3.2. The covalent bond

Covalent bonds are formed by the sharing of electrons between atoms. The electron density in a molecule can be approximately calculated using one of two approaches based on quantum mechanics: valence bond theory and molecular orbital theory.

3.2.1. Valence bond theory

The basic postulates of the valence bond theory can be stated as follows:

• When two atoms form a chemical bond, an atomic orbital on one of them overlap with an atomic orbital of the other. (Two orbitals overlapped share some common region in space).

• Two electrons with opposite spins can be shared between two overlapping orbitals, the electron density being concentrated between the nuclei of the bonded atoms.

• The greater the degree of overlap is the stronger the bond is. Therefore atoms tend to position themselves so that a maximum orbital overlap is achieved.

• Bond order is the number of covalent bonds that exist between two atoms. A pair of atoms can be bonded by one, two, three or (rarely²) four covalent bonds. The bond order increase causes decrease of the bond length and increase of the bond energy.

• There are three kinds of covalent bonds:  (sigma),  (pi) and  (delta). A bond that concentrates electron density along the line joining the bound nuclei is called a  bond (sigma bond). A bond that is formed by the sideway overlap of: (a) two p orbitals, (b) p and d orbitals, (c) two d orbitals is called a pi bond ( bond). A bond that is formed by the sideway overlap of four lobes of two d orbitals is called a delta bond ( bond). A single bond is always  bond, because for  bond repulsion between nuclei is minimal. Double and triple bonds result from additional  bonding ( + ) and ( + 2 ), respectively; and quadruple bond consist of ( + 2  + ) bonds.

• To explain geometry of molecules, it was supposed, that atoms may form hybrid orbitals by mixing appropriate combinations of atomic orbitals.

The covalent bond in elementary substances: several single bonds or one multiply bond? The relative energy of a multiply bond (double or triple) and several (two or three) single bonds depends on the electronic structure and size of the atoms forming the bond. Small atoms of elements in period 2 (C, N and O) are able to approach each other closely. As a result, effective sideways overlap of their p orbitals can occur, and these atoms form strong  bonds. One double bond O=O is stronger, than two single bonds O-O, therefore oxygen exist in nature as stable diatomic molecule O₂ containing a double bond. For similar reason nitrogen molecule contains a triple bond; this molecule is extremely strong and nonreactive.

When the atoms are large (periods 3, 4 and so on), -type overlap between their p orbitals is relatively uneffective. One double bond S=S is weaker than two single bonds S-S, therefore at ambient conditions sulfur forms stable ring molecule S₈; molecule S₂ exists only in vapor at high temperature.

On the other hand, large atoms starting from period 3 are able to form rather strong d-p and d-d  bonds due to diffuse character of d orbitals. Multiply bonds, involving d-p bonding exist in numerous oxygen-containing acids and their anions (ClO₄^, SO₄^2, PO₄^3…). Extremely high melting point of Ta, W, Re and some other transition elements are caused by strong d-d  bonds; similar bonds exist in some compounds of transition elements, for example Re₃Cl₉ and Mo₆Cl₁₂.

Coordinate (donor-acceptor) covalent bond is formed when a pair of electrons from one atom is shared by two atoms; the second atom should have an empty orbital. For example, in the ammonium ion NH₄⁺, nitrogen atom donates a pair of electrons to H⁺ (free 1s orbital). It should be noted, that all four bonds in NH₄⁺ are identical, so coordinate bonds don’t differ from other covalent bonds formed by sharing electrons from two atoms. Another example is an addition compound Cl₃B:NH₃, where nitrogen atom donates a pair of electrons to boron (free 2p orbital).

The shape of hybrid orbitals is different from that of initial orbitals. They consist of two “lobes” that points in opposite directions, one lobe is much larger, than the other. This large lobe extends father from the nuclei than initial orbitals and is able to overlap with orbitals of other atoms much more effectively. Hybrid orbitals are identical, except for the directions in which they point. Types of hybrid orbitals are given in the table 3.2.

Hybridization of atomic orbitals was supposed basing on experimental data, such as equivalence of all four C-H bonds in CH₄ molecule. These bonds are formed by orbitals of different shape and energy (three p orbitals and one s orbital of carbon atom) and unequivalence of bonds might be expected. However, there is no direct confirmation of the hybridization process itself.

The geometry (shape, structure) of a molecule or an ion is arrangement of atoms in the space relative to each other. One of atoms of a molecule or an ion is considered as a central atom, other atoms (or groups of atoms) are called ligands. Although the number of different molecules is enormous, the number of different ways in which nearest neighboring atoms are arranged around a particular atom is rather limited. Because of this, understanding and describing molecular shapes are not as complicated as they might otherwise be.

Table 3.2. Hybrid orbitals

Hybrid orbitals	Number of orbitals	Orientation
sp	2	Linear
sp2	3	Planar triangle
sp3	4	Tetrahedral
sp3d	5	Trigonal bipyramidal or tetragonal pyramidal
sp3d2	6	Octahedral
sp3d3	7	Pentagonal bipyramidal

Valence shell electron pair repulsion theory (VSEPR theory) is a simple theory able to predict geometry of a molecule or an ion accurately. The VSEPR theory proposes that the geometric arrangement of ligands around the central atom (c.a.) is determined solely by the repulsion among the electron pairs in the valence shell of the central atom. The VSEPR theory distinguishes two kinds of electron pairs in a valence shell which affect the geometry of a molecule (or ion): unshared pairs of the central atom called lone pairs and electron pairs of  bonds. Electron pairs of  bonds are not associated with a separate area of space around the central atom, they merely increase electron density around  bonds when a multiply bond is formed. So, the coordination number of the central atom (equal to the number of ligands) and number of lone pairs of the c.a. are the two quantitative characteristics used by the VSEPR theory to predict the shape of a molecule or ion.

Most molecules have shapes that can be considered as derived from a basic set of just five different geometries. These geometries are presented in table 3.3, all of them are highly symmetrical.

Table 3.3. Basic geometries of molecules AX_n*

Coordination number (n)	2	3	4	5	6
Arrangements of electron pairs	linear	flat triangle**	tetrahedral	trigonal bipyramidal	octahedral
Bond angle	180º	120º	109.5º	90º and 120º	90º
Example	BeCl2	BCl3	CH4	PCl5	SF6
Structure

*Central atom of the molecule doesn’t have lone pairs

** Central atom is located in the centre of a triangle formed by ligands (in the same plane)

When the central atom of the molecule has lone pairs, the molecule can be presented by a formula AX_nE_m (where n - coordination number, m - number of lone pairs). Similarly, an ion can be presented by a formula AX_nE_m^y–. The arrangement of electron pairs around the central atom depends on the steric number of the central atom:

Steric number = (Coordination number + Number of lone pairs)

The basic arrangements of electron pairs around the central atom are given in the table 3.4. As one can see, they are identical to the basic geometries presented in the table 3.3. However, when we give the shape of a molecule or an ion, we always describe how the atoms in the molecule are arranged around the central atom, but not how the electrons are arranged. Due to replacement of one, two or three ligands by lone pairs, the shapes of the AX_nE_m molecules and AX_nE_m^y– ions (presented in the table 3.4) usually are less symmetrical then that of AX_n ones.

_{Table 3.4. Geometries of molecules AXnEm and ions AXnEmy– predicted using VSEPR theory}

Steric number	Arrangements of electron pairs	Coordination number (n)	Number of lone pairs (m)	Geometries of molecule	Examples
3	triangle	2	1	Bent: AX2E	SO2 , NO2
4	tetrahedral	3	1	Trigonal pyramid: AX3E	NH3, SO32
		2	2	Bent: AX2E2	H2O, ClO2
5	trigonal bipyramidal	4	1	Distorted tetrahedron: AX4E	SF4
		3	2	T-shaped: AXnEm	BrF3
		2	3	Linear: AX2E3	XeF2 , ICl2
6	octahedral	5	1	Tetragonal pyramid: AX5E	ICl5, TeF5
		4	2	Square: AX4E2	XeF4, ICl4

The AX₂E molecule. Three electron pairs will have minimum repulsion if they are arranged at the corners of triangle. The structure of an AX₂E molecule (derived from flat triangle by replacement of one ligand by the lone pair) can be described as angular or bent.

Molecules AX₃E and AX₂E₂. If an atom has four pairs of electrons in its valent shell, minimum repulsion occur if they are arranged tetrahedrally. When one lone pair is present, a trygonal pyramidal AX₃E molecule is formed. Shape of this molecule can be described as a pyramid with triangular base. When two lone pairs are present, an angular AX₂E₂ molecule is formed.

Repulsion between the lone pair and a bond. The lone pair is larger in volume than a pair of electrons in a bond (because in a lone pair electrons are attached to only one nucleus, while the electrons of a bond are under the influence of two positive nuclei). Therefore, the lone pair exerts a greater repulsion towards other pairs in the valent shell, than pairs of electrons in bonds. This extra repulsion of the lone pair should be taken into account to predict location of the lone pair(s) in molecules with five or six electron pairs in a valence shell of the central atom.

Prediction of the lone pair(s) location for the AX₄E, AX₃E₂ and AX₂E₃, molecules.

Five electron pairs will have minimum repulsion if they are arranged at the corners of trigonal bipyramid (see table 3.3.). There are two unequivalent positions in the trygonal bipyramid: three equatorial ones in the central triangular plane, and two axial positions, perpendicular to this plane.

The AX₄E molecule. If one of five electron pairs is a lone pair, one might suspect that there are two possible molecular structures, one with the lone pair in the equatorial position and the other with the lone pair in axial position. The lone pair in the equatorial position of the trigonal bipyramid is alongside two bonds that are at angles 90 and two bonds that are at angles 120. The lone pair in the axial position of the trigonal bipyramid is alongside three bonds that are at angles 90 and the fourth bond is at an angle 180. If we consider only the strongest repulsions due to the nearest neighbors (at angles 90), the lone pair in the equatorial position has only two nearest-neighbor bonds, but the lone pair in the axial position has three. As structure with the lone pair in the equatorial position has the lesser total amount of repulsion, it is preferred. In fact, it is always found, that the lone pair prefers to be in the equatorial plane of the trigonal bipyramid, even if there are two or three lone pairs.

The shape of the AX₄E molecule is difficult to describe. Often a term distorted tetrahedral is used, sometimes AX₄E structure is said to be a “seesaw” structure (it resembles a children’s apparatus of the same name).

Molecules AX₃E₂ and AX₂E₃. The molecule AX₃E₂ has two lone pairs in equatorial plane, the atoms arrangement is described as T-shaped. The molecule AX₂E₃ has three lone pairs in equatorial plane, the atoms are in a straight line; the structure is described as linear.

Six electron pairs will have minimum repulsion if they are arranged octahedrally (see table 3.3.). Except AX₆ molecules, only AX₅E and AX₄E₂ ones are observed to occur (not AX₃E₃ or AX₂E₄). The atoms in the AX₅E molecule are at the corners of a pyramid, having a square base, so this structure is described as being square pyramidal. There are two lone pairs in AX₄E₂ molecule, minimal repulsion occurs if they are as far apart as possible. This gives an arrangement of atoms that we describe as square planar.

Deviation of the valent angles from regular ones. Valent angles are equal to 180º, 120º, 109.5º and 90º only in molecules AX_n (all ligands are identical and there are no lone pairs in the valent shell of the central atom). Valent angles usually deviate from regular ones: (a) if the central atom is bonded to different atoms; (b) when there is lone pair(s) in the valent shell of the central atom; (c) when there is double bond(s) along with single one(s).

The repulsion between the lone pair and a single bond is greater than repulsion between two single bonds, as shown in fig 3.2. As double bond is larger in volume than single bond, the double bond exerts a greater repulsion towards other pairs in the valent shell, than single bond (see fig.3.3). The repulsion of a double bond and the lone pair towards other pairs in the valent shell are of similar value (see fig.3.4.(a)). The repulsion between unpaired electron and a bond or lone pair is small because of lower electron density (lower charge) (see fig. 3.4.(b)). So, the repulsion value decreases in a raw:

lone pair  double bond  single bond  single electron

109.5º	109.5º	120º, 180º	90º

Fig 3.2. The repulsion between the lone pair and a single bond

109.5º	109.5º	109.5º

Fig 3.3 The repulsion between double bond and single bond

120º	120º
(a)	(b)

Fig 3.4 The repulsion between (a) double bond and the lone pair (b) double bond and the single electron

To predict geometry of a molecule or an ion using the VSEPR theory you should:

1. Determine the central atom (c.a.) and its valent shell configuration. Write down (or draw) valent shells of the central atom and place the electrons in orbitals. Note if unfilled subshells are available.

2. Determine quantity of ligands, i.e. coordination number (c.n.).

3. Write down (or draw) the valent shells of the ligands and place the electrons in orbitals

4. If the particle examined is an ion, electrons should be added to the most electronegative atom(s) or removed from the less electronegative atom(s) until the net charge of the ion will be achieved.

5. Examine how  bonds are formed. (Quantity of  bonds formed by the central atom is equal to the coordination number). Take into account, that some electrons may become unpaired, if vacant orbital(s) are available.

6. Examine, if multiply bond(s) can be formed.

7. Determine the number of lone pairs of the central atom remained (m), write down the type of molecule AX_nE_m or ion AX_nE_m^y–

8. Calculate the steric number (s.n.) of the central atom:

Steric number = (Coordination number + Number of lone pairs)

9. Determine hybridization of the orbitals of the central atom. Select the basic geometry, i.e. arrangement of ligands and lone pairs around the central atom which gives minimal repulsion between them (see table 3.3 and 3.4, if necessary).

10. Draw the molecule (ion) and describe its shape. Lone pairs should be drawn in your picture, thus, consider the atoms only, when you describe the shape of a molecule (ion)

11. Examine possible deviation of the valent angles from regular ones. Mark these expected deviation of the valent angles in your picture.

Example 3.1. Predicting the shape of an ion by the VSEPR theory

Problem: What is the shape of the tetrafluoroborate ion, BF₄^?

Solution:

1. Boron is the central atom of the ion. Its valent shell configuration is:

B 2s²2p¹

2. There are four ligands bound to the boron atom, so its coordination number is 4:

c.n. (B) = 4.

3. The valent shell configuration of the fluorine atom is:

F 2s²2p⁵

4. The particle examined is an ion, so, one electron should be added. The electronegativity of the fluorine atom is significantly higher, than that of boron atom (4.1 and 2.0, respectively), therefore the electron should be added to one of four F atom:

F^ 2s²2p⁶

5. If 2s electrons of boron become unpaired, 3  bonds can be formed by sharing of electrons between B and F atoms. The B atom has vacant 2p orbital, thus a donor-acceptor covalent  bond is formed additionally by sharing pair of electrons from F^ ion.

6. No multiply bonds can be formed (all electrons of the B atom used to form  bonds).

7. Boron atom doesn’t have lone pairs; the BF₄^ ion belongs to the AX_n^– type.

8. The steric number of the boron atom:

s.n.(B) = 4 (c.n.) + 0 (number of lone pairs) = 4

9. The sp³ hybridization is observed for B. A tetrahedral arrangement of ligands around the boron atom gives minimal repulsion between them

10. Tetrahedral ion

11. As all ligands are identical, negative charge of the BF₄^ion will be equally distributed between them. No deviation of the valent angles from 109.5º is expected.

Answer: The tetrafluoroborate ion, BF₄^ is expected to have a regular tetrahedral shape.

Example 3.2. Predicting the shape of a molecule by the VSEPR theory

Problem: What is the shape of the chlorine trifluoride, ClF₃?

Solution:

1. Chlorine is the central atom of the molecule. Its valent shell configuration is:

Cl 3s²3p⁵3d⁰

2. There are three ligands bound to the chlorine atom, so its coordination number is 3:

c.n. (Cl) = 3

3. The valent shell configuration of the fluorine atom is:

F 2s²2p⁵

4. –

5. If 3p electrons of chlorine become unpaired, 3  bonds can be formed by sharing of electrons between Cl and F atoms.

6. No multiply bonds can be formed.

7. Chlorine atom has two lone pairs; the ClF₃ molecule belongs to the AX₃E₂ type.

8. The steric number of the chlorine atom:

s.n. (Cl) = 3 (c.n.) + 2 (number of lone pairs) = 5

9. The sp³d hybridization is observed for Cl. A trigonal bipyramidal arrangement of three fluorine atoms and two lone pairs around the chlorine atom gives minimal repulsion between them. The lone pairs are located in equatorial positions of trigonal bipyramid.

10. T-shaped

11. The valent angles in the ClF₃ molecule are expected to be slightly smaller, than 90 because repulsion between the lone pair and chlorine atom is greater than between two chlorine atoms.

Answer: The molecule of chlorine trifluoride, ClF₃ is expected to be T-shaped with valent angles slightly smaller, than 90.

Example 3.3. Predicting the shape of a molecule by the VSEPR theory

Problem: What is the shape of the thionyl chloride, SOCl₂?

Solution:

1. Sulfur is the central atom of the molecule. Its valent shell configuration is:

S 3s²3p⁴3d⁰

2. There are three ligands bound to the sulfur atom, so its coordination number is 3:

c.n. (S) = 3

3. The valent shell configurations of the chlorine and oxygen atoms are:

Cl 3s²3p⁵, O 2s²2p⁴

4. –

5. If 3p electrons of sulfur become unpaired (3d orbital should be used), S atom can form two  bonds with Cl atoms and one with O atom.

6. Additionally one bond of  type can be formed between S and O atoms.

7. Sulfur atom has a lone pair, so the SOCl₂ molecule belongs to the AX₃E type.

8. The steric number of the sulfur atom:

s.n.(S) = 3 (c.n.) + 1 (number of lone pairs) = 4

9. The sp³ hybridization is observed for S; a tetrahedral arrangement of ligands and lone pair around the sulfur atom gives minimal repulsion between them. The lone pair is located in one of four vertex of tetrahedron.

Trigonal pyramidal structure

11. The repulsion between lone pair and bonds is greater than repulsion between bonds. The repulsion between the double bond and single ones is greater than repulsion between single bonds. Therefore the valent angles in the SOCl₂ molecule are expected to be smaller, than 109.5; the Cl-S-Cl bond angle should be smaller, than Cl-S-O ones.

Answer: The molecule of thionyl chloride, SOCl₂ is expected to be trigonal pyramidal with valent angles smaller, than 109.5; the Cl-S-Cl bond angle should be smaller, than Cl-S-O ones.