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Advances in Physical Chemistry

Volume 2012 (2012), Article ID 720197, 17 pages

http://dx.doi.org/10.1155/2012/720197

## Potential Energy Surfaces for Reactions of X Metal Atoms (X = Cu, Zn, Cd, Ga, Al, Au, or Hg) with YH_{4} Molecules (Y = C, Si, or Ge) and Transition Probabilities at Avoided Crossings in Some Cases

^{1}Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364, 01000 Ciudad de México, DF, Mexico^{2}División de Estudios de Posgrado e Investigación, Instituto Tecnológico de Toluca, A.P. 890, 52140 Metepec Edo. Mex, Mexico

Received 14 July 2011; Revised 4 October 2011; Accepted 22 November 2011

Academic Editor: Laimutis Bytautas

Copyright © 2012 Octavio Novaro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We review *ab initio* studies based on quantum mechanics on the most important mechanisms of reaction leading to the C–H, Si–H, and Ge–H bond breaking of methane, silane, and germane, respectively, by a metal atom in the lowest states in symmetry: X(2nd excited state, 1st excited state and ground state) + YH_{4} H_{3}XYH H + XYH_{3} and XH + YH_{3}. with X = Au, Zn, Cd, Hg, Al, and G, and Y = C, Si, and Ge. Important issues considered here are (a) the role that the occupation of the d-, s-, or p-shells of the metal atom plays in the interactions with a methane or silane or germane molecule, (b) the role of either singlet or doublet excited states of metals on the reaction barriers, and (c) the role of transition probabilities for different families of reacting metals with these gases, using the H–X–Y angle as a reaction coordinate. The breaking of the Y–H bond of YH_{4} is useful in the production of amorphous hydrogenated films, necessary in several fields of industry.

#### 1. Introduction

Here, an overview of potential energy surface (PES) calculations for reactions of a metal atom with a gas molecule has been compiled. Among metal atoms, we consider cadmium, copper, zinc, gallium, aluminum, mercury, and gold, and among gas molecules methane, silane, and germane. The potential energy surfaces of an YH_{4} molecule with a metal atom were determined using *ab initio* Hartree-Fock Self-Consistent Field (HF-SCF) calculations, where the atom core is represented by relativistic effective core potentials (RECPs) [1–5]. These calculations are followed by a Multiconfigurational Self Consistent Field (MC-SCF) study [6]. The correlation energy is accounted for through extensive variational and perturbative second-order multireference Moller-Plesset configuration interaction (MR-CI) analysis of selected perturbations obtained by iterative process calculations using the CIPSI program package [7]. The reference (S) spaces used for the variational CI of the molecular states arising from the three X + CH_{4} asymptotes contain between 108 and 428 determinants, which generate between 7 million and 111 million perturbed MP2 determinants near the region of the reactants and the equilibrium geometry of the methyl-metal-hydride intermediate, respectively. This methodology is particularly useful in the study of systems constituted by a few atoms.

Transition probabilities for the interaction of the lowest excited states of the metal X with tetrahedral gas molecules are studied through one-dimensional Landau-Zener theory. The strategy for obtaining the reaction pathways for X + YH_{4} interactions has been extensively used in references [8–14] based on the original proposal by Chaquin et al. [15]. The initial approach (starting from 20 a.u.) of the X atom to one Y–H bond was done in a C_{s} symmetry in a plane containing the X, Y, and two hydrogen atoms. All the angles and distances obtained by a self-consistent field approach to find the molecular geometry of the HXYH_{3} intermediate are exhibited in Figure 1. From these parameters, = H–X–Y is the first angle fixed in order to optimize the other angles and distances for each point on a potential energy curve, and taking steps of 10 degrees for the angle. When two energy levels cross each other as a function of time, the semiclassical theory of time-dependent Landau-Zener theory [16, 17] for nonadiabatic transition can be utilized. Zener [17] proposed a transition probability from one potential energy curve to another as a linear velocity function, using the distance as the reaction coordinate. Whereas for an angular velocity using the angle as the reaction coordinate, all developments established by Zener are still valid, given that for determining a time-dependent transition probability, the Schrödinger equation must be used. Thus, the transition probability from one-potential energy curve to another is an angular velocity function, using the angle as the reaction coordinate. Our theoretical results predict the formation of XH + YH_{3} or H + XYH_{3} products after breaking the intermediate HXYH_{3}. Some of these products are useful in the industry to get hydrogenated amorphous films (*a*-Y : H) among the other applications [18–30].

The complexity of the organometallic systems, aggregates, and metallic surfaces makes it difficult to determine the fundamental mechanisms which govern their interactions with hydrocarbons. In this situation, one begins with a study of the interaction of a single metallic atom with a hydrocarbon molecule as a model to understand the conditions that the metallic center in these activations requires. Thus, the study of the interaction of the different metallic atomic states with the methane, silane, or germane molecule is fundamental in elucidating the role of various d-, s-, or p-orbital occupations in a metal atom.

Experimental studies [31–38] carried out on the photolysis induced by the metallic atom excitation in a matrix of methane at low temperatures show that when the C–H bond of methane activates through photoexcited metallic atoms deposited in a matrix at low temperatures, the process only happens with radiation wavelengths previously known to be absorbed in the free atom, allowing it to achieve a transition to an excited state. The very low temperature of the matrix fixes the position and also reduces the internal movements of methane molecules contained in it, allowing the interaction of C–H bonds with deposited metallic atoms.

The photochemical studies on the metal-alkane interaction have attracted the attention of researchers for long time. It is worth mentioning that in 1985 the work entitled “Activation of Methane by Photoexcited Copper Atoms and Photochemistry of Methylcopper Hydride in Solid Methane Matrices,” developed in Lash Miller laboratories of the University of Toronto in Canada by the G.A Ozin group [32] was considered in the publication analysis of Scientific Information Institute (current contents) as the most promising work of that year.

At room temperature and standard pressure, methane (CH_{4}) is a colorless, odorless, and flammable gas [39], in fact the simplest hydrocarbon. It is the major constituent of natural gas and is released during the decomposition of plants or other organic compounds, as in marshes and coal mines. Methane is the first member of the alkane series. The strength of the carbon hydrogen covalent bond in methane is perhaps the strongest in all hydrocarbons, and, thus, its use as a chemical feedstock is limited. Despite the high activation barrier for breaking the C–H bond, CH_{4} is still the main starting material for manufacture of hydrogen in steam reforming. The search for catalysts which can facilitate C–H bond activation in methane (105 kcal mol^{−1} to break it) and other low alkanes is an area of research with considerable industrial significance [40].

Silane is a chemical compound (SiH_{4}) analogous to methane, and it is also a gas at room temperature which undergoes spontaneous combustion in air. The name “silane” is also given to a family of compounds that are silicon analogs of alkane hydrocarbons. The radical SiH_{3} is termed silyl. The nomenclature parallels that of alkyl radicals. Silane may also carry certain functional group, just as alkanes do. There is (at least in principle) silicon analog for each carbon alkane. Silanes are useful for several industrial and medical applications. For instance, they are used as coupling agents to adhere glass fibers to a polymer matrix, stabilizing the composite material. They can also be used to couple a bioinert layer on a titanium implant. Other applications include water repellents, masonry protection, control of graffiti, applying polycrystalline silicon layers on silicon wafers when manufacturing semiconductors, and sealants. In addition, silane and similar compounds containing Si–H bonds are used as reducing agents in organic and organometallic chemistry [41–47].

Germane is the chemical compound with the formula GeH_{4}, and an analog of methane. It is the simplest germanium hydride and one of the most useful compounds of germanium. Like the related compounds silane and methane, germane is tetrahedral. It burns in the air to produce GeO_{2} and water. Some processes for the industrial manufacture of germane [48], in which our calculations might be useful, are (a) chemical reduction method, (b) an electrochemical reduction method, and (c) a plasma-based method. The gas decomposes near 600 K to germanium and hydrogen. Germane is used in the semiconductor manufacturing for epitaxial growth of germanium [49]. Organogermanium precursors have been examined as less hazardous liquid alternatives to germane for deposition of Ge-containing films [50]. Germane is flammable and toxic.

The quantum chemistry studies presented here provide valuable information about the activation of methane or silane or germane molecules with metal atoms. Products of these reactions are methyl CH_{3} or silyl SiH_{3} or germyl GeH_{3} radicals, which are key in surface growth of amorphous hydrogenated carbon or silicon or germane films (thin films).

Methane CH_{4}, silane SiH_{4}, and Germane GeH_{4} species turn out to be valuable substances in the industry of the semiconductors since the germane or silane or methane dehydrogenation in gaseous phase is one of the most current methods to obtain semiconductors in the form of amorphous hydrogenated carbon (*a*-C : H) [51] or silicon (*a*-Si : H) [52] or germane (*a*-Ge : H) [53] thin films. The interaction of CH_{3}, SiH_{3}, or GeH_{3} radicals and atomic hydrogen with the surfaces of carbon, silicon or germane films *a*-C : H, *a*-Si : H, or *a*-Ge : H plays a fundamental role in the understanding of the growth of these plasma films at low temperature. Street [54] says that the dehydrogenated material has a very high defect density which prevents doping, photoconductivity, and the other desirable characteristics of a useful semiconductor. While a real crystal contains defects such as vacancies, interstitial, and dislocations, the elementary defect of an amorphous semiconductor is the coordination defect, when an atom has too many or too few bonds [54]. Defect equilibrium is in general described by a reaction of the type
where A–D are different configurations of point defects, dopants, electronic charges, and so forth. The properties of interest are the equilibrium state and the kinetics of the reaction [54]. Calculations as those accomplished by us mentioned previously might help to find these properties. As an example, our calculated energy of the intermediate corresponds to the defect formation energy which determines the equilibrium defect density, as part of the kinetics of reaction.

Methyl radical absorption on carbon or hydrocarbon thin films is key in thin-film growth at low-temperature plasmas (<) using hydrocarbon precursor gases.

The current production procedure of amorphous hydrogenated carbon (*a*-C : H) or silicon (*a*-Si : H) films is the deposition by means of the decomposition of methane or silane through glow discharges produced by radio frequency (RF). This method is known as plasma-enhanced chemical vapor deposition (PECVD) [55].

The breaking of the C–H bond of CH_{4} is useful for generating amorphous hydrogenated carbon (*a*-C : H) films, which represent a class of high-technology materials with mechanical, optical, chemical, and electrical properties among polymeric, graphite, and diamondoid films.

The (*a*-C : H) is of interest to the electronic industry as a viable and cheap semiconductor that can be prepared in an ample rank of layers. The fine layer of *a*-C : H, also well-known as diamondoid carbon, is used as a revesting material of hard and low friction. The polymeric film *a*-C : H has a strong photoluminescence and is being developed as an electroluminescent material. It is also used as dielectric in metal-insulator-metal switch in screens of active matrix. The ability of deposition near to room temperatures using cheap methods of chemical vapor deposition (CVD) makes this material useful for the industry. The amorphous nature of these materials and their relative facility of deposition make them ideals for its use in a great amount of applications such as in panels of flat screen and diamondoid technology [56].

The *a*-C : H films are prepared through a glow discharge of RF in a pure methane atmosphere at different gas pressures in which the methane decomposition generates the methyl CH_{3} radical, which plays a preponderant role in the generation of amorphous hydrogenated carbon surfaces [57, 58]. The electronic industry takes profit of the previous proceeding on the formation of diamondoid films.

Dense amorphous hydrocarbons have some of the highest densities among hydrocarbon and fall between crystalline diamonds and adamantanes, according to Angus [59]; this is the property that makes it so attractive to the electronic industry.

In case of diamondoid formation,(1)some excited metallic atoms break the C–H bond of methane with the consequent production of methyl CH_{3} radicals [8, 60, 61]. Knowing that a dangling bond is an unsatisfied valence on an immobilized atom, the methyl CH_{3} radicals undergo a chemisorption process controlled by the creation of dangling bonds in the *a*-C : H surface by an atom H. This is a proper process in the formation of diamondoid films in the electronic industry [62],(2)Ramĺrez and Sinclair [63] (and Velasco-Santos et al., [64]) affirm that carbonaceous natural products can have different allotropic forms of the carbon, since the amorphous carbon has graphite transitions at different temperatures when some specific metals are in contact with carbon and act as catalysts,(3)methyl CH_{3} radicals can also be generated by photodissociation of aromatic hydrocarbon, consequently amorphous hydrogenated carbon thin films can be produced. Taguchi et al. [65] used two classes of laser (ArF or KrF) and two types of aromatics (benzene C_{6}H_{6} or toluene C_{6}H_{5}–CH_{3}) to generate diamondoid films.

The breaking of Si–H bond of SiH_{4} is useful in manufacture of semiconductor films of amorphous hydrogenated silicon (*a*-Si : H). Before the *a*-Si : H material has been developed, the research was on amorphous silicon without hydrogen, prepared by sputtering or by thermal evaporation.

There are at least two models for the generation of amorphous hydrogenated silicon (*a*-Si : H).(i) In the thermodynamic approach [62, 66, 67], the formation of dangling bonds, the principle defect in *a-*Si:H, is attributed to the breaking of weak Si–Si bonds caused by mobile H that is released from Si–H bonds [68]:(ii) In the Matsuda-Gallagher-Perrin MGP model [66, 67], SiH_{3} *is assumed to be the only growth precursor. This assumption is based on the (presumed) dominance of this radical in plasmas leading to the device-quality a-*Si : H.

The central assumption in the MGP model is that the SiH_{3} reaching the *a-*Si : H can go to a weakly adsorbed (physisorbed) state forming a three-center Si–H–Si bond on a surface Si–H site. The activation of silane molecules has received a lot of attention as much in the experimental aspect [61, 69–72] as in the theoretical [9, 73]. The activation of the Si–H bond is important as much in the processes of polymerization of silane [74, 75], as in organometallic catalytic reactions [76–78].

The growth of amorphous hydrogenated silicon films *a*-Si : H in silicon substrata through PECVD in silane is widely used in the manufacture of electronic, optoelectronic, and photovoltaic devices. Amorphous semiconductors of thin films are used in an ample variety of applications such as solar cells, TFT, photoreceptors, and apparatus of images [79].

The *a*-Ge : H films are prepared through a glow discharge of RF in a pure germane atmosphere at different gas pressures in which the germane decomposition generates the germyl GeH_{3} radical, which plays a preponderant role in the generation of amorphous hydrogenated germane surfaces [53].

Here we stress some important results at metal-methane, metal-silane, and metal-germane interactions grouped in three different families (coinage metals: Cu and Au. Pseudotransition metals: Zn, Cd, and Hg, and metals: Al and Ga). Castillo et al. [8, 60, 80] carried out calculations of potential energy surfaces of the interactions copper methane and zinc methane with the aim of determining the mechanisms of reaction that involve the three lowest states of the copper atom (^{2}S, ^{2}D, and ^{2}P) as well as to determine the reaction routes that govern the interaction of the three lowest states of the zinc atom (^{1}S, ^{3}P, and ^{1}P) in the process of the C–H bond activation of the methane molecule. Luna-García et al. [11, 12, 81] found the interaction potential curves of the mercury-Germane, cadmium-Germane, copper-silane, and copper-germane in the three lowest states of each metal; he improved a computational methodology to get the products of the breaking of the intermediate. Pacheco-Sánchez et al. [13, 14, 40] achieved the calculation of gallium-methane and gallium-silane interactions as much in the ground state as in the two lowest excited states of gallium; he extended Landau-Zener theory [82–85] to use the angle instead of the distance as reaction parameter in transition probability calculations at avoided crossings. Transition probability theory is described here when the reaction parameter is distance or angle. In addition, our group has considered also the following interactions: cadmium methane [86], zinc silane [9], cadmium silane [10], mercury silane [9], gold silane [87], and recently aluminum methane [88].

#### 2. Transition Probability Theory

When two potential energy curves are very near to each other, it seems that they crossover. Actually, in the apparent crossover point, the system is degenerate, since the two different electronic configurations have the same energy [89]. This introduces a resonance energy that separates the surfaces slightly, in such way that they never intersect but only closely approach to each other before repelling. Whereas the wave function of the molecule in a given curve is of one character before the crossing point (CP) and of another character after it, the wave function of the other curve is reversed in turn; an example is given in Figure 2, where we can see that while the wavefunction has 6p^{1} character before CP at and 6s^{1} character after CP at , the wavefunction has 6s^{1} character before CP at and 6p^{1} character after CP at . If changes with a finite velocity, the probability that the molecule changes of a wave function to the other when passing through the crossing point is in such way that its electronic state is represented by a linear combination of the type
where or according to the reaction coordinate in study. By convenience in the calculation of and , the eigenfunctions and are expressed in terms of two other wave functions and with energies and , respectively, which intersect when they are plotted as functions of the internuclear distance (or angle). Due to the fact that and are not exact eigenfunctions of the whole Hamiltonian at the crossing point, the interaction energy between the two states and have to be included, then and , where is the difference between the exact eigenvalues and and the approximate eigenvalues , : . Normalizing and orthogonalizing all the wavefunctions involved, and following the one-dimensional (distance as reaction coordinate) developments established by Zener [17] for a time-dependent Schrödinger equation, we obtain the transition probability as
where
The denominator can be expressed as
where is the velocity ( or , means linear and means angular velocity) at which the system crosses , and is the difference of the slopes of the two potential surfaces crossing at . Finally, we have
for the transition probability of nonadiabatic behavior. The probability for a system remaining in the initial energy surface is then
Rosenkewitsch [90] states that Zener [17] has obtained a similar formula to that one obtained by Landau [16] (or (7) above)
where is the relative velocity, and and are “forces” acting on the two states. Making the identification which corresponds to the change of pure kinetics energy with time, we can almost have the equation found by Zener ((7) when ), because the exponent of the Landau formula also has a factor of .

Explicit calculations of transition probabilities of nonadiabatic behavior using (7) are straightforward when the reaction coordinate is the distance [91, 92] (in these cases gas is hydrogen and metal is ruthenium [91] and platinum [92]); however, when the latter is the angle, it will be necessary to calculate the angular velocity and the moment of inertia as accomplished in references [40, 82–85, 87, 88] for tetrahedral molecules interacting with metals.

##### 2.1. Interactions of the Cu, Au Coinage Metals with YH_{4}

Castillo et al. [60] found that copper in its second excited state (^{2}P: 3d^{10} 4p^{1}) breaks the C–H bond of methane, and its avoided crossing with the first excited state allows (^{2}D: 3d^{9} 4s^{2}) to surpass a barrier of 48 kcal mol^{−1}. We have a transition energy of ~83 kcal mol^{−1} between the states ^{2}P and ^{2}D and transition energy of ~43 kcal mol^{−1} between the states ^{2}D and ^{2}S (see Table 1 and Figure 3). Transition probability (TP) values at four avoided crossings [82] are obtained when copper is photoexcited to this second excited state Cu (^{2}P: 3d^{10} 4p^{1}), given that this copper atom is inserted in the methane molecule. It is also noteworthy [60] that the ground state is initially repulsive, but due to the avoided crossings, it became attractive after surmounting a rather high barrier. This behavior is initially exhibited at the first minimum (69 kcal mol^{−1}, 26°) of the 5 ^{2}A′ state forming the HCuCH_{3} intermediate, which when continuing its path has a probability of crossing towards the 4 ^{2}A′ state with 4p^{1} character and a probability of continuing by the same 5 ^{2}A′ state with 4s^{2} character. When following via 4 ^{2}A′ has a probability of crossing towards the 3 ^{2}A′ state with 4p^{1} character and a probability of continuing by the same 4 ^{2}A′ state with 4s^{2} character. When continuing via 3 ^{2}A′, a very small potential barrier is surmounted with a probability of crossing towards the 2 ^{2}A′ state with 4p^{1} character, and a probability of continuing by the same 3 ^{2}A′ state with 4s^{2} character. When continuing via 2 ^{2}A′, it descends until reaching X ^{2}A′, then it has a probability of crossing towards the X ^{2}A′ state with 4p^{1} character and a probability of continuing by the same 2 ^{2}A′ state (see Figure 3) with 4s^{1} character. Finally, when continuing via X ^{2}A′ state (with 4p^{1} character), while the theoretical energy of the HCuCH_{3} intermediate is 14 kcal mol^{−1}, the experimental energy is between 15 and 25 kcal mol^{−1}. From here we can build the corresponding potential energy surfaces leading to the products by calculating the energy against the distance [60]. These products are H + CuCH_{3} and HCu + CH_{3}, with theoretical energies 52 and 41 kcal mol^{−1}, respectively, and experimental energy values 46 and 40 kcal mol^{−1}, respectively, as mentioned in Table 1.

Pacheco-Sánchez et al. [87] have found that Gold in its second excited state (^{2}P: 5d^{10} 6p^{1}), through a series of avoided crossings that diminish the barriers for the ground state and first excited state, breaks the Si–H bond of silane, finally overcoming a barrier of 24.0 kcal mol^{−1} of the ground state. The experimental transition energy between the excited state (^{2}P: 5d^{10} 6p^{1}) and the ground state (^{2}S: 5d^{10} 6s^{1}) is 114.28 kcal mol^{−1}, which is comparable to the transition energy of 109.8 kcal mol^{−1} calculated by sPacheco-Sánchez et al. (Table 1). Then, the HAuSiH_{3} intermediate encounters four avoided crossings between trajectories C ^{2}A′, B ^{2}A′, A ^{2}A′, and X ^{2}A′ of the energy surfaces. When gold is photoexcited to its second excited state ^{2}P: 5d^{10} 6p^{1}, it initially passes through the avoided crossing around (77 kcal mol^{−1}, 28°) of D ^{2}A′ state, and there are two possibilities of the intermediate formed for following a trajectory. Taking it to pass the avoided crossing, it has a probability [87] for crossing towards C ^{2}A′ maintaining the 6p^{1} character, where it finds another avoided crossing, and there is a probability of 0.737 for crossing toward B ^{2}A′ also maintaining the 6p^{1} character. Here it finds another avoided crossing, and there is a probability of 0.803 of crossing toward the A ^{2}A′ still maintaining the 6p^{1} character (see Figure 4). Henceforth, it finds the last avoided crossing, and there is a probability of 0.541 of crossing to the X ^{2}A′ maintaining the 6p^{1} character too, where the products are reached at 120°. Finally, the intermediate with energy 3.34 kcal mol^{−1} evolves toward the HAu + SiH_{3} and AuSiH_{3} + H products, whose energies are 21.9 and 30.91 kcal mol^{−1}. These products are of greater energy than the reactants, something typical in an endergonic reaction in which the reaction requires absorption of energy.

Luna-García et al. [12] have found that copper in its second excited state (^{2}P: 3d^{10} 4p^{1}) breaks the Si–H bond of silane, in a manner quite similar to the previous cases surpasses a barrier of 26.0 kcal mol^{−1} of the ground state. The experimental transition energy between the excited state (^{2}P: 3d^{10} 4p^{1}) and the ground state (^{2}S: 3d^{10} 4s^{1}) is 87.8 kcal mol^{−1}, which agrees with the transition energy of 87.7 kcal mol^{−1} theoretically calculated (Table 1). The initial formation of the HCuSiH_{3} intermediate starts at the minimum (70 kcal mol^{−1}, 20°) with 4p^{1} character and encounters four avoided crossings among trajectories 4 ^{2}A′, 3 ^{2}A′, 2 ^{2}A′, and X ^{2}A′ of the energy surfaces (see Figure 5). Finally, due to the endergonic nature of this reaction, the intermediate with energy 5.8 kcal mol^{−1} (and 4p^{1} character) is divided into the HCu + SiH_{3} and CuSiH_{3} + H products with theoretical energy values of 33.2 and 36.0 kcal mol^{−1}, respectively (see Table 1). In the same Table, we could only report the experimental energy value 27.7 kcal mol^{−1} of HCu + SiH_{3} products.

Luna-García et al. [12] have found that again copper in its excited state (^{2}P: 3d^{10} 4p^{1}) breaks a Ge–H bond of germane, overcoming a barrier of 27.0 kcal mol^{−1} of the ground state. The experimental transition energy between the excited state (^{2}P: 3d^{10} 4p^{1}) and the ground state (^{2}S: 3d^{10} 4s^{1}) is of 87.8 kcal mol^{−1}, which agrees with the transition energy of 87.7 kcal mol^{−1} theoretically calculated (Table 1). The initial formation of the HCuGeH_{3} intermediate starts at the minimum (72 kcal mol^{−1}, 17°) with 4p^{1} character and passes through four avoided crossings with the trajectories 4 ^{2}A′, 3 ^{2}A′, 2 ^{2}A′, and X ^{2}A′ of the energy surfaces (see Figure 6). Finally the intermediate with energy 1.6 kcal mol^{−1} (and 4p^{1} character) evolves toward HCu + GeH_{3} and CuGeH_{3} + H products with 27.1 and 33.2 kcal mol^{−1}, respectively (see Table 1). The latter is due to the endergonic nature of the reaction. In the same table, there is only the experimental energy value 21.7 kcal mol^{−1} of HCu + GeH_{3} products.

##### 2.2. Interactions of the Zn, Cd, and Hg Pseudotransition Metals with YH_{4}

Castillo et al. [8] found that Zinc in its first excited state (^{1}P: 3d^{10} 4s^{1} 4p^{1}) lying 141 kcal mol^{−1} high breaks the C–H bond of methane, while the Zn ground state presents a quite high activation barrier of 90.5 kcal mol^{−1} (see Figure 7, Table 2(a)). The TP of the potential energy surfaces (2 ^{1}A′ → X ^{1}A′) is obtained for the reaction: Zn (^{1}P: 4s^{1} 4p^{1}) + CH_{4} → Zn (^{1}S: 4s^{2}) + CH_{4} through one avoided crossing; thenceforth, the products of the reaction Zn + CH_{4} are reached. This happens when zinc is photoexcited to the second excited state Zn (^{1}P: 3d^{10} 4s^{1} 4p^{1}), which allows this zinc atom to be inserted in the methane molecule. The latter effect is initially exhibited at the minimum (93 kcal mol^{−1}, 59°) of the 2 ^{1}A′ state forming the intermediate HZnCH_{3}, which when continuing its path has a probability [82] of crossing towards the X ^{1}A′ state and a probability of continuing by the same 2 ^{1}A′ state. When following via X ^{1}A′ (see Figure 7), at the minimum of this state, the product formation can be obtained [82]. Finally, acquiring the formation of the HZnCH_{3} intermediate with a calculated energy of ~25 kcal mol^{−1} that passes through the avoided crossing of 2 ^{1}A′ and X ^{1}A′ sates, with the subsequent formation of HZn + CH_{3} and ZnCH_{3} + H final products, with the corresponding energies ~82 and ~93, respectively; as it can be seen in Table 2(a).

Ramírez-Solís and Castillo [86] compared the C_{3v} versus C_{2v} symmetries of Cd (^{1}S, ^{3}P, ^{1}P) + CH_{4} interactions in the edge on geometry orientation. For C_{3v} symmetry, they calculated two more geometry orientations. Before starting their molecular calculations, they successfully reproduced the lowest states of the atomic spectra of cadmium as mentioned in Table 2(a) which corresponds to 125.6 kcal mol^{−1} for the second excited state. They did not construct the potential energy surfaces; however, they calculated the breaking of the C–H bond of methane with the consequent formation of an intermediate with energy of 31.9 kcal mol^{−1} and its decomposition in possible products: HCd + CH_{3} and H + CdCH_{3} with energy of 56.6 and 67.9 kcal mol^{−1}, respectively (see Table 2(a)).

Luna-García et al. [9] found that Zinc in its singlet second excited state (^{1}P: 3d^{10} 4s^{1} 4p^{1}) breaks the Si–H bond of silane, by reversing the initially repulsive ground state curve, thus, overcoming a barrier of ~80 kcal mol^{−1} of the latter. The experimental transition energy between the excited state (^{1}P: 3d^{10} 4s^{1} 4p^{1}) and the singlet ground state (^{1}S: 3d^{10} 4s^{2}) is 133.7 kcal mol^{−1}, which is comparable to the calculated transition energy of 141.0 kcal mol^{−1} of the second excited state of Zinc calculated theoretically (Table 2(b), Figure 8). The formation of the HZnSiH_{3} intermediate arising at the minimum (81 kcal mol^{−1}, 60°) comes with 4p^{1} character and approaches the avoided crossing between 2 ^{1}A′ and X ^{1}A′ states keeping 4p^{1} character until the end of the latter state with an energy 11.5 kcal mol^{−1}. Finally, due to the endergonic nature of this reaction, the intermediate leads toward the HZn + SiH_{3} and ZnSiH_{3} + H products with calculated energies of 74.0 and 84.0 kcal mol^{−1}, respectively (see Table 2(b)). The experimental energy for HZn + SiH_{3} products is 70.4 kcal mol^{−1} in agreement with that calculated theoretically. We have not calculated transition probabilities yet.

Luna-García et al. [10] found that cadmium in its singlet second excited state (^{1}P: 4d^{10} 5s^{1} 5p^{1}) breaks the Si–H bond of silane, creating a barrier of ~89 kcal mol^{−1} for the ground state. The experimental transition energy between the excited state (^{1}P: 4d^{10} 5s^{1} 5p^{1}) and the ground state (^{1}S: 4d^{10} 5s^{2}) is 124.9 kcal mol^{−1}, which is comparable to the transition energy of 128.5 kcal mol^{−1} calculated theoretically (Table 2(b), Figure 9). The formation of the HCdSiH_{3} intermediate at the minimum (92 kcal mol^{−1}, 45°) comes with 5p^{1} character, passes through the avoided crossing between 2 ^{1}A′ and X ^{1}A′ pathways, and keeps 5p^{1} character until the minimum (20.7 kcal mol^{−1}, 180°) of the X ^{1}A′ state is reached (see Figure 9). Finally, the intermediate is broken reaching the HCd + SiH_{3} and CdSiH_{3} + H products with calculated energies of 75.5 and 89 kcal mol^{−1}, respectively (see Table 2(b)). The experimental energy for HCd + SiH_{3} is 74.0 kcal mol^{−1}. We have not calculated transition probabilities yet.

Luna-García et al. [9] found that mercury in its singlet second excited state (^{1}P: 5d^{10} 6s^{1} 6p^{1}) breaks the Si–H bond of silane, leading to a ground state barrier of ~102 kcal mol^{−1}. The experimental transition energy between the excited state (^{1}P: 5d^{10} 6s^{1} 6p^{1}) and the singlet ground state (^{1}S: 5d^{10} 6s^{2}) is 154.6 kcal mol^{−1}, which agrees to the transition energy of 157.6 kcal mol^{−1} theoretically calculated (see Figure 10 and Table 2(b)). The formation of the HHgSiH_{3} intermediate arising at the minimum (106 kcal mol^{−1}, 70°) comes with 6p^{1} character and passes through an avoided crossing between trajectories 2 ^{1}A′ and X ^{1}A′ of the energy surfaces maintaining 6p^{1} character. The transition probability [84] at the avoided crossing among the latter states is obtained when mercury is photoexcited to the Hg ^{1}P: 5d^{10} 6s^{1} 6p^{1} excited state, allowing it to be inserted in silane (SiH_{4}) molecule. When the reaction pathway passes the crossing point and continues its way has a probability of crossing toward curve X ^{1}A with 6p^{1} character and a probability of of continuing in the same curve 2 ^{1}A′ (see Figure 10) with 6s^{2} character. When going toward the X ^{1}A′ state, it allows the formation of a stable intermediate at the minimum (28 kcal mol^{−1}, 180°) of the state. Finally, the latter is broken and reaches either the HHg + SiH_{3} or HgSiH_{3} + H products with energies 87.6 and 95.4 kcal mol^{−1}, respectively.

Luna-García et al. [11] have found that Cadmium in its singlet second excited state (^{1}P: 4d^{10} 5s^{1} 5p^{1}) breaks the Ge–H bond of germane, while producing a ground state barrier of 83.6 kcal mol^{−1} (see Figure 11). The experimental transition energy between the excited state (^{1}P: 4d^{10} 5s^{1} 5p^{1}) and the ground state (^{1}S: 4d^{10} 5s^{2}) is 124.9 kcal mol^{−1}, which agrees with the transition energy of 128.5 kcal mol^{−1} calculated theoretically (Table 2(c)). The formation of the HCdGeH_{3} intermediate at (89 kcal mol^{−1}, 50°) comes with 5p^{1} character and goes by an avoided crossing between the 2 ^{1}A′ and X ^{1}A′ states maintaining a 5p^{1} character until the next minimum at (13.6 kcal mol^{−1}, 180°) (see Figure 11). Finally the intermediate is broken in the HCd + GeH_{3} and CdGeH_{3} + H products with calculated energies 70.3 and 79.9 kcal mol^{−1}, respectively. The experimental energy for HCd + GeH_{3} is 68.9 kcal mol^{−1}. We have not calculated the transition probabilities yet.

Luna-García et al. [11] have found that mercury in its excited state (^{1}P: 5d^{10} 6s^{1} 6p^{1}) breaks the Ge–H bond of germane, while forming a ground state barrier of 86.1 kcal mol^{−1}. The experimental transition energy between the excited state (^{1}P: 5d^{10} 6s^{1} 6p^{1}) and the ground state (^{1}S: 5d^{10} 6s^{2}) amounts to 154.6 kcal mol^{−1}, in good agreement with the transition energy of 157.6 kcal mol^{−1} theoretically calculated (Table 2(c)). The transition probability [84] between the PES is obtained for the reaction Hg ^{1}P(5d^{10} 6s^{1} 6p^{1}) + GeH_{4} → Hg ^{1}S(5d^{10} 6s^{2}) + GeH_{4}. When mercury is photoexcited to the Hg (^{1}P:5d^{10} 6s^{1} 6p^{1}) second excited state, the mercury atom is inserted in germane (GeH_{4}) molecule. This effect is observed in the minimum (90 kcal mol^{−1}, 60°) of the 2 ^{1}A′ state arriving with 6p^{1} character, where the HHgGeH_{3} intermediate is formed. When this intermediate continues its way, it has a probability of crossing toward X ^{1}A state with 6p^{1} character and a probability of of continuing in the same 2 ^{1}A′ state (see Figure 12) with 6s^{2} character. When going toward the X ^{1}A′ state, it leads to the formation of the product at the minimum (21 kcal mol^{−1}, 180°) of the state. Finally, the intermediate evolves towards the HHg + GeH_{3} and HgGeH_{3} + H products with calculated energies 80 and 87.7 kcal mol^{−1}, respectively, while the experimental energy for HHg + GeH_{3} is 75.3 kcal mol^{−1}.

##### 2.3. Interactions of the Al, Ga Metals with YH_{4}

Pacheco-Blas et al. [88] found that the aluminum in its doublet second excited state (^{2}D: 3s^{2} 3d^{1}) breaks the C–H bond of methane, as does the 2 ^{2}A′ state after surpassing an activation barrier of 14.0 kcal mol^{−1} (see Figure 13). The experimental transition energy between this second excited state (^{2}D: 3s^{2} 3d^{1}) and the ground state (^{2}P: 3s^{2} 3p^{1}) is 92.5 kcal mol^{−1} and agrees with the transition energy of 93.9 kcal mol^{−1} theoretically calculated (see Table 3(a)). The transition probability [88] among the corresponding PES for the reaction Al ^{2}D(3s^{2} 3d^{1}) + CH_{4} → Al ^{2}P(3s^{2} 3p^{1}) + CH_{4} starts with the formation of the HAlCH_{3} intermediate and goes through two avoided crossings with the 3 ^{2}A′, 2 ^{2}A′, and X ^{2}A′ states. When aluminum is photoexcited to its Al ^{2}D: 3s^{2} 3d^{1} second excited state, it is inserted into a C–H bond of methane at the minimum (87 kcal mol^{−1}, 20°) in the 3 ^{2}A′ state with 3d^{1} character. Under these conditions, the HAlCH_{3} intermediate is formed. As the latter continues its path, it has a probability of 0.85 of crossing toward the 2 ^{2}A′ state with 3d^{1} character going to the Al ^{2}D: 3s^{2} 4s^{1} first state and a probability of 0.15 of continuing in the same 3 ^{2}A′ state with 4s^{1} character. When the intermediate continues its pathway, there is a probability of 0.89 for crossing from the 2 ^{2}A′ state to the X ^{2}A′ with 3d^{1} character (see Figure 13), where the most stable intermediate is found at = 120° and 6.5 kcal mol^{−1}. When this occurs, the final products are reached from the Al + CH_{4} reaction, that is, to say, the intermediate is broken in the HAl + CH_{3} and AlCH_{3} + H products with calculated energies 42.8 and 48.2 kcal mol^{−1}, respectively. In this case, the corresponding experimental [36] energy is 45.0 kcal mol^{−1} in both cases.

Pacheco-Sánchez et al. [14] have found that gallium in its doublet first excited state (^{2}S: 4s^{2} 5s^{1}) breaks the C–H bond of methane, producing in the lowest state an activation barrier of 53.7 kcal mol^{−1}. The experimental transition energy among the excited state (^{2}S: 4s^{2} 5s^{1}) and the ground state (^{2}P: 4s^{2} 4p^{1}) is 69.3 kcal mol^{−1}, in agreement with the transition energy 72.7 kcal mol^{−1} theoretically calculated (see Table 3(a)), within the 3 kcal mol^{−1} of tolerance. The transition probability [14] among the corresponding PES for the reaction Ga ^{2}S(4s^{2} 5s^{1}) + CH_{4} → Ga ^{2}P(4s^{2} 4p^{1}) + CH_{4} starts with the formation of the intermediate HGaCH_{3} and passes through two avoided crossings with 3 ^{2}A′, 2 ^{2}A′, and X ^{2}A′ states. When gallium is photoexcited to its Ga ^{2}S: 4s^{2} 5s^{1} first excited state, it is inserted into a C–H bond of methane at the minimum (60 kcal mol^{−1}, 25°) in the 3 ^{2}A′ state with 5s^{1} character. Under these conditions, the intermediate product HGaCH_{3} is formed. As the latter continues its path, it has a probability of 0.715 [83] (the branching fraction is 0.73 [96]) of crossing toward the 2 ^{2}A′ state with 5s^{1} character going to the Ga ^{2}P: 4s^{2} 4p^{1} ground state and a probability of 0.285 of continuing by the same 3 ^{2}A′ state with 4p^{1} character. When the intermediate continues its pathway, there is a probability of 0.46 for crossing from the 2 ^{2}A′ state to X^{2}A′ with 5s^{1} character (see Figure 14), where the most stable intermediate is found at and 6.1 kcal mol^{−1}. Finally, this intermediate is broken in the products HGa + CH_{3} and GaCH_{3} + H with calculated energies 40.2 and 45.7 kcal mol^{−1}, respectively. The corresponding experimental energy is 38.9 kcal mol^{−1} in HGa + CH_{3} case.

Pacheco-Sánchez et al. [13] have found that gallium in its doublet first excited state (^{2}S: 4s^{2} 5s^{1}) breaks the Si–H bond of silane, overcoming a barrier of 51.0 kcal mol^{−1} of the ground state. The experimental transition energy between the excited state (^{2}S: 4s^{2} 5s^{1}) and the ground state (^{2}P:4s^{2} 4p^{1}) is 69.3 kcal mol^{−1}, which agrees to the transition energy of 71.7 kcal mol^{−1} theoretically calculated (Table 3(b)). The transition probability [13] among the corresponding PES for the reaction Ga ^{2}S(4s^{2} 5s^{1}) + SiH_{4} → Ga ^{2}P(4s^{2} 4p^{1}) + SiH_{4} starts with the formation of the intermediary molecule HGaSiH_{3} passing through two avoided crossings among 3 ^{2}A′, 2 ^{2}A′, and X ^{2}A′ states. The probability [85] that it crosses from 3 ^{2}A′ state to 2 ^{2}A′ state is and a probability of continuing along the same 3 ^{2}A′ state. When crossing towards the 2 ^{2}A′ state in its descent, it has a probability of crossing towards the X ^{2}A′ state and a probability of continuing along the same 2 ^{2}A′ state (see Figure 15). Finally at the minimum (6.1 kcal mol^{−1}, 120°) of the X ^{2}A′ state, the intermediate is broken into the HGa + SiH_{3} and GaSiH_{3} + H products with energies 26.1 and 45.3 kcal mol^{−1}, respectively. The corresponding experimental energy is 24.5 kcal mol^{−1} in HGa + SiH_{3} case.

Lefcourt and Ozin [100] optimized geometry of Al–SiH_{4} interaction for ^{2}A_{1} state in C_{2v} symmetry with the single point SCF energy and calculated that the geometry-optimized structure does not differ very much at all from the initial geometry, a situation that is reflected in the converged energy which is only approximately 1/100 of a hartree lower (~6.21 kcal/mol) than the single-point value associated with the starting structure. Calculations carried out on systems having the equivalent starting geometry but inhabiting the ^{2}B_{1} and ^{2}B_{2} electronic states yielded final energies that were considerably higher (0.17 and 0.31 hartree, resp.) than the energy obtained in the ^{2}A_{1} optimization.

#### 3. Discussion

We have presented here a series of studies on X + YCH_{4} (X = Cu, Zn, Cd, Ga, Al, Au, Hg, and Y = C, Si, Ge) reactions, attempting to identify possible general patterns. The very first study presented, the Cu + CH_{4} reaction was simultaneously being studied experimentally at Toronto University in Canada [32, 33, 101–104], allowing us to make a very close comparison and correlation between their cryogenic experiments and our quantum mechanical calculations. This was due to the fact that Jaime Garcia-Prieto (a member of our group in Mexico) was working with Professor Geoffrey Ozin at Lash Miller Lab. in Toronto on the photochemical activation of Cu in methane matrices at near-absolute zero conditions. Originally Ozin expected the open shell Cu: ^{2}D excited state to be responsible for methane activation, but our calculations proved that the higher lying ^{2}P was the real culprit, albeit through a series of avoided crossings a fact which was finally acknowledged in the experimental and theoretical papers [32, 60]. Since then our group, mainly through the late Sidonio Castillo and his collaborators, studied a series of transition metals. We have presented here an overview of this line of research although we evidently must accept that the list of metals studied to date is far from exhaustive. This notwithstanding, we provide here enough examples of the mechanism first advanced in our explanation of the Lash Miller Lab experiments of the activation of methane by copper [60, 82]. In short we have advanced a daring hypothesis avoided crossings and transition probabilities for curves of the same symmetry play a systematic and clear-cut role in the activation of methane, silane, and germane. Indeed, we show the evidence that in general, the most stable intermediate has a rather weak binding, thus leading to the products. The potential energy surfaces lead from the reactants climbing to this transition state which requires moderate energies, normally derived from photoactivation processes and finally yielding the final products (XH + YH_{3} or H + XYH_{3}). To document this, we report the configurations and bond distances as they evolve in each specific case. We kindly refer our readers to access our original papers for details. In any case, our reported energies are compatible with the spectra of reactants, intermediates, and products. For instance, the probabilities for the Ga + CH_{4} reaction agree quite well with experimental results [83, 96], whereas those obtained for other systems reported here were widely discussed in the original papers [82–88].

The potential energy surfaces for the coinage metals Cu and Au with silane are rather similar. The main difference lies in the energies of their respective second excited states, making their potential energy wells differ from their respective intermediate complexes onward to the final products. To wit, in the Au + SiH_{4} reaction, the minimum at 120 degrees has only a depth of 3.34 kcal/mol, while for Cu + SiH_{4} the minimum lies at 100 degrees and is 5.58 kcal/mol deep. These results would make the study of the Ag + SiH_{4} reaction all the more desirable, a valid proposition for several other metals. We explicitly state here that our work on these systems is far from complete. Take for instance our results involving Hg which are rather different from the other metals, thus, naturally deserving closer scrutiny. Perhaps we may tentatively relate this peculiarity of mercury to its particularly stable ground state ^{1}S_{0} (with its closed 5d^{10} 6s^{2} valence shell), from which stem the unique dense metal liquid character of mercury, in dire contrast with its solid metal neighbors with stable electronic arrangements. In effect we dare propose that this line of research is both valid and promising and needs much more work and attention, especially so with a close collaboration with cryogenic experiments, as we hope to establish.

Difference of PES of coinage metals interacting with silane is due to the second excited state leading to the reported well depths for the potentials for the ground state. Analogously, the PES of pseudotransition metals interacting with silane is very similar, and the main difference is the ^{1}P singlet energy of the pseudotransition metals at the second excited state. This causes the depth wells of potential (11.5 kcal mol^{−1}, 180°) for Zn + SiH_{4}, while for Cd + SiH_{4} it is (20.7 kcal mol^{−1}, 180°), and for Hg it is (28 kcal mol^{−1}, 180°), since the initial formation of the intermediate until the product formation.

The methane complexes with Zn and Cd in the second excited state ^{1}P at has energies ~141 and 125.6 kcal mol^{−1}, respectively. The germane complexes with Cd and Hg in the second excited state ^{1}P at have energies 128.5 and 157.6 kcal mol^{−1}, respectively. The initial excitation energy of the pseudotransition metals yields the depth wells of potential, which is ~25 kcal mol^{−1} for Zn + CH_{4} and 31.9 kcal mol^{−1} Cd + CH_{4}. The initial excitation energy of the pseudotransition metals yields the depth wells of potential, which is 13.6 and 21.0 kcal mol^{−1} for Cd + GeH_{4} and Hg + GeH_{4}, respectively.

Finally, PES of metals Al ^{2}D doublet and Ga ^{2}P doublet in the second excited state interacting with methane for has energies 93.9 and 93.7 kcal mol^{−1}, respectively. The initial energy of the metals yields the depth wells of potential, of 6.5 and 6.1 kcal mol^{−1} for Al + CH_{4} and Ga + CH_{4}, respectively, and 6.21 taken from reference [100] and 6.1 kcal mol^{−1} for Al + SiH_{4} and Ga + SiH_{4}, respectively. While the latter result is as expected, in the previous one there is a very small deviation. This is due to the very small energy difference at 2nd excited state between Al and Ga, and to the approximation error carried out by the numerical solutions of the system. In defense of this argumentation, it is the fact that the dissociation energy of the ionic Al^{+}−CH_{4} complex has been experimentally reported to be 2120 ± 105 cm^{‒1} (6.06 ± 0.3 kcal mol^{−1}) [99]. Somewhat coincidentally we have obtained the value of 6.5 kcal mol^{−1} for the dissociation energy of Al + CH_{4} complex. It is quite remarkable that the dissociation energies for Al^{+}−CH_{4} and Al + CH_{4} are quite close, and that the experimental value for this ionic Al^{+}–CH_{4} reaction is the expected value for the dissociation energy of Al–CH_{4}.

A very important issue is to get the most stable intermediate with the lowest energy on which the breaking of the intermediate is achieved. The products of the reaction emerge at the end from the X A′ state, calculated using the parameter values obtained at this minimum.

The intermediate has a much lower energy than the final products due to the endergonic nature of these reactions. Initially, the metal needs a photoexcitation for taking it to an excited state for being inserted in a Y–H bond of the gas molecule, in order to surpass the activation barrier of the ground state for the reaction. After the avoided crossings, the deformed metastable intermediate arrives to the most stable arrangement, having the lowest energy with the optimal geometry. At this low energy arrangement, its internal bond orientation produces a still strong electrostatic repulsion due to the energy gained by the photoexcitation of the metal, greater than the energy of the intermediate. The metal atom also shares its energy with the carbon and hydrogens. Thus, the new energies of the products cannot be greater than the initial excitation of the metal or lower than the energy of the reactants in their ground state.

The resulting probabilities for the case of Ga + CH_{4} are in agreement with previous experimental results [83], whereas those obtained for the other cases are only theoretical predictions.

#### 4. Conclusions

From the theory of transition probabilities at avoided crossings between curves of the same symmetry, we can deduce that the character of the wave function is the same at the beginning of the highest state path than at the minimum of the ground state path after all avoided crossings.

The energy value of the initial state of the metal which eventually leads to the intermediate, as well as the character of the wavefunction is crucial for the product formation. This character corresponds to the dominant excited state of the metal inserting in the gas molecule and is independent of the gas in consideration (methane, silane, or germane).

As expected the wells’ potential depth is dependent on the energy level of the metal excited state in consideration, and the pathway is reflected in the character of the wavefunction revealing the electronic configuration of the corresponding excited state. This means that the higher the energy of the metal atom excited state, the deeper the potential energy well at the ground state minimum.

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