The Role of Quadruple Bonding in the Electron Transport through a Dimolybdenum Tetraacetate Molecule

A dimolybdenum tetraacetate (Mo2(O2CCH3)4) molecule is embedded between two electrodes formed by semi-infinite 1D monatomic chains of lithium, aluminum, and titanium atoms. Electron transport through the Mo2(O2CCH3)4 molecule is calculated. The role of quadrupole bonding in the transport properties of the studied systems is analyzed.


Introduction
Dimolybdenum tetraacetate Mo 2 (O 2 CCH 3 ) 4 (DMT) is a molecule whose electronic structure is characterized by a quadruple Mo-Mo bonding with the σ 2 π 4 δ 2 configuration where eight electrons are 4d valence electrons of two molybdenum atoms [1][2][3][4][5]. Theoretical chemistry pays particular attention to the properties of δ bonding, which is weaker than σ and π bonds but nonetheless affects the electronic absorption spectra and the spatial structure of the molecules with quadruple bonds [6,7]. Since DMT molecules can form ordered structures on various surfaces, they are currently considered as promising molecular units for the design of nanoscale electronic devices [8][9][10]. The role of σ, π, and δ bonds in the electron transport through a single molecule can also be discussed, since these bonds are formed by different overlaps of d-orbital lobes.
Electron transport through a single molecule is commonly described using a method based on the nonequilibrium Green's function (NEGF) formalism [11][12][13]. The NEGF technique allows calculating the transmission function T(E) showing the sum probability for an electron with energy E to pass from the source to the drain through channels represented by the molecular orbitals (MOs) of the electrode-molecule-electrode system. T(E) depends both on the electronic structure of the electrodes and the molecule and on the interactions between them. In the case of weak contacts between the molecule and the electrodes, T(E) reflects the structure of energy levels of the MOs of the isolated molecule, and its shape is represented by a series of peaks (resonances) in the vicinity of MO energies [14]. Therefore, we assume that varying the distance between the contacts and the molecule can reveal T(E) peaks corresponding to individual MOs of the embedded molecule that can be used to analyze the participation of σ, π, δ bonds in the electron transport.
In this work, we consider electron transport through a DMT molecule supplemented by two M atoms on both sides and embedded into a system of semi-infinite electrodes. The electrodes were represented by 1D chains of four M atoms (M = Li, Al, Ti) in the left (source) and right (drain) parts of the system, respectively. These s-, p-, and d-block metals are convenient models to study Fermi level shifts (E F ). The 1D chain is the simplest model of electron container needed to calculate transmission function in a molecular system. The central part of the system was represented in two ways (

Quadruple Bond in Mo2(O2CCH3)4 Molecule
The optimized interatomic distances of the DMT molecule are Mo-Mo = 2.134 Å , Mo-O = 2.111 Å , and Mo-Mo-O = 91.62° (see also Tables S1 and S2). These parameters agree with the corresponding experimental values Mo-Mo = 2.093 Å , <Mo-O> = 2.119 Å , and <Mo-Mo-O> = 91.8° [1]. Figure 2 shows MOs characterized by bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions of Mo 4d orbitals, in agreement with the calculation results reported previously for the electronic structure of the DMT molecule [4]. Therefore, we assume that these MOs can be used to estimate the role of σ/σ*, π/π* and δ/δ* interactions in the electron transport through the molecule. The calculated HOMO-LUMO gap is 2.002 eV, in good agreement with the experimental value ~2.69 eV [7].    Figure 2 shows MOs characterized by bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions of Mo 4d orbitals, in agreement with the calculation results reported previously for the electronic structure of the DMT molecule [4]. Therefore, we assume that these MOs can be used to estimate the role of σ/σ*, π/π* and δ/δ* interactions in the electron transport through the molecule. The calculated HOMO-LUMO gap is 2.002 eV, in good agreement with the experimental value~2.69 eV [7]. atoms for monoatomic wires (M6-M6) and by a DMT molecule embedded between M atoms of the monoatomic wires (M6-DMT-M6). System M6-M6 was used as a reference when studying transmission functions in systems (M6-DMT-M6).

Quadruple Bond in Mo2(O2CCH3)4 Molecule
The optimized interatomic distances of the DMT molecule are Mo-Mo = 2.134 Å , Mo-O = 2.111 Å , and Mo-Mo-O = 91.62° (see also Tables S1 and S2). These parameters agree with the corresponding experimental values Mo-Mo = 2.093 Å , <Mo-O> = 2.119 Å , and <Mo-Mo-O> = 91.8° [1]. Figure 2 shows MOs characterized by bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions of Mo 4d orbitals, in agreement with the calculation results reported previously for the electronic structure of the DMT molecule [4]. Therefore, we assume that these MOs can be used to estimate the role of σ/σ*, π/π* and δ/δ* interactions in the electron transport through the molecule. The calculated HOMO-LUMO gap is 2.002 eV, in good agreement with the experimental value ~2.69 eV [7].   The T DMT (E) at the most indicative distances (the shortest, the distance where the peaks narrowing sharply, the longest, and points between them) are given at SI (Figures S1-S3).

System Li 6 -(Mo 2 (O 2 CCH 3 ) 4 )-Li 6
Function T wire (E) for the Li wire has three plateaus ( Figure 3). The embedding of a DMT molecule into a wire of lithium atoms decreases the T DMT (E) values compared to the T wire (E) values. The T DMT (E) for d(Li-Mo) = 2.90 Å has three humps in the energy range where function T wire (E) for the Li wire has three plateaus. The peaks of these humps are located at −2.45 eV, −0.92 eV, and 0.84 eV energies. Function T DMT (E) begins narrowing sharply at the d(Li-Mo) distance equal to 3.10 Å ( Figure S1). The T DMT (E) function for d(Li-Mo) = 3.10 Å has peaks in the same regions at −3.43 eV, −3.28 eV, −2.48 eV, −0.97 eV, and −0.62 eV as well as some peaks above 0 eV.
DMT molecule into a wire of lithium atoms decreases the TDMT(E) values compared to the Twire(E) values. The TDMT(E) for d(Li-Mo) = 2.90 Å has three humps in the energy range where function Twire(E) for the Li wire has three plateaus. The peaks of these humps are located at −2.45 eV, −0.92 eV, and 0.84 eV energies. Function TDMT(E) begins narrowing sharply at the d(Li-Mo) distance equal to 3.10 Å ( Figure S1). The TDMT(E) function for d(Li-Mo) = 3.10 Å has peaks in the same regions at −3.43 eV, −3.28 eV, −2.48 eV, −0.97 eV, and −0.62 eV as well as some peaks above 0 eV.
The LUMO and LUMO+5 (corresponding to δ* and σ* interactions, respectively) of the DMT molecule are the only orbitals that fall into non-zero value regions of both functions Twire(E) and TDMT(E). On the one side, the LUMO+1 and LUMO+2 are close to the dip (−2; −1.5 eV) of function Twire(E). On the other, these two orbitals are close to the TDMT(−1.5eV) peak at d(Li-Mo) = 3.1 Å . Thus, the participation of LUMO+1 and LUMO+2 (corresponding to two π* interactions) in electron transport is ambiguous. The HOMO, HOMO-1, HOMO-2, and HOMO-3 are located beyond the transmission bands and apparently do not participate in conduction. As can be seen, the 1D chain can't be considered as a simple bulk electron container, since the transmission function of unperturbed by molecule 1D chain have some structure with humps, dips and plateau [16]. Thus, the DMT can provide channels for electrons only in the region where the transmission of the reference 1D chain of Li is non-zero.  The LUMO and LUMO+5 (corresponding to δ* and σ* interactions, respectively) of the DMT molecule are the only orbitals that fall into non-zero value regions of both functions T wire (E) and T DMT (E). On the one side, the LUMO+1 and LUMO+2 are close to the dip (−2; −1.5 eV) of function T wire (E). On the other, these two orbitals are close to the T DMT (−1.5eV) peak at d(Li-Mo) = 3.1 Å. Thus, the participation of LUMO+1 and LUMO+2 (corresponding to two π* interactions) in electron transport is ambiguous. The HOMO, HOMO-1, HOMO-2, and HOMO-3 are located beyond the transmission bands and apparently do not participate in conduction. As can be seen, the 1D chain can't be considered as a simple bulk electron container, since the transmission function of unperturbed by molecule 1D chain have some structure with humps, dips and plateau [16]. Thus, the DMT can provide channels for electrons only in the region where the transmission of the reference 1D chain of Li is non-zero. The TDMT(E) function of the Al6-DMT-Al6 system for d(Al-Mo) = 2.70 Å has close to unity values over the largest part of the calculated energy range (Figure 4). The only exception is the energy range from −2.7 eV to −0.27 eV, where TDMT(E) is close to 3. Function TDMT(E) deviates from Twire(E) in the energy region from −7.8 eV to −6.9 eV and above the Fermi level (−3.7 eV) up to 0.3 eV (Figure 4). The transmission function became more resolved at d(Al-Mo) = 4.30 Å , still keeping most of features of short distance TDMT(E). Function TDMT(E) for d(Al-Mo) = 4.30 Å is represented by peaks at −6.69 eV, −5.87 eV, −1.38 eV, and −0.36 eV. The DMT molecule suppresses electron transport compared to the reference Al wire, similar to the system with Li. The energy regions of functions Twire(E) and TDMT(E) (for d(Al-Mo) = 2.70 Å) contain DMT MOs exhibiting bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions. At d(Al-Mo) = 4.30 Å , HOMO-3 (σ-bond), HOMO-2 and HOMO-1 (two π-bonds), LUMO+1 and LUMO+2 (two π*-interactions), LUMO+5 (σ*-interaction) can contribute to TDMT(E). The δ/δ*-interactions (HOMO and LUMO) don't participate in transmission at this distance.

System Ti 6 -(Mo 2 (O 2 CCH 3 ) 4 )-Ti 6
Function T DMT (E) is smaller than T wire (E) over the largest part of the calculated energy range, meaning that the embedded molecule significantly affects the electron transport through the Ti wire (

Discussion
The Mo-Mo distance in DMT (2.13 Å ) is smaller than interatomic distances in 1D chains (2.86 Å -3.04 Å ). The DMT molecule can be viewed as a defect disturbing the electronic structure of the monatomic wire. In addition, the embedded DMT molecule can be considered as a kind of filter that blocks some of the electron transport channels originally present in the 1D monatomic chains. This idea can explain the fact that the transmission functions of systems with an embedded DMT molecule are smaller than those of the wires and that not all MOs exhibiting bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions of Mo 4d orbitals can participate in the electron transport through the DMT molecule.

Theoretical Calculations
The geometry of the dimolybdenum tetraacetate (DMT) molecule was optimized by the ADF engine of the AMS2020 package [17,18] using the BP86 density functional [19] and the TZP all-electron basis set for all atoms [20]. All of the calculated vibrational frequencies were real, indicating that the optimized structure of the DMT molecule corresponds to the energy minimum (Table S1). Scalar relativistic effects were considered within the ZORA approach [21].

Discussion
The Mo-Mo distance in DMT (2.13 Å) is smaller than interatomic distances in 1D chains (2.86 Å-3.04 Å). The DMT molecule can be viewed as a defect disturbing the electronic structure of the monatomic wire. In addition, the embedded DMT molecule can be considered as a kind of filter that blocks some of the electron transport channels originally present in the 1D monatomic chains. This idea can explain the fact that the transmission functions of systems with an embedded DMT molecule are smaller than those of the wires and that not all MOs exhibiting bonding (σ, π, δ) and antibonding (σ*, π*, δ*) interactions of Mo 4d orbitals can participate in the electron transport through the DMT molecule.

Theoretical Calculations
The geometry of the dimolybdenum tetraacetate (DMT) molecule was optimized by the ADF engine of the AMS2020 package [17,18] using the BP86 density functional [19] and the TZP all-electron basis set for all atoms [20]. All of the calculated vibrational frequencies were real, indicating that the optimized structure of the DMT molecule corresponds to the energy minimum (Table S1). Scalar relativistic effects were considered within the ZORA approach [21].
The GGA-BP86 density functional was chosen due to its time cost advantages in T(E) calculations [31]. For efficiency reasons, the radial part of the basis set functions is multiplied by a Fermi-Dirac function controlled by two main parameters: distance from the center R = 7 Å and decay width ∆ = 0.7 [23]. Function T(E) was calculated using the self-consistent field method in the energy range −5 eV < E − E F < 5 eV at a temperature of 316 K (corresponding to the kT energy of 0.001 Hartree).

Conclusions
It was shown on the example of M 6 -DMT-M 6 systems (M = Li, Al, Ti) that an embedded DMT molecule can be considered as a defect decreasing the transmission function. It was revealed that the transmission function depends on d(Mo-M) distances. It was shown that increasing the d(Mo-M) values affects not only the shape of the transmission function but also the number of DMT MOs involved in the electron transport through the molecule. No dependence of the transmission function on different overlaps of d-orbital lobes of Mo was revealed. For M 6 -DMT-M 6 (M = Al, Ti) systems, the highest values of transmission functions were found in the region of LUMO+1 and LUMO+2 characterized by the π* interaction. However, these values can be explained not by the overlaps of d-orbital lobes but by the presence of two such MOs.
Although the results of this work may seem purely theoretical, they provide a first insight into the role of overlapping d-orbital lobes in the electron transport through a molecule. The relevance of the present study is substantiated by the success that has been achieved in experimental works devoted to the fabrication of 1D monatomic chains of metals [32]. The greatest progress in this field was achieved for gold wires [33]. For many other metals, nanowires with a diameter smaller than 20 nm were obtained, and the work continues to reduce their thickness still further [34]. In this regard, the transition to monatomic wires can be viewed as the next step towards the design of single molecule devices.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
There are no conflicts of interest to declare.