Structural aspects of FeSi

Transcription

1 Comenius university in Bratislava Faculty of mathematics, physics and informatics Structural aspects of FeSi bachelor s thesis 2015 Andrej Vlček

2 Comenius university in Bratislava Faculty of mathematics, physics and informatics Structural aspects of FeSi bachelor s thesis Study program: Field of study: Department: Supervisor: Physics (Single degree study, bachelor I. deg., full timeform) 1160 Physics FMFI.KEF - Department of Experimental Physicss doc. RNDr. Richard Hlubina, DrSc. Bratislava, 2015 Andrej Vlček

3 Comenius University in Bratislava Faculty of Mathematics, Physics and Informatics THESIS ASSIGNMENT Name and Surname: Study programme: Field of Study: Type of Thesis: Language of Thesis: Secondary language: Andrej Vlček Physics (Single degree study, bachelor I. deg., full time form) Physics Bachelor s thesis English Slovak Title: Aim: Structural aspects of FeSi The goal of this bachelor thesis is to find a qualitative explanation of the observed crystal structure of FeSi. Supervisor: Department: Head of department: Assigned: doc. RNDr. Richard Hlubina, DrSc. FMFI.KEF - Department of Experimental Physics prof. Dr. Štefan Matejčík, DrSc. Approved: prof. RNDr. Jozef Masarik, DrSc. Guarantor of Study Programme Student Supervisor

4 Univerzita Komenského v Bratislave Fakulta matematiky, fyziky a informatiky ZADANIE ZÁVEREČNEJ PRÁCE Meno a priezvisko študenta: Študijný program: Študijný odbor: Typ záverečnej práce: Jazyk záverečnej práce: Sekundárny jazyk: Andrej Vlček fyzika (Jednoodborové štúdium, bakalársky I. st., denná forma) fyzika bakalárska anglický slovenský Názov: Cieľ: Structural aspects of FeSi Štruktúrne aspekty FeSi Cieľom bakalárskej práce je nájsť kvalitatívne zdôvodnenie pozorovanej kryštálovej štruktúry materiálu FeSi. Vedúci: Katedra: Vedúci katedry: Dátum zadania: doc. RNDr. Richard Hlubina, DrSc. FMFI.KEF - Katedra experimentálnej fyziky prof. Dr. Štefan Matejčík, DrSc. Dátum schválenia: prof. RNDr. Jozef Masarik, DrSc. garant študijného programu študent vedúci práce

5 I would like to express my gratitude to my thesis advisor doc. RNDr. Richard Hlubina, DrSc for his invaluable guidance and support throughout this project.

6 Abstract This thesis elaborates on Pauling s hypothesis that resonating valence bonds can explain the strange 7-fold coordination of iron and silicon atoms in the crystal structure of FeSi. Our goal is to obtain a deeper understanding of the resonance phenomenon in FeSi. We start by constructing the space group of FeSi from its generators and we prove its closure by constructing the multiplication table of the group. Next we review basic quantum chemistry concepts such as hybridization and the molecular orbital vs. valence bond description of the chemical bond, as well as the notion of resonating valence bonds. The thesis reports on two main results: (1) The possible bond configurations in FeSi were enumerated by brute force colorings of the periodically repeated unit cell. The minimal length of the local resonating ring is 4 and there are 12 colorings with the maximal number of resonating rings, that is 16 out of 24 rings of length 4. (2) We present an explicit construction of the sp d hybridization for iron where the hybrid orbitals are chosen so that they are oriented towards the 7 nearest neighbours of the iron atom in FeSi. Keywords: FeSi, resonating valence bond theory, sp d hybridization

7 Abstrakt Táto práca rozvíja Paulingovu hypotézu, že rezonujúce valenčné väzby by mohli byt schopné vysvetlit zvláštnu sedemnásobnú väzbovost železa a kremíka v kryštalickej štruktúre FeSi. Našim ciel om je dosiahnut hlbšie porozumenie rezonancie vo FeSi. Začíname konštrukciou priestorovej grupy FeSi z jej generátorov a dokazujeme jej uzavretost zhotovením tabul ky násobenia v grupe. Ďalej prinášame prehl ad základných kvantových konceptov v chémii, ako hybridizácia, chemická väzba a jej opis pomocou molekulových orbitálov vs. popis pomocou valečných väzieb, ako aj pojem rezonujúcich valenčných väzieb. Táto práca prináša dva hlavné výsledky: (1) Možné konfigurácie boli nájdené ofarbovaním hrubou silou periodicky opakovanej jednotkovej bunky FeSi. Najkratší lokálny rezonujúci cyklus je dĺžky 4 a je 12 ofarbení s najväčším počtom rezonujúcich cyklov, čo je 16 z 24 cyklov dĺžky 4. (2) Uvádzame explicitnú konštrukciu sp d hybridizácie železa, prčom hybridizované orbitály sú zvolené tak, že mieria k 7 najbližším susedom atómu železa vo FeSi. Kl účové slová: FeSi, rezonujúce valenčné väzby, sp d hybridizácia

8 Contents Preface 1 1 Basic properties of FeSi Geometric structure of FeSi Space group of FeSi Chemistry and local environment Quantum-mechanical concepts in chemistry Atomic orbitals Chemical bond Hybridization Resonance in benzene Pauling s RVB description of FeSi 19.1 Geometric problem: edge coloring Iron hybridization Pivoting resonance Conclusion 2 Záver A Geometric structure of the resonating bonds in FeSi 4 iv

9 List of Figures 1.1 The unit cell of the FeSi crystal The coordination polyhedron of a Fe atom in FeSi The unit cell of FeSi, bonds coloured according to length The molecule H Angular variation of the probability density for sp orbitals Angular variation of the probability density for sp 2 orbitals sp 2 hybridization of the carbon atoms in benzene Energy of 2p z electrons in benzene in MO theory model Superposition of Kekulé structures for benzene The unit cell of FeSi without the shortest bonds The resonance ring in FeSi The colored graphs with the maximum number of the resonating cycles The resonant rings in the best coloring of FeSi Angular variation of the optimal hybrid orbitals, dependence on θ Angular variation of the optimal hybrid orbitals, dependence on φ Hybrid orbital 0, angular variation of the probability density Hybrid orbitals, angular variation of the probability density All hybrid orbitals, angular variation of the probability density Pivoting resonance in FeSi, view along one of the sp orbitals v

10 Preface Crystalline structures of FeGe and MnSi share the same space group as FeSi. Recently, these materials have been intensively investigated because they exhibit several anomalous phenomena. In fact, FeSi as a paramagnetic insulator intrigues by its dependence of magnetic susceptibility on temperature - local magnets appear at higher temperatures, not at lower as usual. On the other hand, FeGe and MnSi are magnetic, and the orientation of their magnetization displays an interesting helical spatial dependence. Finally, the change of electronic properties from FeSi to FeGe is anomalous, because it might be expected that FeSi should be metallic and FeGe should be an insulator, exactly opposite to what is observed experimentally. In modern electronic structure calculations, atomic positions are often taken as an input to be determined by experiment. However, this is just a subproblem of the question what properties a substance of a certain chemical composition will exhibit. One should also ask why does a given substance crystallize in a given pattern. This question is particularly interesting in the FeSi crystal structure, since all atoms are surrounded by seven neighbours in this crystal, implying a weird pattern of bonds. In this thesis we revisit an old hypothesis due to Pauling from [5] about the origin of the FeSi crystal structure. According to this hypothesis, the crystal structure is stabilized by resonance of valence bonds between pairs of nearest-neighbor atoms. In chapter 1 we start by describing the geometric features of the FeSi crystal lattice by mathematical methods centered around the notion of symmetry. In chapter 2 we next review phenomena associated with the quantum-mechanical superposition principle such as the chemical bond, hybridization, and resonance. Chapter is the main part of this thesis. We start by using a graph theoretic approach to answer questions about resonant cycles in a small lattice. We reduce the problem to a finite graph coloring and search for maximally resonant configurations computationally. Next we construct hybrid orbitals of the sp d type at the Fe site which are compatible with the local structure of the lattice. We have used VESTA [4], a D visualization program, for some of the figures and calculations in this thesis. 1

11 Chapter 1 Basic properties of FeSi Materials such as FeSi, FeGe, MnSi and many other crystallize in the same, so-called B20 crystal structure. In recent years, these materials have attracted a lot of interest. The reasons are at least three-fold: (1) FeSi is a paramagnetic insulator with unexpected magnetic behaviour under normal circumstances. (2) FeGe, MnSi and MnGe are ferromagnetic metals exhibiting interesting helical spatial dependence of their magnetization. () Transition insulator metal in FeSi 1 x Ge x with increasing concentration of the Ge atoms x is anomalous, because, according to standard theory, the transition should happen with decreasing concentration x. 1.1 Geometric structure of FeSi The Bravais lattice is a lattice generated by three linearly independent vectors a 1, a 2, a, called primitive vectors, such that every point R in the lattice is a linear combination over the integers. A point R in the lattice satisfies: R = n1 a1 + n 2a2 + n 2a, (1.1) where n 1, n 2, n are integers. Sites in a crystal might be shifted from the Bravais lattice positions by a vector from the set of vectors called basis. Positions of sites in a crystal can be always expressed as a sum of a Bravais lattice position and a basis vector. The FeSi crystal has a simple cubic Bravais lattice, so that the primitive vectors of the lattice are three mutually orthogonal vectors with the same length. We can express the 2

12 CHAPTER 1. BASIC PROPERTIES OF FESI Figure 1.1: The unit cell of the FeSi crystal, Fe atoms in red, Si in blue. Note that the unit cell contains 4 formula units of FeSi, i.e. 4 Fe and 4 Si atoms. The lower left corner of the cube is the coordinate origin. position coordinates of its basis in terms of a unit lattice parameter, u, as follows: (u, u, u), (1/2 + u, 1/2 u, u), ( u, 1/2 + u, 1/2 u), (1/2 u, u, 1/2 + u). (1.2) The parameter u takes the values uf e = 0.158, usi = for the Fe and Si sites, respectively. The actual lattice parameter of FeSi is actually a = 4.49 A at liquid nitrogen temperature. [] 1.2 Space group of FeSi The symmetry-preserving operations of a crystal s lattice form a group, called the space group, under usual composition of mappings. The point group is a space subgroup that fixes at least one site in the lattice. Obviously, any non-zero translation of the crystal lattice does not belong to the point group. The space group of FeSi, P21, contains 12 distinct symmetry operations that are not translations by a lattice vector.[] Here it is understood that two operations are considered equivalent iff one of them can be expressed as a product of the other one with a pure translation. Theorem The point group of FeSi contains a three-fold rotation. Proof. The three-fold rotation R of a vector with coordinates (x, y, z) around the [111] axis can be expressed as R: (x, y, z) 7 (z, x, y). Obviously, it keeps the point (u, u, u) fixed and transforms other members of the basis cyclically: (1/2+u, 1/2 u, u) ( u, 1/2 + u, 1/2 u) (1/2 u, u, 1/2 + u). Note that the rotation, R, around the [111] axis is also a symmetry operation of the simple cubic Bravais lattice. We can write any crystal site s as s = b + u where b is a Bravais

13 CHAPTER 1. BASIC PROPERTIES OF FESI 4 lattice vector and u is a basis vector. It follows by linearity of R that R(s) = R(b)+R(u). In other words, the rotation of an arbitrary site is the sum of rotations of the Bravais vector and of the basis. We have already shown that R preserves the Bravais lattice and the basis. Since R also fixes at least one point, namely (u, u, u), it belongs to the point group of FeSi. Note that for a three-fold rotation, R = I = R 2 R = RR 2, where I is the identity rotation. Hence R 1 = R 2 and the point group contains I, R, R 1. [] Theorem The non-symmorphic group of FeSi contains a two-fold screw axis. Proof. We first show that the transformation defined by S x S : (x, y, z) (1/2 + x, 1/2 y, z) (1.) is a symmetry operation and then proceed to identify it as a screw-axis. Let (m 1, m 2, m ) denote the basis vector of FeSi and let represent the action of S. Then we have (m 1, m 2, m ) (m 1, m 2, m ), explicitly: (u, u, u) (1/2 + u, 1/2 u, u), (1/2 + u, 1/2 u, u) (u, u, u) + (1, 0, 0), (1.4) ( u, 1/2 + u, 1/2 u) (1/2 u, u, 1/2 + u) + (0, 0, 1), (1/2 u, u, 1/2 + u) ( u, 1/2 + u, 1/2 u) + (1, 0, 1). Observe that the two-fold rotation around the x-axis takes (x, y, z) to (x, y, z). Note that the translation by (1/2, 0, 0) adds 1/2 to the x-coordinate, but the addition of 1/2 to the y-coordinate cannot be achieved in a similar manner, because the vector (0, 1/2, 0) is not parallel to the axis of rotation. We can circumvent this issue by choosing a different screw axis, namely the line y = 1/4, z = 0. Equation 1.4 is now fulfilled which allows us to interpret S as a two-fold screw axis, i.e. displacement along the line y = 1/4, z = 0 followed by a translation of 1/2 in the axis direction. The screw axis displacement S x takes an arbitrary point with a Bravais lattice position (n 1, n 2, n ) and a basis position (m 1, m 2, m ) to (n 1, n 2, n ) + (m 1, m 2, m ). Thus a Bravais lattice vector is rotated around the x-axis and the claim follows. Note that S 1 x = S x. The remaining operations can be expressed as compositions of R and S x [1]. discovery of the non-trivial identity SRSR = R 2 S enabled us to show that the space group is finite and to express all its elements, see Table 1.2. The finiteness follows directly from the identity, because elements such as SR n SR n or R n SR n S where n = 1, 2 can be simplified to elements we have already found - consider the products of RS, SR, R 2 S, SR 2 in Table 1.2. Now all group elements are listed, and their interpretation is in Table 1.1. The

14 CHAPTER 1. BASIC PROPERTIES OF FESI 5 Importantly, the two other two-fold screw displacements are S y RSR 2 and S z R 2 SR. Their screw axes can be identified similarly as in proof of Theorem 1.2.2, see Table 1.1. These screw axes are parallel to y- and z-axes, respectively. Furthermore, we have recognized six other operations. They correspond to three threefold rotations and their powers. Their rotation axes are parallel to body diagonals different from [111]. We demonstrate how these operations were recognized as rotations by means of an example. Consider SR : (x, y, z) (1/2 + z, 1/2 x, y) and the rotation ρ around the [1-11] axis given by ρ : (x, y, z) (z, x, y). Define T to be the translation by (0, 1/2, 1/2) and observe that T 1 ρt and SR are identical: T 1 ρt (x, y, z) = T 1 ρ(x, y 1/2, z + 1/2) = T 1 (z + 1/2, x, y + 1/2) = thereby proving our interpretation. = (1/2 + z, 1/2 x, y) = SR(x, y, z), It is worth pointing out that translations by a unit lattice vector can be written as second powers of screw axis displacements, thus any lattice vector translation is also generated by S and R. To sum up, the space group consists of a translation by the Bravais lattice vector and these following 12 operations: the identity operation 1, three screw axis displacements S x, S y, S z and eight three-fold rotations around 4 different axes R 0, R0 1, R 1, R1 1, R 2, R2 1, R, R 1.

15 CHAPTER 1. BASIC PROPERTIES OF FESI 6 formula type of operation S (1/2 + x, 1/2 y, z) S x : 2-fold scr. axis (t, 1/4, 0), transl. 1/2 RSR 2 ( x, 1/2 + y, 1/2 z) S y : 2-fold scr. axis (0, t, 1/4), transl. 1/2 R 2 SR (1/2 x, y, 1/2 + z) S z : 2-fold scr. axis (1/4, 0, t), transl. 1/2 R (z, x, y) R 0 : rot. axis [111], +120 R 2 (y, z, x) R0 1 : rot. axis [111], +240 RSR ( y, 1/2 + z, 1/2 x) R 1 : rot. axis (t, t, 1/2 t), +120 SRS (1/2 z, x, 1/2 + y) R1 1 : rot. axis (t, t, 1/2 t), +240 RS ( z, 1/2 + x, 1/2 y) R 2 : rot. axis ( t, 1/2 t, t), +120 SR 2 (1/2 + y, 1/2 z, x) R2 1 : rot. axis ( t, 1/2 t, t), +240 R 2 S (1/2 y, z, 1/2 + x) R : rot. axis (1/2 t, t, t), +120 SR (1/2 + z, 1/2 x, y) R 1 : rot. axis (1/2 t, t, t), +240 Table 1.1: Table of the FeSi space group operations applied to the point (x, y, z). The axes are given in a parametric form with t R, the orientation angle for rotation axes aiming to positive z values. Every site in FeSi lies on exactly one of the rotation axes, as it can be found from the parametric formulas for the axes in this table.

16 CHAPTER 1. BASIC PROPERTIES OF FESI 7 1 S R R 2 SR RS SR 2 R 2 S SRS RSR R 2 SR RSR S R R 2 SR RS SR 2 R 2 S SRS RSR R 2 SR RSR 2 S S 1 SR SR 2 R SRS R 2 RSR RS R 2 S RSR 2 R 2 SR R R RS R 2 1 RSR R 2 S RSR 2 S SR 2 R 2 SR SR SRS R 2 R 2 R 2 S 1 R R 2 SR S SRS RS RSR 2 SR RSR SR 2 SR SR SRS SR 2 S R 2 S RSR R 2 SR 1 R 2 RSR 2 R RS RS RS R RSR RSR 2 R 2 SR 2 1 R 2 SR R 2 S S SRS SR SR 2 SR 2 RSR S SR RSR 2 1 RS SRS R 2 SR R R 2 S R 2 R 2 S R 2 S R 2 R 2 SR SRS 1 RSR 2 R SR S RS SR 2 RSR SRS SRS SR R 2 S R 2 SR SR 2 R 2 S RSR 2 RSR 1 RS R RSR RSR SR 2 RSR 2 RS S R 2 SR SR R 1 SRS R 2 R 2 S R 2 SR R 2 SR RSR 2 SRS R 2 S RS SR RSR R 2 R SR 2 1 S RSR 2 RSR 2 R 2 SR RS RSR SRS R R 2 S SR 2 SR R 2 S 1 Table 1.2: Multiplication table of the FeSi space group. The group elements are expressed in terms of the generators R and S.

17 CHAPTER 1. BASIC PROPERTIES OF FESI 8 1 R0 R 0 1 R1 R 1 1 R2 R 2 1 R R 1 Sx Sy Sz 1 1 R0 R 0 1 R1 R 1 1 R2 R 2 1 R R 1 Sx Sy Sz R0 R0 R Sz R 2 1 R Sy Sx R1 R2 R 1 1 R 1 R 0 1 R R0 R 1 Sy Sx R 1 1 R2 Sz R R 1 2 R1 R1 R1 Sy R2 R Sz R 1 R0 Sx R 2 1 R R 0 1 R 1 1 R 1 1 R Sz 1 R1 R 0 1 Sx Sy R 2 1 R 1 R0 R2 R2 R2 R1 Sy Sx R R Sz R 0 1 R0 R 1 R 1 1 R 2 1 R 2 1 Sx R 1 R0 Sz 1 R2 R 1 1 Sy R1 R 1 0 R R R Sz R 1 1 R2 Sx Sy R0 R 1 1 R 0 1 R1 R 2 1 R 1 R 1 R 2 1 Sx Sy R 0 1 R1 Sz 1 R R 1 1 R2 R0 Sx Sx R 1 R 2 1 R R2 R 1 1 R 0 1 R1 R0 1 Sz Sy Sy Sy R2 R1 R 0 1 R 1 R0 R R 2 1 R 1 1 Sz 1 Sx Sz Sz R 1 1 R R 2 1 R0 R 1 R1 R 0 1 R2 Sy Sx 1 Table 1.: Multiplication table of the FeSi space group.

18 CHAPTER 1. BASIC PROPERTIES OF FESI 9 1. Chemistry and local environment Every site in the FeSi crystal is lying on a three-fold rotation axis, as we have mentioned in Table 1.1. Therefore the local surroundings of the site must exhibit three-fold symmetry. The local surroundings of every Fe atom are such that the coordination polyhedron of the Fe atom in FeSi is a nonahedron, Figure 1.2. (a) view along the shortest bond (b) perpendicular view Figure 1.2: The coordination polyhedron of a Fe atom in FeSi The Fe atom has seven Si neighbours: one at 2.27Å, three at 2.5Å and three at 2.52Å [], Figure 1.. The nearest neighbour lies on the local rotation axis of the site, and the neighbours with the same distance from the site are transformed to each other by the three-fold rotation. Because of that, the angles α between the rotation axis and the equidistant neighbours of the studied site are the same and summarized in Table 1.4. atom 2.5Å 2.52Å Si Fe Table 1.4: Angles between the nearest and more distant neighbours of Fe and Si Morever, when viewed along the the shortest bond acting as the rotation axis of the site, three neighbours at the same distance enclose a 120. These two triplets do not overlap completely, owing to the angular shift of 22.8, Figure 1.2a. Analogously, every Si atom has seven Fe neighbours at the same distances as Fe sites, but the angles between the triplets and the rotation axis differ from those for the Fe sites. Chemical properties of an atom arise from its electron configuration. The configurations of iron and silicon are [Ar] d 6 4s 2 and [Ne] s 2 p 2, respectively. Iron tends to be at most hexavalent in inorganic compounds, but in a crystal it might theoretically bind with all eight of its valence electrons. On the other hand, silicon, a predominantly tetravalent element, has a configuration which does not allow for a similar valency increase.

19 CHAPTER 1. BASIC PROPERTIES OF FESI 10 Figure 1.: The unit cell of FeSi with the network of neighbours. Fe atoms in red, Si atoms in blue. Red bonds are the shortest ones, yellow the longest, and orange those in between. Therefore we find ourselves in a seemingly paradoxical situation: on Si the number of the valence electrons is smaller than the number of nearest neighbours (7), and also on Fe the number of nearest neighbours surpasses the usual hexavalency. Interestingly, the mean number of electrons per nearest-neighbour bond is 12/7, i.e. not an integer, although FeSi is insulating. Pauling sketched a possible solution to this peculiarity in [5]. The aim of this thesis is to make some of his claims more quantitative.

20 Chapter 2 Quantum-mechanical concepts in chemistry 2.1 Atomic orbitals Since atomic orbitals are the keystone of quantum insight on chemistry, it is best to start by getting acquainted with them, especially with the angular components of wave functions. The time-independent Schrödinger equation for a spherically symmetric atom is usually solved by separation of variables in spherical coordinates, r, θ and φ. Spherical harmonics are known as the set of functions that satisfy the angular parts of the equation. These functions depend on two integers l and m, l > m, where l is an eigenvalue of the total angular momentum operator and m is an eigenvalue of its z component. Explicit formulas for spherical harmonics are well known. Up to l = 2, they read: [6] 15 Y 2 2 = 2π sin2 θe 2iφ 15 Y 1 1 = sin 8π θe iφ Y 2 1 = sin θ cos θe iφ 8π Y 00 = 1 ( 5 Y 10 = cos θ Y 4π 4π 20 = 4π 2 cos2 θ 1 ) 2 15 Y 11 = sin θeiφ Y 8π 21 = sin θ cos θeiφ 8π 15 Y 22 = 2π sin2 θe 2iφ In elementary chemistry, for atomic orbitals we take real linear combinations of spherical harmonics with fixed l. Letters s, p, d, f denote atomic orbitals with l = 0, 1, 2,, 11

21 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 12 respectively. The degeneracy of atomic orbitals grows as 2l + 1, so e.g. for l = 1 we have three orbitals p x, p y and p z. The indices denote the dependence on Cartesian coordinates. We will use x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. The first atomic orbitals up to! l = 2 normalized so that Y 00 = 1 are given by: 5 1 d z 2 = r 2 2 (z2 r 2 ) p x = r x d xy = s = 1 p y = r y d xz = p z = r z d yz = 5 xy r 2 5 xz r 2 5 yz r 2 d x 2 y 2 = 5 r 2 2 (x2 y 2 ) The angular component ψ o (θ, φ) of atomic and hybrid orbitals is commonly plotted as a surface in D, i.e. as a manifold of points (r, θ, φ). The angles θ, φ are taken as independent variables and the radius r is set to r(θ, φ) = ψ o (θ, φ) 2 to avoid negative values. One should keep in mind that this plot does not represent the spatial probability distribution of an electron, only its angular preference. This approach is convenient to study the directions of bonds, and we adopt it later in this thesis to visualize hybridization, see Figures 2.2a, 2.2, 2., etc. 2.2 Chemical bond Consider a molecule of H it has two single protons 1,2 and an electron around them. Without an electron, the protons would only interact through Coulomb repulsion and they would never bind. However, adding in an electron creates a ground state in which the hydrogen nuclei are at a finite distance. The wave function of the electron is assumed to be a linear combination of atomic orbitals (LCAO technique): ψ(r) = c 1 a(r) + c 2 b(r), (2.1) where a(r), b(r) are the ground states of the hydrogen atoms 1,2. The problem of choosing the coefficients c 1, c 2 can be simplified thanks to the problem s inherent symmetry. The proposed H + 2 molecule is reflectionally symmetric with respect to the y axis, Fig 2.1. A mirror symmetry operator P, P ψ(x, y, z) = ψ( x, y, z) can be introduced and it can be shown that P commutes with the Hamiltonian of the electron

22 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 1 Figure 2.1: The molecule H + 2, 1 and 2 denote the protons. The electron s distance from 1 and 2 is r 1 and r 2, respectively. in a H + 2 molecule. Its eigenvalues are ±1 and its eigenfuctions are either symmetric or antisymmetric with respect to the mirror symmetry. Since P and the Hamiltonian commute, they also share eigenfunctions. Hence we can narrow down the set of coefficients c 1, c 2 to the symmetric (S) or antisymmetric (A) combination of orbitals: ψ S (r) = a(r) + b(r), ψ A (r 1, r 2 ) = a(r) b(r). (2.2) We can use the total energy of the system to determine whether the protons are bonded as follows. When the protons are infinitely far apart and the electron is in the ground state around one of the protons, the total energy of the system equals the ground state energy of the hydrogen atom. The symmetric state energy turns out to be lower than the hydrogen ground state energy. Hence, the symmetric orbital is a bonding orbital, while the antisymmetric orbital is an antibonding one. In the antisymmetric orbital, the likelihood of finding an electron between the protons is much smaller than in the symmetric case. Moreover, the antisymmetric wave function is zero exactly at the midpoints of the protons. The electron in the symmetric state is more delocalized and benefits from the potential of both protons compared to the electron in the antibonding state which is less delocalized. Similarly, another electron can occupy the symmetric orbital to create the H 2 molecule. The first guess for the electrons wave function, which turns out to be wrong, might be ψ wrong (1, 2) = 1 [a(1) + b(1)] 1 [a(2) + b(2)], (2.) 2 2 }{{}}{{} bonding orbital bonding orbital where e.g. a(1) signifies occupation of the orbital a by electron 1. The problem is that ψ wrong remains unchanged when the electrons are swapped (symmetric). A correct wave function for a system of (two) electrons must be antisymmetric, with respect to swapping 1 2, to insure that the electrons obey the Fermi exclusion principle. Luckily, one can easily introduce antisymmetry to ψ wrong by including a spin

23 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 14 term in the product: ψ MO (1, 2) = 1 [a(1) + b(1)] 1 [a(2) + b(2)] 1 ( ). (2.4) }{{}}{{}}{{} bonding orbital bonding orbital antisym. spin part Equation 2.4 forms the basis of the so-called molecular orbital (MO) theory of the chemical bond, by Hund and Mulliken. Multiplication of the first two terms in 2.4 yields contributions of the type a(1)a(2) and b(1)b(2), where both electrons occupy the same orbital. These states are not likely to present whenever the Coulomb interaction is sufficiently strong. In such cases, the wave function becomes: ψ V B (1, 2) = 1 2 [a(1)b(2) + b(1)a(2)] 1 2 ( ), (2.5) which can be written as where: ψ V B (1, 2) = 1 2 [ a b a b ] (ab), (2.6) a b = 1 2 [a (1)b (2) b (1)a (2)], (2.7) a b = 1 2 [a (1)b (2) b (1)a (2)]. (2.8) Equation 2.6 tells us that electrons in the bond are spin correlated. This description of the bond, introduced by Heitler and London and further developed by Pauling, is called valence-bond theory. Most of the present-day literature in solid-state physics makes use of the molecular orbital theory. Pauling suggested the valence-bond theory might bring deeper insight to FeSi. [5] 2. Hybridization Atoms naturally seek the lowest energy state. Thus the energy decrease that accompanies the creation of a chemical bond motivates atoms to bind. To form a bond, the orbitals are often linearly combined to yield a more directional orbital coordinating the bonded atoms. The linear combination of orbitals being utilized to form other, often more directional and more bonding accessible orbitals is called hybridization. It frequently mixes in orbitals with higher energies than those occupied in the ground state, because the energy increase can be compensated by the energy released after forming new bonds. The best known examples of hybridization are carbon sp and carbon sp 2 present in organic compounds. Silicon belongs to the same group as carbon and hybridizes similarly.

24 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 15 The ground state electron configuration of a carbon atom is [He] 2s2 2p2, which makes it seem that carbon only forms two bonds. This fact strongly disagrees with the well established tetravalence of carbon in organic chemistry. The solution of this discrepancy is due to hybridization of the 2s orbital and the three 2p orbitals: 2px, 2py, 2pz. This occurs even though only two out of the three p orbitals are occupied in the ground state. In sp hybridization, the resulting hybrid orbitals assume tetrahedral arrangement (Fig. 2.2b): 1 si + pz i sp0 i = 2 2 r sp1 i = si pz i px i 2 2 r si pz i px i py i sp2 i = r si pz i px i + py i sp i = sp hybridization is usually presented in a different form. We purposefully rotated it so that one of the orbitals is oriented along z- axis ( sp0 i) and another in the xz plane ( sp1 i). Every sp hybrid orbital has mean energy 14 Es + 4 Ep, where Es, Ep is the energy of an electron in s and p orbital, respectively. All the hybrid orbitals have the same shape - a large positive lobe and an often neglected smaller negative one in the opposite direction, Figure 2.2a. This hybridization accounts for the tetrahedral coordination of neighbouring atoms around the central carbon atom, for instance in diamond. (a) The sp1 orbital (b) All four sp orbitals Figure 2.2: Angular variation of the probability density for sp orbitals. In sp2 hybridization, formation of three planar hybrid orbitals is desirable. Therefore one of the three p orbitals pointing along the orthogonal directions is unmixed with the other p orbitals. In our choice, pz is left out. The shape of the orbitals sp21, sp22 and sp2 is similar to the shape of sp orbitals - one large positive lobe and one smaller negative lobe

25 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 16 in the opposite direction. Similarly as in sp hybridization, we have purposefully oriented one of the orbitals, ( sp21 i), along the x-axis: r sp1 i = si + px i (2.9) sp22 i = si px i py i (2.10) sp2 i = si px i + py i (2.11) 6 2 This type of hybridization occurs in benzene and graphite. The orbitals sp21, sp22, sp2 form a 120 angle, see Fig 2.b. (a) The sp21 orbital (b) All sp2 orbitals Figure 2.: Angular variation of the probability density for sp2 orbitals 2.4 Resonance in benzene The benzene molecule, C6 H6, consists of six carbon atoms joined in a ring. The hexagonal pattern can be explained by sp2 carbon hybridization, wherein a carbon atom binds to each of his two neighbours and one hydrogen atom. The resulting arrangement is shown in Figure 2.4. Figure 2.4: sp2 hybridization of the carbon atoms in benzene Since the hybridization involves only three electrons from each carbon, the remaining six electrons are expected to occupy the six carbon 2pz orbitals. The bond formation

26 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 17 between these electrons further contributes to the stability of benzene and it can be explained in the theory of molecular orbitals be delocalization of electrons or as a resonance in valence bond theory. Within MO theory, the electrons delocalize by hopping between the neighbouring 2p z orbitals. The process of an electron hopping to an adjacent 2p z is therefore described by tight-binding model in a ring of six sites. This enables us to use the energy dispersion formula that comes with this model [2]: ε k = 2t cos ka, (2.12) where ka = 2π n, n =, 2,..., 1, 2 and t is the hopping amplitude. Note that every 6 Figure 2.5: The 2p z electron s state in benzene depends on its reciprocal lattice vector k. Full dots represent occupied states, empty unoccupied. The zero energy level on the y- axis equals to the ground state energy of the electron in the isolated 2p z orbital. occupied state in Figure 2.5 has a lower energy than the energy of the electron in an isolated 2p z orbital, thereby further lowering the energy, as expected. On the other hand, the valence-bond theory develops the original idea that single bond and double bonds are present in benzene, forming two possible Kekulé structures, Figure 2.6: Figure 2.6: Two Kekulé structures for benzene. Double bonds show the positions of the 2p z valence bonds. The contributions of the 2p z orbitals to these structures are (12)(4)(56) and (2)(45)(61). Pauling has suggested that the energy of the molecule decreases further, if we allow for their linear combinations.

27 CHAPTER 2. QUANTUM-MECHANICAL CONCEPTS IN CHEMISTRY 18 In valence-bond theory, a molecule s wave function is a superposition of wave functions that corresponding to valid molecular diagrams. This allows us to (intuitively) express the electrons state in benzene as in Figure 2.6. This linear combination of valence-bond configurations has been called resonance by Pauling. Ψ = α(2)(45)(61) + β(12)(4)(56) (2.1) It might come naturally from Figure 2.6 to think that electrons in the double bond shift along the benzene ring in the process of resonance. However, this disagrees with the viewpoint of valence-bond theory. On the contrary - the position of electrons are fixed, but their correlations change.

28 Chapter Pauling s RVB description of FeSi Pauling views the FeSi structure in the following way: every nearest-neighbour bond is formed by a bonding orbital, which is made of the relevant orbitals on Fe and Si (to be described later). Furthermore, he assumes that each bond is fractionally occupied. [5] He assumes the following bond occupations: 1 for the closest neighbour, 2/ for neighbours in the closer triplet and 1/ for the farther triplet of neighbours. This does not change the total valency of the Si atom, since = 4. He further assumes that Fe atoms have the same bond occupations. However, Fe has valency 8, so the 4 leftover electrons need to be accounted for. According to Pauling, these electrons fully occupy two d orbitals. The bond occupation 1 of the shortest bond in FeSi means that the bond consists of 2 electrons, one from silicon and the other one from iron. Note that the shortest bond is fully occupied. The remaining bonds are weaker. Pauling attributes the fractional occupation (2/ and 1/) of the orbitals to the resonance between occupied and empty orbitals. This is analogous to the case of benzene described in 2.4. Similarly to that, the resonance between different configurations of full and empty bonds should lower the energy of the system..1 Geometric problem: edge coloring The abundance of bonds in FeSi enables resonance to occur. According to Pauling, the shortest bond should always be filled by two electrons, and thus not involved in the resonance. On the other hand, we expect the three remaining electrons (per site) to play a role. A snapshot of FeSi at a given moment of time will reveal that three of the six bonds are filled with just two electrons, leaving the other three orbitals empty. We disregard snapshots with only one electron in the bonding orbital. (We will discuss this assumption later in.). 19

29 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 20 In the rest of this section, we will disregard the shortest bonds, since they do not take part in the resonance. In this case every atom (both Fe and Si) is connected to its neighbours by only 6 bonding orbitals, as shown in Figure.1. Figure.1: The unit cell of FeSi without nearest bonds: Fe atoms in red, Si in blue; orange bonds are shorter Under the assumption of the empty/occupied orbital resonance, it is natural to ask whether pairs of different snapshots can resonate. Theoretically speaking, the FeSi crystal is infinite. We must establish some reasonable boundary condition to get around the theoretical infinite number of possibilities. We decided to narrow down to the repeated pattern generated by a single unit cell. If this yields a reasonable number of possibilities, larger patterns will not be considered. Let us start by enumerating the number of possible valence-bond configurations. The problem of finding the possibilities can be expressed in graph-theoretic terms as the following graph-coloring problem. Atoms correspond to vertices, bonds to edges. The graph has 8 vertices and six edges enter every vertex, representing hexavalency. The boundary condition implies that some edges must lead to vertices outside the unit cell. The task is to color the edges in such a way that every vertex is connected by precisely three red and three blue edges, where red and blue corresponds to occupied and unoccupied

30 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 21 bonding orbitals. Note that here we disregard the difference between the shorter and longer bonds. In Appendix A, we have proven that the FeSi structure, neglecting nearest neighbour bonds, is isomorphic to the simple cubic structure (if we disregard the difference between the Fe and Si atoms, which is irrelevant in the coloring problem). Hence we are justified in solving the problem on the simple cubic lattice. We solved this problem by brute force. The algorithm we used to try out the combinations is as follows: At first, we color only edges starting from the Si sites so that each Si site has three red and three blue edges. Note that this will color all edges. Then, we check whether there are exactly three red edges entering every Fe site. The coloring of one Si site does not depend on the coloring of other Si sites. Since there are ( 6 ) possibilities how to color edges entering one site and we need to color all 4 Si sites to color the whole crystal, the number of tested combinations was ( 6 ) 4 = It was found that only 9600 of these combinations are a valid solution to our coloring problem. Since the resonance accounts for the energy decrease, its occurrence is crucial to our problem. We only search for local resonance - the likelihood of resonance that involves a large number of orbitals should be negligible. Inspired by the resonant benzene ring discussed in section 2.4, we sought after cycles in FeSi that would resonate in a similar way, making use of the graph theoretic approach discussed above. Cycles of length 2 are present in the unit cell only due to the boundary condition. The color inversion of such cycles would cause a global resonance, therefore it is disregarded. There are no cycles of length, as per earlier argument where we proved the equivalence to a simple cubic lattice. The smallest and therefore most local resonance in this graph is the color inversion of a length 4 cycle with alternating colors: Figure.2: The resonance ring in FeSi It is clear from the cubic representation that there are at most 24 cycles of length 4 per unit cell in the FeSi crystal. To find the number of resonating cycles for each of the colored graphs, we tested each of the 24 options and identified colored graphs with the greatest numbers of resonant cycles. The maximum number of resonant cycles in the unit cell is 16. We identified the 12

31 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 22 colored graphs that attain this maximum. Six of them are shown in Figure., the rest can be found by color-inverting these graphs. We observed that there is only one pattern of best coloring, which is rotated around coordinate axes or having inverted colors, producing all 12 colorings. Thus we can consider any of the 12 colorings to illustrate the maximal number of resonant rings per unit cell without loss of generality. Figure.4 demonstrates the resonant rings from Figure.a. Figure.4 shows all the resonating rings. There is some freedom in choosing which rings belong to the unit cell, for example only the rings in the 2 2 top right square. Noting that the edges in parallel planes have the same color, the total number of resonant rings per unit cell is given by 2 ( ) = 16. (a) (b) (c) (d) (e) (f) Figure.: The colored graphs with the maximum number of the resonating cycles.2 Iron hybridization In this section, we offer insight into the nature of the resonant orbitals by treating iron hybridization from the viewpoint of quantum physics. Pauling suggested, that the sp d hybridization might be realized in [5]. Furthermore, we assume that orbitals d xy and

32 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 2 (a) xz plane (b) yz plane (c) xy plain Figure.4: The resonant rings in the best coloring from Figure.a, resonant rings in grey, the conventional orientation of axes d x 2 y2 are fully occupied and do not hybridize. The remaining 4 electrons of the Fe atom will be assumed to occupy linear combinations of 7 atomic orbitals 4s, 4p x, 4p y, 4p z, d z 2, d xz and d yz. In what follows we will assume that the z axis is chosen along the triple axis of the Fe site. Our goal will be to find such linear combinations of these orbitals which point in the direction of the 7 nearest neighbours of the Fe atoms in FeSi. The standard sp d hybridization leads to bonds which form a pentagonal bi-pyramid. This coordination strongly disagrees with the coordination of bonds in the FeSi crystal, implying the necessity to construct a different sp d hybridization for FeSi. Mathematically, we have to solve the following problem. The original basis of the 7-dimensional sp d Hilbert space is given by the following vector of atomic orbitals Ψ: Ψ = ( s, pz, d z 2, p x, p y, d xz, d yz ) (.1) Our task is to change the basis of this Hilbert space so that the new basis vectors (hybridized orbitals) point in the directions of the nearest neighbours of the Fe atom. The new hybridized orbitals are denoted as follows: the hybrid orbital of the shortest bond is 0, the nearer triplet is a1, a2, a, and the farther triplet are b1, b2, b. The orbitals a1, b1 are arranged such that they lie in the xz plane. The hybridized orbitals can be expressed as a scalar product of a complex number vector with the vector Ψ. The chosen orientation of orbitals makes the hybrid orbital coefficients a1, b1 corresponding to p y, d yz equally zero. The same is true for coefficients corresponding to p x, p y, d xz, d yz for the hybrid orbital 0. As the coordination polyhedron possesses a three-fold axis, the state triplets are symmetric under three-fold rotations. Therefore if any one orbital from the triplet is given, we can find the other two by performing a rotation around the shortest bond. The coefficients corresponding to s, p z, d z 2 are preserved by such rotation. On the other hand, coefficients

33 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 24 corresponding to p x, p y and d xz, d yz are altered by the rotation. They transform as 2D vectors under three-fold rotation R 1/ : ( R 1/ (1, 0) 1 2, 2 In general, the hybrid orbitals are of the form: ) (, R1/(1, 2 0) 1 ) 2, 2 (.2) a1 =( c, α, 0, β, 0) Ψ a2 =( c, 1 2 α, 2 α, 1 2 β, 2 β) Ψ a =( c, 1 2 α, 2 α, 1 2 β, 2 β) Ψ b1 =( c, α, 0, β, 0) Ψ (.) b2 =( c, 1 2 α, 2 α, 1 2 β, 2 β ) Ψ b =( c, 1 2 α, 2 α, 1 2 β, 2 β ) Ψ 0 =( c, 0, 0, 0, 0) Ψ where c, c c are three-component vectors representing weights of orbitals unaffected by the rotation around the shortest bond: s, p z, d z 2. Next, we analyze the implications of orthonormality of the hybrid orbitals. The symmetry of the hybrid orbitals causes many inner products between hybrid orbital to be equal. Independent information can be obtained from the following inner products: a1 a1 = 1 c 2 + α 2 + β 2 = 1 a1 a2 = 0 c ( α 2 + β 2 ) = 0 c 2 = 1, α 2 + β 2 = 2 (.4) b1 b1 = 1 c 2 + α 2 + β 2 = 1 b1 b2 = 0 c ( α 2 + β 2 ) = 0 c 2 = 1, α 2 + β 2 = 2 (.5) a1 b1 = 0 c c + αα + ββ = 0 a1 b2 = 0 c c 1 c c = 0, αα + ββ = 0 (.6) 2 (αα + ββ ) = = 1 c 2 = 1 (.7) 0 a1 = 0 c c = 0 (.8) 0 b1 = 0 c c = 0 (.9) To sum up, the orthonormality makes the three vectors c, c c mutually orthogonal, the vectors (α, β), (α, β ) should also be orthogonal and the norms of all the above, too, follow

34 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 25 from the orthonormality condition. These conditions leave 4 free parameters γ, δ, σ, ω, if we use the following parametrization, where d 1 and d 2 are auxiliary vectors: c = 1 (sin γ cos δ, sin γ sin δ, cos γ) (.10) d1 = 1 ( sin δ, cos δ, 0) (.11) d2 = 1 (cos γ cos δ, cos γ sin δ, sin γ) (.12) c = sin σ d 1 + cos σ d 2 (.1) c = (cos σ d 1 sin σ d 2 ) (.14) 2 (α, β) = (cos ω, sin ω) (.15) 2 (α, β ) = (sin ω, cos ω) (.16) In this parametrization, the parameter domains are: γ (0, π), δ (0, 2π), σ (0, 2π), ω (0, 2π). (.17) It is important to note that the square moduli of the hybrid orbitals are unchanged by the transformation: γ π γ, δ π + δ, σ π σ, ω π + ω. (.18) In order to find the best hybrid orbitals, we adjusted the positions of the squared modulus maxima of the wave functions to point in the same direction as the bonds. As the situation exhibits three-fold symmetry, we only have to examine the maxima of a1, b1 as functions of θ. The direction of the maximum of 0 is always along the z- axis. At first, we chose a set of trial equidistant points in the 4D parameter space (γ, δ, σ, ω) with a step of π/10 and tested the = parameter combinations. For a given point in the parameter space, we tested 50 trial equidistant values of θ to localize the maxima theta a1, theta b1 of a1, b1, that should be θ 0 a1 = 7.9, θ 0 b1 = The optimal orbitals were constructed by minimization of the cost function L(θ a1, θ b1 ), where L(θ a1, θ b1 ) = (θ a1 θ 0 a1) 2 + (θ b1 θ 0 b1) 2. (.19) This test revealed two large areas in the 4D parameter space which fit the bond directions. One of them lies in the vicinity of: (γ, δ, σ, ω) = ( 7, 2, 1, ) π (.20)

35 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 26 The other region with favorable bond directions obtains by applying the transformation.18. Next, we investigated the surroundings of the point.20, with functions of θ being tested tested in 500 trial points instead of the previous 50. We discovered that the following point in the parameter space: (γ hyb, δ hyb, σ hyb, ω hyb ) = (2.059, 0.58, 4.614, 0.15) (.21) yields the hybridization with the required maxima positions to two decimal places, Figure.5. Figure.6 illustrates the angular distribution of the optimal hybrid orbitals at the values θa1, 0 θb1 0 as functions of φ. This parametrization results in the following hybrid orbitals: 0 = 0.6 s 0.8 p z d z 2 (.22) a1 = s p z d z p x d xz (.2) b1 = 0.7 s p z d z p x d xz (.24) It should be pointed out that our set of hybrid orbitals is not unique. There exist other sets that satisfy the maxima position criterion to a given accuracy. This arises from the fact that the number of parameters exceeds the number of conditions for the hybrid orbital maxima. The D view of hybrid orbital 0 is shown in Figure.7. It contains two lobes of different size, one of which is clearly dominant. Figure.8a reveals the prominent but not very directional lobe of the hybrid orbital a1 in the required direction accompanied by a much smaller lobe pointing in a different direction. The hybrid orbital b1 illustrated in Figure.8b exhibits a significantly more directional lobe aiming in the desired direction, albeit it is surrounded by three smaller lobes pointing elsewhere. All hybrid orbital are visualized in Figure.9.

36 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 27 (a) a1 (b) b1 Figure.5: Angular variation of the optimal hybrid orbitals, dependence on θ. The dotted lines show the experimentally observed bonding angles.

37 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 28 (a) a1 (b) b1 Figure.6: Angular variation of the optimal hybrid orbitals, dependence on φ.

38 CHAPTER. PAULING S RVB DESCRIPTION OF FESI (a) Global view (b) Dependence on θ. Figure.7: Hybrid orbital 0i, angular variation of the probability density (a) a1i, global view (b) b1i, global view Figure.8: Hybrid orbitals, angular variation of the probability density 29

39 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 0 Figure.9: All hybrid orbitals, global view, angular variation of the probability density. Pivoting resonance This section discusses hybridization of the silicon atom. Unlike, in the case of iron, sp d hybridization does not feasibly account for the 7 possible bonding orbitals - silicon has no d valence orbitals. Let us revisit sp hybridization, mixing s and three p orbitals, in the case of silicon. Note that if we rotate arbitrarily any of the silicon sp hybrid orbitals, a new orbital can be still expressed as a linear combination of the unmixed s and p orbitals. Therefore the resonance might happen following way illustrated by Figure.10. A silicon atom with three hybrid sp orbitals 1, 2 and is bound to three surrounding iron Figure.10: Pivoting resonance in FeSi, view along one of the sp orbitals atoms. The presence of an extra iron atom that is unbound to the silicon results in the following: the sp hybrid orbital 1 unbinds from its current bonding partner and rotates, forming a new orbital 2 to bind with the previously unbound iron atom. This back and

40 CHAPTER. PAULING S RVB DESCRIPTION OF FESI 1 forth swapping of bonding partners is referred to as resonance pivoting. The ability to (always) express the new rotated sp orbitals as a linear combination of s and p orbitals results in the probability 1 2 being almost certainly non-zero. As a special type of resonance, the decrease of the energy is associated with pivoting resonance. However, the energy decrease cannot be predicted without calculations of orbital overlaps such as in 1 2 in the case of Figure.10 It is interesting to mention Pauling s argument justifying his assumption that occupation of the orbital involved in bonding to a neighbour from a closer triplet, (2/), is larger than that of an orbital bonding to a farther neighbour, (1/), in the perspective of sp hybridization. When the nearest neighbour coordinates with the distant triplet, the coordination shape is closer to a tetrahedron than when it coordinates with a nearer triplet neighbour, see Table 1.4.