A simple geometrical path towards hybrid orbitals

ARTICLES PRESENTED IN THE 4TH TECHNICAL METTING OF MATERIALS SCIENCE AND TECHNOLOGY POST-GRADUATION PROGRAM OF UNESP, POSMAT

A simple geometrical path towards hybrid orbitals

Denis Rafael Nacbar^{I, II}; Allan Victor RibeiroII, III, ^*; Alexys Bruno-Alfonso^IV

^IInstituto Federal de Educação, Ciência e Tecnologia de São Paulo - IFSP, CEP 12929-600, Bragança Paulista, SP, Brazil

^IIPrograma de Pós-Graduação em Ciência e Tecnologia de Materiais - POSMAT, Universidade Estadual Paulista - Unesp, CEP 17033-360, Bauru, SP, Brazil

^IIIInstituto Federal de Educação, Ciência e Tecnologia de São Paulo - IFSP, CEP 16201-407, Birigui, SP, Brazil

^IVDepartamento de Matemática, Faculdade de Ciências, Universidade Estadual Paulista - Unesp, CEP 17033-360, Bauru, SP, Brazil

ABSTRACT

It is shown that the standard spⁿ hybrid orbitals are orthogonal orbitals that minimize the total quadratic spread. This is done in a concise way that may improve the understanding of hybrid orbitals. The fact that maximally localized Wannier functions of crystalline materials may resemble hybrid orbitals is discussed.

Keywords: hybrid orbitals, localization, Wannier functions

1. INTRODUCTION

Hybrid atomic orbitals were introduced, in 1931, by the Nobel Prize winner Linus Pauling, in order to explain the nature of the chemical bond on quantum mechanical grounds¹. Since then, these orbitals have been successfully used to describe single and multiple bonds between atoms^2,3. Although a thorough understanding of the hybrid orbitals rely on energy calculations, geometrical considerations⁴, the maximization of the overlap integrals⁵, and group-theoretical approaches⁶ have been proposed.

The aim of this work is to present an alternative approach motivated by modern methods of Materials Science involving Wannier functions⁷. Such functions were introduced in solid-state physics by G. H. Wannier, in 1937. They form a complete basis of the space of electronic states of a crystal, just like the Bloch functions do. However, the former ones are localized, in contrast with the extended character of the latter. The Wannier functions lack unicity and some effort has been devoted to the development of analytical and numerical techniques that optimize their degree of localization⁷. We show that the hybrid spⁿ orbitals are the most localized among the orthogonal linear combinations of one s and np orbitals. This is helpful to understand why maximally localized Wannier functions may resemble the hybrid orbitals^7,8.

A similar path towards the hybrid orbitals has been used in previous works. In 1952, Hurley⁹ addressed this problem in his Doctoral Thesis. A decade later, Boys^10,11 used the same localization criterion, while Edmiston and Ruedenberg^12,13 dealt with the minimization of the electrostatic energy. Since the mentioned works are hardly available or give no mathematical details on the hybrid orbitals, we present an independent approach. In Section 2, we derive the hybrid orbitals as maximally localized states, according to the Foster-Boys criterion. The concluding remarks are given in Section 3.

2. MINIMIZATION OF THE TOTAL QUADRATIC SPREAD

The degree of localization of a wave function may be expressed by the variance of the associated probability distribution⁷. In this sense, a state is more localized if its distribution has less variance. The variance of an electronic state ψ is given by:

where r = (x₁,x₂,x₃) is the particle position.

Let us consider an orthonormal set of states φ₀,φ₁,...,φ_n. These states can be linearly combined to produce a new set of orthonormal states ψ₀,ψ₁,...,ψ_n, i.e.,

with j = 0,...,n. The coefficients of the linear combination fulfill

where δ_j,j' is the Kronecker symbol. This is equivalent to

with

The total quadratic spread of each set of states is a Foster-Boys functional⁷. For the sets φ₀,φ₁,...,φ_n and ψ₀,ψ₁,...,ψ_n, the functional is given by

and

respectively. Taking Equation 5 into account, the difference between Equations 7 and 6 simplifies to

where

When one considers the states φ₀,φ₁,...,φ_n as atomic orbitals whose probability distribution presents inversion symmetry, the condition applies for k,k' = 0,...,n. Hence,

This should be minimized, taking into account the n(n + 1)/2 independent conditions given by Equation 5.

To deal with the hybridization of one s and n p orbitals of a given atomic shell, we take n = 1,...,3,

and

for k = 1,...,n. R₀(r) and R₁(r) are the radial parts of the respective orbitals.

By taking into account the symmetries of orbitals in Equations 11 and 12, Equation 10 simplifies to

where

From the physical point of view, no generality is lost by assuming

for every j. Hence, according to Equation 3, we obtain

This will be minimized, satisfying the conditions in Equations 5 and 15.

The conditions in Equation 5 may be stated in terms of the vectors , with k = 0,...,n. They form an orthonormal set in , where denotes the set of complex numbers. In particular, for a given unit vector , it is required the existence of an orthonormal set of vectors that are orthogonal to . Since is real, this requirement is always fulfilled by a set of real vectors . This means that a set of real hybrid orbitals will minimize the total quadratic spread.

For real unit vectors , Equation 16 takes the form

with

The minimization of should be performed under the restrictions (Equation 15) and G = 1, where

The global minimum of f under the condition (Equation 19) may be found by using the method of Lagrange multipliers. The existence of the global minimum is guaranteed by the fact that the objective function is continuous and the restrictions lead to a closed limited subset of , where is the set of real numbers¹⁴. The values of c_0,0,...,c_n,0 must satisfy the equations

for j = 0,...,n, where λ is the Lagrange multiplier. After computing the partial derivatives, Equation 20 simplifies to

The system of equations in Equation 21 has 2ⁿ⁺¹ - 1 meaningful solutions. In fact, each c_j,0 is either 0 or , and at least one of the coefficients c_j,0 is not zero. The solutions may be classified in terms of the number m + 1 of non-null coefficients c_j,0, with m = 0,...,n. Since G = 1, we obtain , for m + 1 values of j, and c_j,0 = 0 for the other n - m values of j. For each m, the value of f is given by f_m = 1/(m + 1), and the global minimum of f corresponds to m = n. Therefore, the minimum value of the total quadratic spread corresponds to

.

To deal with the shape of the states ψ₀,ψ₁,...,ψ_n, we introduce the vectors ν_j = (c_j,1,..., c_j,n). From Equation 3, we obtain

and

for j' ≠ j. Hence, the angle between different vectors ν_j and ν_j' is given by

For the cases n = 1,2 and 3, this angle equals θ₁ = arccos( - 1) = 180°, and , respectively. These values correspond to the hybrid orbitals sp, sp² and sp³, respectively.

We also note that each real and normalized linear combination of the p_xk orbitals may be obtained by rotating p_x1. Hence, Equation 2 may be written as

where p_νj is the real p orbital along ν_j. Since the s orbital is isotropic, all the spⁿ orbitals, for a given n, have the same shape.

3. CONCLUSIONS

We have shown that the standard spⁿ hybrid orbitals are the orthogonal linear combinations of one s and n real p orbitals that minimize the total quadratic spread. This helps to understand why maximally-localized Wannier functions may resemble hybrid orbitals in crystalline materials^7,8. It is the case when the set of energy bands under consideration is essentially reproduced by using a linear combination of atomic orbitals. In such cases, the multiband Wannier functions of maximal localization should mimic the atomic orbitals. Because of the requirement of maximal localization, such orbitals are hybrid orbitals instead of pure s or p orbitals. We also note that the maximally-localized orbitals are useful when the individual contribution of each constituent atom is under investigation^15,16.

Finally, we would like to point out that it has been proposed that hybrid orbitals should better be excluded from the Chemistry curriculum¹⁷. Reinforcing several claims against such a proposal^18,19, we stress that the growing importance of Wannier functions in Materials Science brings renewed interest on hybrid orbitals. Furthermore, we believe the present approach also applies to hybrid orbitals that include d and/or f states, such as the icosahedral ones²⁰.

ACKNOWLEDGEMENTS

We thank the Brazilian Agencies CNPq and FAPESP for financial support.

Received: April 18, 2014

Revised: September 10, 2014

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Publication Dates

• Publication in this collection
  10 Feb 2015
• Date of issue
  Dec 2014

History

• Accepted
  10 Sept 2014
• Received
  18 Apr 2014