6s1]	a5D[5d66s2]	a5D[5d66s2]	0.638(0.757)
IrB	Χ3Δ	1σ22σ23σ11π21π21δ21δ1	σ2πx2πy2	a4F[5d76s2]	a4F[5d76s2]	a4F[5d76s2]	0
PtB	Χ2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	3D[5d96s1]	3D[5d96s1]	3D[5d96s1]	0
AuB	Χ1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[5d106s1]	2S[5d106s1]	2S[5d106s1]	0
HgB	Χ2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[5d106s2]	1S[5d106s2]	1S[5d106s2]	0
	Χ2Σ+	1σ22σ23σ24σ11π21π21δ21δ2	(π2π2)	1S[5d106s2]	1S[5d106s2]	1S[5d106s2]	0

^a Ref. [21]. ^b Expt values of the energy separation between the term with the lowest in energy spin–orbit coupling angular momentum quantum number J (average term).

Table 6. Atomic states of transition metals forming the bonding (in situ M) of the ground MB state (X) and the atomic state of M in R_M-B infinity; and of the atomic ground state X_M and the energy difference between the ground atomic state and the atomic state forming the bonding T_e (eV) (M_J-averaged experimental).

MB	X	Configuration	Bond	In Situ M	In RM-B Infinity	XM	Τe b
		1st row		MB		M
ScB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[3d24s1] a	4F[3d24s1]	2D[3d14s2]	1.428(1.427)
TiB	X6Δ	1σ22σ13σ11π11π11δ1	σ1πx1πy1	5F[3d3(4F)4s1] a	5F[3d3(4F)4s1]	a3F[3d24s2 ]	0.813(0.806)
VB	X7Σ+	1σ22σ13σ11π11π11δ11δ1	σ1πx1πy1	6D[3d4(5D)4s1] a	6D[3d4(5D)4s1]	a4F[3d34s2]	0.262(0.245)
CrB	X6Σ+	1σ22σ23σ11π11π11δ11δ1	σ2πx1πy1	7S[3d5(6S)4s1] a	7S[3d5(6S)4s1]	7S[3d5(6S)4s1]	0
MnB	X5Π	1σ22σ23σ11π21π11δ11δ1	σ2πx2πy1	6S[3d54s2] a	6S[3d54s2]	a6S[3d54s2]	0
FeB	Χ4Σ−	1σ22σ23σ11π21π21δ11δ1	σ2πx2πy2	a5F[3d7(4F)4s1] a	a5D[3d64s2]	a5D[3d64s2]	0.859(0.875)
CoB	Χ3Δ	1σ22σ23σ11π21π21δ21δ1	σ2πx2πy2	b4F[3d8(3F)4s1] a	a4F[3d74s2]	a4F[3d74s2]	0.432(0.417)
NiB	Χ2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	a3D[3d9(2D)4s1] a	a3F[3d8(3F)4s2]	a3F[3d8(3F)4s2]	0.025(-0.030)
CuB	Χ1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[3d10(1S)4s1] a	2S[3d10(1S)4s1]	2S[3d10(1S)4s1]	0
		2nd row
ZnB	Χ2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[3d104s2]	1S[3d104s2]	1S[3d104s2]	0
YB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[4d25s1]	a4F[4d2(3F)5s]	a2D[4d5s2]	1.356(1.359)
ZrB	X6Δ	1σ22σ13σ11π11π11δ1	σ1πx1πy1	5F[4d3(4F)5s1]	a5F[4d3(4F)5s1]	a3F[4d25s2]	0.604(0.588)
NbB	5Π/5Φ	1σ22σ13σ11π21π11δ1	σ1πx2πy1	a6D[4d4(5D)5s1]	a6D[4d4(5D)5s1]	a6D[4d4(5D)5s1]	0
	3Σ+	1σ22σ13σ11π21π2	σ1πx2πy2	a4D[4d45s1]	a4F[4d35s3]	a6D[4d4(5D)5s1]	1.043(1.049)
MoB	X6Π	1σ22σ13σ11π21π11δ11δ1	σ1πx2πy1	a7S[4d5(6S)5s]	a7S[4d5(6S)5s]	a7S[4d5(6S)5s]	0
TcB	X3Σ−	1σ22σ21π21π21δ11δ1	σ2σ2πx2πy2	4F[4d7]	4D[4d6(5D)5s]	6S[4d55s2]	1.827(2.332)
	5Σ−	1σ22σ13σ11π21π21δ11δ1	σ1πx2πy2	6D[4d65s1]	6S[4d55s2]	6S[4d55s2]	0.319(0.406)
	7Σ−	1σ22σ13σ11π21π12π11δ11δ1	σ1π1	6D[4d65s1]	6S[4d55s2]	6S[4d55s2]	0.319(0.406)
RuB	X2Δ	1σ22σ21π21π21δ21δ1	σ2σ2πx2πy2	b3F[4d8]	a3F[4d7(a4F)5s]	a5F[4d7(a4F)5s]	1.131(1.092)
RhB	X1Σ+	1σ22σ21π21π21δ21δ2	σ2σ2πx2πy2	a2D[4d9]	a2D[4d9]	a4F[4d8(3F)5s]	0.410(0.342)
PdB	X2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	1S[4d10]	1S[4d10]	1S[4d10]	0
AgB	X1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[4d105s]	2S[4d105s]	2S[4d105s]	0
CdB	X2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[4d105s2]	1S[4d105s2]	1S[4d105s2]	0
		3rd row
LaB	X5Σ−	1σ22σ13σ11π11π1	σ1πx1πy1	4F[5d2(3F)6s]	4F[5d2(3F)6s]	2D[5d6s2]	0.331(0.355)
	3Π	1σ22σ11π21π1	σ1πx2πy1	b4F[5d3]	2D[5d6s2]	2D[5d6s2]	1.541(1.560)
HfB	X4Σ−	1σ22σ23σ11π11π1	σ1πx1πy1	a3F[5d26s2]	a3F[5d26s2]	a3F[5d26s2]	0
TaB	X5Δ	1σ22σ13σ11π21π11δ1	σ1πx2πy1	a6D[5d46s1]	a4F[5d36s2]	a4F[5d36s2]	1.210(1.038)
	3Σ+	1σ22σ13σ11π21π2	σ1πx2πy2		a4F[5d36s2]	a4F[5d36s2]
WB	X6Π	1σ22σ13σ11π21π11δ11δ1	σ1πx2πy1	7S[5d56s1]	5D[5d46s2]	5D[5d46s2]	0.366(−0.187)
	6Σ+	1σ22σ23σ11π11π11δ11δ1	σ2πx1πy1	7S[5d56s1]	5D[5d46s2]	5D[5d46s2]	0.366(−0.187)
ReB	X5Σ−	1σ22σ13σ11π21π21δ11δ1	σ1πx2πy2	a6D[5d66s1]	a6S[5d56s2]	a6S[5d56s2]	1.457(1.759)
	3Σ−	1σ22σ21π21π21δ11δ1	σ2πx2πy2	a4P[5d56s2]	a4P[5d56s2]	a6S[5d56s2]	1.436(1.603)
OsB	Χ4Σ−	1σ22σ23σ11π21π21δ11δ1	σ2πx2πy2	a5F[5d7(4F)6s1]	a5D[5d66s2]	a5D[5d66s2]	0.638(0.757)
IrB	Χ3Δ	1σ22σ23σ11π21π21δ21δ1	σ2πx2πy2	a4F[5d76s2]	a4F[5d76s2]	a4F[5d76s2]	0
PtB	Χ2Σ+	1σ22σ23σ11π21π21δ21δ2	σ2πx2πy2	3D[5d96s1]	3D[5d96s1]	3D[5d96s1]	0
AuB	Χ1Σ+	1σ22σ23σ21π21π21δ21δ2	σ2πx2πy2	2S[5d106s1]	2S[5d106s1]	2S[5d106s1]	0
HgB	Χ2Π	1σ22σ23σ21π21π22π11δ21δ2	σ2π2	1S[5d106s2]	1S[5d106s2]	1S[5d106s2]	0
	Χ2Σ+	1σ22σ23σ24σ11π21π21δ21δ2	(π2π2)	1S[5d106s2]	1S[5d106s2]	1S[5d106s2]	0

^a Ref. [21]. ^b Expt values of the energy separation between the term with the lowest in energy spin–orbit coupling angular momentum quantum number J (average term).

On the contrary, the NPA charge on the metals as the M changes from left to right has the same shape for all three rows apart, from the MnB of the first row; see Figure 4c. Finally, the change in the vibrational frequencies with respect to M for the three rows has the same trend; see Figure 4f. In most MB molecules, the second- and third-row MBs have larger ω_e values than the MBs of the first row.

The dissociation energy per bond, i.e., per two bonding electrons, and the number of the formed bonds for all MBs of the first, second, and third rows are depicted in Figure 5. Regarding the diagram of the D_e/bond, the general shape is the same for all three rows. The largest D_e/bond, except for the first two and the last MBs, is observed for the MBs of the third row. Thus, the fact that the 4d series exhibits greater intrinsic bond energies [26] is explained with the formation of more multiple bonds than in some MBs of the 3d and 5d series, while the D_e/bond is the highest for the third row, as expected.

Regarding the number of formed bonds, it seems that the MB molecules of the first and third rows present the same multiple bonds, except the (n)d³(n + 1)s² atoms, where, for VB, the ground state is X⁷Σ⁺ and for ΤaB it is X⁵Δ. The difference in spin multiplicities results in different orders of bonding. The largest observed difference between MBs of the second row with the corresponding MBs of the first and third rows is for the fifth MB, i.e., TcB (quadruple bond) vs MnB and ReB (two and a half bonds). Finally, the X states of the RuB and RhB molecules of the second row also have quadruple bonds, while the corresponding MB molecules of the other rows have triple bonds.

The formation of quadruple bonding in the MB part of the complexes has been reported in the literature. For instance, in the anionic complex FeB(CO)₃⁻, a quadruple bond is formed in the FeB part. The BFe(CO)₃⁻ anion was calculated via DFT and DLPNO-CCSD(T) methodologies and identified using mass-selected infrared photodissociation spectroscopy in the gas phase [63].

Finally, it should be noted that in the transition-metal molecules, highly correlated electrons are involved in the spin−orbit interactions, and relativistic effects exist. The use of basis sets with pseudopotentials of course simplified the complicated effects of the motion of the core (non-valence) electrons. Furthermore, 5d MB molecules suffer from strong spin−orbit effects, which here have been neglected. The use of atomic spin−orbit stabilization [64] may lead to reductions in the calculated dissociation energies [25]. However, our calculated values are in very good agreement with the experimental ones, with the exception of TaB, since both M and MB species are subject to the spin–orbit effects; thus, via the cancellation of errors, the D_e results are good. Specifically, our B3LYP/aug-cc-PVQZ(-PP) values are in better agreement with the experimental ones, and the energy differences between experimental and calculated values range from 0.046 (OsB) to 0.410 eV (LaB), apart from TaB, where the calculated value overestimates the experimental D₀ by 0.749 eV. On the contrary, for the second-row MBs, the energy differences between experimental and calculated values range from 0.002 eV to 0.343 eV. Finally, note that the main aim of the present study is to study the periodic bonding schemes and trends for MB molecules over three rows, providing a quantitative and qualitative picture of the chemical bonding in the MB species, which is presented here adequately.

4. Computational Details

The ground states of the ZnB and the second- and third-row-transition-metal monoborides, MBs, were calculated employing the B3LYP [55,65], MN15 [56], and TPSSh [57] functionals, along with the correlation-consistent basis sets of Dunning et al., i.e., the aug-cc-pVQZ-PP basis set for all M samples except La, and the aug-cc-pVQZ basis set for B [58,59,60,61]. For the La atom, the def2-QZVPPD basis set was used [62]. Additionally, low-lying excited states were calculated for the NbB, TcB, LaB, TaB, ReB, and HgB molecules. Bond lengths, dissociation energies with respect to the adiabatic products and with respect to ground state products, frequencies, and dipole moments were calculated. The charges were obtained via the natural population analysis, NPA. In the case of the NbB molecule, additional MRCISD (MRCISD+Q)/aug-cc-pVQZ(-PP) calculations were carried out to clarify the ground state. In their reference CASSCF calculations, the eight valence electrons were distributed in ten orbitals (5s4d + 2s2p), resulting in 2060 configuration state functions (CSFs), while the number of MRCISD CSFs were 5.9 × 10⁷ and they were reduced to 1.1 × 10⁷ after the internally contracted approach. All DFT calculations were performed using the GAUSSIAN package [66]. The multireference calculations were carried out using the MOLPRO package [67].

5. Summary and Conclusions

Boron plays an important role in chemistry, biology, and materials science. Diatomic transition-metal borides are important building blocks of many complexes and materials and thus the knowledge of their dissociation energy, bond distances, and bonding analysis dipole moments are very useful. It is interesting that boron forms a variety of orders of bonding from single to quadruple bonds and bonds of different types, i.e., covalent, dative, and ionic bonds. In the present paper, the diatomic borides of transition metals are reviewed and studied.

In the first part, a review on the available experimental and theoretical studies on the first-row-transition-metal borides, i.e., ScB, TiB, VB, CrB, MnB, FeB, CoB, NiB, CuB, and ZnB; the second-row-transition-metal borides, i.e., YB, ZrB, NbB, MoB, RuB, RhB, PdB, AgB, and CdB; and the third-row-transition-metal borides, i.e., LaB, HfB, TaB, WB, ReB, OsB, IrB, PtB, AuB, and HgB, is presented. There was a gap in the literature regarding TcB, which is studied here for the first time in detail. While where there were doubts regarding which state is the ground state for some MBs, here, it is clarified.

In the second part, the second- and third-row-transition-metal borides, MBs, are studied via DFT calculations using the B3LYP, TPSSh, and MN15 functionals in conjunction with the aug-cc-pVQZ-PP_M/aug-cc-pVQZ_B basis sets. In the case of the NbB molecule, additional multireference calculations, i.e., MRCISD(MRCISD+Q)/aug-cc-pVQZ(-PP), were carried out to clarify which state is the ground state, because DFT could not decipher it. Bond distances, dissociation energies, frequencies, dipole moments, and natural NPA charges are presented. Comparisons between the MB molecules of all three rows and their bonding are presented. All differences and similarities are analyzed. Both result from the differences in bonding schemes.

It was found that, apart from RhB which was recently reported to form quadruple bonds [2,4], RuB and TcB form quadruple bonds in their ground states as well. The X states of these three molecules present the same quadruple bonding, σ²σ²π_x²π_y², and as the metal is escalated from Tc to Rh, the additional valence electron is added to the single occupied d orbitals, while the X states change from X³Σ⁻ (TcB) to X²Δ (RuB) and then to X¹Σ⁺ (RhB). Finally, here, we studied the TcB molecule, i.e., three states were calculated, filling the gap that existed in the literature.

As a final remark, it has been reported that the states of the diatomic and triatomic molecules of sulfides are involved in complexes and solid or 2D materials as building blocks, explaining the variety of their morphologies [19,20]. Similarly, the present data may present a new approach to exploring the properties of solid state and 2D metastable polymorphic materials involving transition-metal borides, while they may assist in explaining the catalytic properties of complexes, including transition-metal boride bonds.