Keywords

1 Introduction

At present time a great interest attracts studying an electrochemical and catalytic activity properties of the different materials, including a development of advanced chemical industry technologies, search and construction of the electrochemical solid-state energy sources, etc. [1,2,3,4,5,6,7,8,9,10,11,12,13]. A great number of the attempts to develop a quantitative theory of electron catalysis for the metal-like and semiconductor systems including the description of processes on electrodes’ surfaces of the electrochemical solid-state energy sources have been undertaken (see [1,2,3,4,5,6,7,8]). One could mention quite good understanding of a mechanism of heterogeneous catalytic process under obligatorily treating intermediate stages, namely, stages of adsorption and desorption [1, 2]. Naturally, any heterogeneous reaction can be treated as the process based on a radical mechanism. The matter is in the fact that the radicals and ion-radicals appear on the surface under chemosorption and provide the corresponding mechanism of these reactions. It is important to note that the reaction path through one-electron charged intermediates (ion-radicals) is not the main mechanism in a heterogeneous catalysis. Nevertheless, it can be possible in some special cases. The detailed review of different approaches to adsorption and catalysis, including the key aspects of different conceptual models (such as the density functional method, ab initio quantum chemistry methods) are presented in [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. The cited methods allowed to get very useful information about the electrochemical and catalytic activity properties of the different materials, nevertheless, different fundamental physical and chemical aspects of the problem remain up to known degree veiled [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. In this chapter we present an advanced quantum-mechanical and electrodynamical approach to description of the catalytic activity for metals, binary metallic alloys and semiconductor materials. Within the approach the quantitative correlation between the electron structure parameters of the materials and their catalytic activity is found on example of simple model reactions.

2 Electrodynamical and Quantum-Mechanical Models for Metals, Alloys and Semiconductors

It is well known that the catalytic properties of different metallic and semiconductors materials are linked with the electronic processes inside and on the surface of the materials. The Fermi level position determines the adsorption and catalytic activity of the surface in relation to molecules of the given kind under other equal conditions. The chemosorption ability of the surface, a degree of its charging, a reactive ability of the chemisorbed particles etc. are dependent upon the Fermi level position on the surface of crystal (speech is about distance between the Fermi level and the conductivity band bottom: \( \left. {E_{F}^{s} } \right) \). The position of the Fermi level inside the crystal can be defined as \( E_{F}^{\nu } \). The corresponding expression linking the values \( E_{F}^{s} \) and \( E_{F}^{\nu } \) is as follows:

$$ \sigma + \int\limits_{0}^{\infty } {\rho (x)dx = 0} $$
(1)

where ρ is a density of the volume charge in the plane, σ is a density of the surface charge. The value of σ can be expressed as function of a pressure P, temperature T: \( \sigma = \sigma \left( {P,T,E_{F} } \right) \) [2, 6, 7]. The second item in (1) is the function of \( E_{F}^{s} \) and \( E_{F}^{\nu } \):

$$ \int\limits_{0}^{\infty } {\rho (x)dx = R\left( {T;E_{F}^{s} ;E_{F}^{v} } \right)} $$
(2)

The relationship between \( E_{F}^{s} \) and \( E_{F}^{\nu } \) (defining the correlation between the surface and bulky properties of the material) can be written as follows:

$$ \sigma \left( {P,T;E_{F}^{s} } \right) + R\left( {T;E_{F}^{s} ;E_{F}^{v} } \right) = 0 $$
(3)

or

$$ E_{F}^{s} = f\left( {P;T;E_{F}^{v} } \right) $$
(4)

Let us further to consider in details the electrodynamical approach to structure of a metallic system. Within the simple approximation [9] with using a set of isotropic s-d energy bands, one could write the next obvious formula for a static dielectric permeability:

$$ \varepsilon = 1 + \varepsilon_{ss} + \varepsilon_{dd} + \varepsilon_{sd} + \varepsilon_{ds} , $$
(5)

In Eq. (5) ε (ij) means the contribution into ε due to the i-j transitions. If one accept here the approximation of the free electrons, then it is easily to write the expression for ε ss component:

$$ \varepsilon_{ss} = 2\pi \nu_{s} \left( {E_{F} } \right)k^{ - 2} \left\{ {1 + \left[ {4\left( {k_{F}^{s} } \right)^{2} - k^{2} } \right]\ln \left| {\left( {2k_{F}^{s} + k} \right)/\left. {2k_{F}^{s} - k} \right)} \right|/4k_{F}^{s} k} \right\}. $$
(6)
$$ \nu_{i} \left( {E_{F} } \right) = N_{i} \left( {E_{F} } \right)a_{B}^{2} e $$

Here the following notations are used: a B is the Bohr radius, k = q·a B , q is the wave number, \( q_{F} = \left( { 3\pi^{ 2} z_{i} /\varOmega } \right)^{ 1/ 3} \); z i is a number of electrons in “i” band; N i (E F ) is a density of states on the Fermi surface in “i” band.

This corresponding ε ds contribution can be represented as follows:

$$ \varepsilon_{ds} = \left[ {2m_{s} k_{d} e^{2} f_{c} /\pi \pi^{2} k^{2} } \right]\left\{ {1 + \left[ {4\left( {k_{d} } \right)^{2} - k^{2} } \right]\ln \left| {\left( {2k_{d} + k} \right)/\left. {2k_{d} - k} \right)} \right|/4k_{d} k} \right\}. $$
(7)

where m s is the effective mass of electron in the conductivity band; k d , f c are the numeral parameters [7, 8, 11]. At last, the corresponding formula for ε dd looks as:

$$ \varepsilon_{dd} = 2\pi \nu_{d} \left( {E_{F} } \right)k^{ - 2} \left| {M_{dd} } \right|^{2} \left\{ {1 + \left[ {4\left( {k_{F}^{d} } \right)^{2} - k^{2} } \right]\ln \left| {\left( {2k_{F}^{d} + k} \right)/\left. {2k_{F}^{d} - k} \right)} \right|/4k_{F}^{d} k} \right\} $$
(8)

where M dd is the matrix element, determined by the superposition of the wave functions for d electrons.

The next key step is in supposing that we can work using the spherical symmetry and the crystal potential is fully screened by the conductivity electrons. Then one could introduce an effective potential, which imitates an influence of the metallic field on the hydrogen atom (for example, for process of the following type: H = H ++ e −). This effective potential can be determined in the following form:

$$ \varPhi (r) = - \frac{{2e^{2} }}{\pi r}\int\limits_{0}^{\infty } {\frac{\sin \,kr}{k\varepsilon (k)}} dk $$
(9)

or after substitution of (5) to (7) resulted in the expression:

$$ \varPhi \left( r \right) = - \left( {e^{2} a/r} \right)\,exp[ - \alpha R]cos[\alpha R] $$
(10)

where

$$ \alpha = \left[ {\pi^{ - 1} 12\left( {k_{F}^{s} } \right)^{2} } \right]^{ - 1/4} \left\{ {\nu_{s} \left( {E_{F} } \right) + \left( {k_{F}^{d} /k_{F}^{s} } \right)\nu_{d} \left( {E_{F} } \right) + f_{c} \left( {k_{F}^{d} /k_{F}^{s} } \right)^{2} \nu_{s} \left( {E_{F} } \right)} \right\}^{1/4} $$
$$ R = 2q_{F}^{s} r, $$
(11)
$$ a = \left( {k_{F}^{s} } \right)^{ - 1} $$

According to Ref. [8, 23] further key idea in the following. One could find the numerical solution of the Schrödinger equation for the hydrogen atom in a field Φ(r) and obtain a spectrum of states, which could be continual or discrete in dependence upon the parameter \( \zeta^{ - 1} = \alpha /a \) [8]. The spectrum is continual, if \( \zeta < \zeta_{0}\,( \zeta_{0} = 0,3 6 2) \) and the corresponding material is a catalyst for the hydrogen ionization reaction. If \( \zeta > \zeta o \), the spectrum is discrete (metal or alloy does not demonstrate catalytic activity for cited reaction). In Refs. [8, 23,24,25] such an approach was successfully applied to studying the metal catalytic properties in relation to reaction of the H ionization and obtained excellent agreement with experiment [2, 5,6,7]. In Ref. [8] there are listed the corresponding parameters α and a for a number of metals.

The analogous consideration can be performed for the binary metallic alloy. Obviously, one should take into account a dependence of the Fermi level position E F as well as the state density ν(E F ), accompanied with the structure parameters α and a upon the change of the admixture concentration c [8, 9]. It is easily to suppose that the admixture’s atoms volume has the spherical form and a radius R c is linked with the concentration as:

$$ \left( {qR_{c} } \right)^{ - 3} = \left( {qr_{s} } \right)^{ - 3} c, $$
(12)

where r s —the electron gas parameter.

Further one could write the standard Poisson equation [8] for the screened potential V(r) near an admixture:

$$ \Delta V(r) = q^{2} \left\{ {V(r) -\Delta E_{F} } \right\} $$
(13)

which has the with the elementary solution of the following type:

$$ V\left( {r,R_{c} } \right) - \Delta E_{F} = \left[ { - Z_{v} e/r} \right]\left\{ {qR_{c} ch\left[ {q\left( {R_{c} - r} \right)} \right] + sh\left[ {q\left( {R_{c} - r} \right)} \right]} \right\}/\left[ {qR_{c} ch\left( {qR_{c} } \right) + sh\left( {qR_{c} } \right)} \right] $$
(14)

Here Zν is a difference of the components valences. Finally, the value ν(E F ) for the binary metallic alloy can be expressed as follows [8]:

$$ \nu \left( {E_{F} } \right) = \nu \left( {E_{F} } \right) +\Delta \nu \left( {E_{F} } \right). $$
(15)

The Fermi level position and respectively the metallic alloy catalytic activity is quantitatively dependent upon the components concentration.

In Refs. [8, 25] there are presented the concrete examples of the modeling the catalytic activity of metallic alloy in relation of the H ionization reaction for different metallic alloys. For example, the corresponding estimates show that the alloy Ni-Cu with a small concentration of Cu (till 16%) is a good catalyst for the hydrogen ionization reaction, however situation is changed in the opposite direction with a growth of ζ. This is in good agreement with experiment [1, 6, 7].

The similar approach can be generalized to find a link between the alloy structure parameters and their catalytic activity for oxygen electro-restoring reaction. One should again to solve the Schrödinger equation for system “O2-electron” in the potential Φ (r). If the negative ion system has the bound state for given values of (6)–(9), then the studied material is a good catalyst for the reaction. In order to solve the Schrödinger equation in a case of diatomic molecule it is necessary to use the two-centre approximation [27,28,29,30,31,32]. In the standard elliptic coordinates:μ = (r a  + r b )/R ab , η = (r a – r b )/R ab (R ab is a distance between nuclei) the variables’ in the corresponding Schrödinger equation are separated if one use the formula: 1/2R ab (μ + η) ≈ 1/2 R ab μ. Then the potential (10) in the molecular approximation can be written as follows:

$$ \varPhi (\mu ,\eta ) = - 2\mu a\,exp\left[ { - \alpha q_{F} R_{ab} \mu } \right]cos\left[ {\alpha q_{F} R_{ab} \mu } \right]/\left[ {R_{ab} \left( {\mu^{2} - \eta^{2} } \right)} \right] = g(\mu )/\left( {\mu^{2} - \eta } \right). $$
(16)

Further one can write the corresponding Schrödinger equation which is reduced to the differential equations system [8, 28]:

$$ \left\{ {d/d\mu \left( {\mu^{2} - 1} \right)d/d\mu - \left[ {\lambda_{ml} + m^{2} /\left( {\mu^{2} - 1} \right) + \mu^{2} c^{2} + R_{ab} g(\mu )/ 2} \right]} \right\}T_{nlm} = 0, $$
$$ \left\{ {d/d\eta \left( { - \eta^{2} + 1} \right)d/d\eta + \left[ {\lambda_{ml} + \eta^{2} c^{2} m^{2} \left( { 1- \eta^{2} } \right)} \right]} \right\}S_{lm} = 0, $$
(17)
$$ \left\{ {d^{2} /d\varphi^{2} + m^{2} } \right\}\varSigma_{m} = 0. $$

where \( \varPsi_{nlm} = {\rm T}_{nlm} (\mu )\,S_{lm} (\eta )\varSigma_{m} (\varphi ) \) is a wave function; \( E=-2c^{2}/R_{ab}^{2} \) is one-electron energy, λ ml is a coupling constant. In order to fine the numerical solution of the Eqs. (17) one could use the standard approaches such as the Numerov method or matrix technique of the Newton-Rafson etc. (see [27,28,29,30,31,32]).

In Refs. [8, 25] the authors has presented the results of application of the described model to estimating the catalytic properties for the alloy Ni-Cu in relation to the oxygen electro-restoring reaction. In a case of the small content of Cu [c(Cu) = 20%; c(Ni) = 80%; E(1π g ) = 1,3 eV] the compound Ni-Cu is an effective catalyst. With changing c, the situation is changing into opposite side. Catalytic activity of material decreases if c(Cu) = 50%, c(Ni) = 50% [E(1π g ) = 1,1 eV]. These facts correlate well to experimental data [6, 7]. If c(Cu) = 90%, c(Ni) = 10% [E(1π g ) = 1,5 eV], the catalytic activity is quite high again.

In order to generalize the above presented approach on a case of description of the catalytic processes on semiconductors and determine a link between their electron structure parameters and the catalytic activity in the relation to model H ionization reaction one could use the known Resta model in the Thomas-Fermi theory [33, 34]. The master Poisson equation has the standard form:

$$ V(r) = q\left\{ {V(r) - A} \right\}, $$
(18)

where q = 4k F /πa B and A is a constant. Beyond the radius R the point charge Z e potential is equal to:

$$ V(R) = - Ze^{2} /\left[ {\varepsilon (0)r} \right],r > R, $$
(19)

where ε(0) is a static dielectric permeability. The general solution with taking into account for the corresponding continuity and boundary conditions looks as:

$$ V(r) = - \left\{ {Ze^{2} /r} \right\}\left\{ {sh\left[ {q\left( {R - r} \right)} \right]/sh\left[ {qR} \right] - Ze/\varepsilon \left( 0 \right)R,r < R } \right. $$
(20)

The link between the screening parameter and ε(0) is determined by the relationship: ε(0) = sh[qR]/qR. Let us note that if ε(0) > 1, R is equal to finite value comparable with distance to the nearest atoms (for example, for NiO, CuO, ZnO, ZnS, ZnTe semiconductors this value is 4.8–6.1 Å) [7, 35]. The next step is the numerical solution of the Schrödinger equation with the potential (20), which allows to determine the corresponding energy spectrum in dependence upon ε(0), k F (E F ) and then to find a link between the semiconductors structure parameters and their catalytic activity likely above described approach. In ref. [8, 24] there are presented the concrete estimates of the catalytic activity for the CuO, ZnO semiconductors in the hydrogen ionization (oxidation) reaction. The general conclusion is that the CuO and ZnO semiconductors are good catalysts for the hydrogen ionization reaction. This fact is in a good agreement with the known experimental data [6, 7].

In conclusion of this section we present a brief generalization of the model to the case of the influence of an external electric field. To determine the corresponding spectrum of the quasi-bound states one needs to carry out the diagonalization of the energy matrice, calculated between states with the same main quantum number (e.g. [36,37,38,39]). Let us note that the corresponding Hamiltonian can be written in the cylindrical coordinates (ρ, z), namely:

$$ H\psi (\rho ,z) = E\psi (\rho ,z) $$
(21)
$$ \begin{aligned} H & = - 1/2\left( {\partial^{2} /\partial \rho^{2} + 1/\rho \partial /\partial \rho + \partial^{2} /\partial z^{2} - 1} \right./\left[ {(z + R/2)^{2} + \rho^{2} } \right]^{1/2} \\ & \quad - 1/\left[ {(z - R/2)^{2} + \rho^{2} } \right]^{1/2} - Fz + V(\rho ,z) \\ \end{aligned} $$

Here F denotes an electric field strength; V is an effective potential (of the type (16), (20)). To solve the Eq. (21) one should use the finite differences method (look [40]). The eigen-values of the Hamiltonian can be computed by means of the inverse iterations method. The corresponding system of inhomogeneous equations can be solved by the Thomas method. However, to date any of the similar theoretical model application are missing.

3 Conclusions

We have presented an advanced quantum-mechanical and electrodynamical approaches in the electron theory of catalysis to determine quantitatively the catalytic activity for metals, binary metallic alloys and semiconductor materials. The solutions of some model tasks associated with stabilization of ionic states of atomic hydrogen and molecular oxygen in effective electron gas are given. Within the approach the quantitative correlation between the electron structure parameters of the materials and their catalytic activity is found on example of simple model reactions. Surely, the catalytic activity properties of different materials are directly connected with the corresponding electronic processes inside and on the surface of the materials. Naturally, to carry out more advanced approach one should take into account a whole number of the additional important factors (for example the electrolyte influence, surface effects, the electrodes potential, electron concentration in the surface layer etc. [1,2,3,4,5,6,7,8,9,10]). A great interest attracts the establishment of correlation between the presented approach and recent theories of the catalysis on the metals and semiconductors. In conclusion let us underline that the presented approach ideas can be quite easily generalized in order to take into consideration a whole number of such additional physical and chemical factors such as a direct electric, magnetic or laser field effect on the catalytic processes and governing by these processes, search and approbation of new classes of nanocluster catalysts and so on [38, 39, 41, 42].