A Simple Geometrical Path Towards Hybrid Orbitals

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 atoms2,3. 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 orbitals7,8. 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, Boys10,11 used the same localization criterion, while Edmiston and Ruedenberg12,13 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.


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 grounds 1 .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 considerations 4 , the maximization of the overlap integrals 5 , and group-theoretical approaches 6 have been proposed.
The aim of this work is to present an alternative approach motivated by modern methods of Materials Science involving Wannier functions 7 .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 localization 7 .We show that the hybrid sp n 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, Hurley 9 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.

Minimization of the Total Quadratic Spread
The degree of localization of a wave function may be expressed by the variance of the associated probability distribution 7 .In this sense, a state is more localized if its distribution has less variance.The variance of an electronic state y is given by: ( ) σ y = y y − y y r r (1)   where r = (x 1 ,x 2 ,x 3 ) is the particle position.Let us consider an orthonormal set of states f 0 ,f 1 ,...,f n .These states can be linearly combined to produce a new set of orthonormal states y 0 ,y 1 ,...,y n , i.e., , 0 with j = 0,...,n.The coefficients of the linear combination fulfill where d j,j' is the Kronecker symbol.This is equivalent to

A Simple Geometrical Path Towards Hybrid Orbitals
Denis Rafael Nacbar The total quadratic spread of each set of states is a Foster-Boys functional 7 .For the sets f 0 ,f 1 ,...,f n and y 0 ,y 1 ,...,y n , the functional is given by ( ) ( ) ( ) respectively.Taking Equation 5 into account, the difference between Equations 7 and 6 simplifies to where When one considers the states f 0 ,f 1 ,...,f n as atomic orbitals whose probability distribution presents inversion symmetry, the condition , 0 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, ( ) ( ) for k = 1,...,n.R 0 (r) and R 1 (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 ,0 0 j c ≥ (15)   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 ( ) 0, , , , , with k = 0,...,n.They form an orthonormal set in 1 n+ C , where C denotes the set of complex numbers.In particular, for a given unit vector The minimization of ∆Ω should be performed under the restrictions (Equation 15) and G = 1, where 2 ,0 0 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 1 n+ R , where R is the set of real numbers 14 .The values of c 0,0 ,...,c n,0 must satisfy the equations ,0 ,0 for j = 0,...,n, where l is the Lagrange multiplier.After computing the partial derivatives, Equation 20 simplifies to 2 ,0 ,0 0 2 The system of equations in Equation 21 has 2 n+1 -1 meaningful solutions.In fact, each c j,0 is either 0 or / 2 l , 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 ,0 1  1 , 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 y 0 ,y 1 ,...,y n , we introduce the vectors n j = (c j,1 ,..., c j,n ).From Equation 3, we obtain for j' ≠ j.Hence, the angle between different vectors n j and n j' is given by , 1 arccos For the cases n = 1,2 and 3, this angle equals and 3 1 arccos 109.471 3 , respectively.These values correspond to the hybrid orbitals sp, sp 2 and sp 3 , 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 s p where p nj is the real p orbital along n j .Since the s orbital is isotropic, all the sp n orbitals, for a given n, have the same shape.

Conclusions
We have shown that the standard sp n 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 curriculum 17 .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 ones 20 .

0 u 1 ,
, it is required the existence of an orthonormal set of vectors 1 is real, this requirement is always fulfilled by a set of real vectors 1 , , n u u …   .This means that a set of real hybrid orbitals will minimize the total quadratic spread.For real unit vectors 0